constitutive models based on dislocation density. formulation and

DOCTORA L T H E S I SDOCTORA L T H E S I S

Luleå University of TechnologyDepartment of Applied Physics and Mechanical Engineering

Division of Computer Aided Design

2005:35|: 02-5|: - -- 05⁄35 --

2005:35

Constitutive models basedon dislocation densityFormulation and implementation

into finite element codes

Konstantin Domkin

DOCTORAL THESIS

Constitutive models based on dislocation density

Formulation and implementation into finite element codes

Konstantin Domkin

September 2005

iii

Abstract

Correct description of the material behaviour is an extra challenge in simulation of the materials processing and manufacturing processes such as metal forming. Material models must account for varying strain, strain rate and temperature, and changing microstructure. Physically based models that can catch the essential phenomena dominating the deformation are expected to have a larger range of usability and validity. Also the models should still be tractable for large scale finite element simulations. This work focuses on the dislocation density models of metal plasticity, their numerical implementation and parameter identification.

The basic concepts of dislocation density modelling are introduced, including the effects of static and dynamic recovery, influence of strain path (PAPER “A”) and modelling of the back-stress (PAPER “B”). Possible mechanisms controlling athermal and thermally activated processes involving dislocations, vacancies and solute atoms are also discussed (PAPER “D”).

Mobile and immobile dislocation densities, vacancy concentrations and other variables are treated as internal state variables. The dislocation models are incorporated in a classical continuum plasticity or viscoplasticity framework by means of the evolution equations for these internal variables which effectively control the hardening behaviour.

Numerical implementation of the models into finite element codes is straightforward. With numerically accurate and computationally efficient standard plasticity stress-update algorithms it is possible to apply the dislocation models in large-scale simulations. In the PAPER “E” a special extended version of a return-map stress-update algorithm and consistent tangent is derived to accommodate the complex coupling effects in the material model. Some of the dislocation models are implemented in user material subroutines for ABAQUS and MSC.MARC and used in simulations of metal forming processes (PAPERS “A” and “C”). Simulations of the cup forming tests shows the capability of the model to capture the main trends in the shape of the forming limit diagrams. Also the models are implemented in a custom toolbox for parameter optimisation in Matlab.

Numerical difficulties of parameter optimisation such as non-uniqueness of the solution, high sensitivity to the starting guess-value and to the choice of the error function, appear to be a common problem with advanced material models (PAPER “B”). Simultaneous curve-fitting of multiple experimental curves of different mechanical testing types is advised to achieve more robust optimisation results. Parameters of dislocation density models usually have clear physical interpretation, and it is possible to obtain values of some of them from sources other than mechanical testing.

The accuracy of physically based models is totally dependent on finding the adequate equations to describe the physical process(es) dominating the material behaviour during deformation. These equations may be more or less accurate than standard engineering models. However, the use of physically significant parameters connected to the microstructure features such as grain size etc gives a natural way to couple them to models for microstructure evolution which is important in simulations of material processing and manufacturing processes.

Keywords: physically based model, dislocation density, finite element method (FEM), return-map algorithm, simulation of metal forming, parameter optimisation

v

Preface. Acknowledgements

This thesis summarises my doctoral studies carried out at the Dalarna University (Högskolan Dalarna – HDa), Borlänge, during the years 2000-2005. During this period I was a registered graduate student at the Luleå University of Technology (Ltu), division of Computer Aided Design. The work was supervised by Professor Lars-Erik Lindgren, both at HDa and Ltu.

The financial support for this research project from the Knowledge Foundation (KK-Stiftelsen) is gratefully acknowledged, as well as support from the industrial partners in this project: Sandvik Materials Technology, Outokumpu Stainless, and SSAB Tunnplåt.

I wish to express my sincere gratitude to my examiner and supervisor, Professor Lars-Erik Lindgren for his guidance throughout my studies, for his support, advice, and critical suggestions to my work, and most importantly for his great patience with me over all these years. Thank you, Lars-Erik!

The research subject was initially thought of as a revival and continuation of the past work of Professor Yngve Bergström, HDa – the work which I now consider outstanding (and even more so in comparison with my own achievements). I very much appreciate the memorable advice he gave me during the spin-off discussions in the first weeks of my studies.

I would like to thank Dr. Lars Troive, HDa, and Dr. Göran Engberg, MIKRAB, who didn’t hesitate to share their expertise and ideas in our discussions of the research problems, and who also contributed to producing the results of some of the appended papers.

I would also like to send warm thanks to all former and present fellow graduate students and staff at Högskolan Dalarna for making it a very pleasant working environment.

Finally, I’d like to thank my friends and my family, especially my parents for their love, support and encouragement.

Konstantin Domkin Borlänge, Sweden. September 2005

vii

Contents1 INTRODUCTION ...........................................................................................................1

1.1 Research questions and approach...................................................................................11.2 Limitations and scope of this study .................................................................................21.3 Empirical and physically based models ..........................................................................31.4 Dislocations in crystals ...................................................................................................41.5 Direct and indirect modelling of the crystal microstructure...........................................4

1.5.1 Discrete dislocation dynamics .......................................................................................................... 4 1.5.2 Crystal plasticity............................................................................................................................... 5 1.5.3 Dislocation density based models..................................................................................................... 6

2 CONSTITUTIVE FRAMEWORK - PLASTICITY.....................................................................62.1 Yield surface and flow rule..............................................................................................72.2 Internal variables and hardening....................................................................................72.3 Rate-dependent plasticity (viscoplasticity)......................................................................8

3 STRESS-UPDATE ALGORITHMS IN FEM.........................................................................83.1 Large deformation aspects of constitutive routines ........................................................93.2 Stress-update algorithms for plasticity – general remarks .............................................93.3 Return-map algorithm. Special cases............................................................................10

4 DISLOCATION DENSITY MODELS. BACKGROUND ..........................................................124.1 Deformation mechanisms. Dislocation density.............................................................124.2 Mobile and immobile dislocations. Evolution equations ..............................................134.3 Further advances in dislocation density modelling ......................................................13

5 STRAIN-RATE AND TEMPERATURE EFFECTS ON HARDENING.........................................145.1 Simplified and semi-empirical models ..........................................................................155.2 Dislocation models with vacancy concentration...........................................................16

6 OTHER ASPECTS OF MATERIAL MODELLING.................................................................176.1 Strain-path dependent hardening..................................................................................176.2 Back-stress and kinematic hardening ...........................................................................176.3 Anisotropy and texture evolution ..................................................................................18

7 PARAMETER OPTIMISATION........................................................................................18

8 SUMMARY OF APPENDED PAPERS AND AUTHOR’S CONTRIBUTION .................................19

9 CONCLUSIONS AND FUTURE WORK.............................................................................22

10 REFERENCES ...........................................................................................................23

APPENDED PAPERS

1

1 Introduction

Computer simulation of materials processing and manufacturing processes such as metal forming is an important part of product design and development in the industry. Simulations using the finite element method (FEM) are commonplace in design of components. There exist commercial finite element codes that have quite advanced features. Also a number of constitutive models of the plastic behaviour of metals are available for finite element modelling. Nonetheless, modelling the material behaviour poses an extra challenge in analysis of manufacturing processes. Correct description of material behaviour is crucial for simulations of material deformation processes. Material models must account for varying strain, strain rate and temperature, and sometimes also changing microstructure. Therefore, it is preferable to use physically based models that can catch the essential phenomena dominating the deformation based on the underlying physics of the deformation coupled to microstructure evolution. However, the models must still be tractable for large scale computations.

The desired features of such models also include the possibilities to obtain model parameters from conventional test data and to implement the model in a finite element code via user-defined material subroutines. Additional advantage of using physically based models is the possibility to rely on information from sources other than mechanical testing to determine the model parameters or verify that their values are in accord with their physical interpretation.

1.1 Research questions and approach

The material modelling process includes these steps: i) obtaining test data, ii) choosing constitutive model, iii) numerical implementation of stress calculation algorithm, and iv) finding material parameters.

This work focuses on the physically based material models of metal plasticity, their numerical implementation and parameter identification. The mechanical testing procedures are not in the focus of the study.

Material models based on consideration of the underlying physical processes are expected to have a larger range of usability and validity than more commonly used empirical models. As can be seen in the literature, there has been some progress made to model the evolution of the microstructure for some limited processes but it is not expected that it will be possible to find one universal model covering all aspects of the behaviour of a given material (McDowell 2000).

The research question(s) can be formulated as:

What physically based models for plastic deformations exist? Do they have any advantages compared with more common empirical models?

The approach used in the work consists of the following parts: Study of the current state and advances in the field of physically based material models for metal plasticity, especially (but not limited to) dislocation density based models. Implementation of some of the dislocation models. Parameter identification. Study of the performance (applicability, accuracy) of a particular dislocation model by Bergström (1983), and research the possibilities to improve the model.

2

Accordingly, the thesis content is arranged as follows. This section, Chapter 1, is an extended introduction to physically based material modelling. In Chapter 2, a framework to constitutive modelling of rate-dependent and rate-independent plasticity is presented as well as some common types of hardening. Next, Chapter 3, a numerical implementation of the models of plasticity with internal variables in the context of finite element solution is discussed. In Chapter 4, the basic concepts of dislocation density modelling are introduced and illustrated with examples from the literature. Further aspects of dislocation models are discussed in subsequent chapters: temperature- and strain-rate dependence in Chapter 5, influence of strain path and modelling of back-stress in Chapter 6. The parameter identification procedures are briefly reviewed in Chapter 7. Finally, the overview of appended papers and discussion conclude the thesis.

1.2 Limitations and scope of this study

The physical basis of inelastic deformations in metals is well described by Stouffer & Dame (1996). The book by Reed-Hill & Abbaschian (1992) gives a more fundamental description of the possible dislocation mechanisms for plastic flow. The physical mechanism involved is mainly the movement of dislocations and interaction between dislocations and crystal structure. This is discussed in more detail in the Chapters 4-6 of this thesis.

The material models intend to capture real material behaviour within a continuum based representation. Thus we deal with different length scales. There are four important length-scales of material modelling, Figure 1-1.

FE analysis

STRUCTURE

Materialproperties Dislocation model

TEST SPECIMEN GRAINS DISLOCATIONS

FE analysis

STRUCTURE

Materialproperties Dislocation model

TEST SPECIMEN GRAINS DISLOCATIONS

Figure 1-1. Length-scales in material modelling of metals.

(1). Structure (Macro) level: e.g. forming process etc, which are typically analysed using the finite element method (FEM). Our aim here is to provide a material model in terms of macroscopic stresses and strains.

(2). Material test level: e.g. uniaxial tensile test, torsion, etc. Material model should be fitted to this experimental data.

(3). Microstructure level: grain structure of polycrystalline metals, etc. It is usually treated only in an average sense in the macroscopic models, for example, introducing metallurgical variables

3

into the formulation. To certain extent, this is based on the experimental results of the studies of crystallographic texture, Transmission Electron Microscopy (TEM) and other studies of dislocation cells and patterns formation due to straining are applied at this level. On the other hand, microstructure problems can also be solved by FEM, using e.g. poly-crystal plasticity theory. This approach will not be used in this study as the focus is on the macro-level material models.

(4). Micro scale (and nano-scale). TEM studies of dislocation systems have provided the experimental evidence to the classical theoretical discussions of physical principles of plastic deformation. The discrete dislocation dynamics can be applied to directly simulate material behaviour at the micro scale.

To summarise the scope and the limitations of the present work: The evaluated material models should account for the basic physics of plastic deformation, but they should still be applicable in finite element analysis on the structure/macro level for the simulations of the metal forming processes.

1.3 Empirical and physically based models

Engineering or empirical models are determined by means of fitting model equations and parameters with experimental data without considering the physical processes causing the observed behaviour. These empirical models are also named engineering models as they are more common in engineering applications than the physically based material models.

Physically based material models, on the other hand, are models where knowledge about the underlying physical process, dislocation processes etc, is used to formulate the constitutive equations. Naturally, the division between these kinds of models is somewhat arbitrarily. Both types of models can be considered “engineering”.

There exist a large number of papers dealing with the short-term properties (yield strength, ultimate strength, ductility etc) and long-term properties (creep, fatigue etc). Many references are found in the book by Marshall (1984). In the majority of FE codes the hardening behaviour of metals is represented either by direct interpolation (e.g. piecewise-linear) of the tabulated stress-strain data, or by empirical relationships, e.g. power laws with strain and strain rate. Such equations for the flow stress and plastic strain are not based on any reasoning about the nature of the deformation. Therefore, most of the equation parameters are lacking physical interpretation. Besides, despite their sometimes remarkably good fit to the measured stress-strain curves within certain range of strains, strain rates and temperatures, empirical relations have no predictive power beyond that range of deformation conditions and material microstructure. Hence, their usage is limited only to the range of deformation conditions at which they were curve-fitted, and the accuracy outside that range is often not satisfactory. The tabulated data approach has the same limitations as interpolation is not possible outside the range of the data.

Two different types of physically based models exist. One option is to explicitly include parameters representing something of the physics of the deformation process. This are computed from evolution equations and are coupled to the constitutive model. The other possibility is to do it somewhat implicitly: determine the form of the constitutive equation based on knowledge about the physical process causing the deformation. Although these equations are laid out as phenomenological, they are based to some extent on physical considerations and experimental observation, which mainly concerns the strain rate and

4

temperature terms. This approach is a so-called “model-based-phenomenology” (Frost and Ashby 1982). The same hardening model can be obtained from empirical models and models of dislocation mechanisms as in the case of power law creep.

1.4 Dislocations in crystals

In crystalline materials, the initiation of plastic flow and subsequent permanent deformation under applied stress is related to the presence of certain crystallographic defects – the dislocations (Figure 1-2). The idea of a dislocation was introduced in the continuum mechanics in early 1900s by Volterra and others, and it was purely geometrical. The physical nature of dislocations and their crucial importance in plastic deformation was demonstrated theoretically in 1930s. The presence of dislocations in real crystals was confirmed for the first time in the mid 1950s by direct observations using transmission electron microscopy.

Edge-dislocation

Burger’s vector b

Edge-dislocation

Burger’s vector b

Figure 1-2. A classical sketch of an edge dislocation in a crystal.

There are four major geometric types of crystal defects: point, line, surface, and volume defects. Point defects represent vacancies and interstitial atoms. Line defects – dislocations and also disclinations – correspond to certain imperfections in the structure of a real crystal. Surface defects are e.g. grain boundaries, twins, internal phase boundaries, stacking faults. Voids and cavities are volume defects. Geometrically, a dislocation is an isolated singular line in a medium, such that displacement of a point is a multi-valued function on any closed circuit that is threaded by this line. In a metal crystal, the dislocation can be constructed by cutting the crystal along a plane, with a dislocation line being an edge of such plane, and then shifting the adjusting parts of the crystal through the distance equal to inter-atomic spacing. The resulting crystal structure has not changed anywhere except in the vicinity of the dislocation line. The direction of the shift is called the Burgers vector (Figure 1-2).

1.5 Direct and indirect modelling of the crystal microstructure

Physically based models may follow different approaches how to describe the changes of the microstructure and the processes which occur in the crystal.

1.5.1 Discrete dislocation dynamics One of the direct approaches to modelling is the discrete dislocation dynamics: motion and interaction of individual dislocation lines is considered, and the stress-strain response of the material is a result of direct simulation of a huge assemble of dislocations. As pointed out by

5

Estrin (1999), “recent developments in computer modelling make it possible, at least in principle, to simulate the dislocation dynamics on the basis of atomistic calculations, while macroscopic deformation can be modelled using mesoscopic dislocation dynamic simulations. This scale-bridging approach is still in its infancy. Yet, one can notice a steady progress and some promising results.” Some good examples can be found in the work of Devincre & Kubin (1994), Rhee et al (1998), Zbib et al (2000). However, the algorithms are extremely demanding computationally, and are not readily available for implementation in standard engineering software.

1.5.2 Crystal plasticity Direct modelling of the crystal structure and deformation of the crystal is the basis for crystal plasticity and its generalised version - the poly-crystal plasticity. It is a very effective approach to deal with material microstructure. It was proposed in the classical work by Asaro & Rice (1977), and since then it has become very popular and proved successful. The material behaviour is studied at the grain level, each grain being a single crystal, and the overall polycrystallineproperties are obtained by averaging procedures, e.g. using classical Taylor-Bishop-Hill method. Recently, the direct FEM modelling of the grain structure has been addressed, but truly realistic models are difficult to create and to solve, and often geometrically simplified models are used.

The crystal plasticity formulation relies on the multiplicative decomposition of the total deformation gradient into inelastic component, associated with pure slip while lattice remains undistorted, and an elastic component, which accounts for elastic stretches and rigid-body rotation.

Figure 1-3. Multiplicative decomposition of the total strain gradient in the crystal plasticity theory (after Asaro and Rice, 1977).

Thermal deformation gradient can also be included as discussed in Meissonnier et al. (2001). From the kinematics of dislocation motion, the rate of the inelastic deformation gradient tensor is assumed as

p

a

aaap FnmF

where a represents the shear strain rate on the slip system a, ma and na are the slip system unit vectors, defining the slip direction and the slip plane normal.

6

To complete the formulation, the flow and evolutionary equations that describe the behaviour of each individual slip system are required. Various phenomenological models are typically employed to describe self- and latent-hardening of the slip system.

1.5.3 Dislocation density based models Alternatively it is possible to model the microstructure indirectly, so that the effects of the micro level processes are somehow accounted for, perhaps in an average way, on the macro level. Such type of approach, using the notion of dislocation density, is the subject of the study in this thesis.

Opposite to the discrete dislocation dynamics approach, the dislocation density based models are formulated at the macro level, i.e. all the quantities (dislocation densities, flow stress etc) are calculated for a representative material volume that can be considered homogeneous. During the 1970´s, in a series of investigations by Swedish researchers, several basic dislocation material models were developed (Bergström 1969, Roberts & Bergström 1973, Bergström 1983). Similar ideas were independently exploited by Kocks (1966, 1976), and later developed by Estrin & Mecking (1984) and Estrin (1998). Many other researchers incorporated dislocation density logic either implicitly (Bammann 1990) or explicitly (Busso 1998) into material models.

With the above indirect approach, dislocation density models provide a bridge between the micro-level phenomena and macro-level continuum quantities, such as stress and strain. The next chapter introduces the constitutive framework of plasticity, in which the dislocation models should be incorporated.

2 Constitutive framework - plasticity

Within the scope of this study, material models are targeted for application in a finite element analysis of manufacturing or material processes such as sheet metal forming or extrusion. Thus a material model should describe thermo-mechanical deformation in terms of the macroscopic stresses and strains. Several phenomenological classes of constitutive models are well-known in the classical continuum mechanics: elasticity, plasticity, creep, viscosity etc. This study is concerned with elasto-plasticity and visco-plasticity only, as these phenomena are most relevant to the deformation of metals (this will be further discussed in the Chapter 4).

From the continuum mechanics point of view, the constitutive equations should satisfy the second principle of thermo-dynamics (dissipation inequality), and also the objectivity principle (frame indifference). The latter is also important in the context of a numerical solution, which is touched upon in the next chapter. The dissipation inequality imposes additional constraints on material parameters. More in-depth information on this subject can be found e.g. in the book by Lemaitre & Chaboche 1990.

Classical formulations of plasticity are well established since the mile-stone work of Hill (1950). Originally developed for small strains case, plasticity theory was also generalised to the case of large deformations, for isotropic as well as for anisotropic materials. The formulations can be found in numerous text-books (e.g. Crisfield 1991, Crisfield 1997, Belytschko et al 2000)

The rate-independent plasticity theory describes the time-independent irreversible (inelastic) deformations. The domain of validity of pure plasticity is restricted by the following limitations: moderate temperature usage and non-damaging loads (Lemaitre & Chaboche 1990).

7

The rate-dependent viscoplasticity theory can describe more complex behaviour, with varying strain rates and higher temperatures.

2.1 Yield surface and flow rule

The yield criterion and yield stress define a surface in the stress space that separates the elastic domain inside the surface and inelastic domain outside the surface:

0),( Yf (2-1)

In most of the appended papers of this thesis, von Mises yield criterion is used (for isotropic materials). In the PAPER “A” a formulation with an anisotropic yield surface is used.

The flow rule establishes relation between the stress and the plastic strain increments (or rates). The associated plasticity uses the yield surface function )(f as a plastic dissipation potential. This has the advantage of wide applicability to different metals, and can be effectively implemented in computations. The associated flow rule

fdd p (2-2)

defines that increments of plastic strain are normal to the yield surface. The scalar d is usually referred to as the ‘plastic multiplier’

In the rate-independent plasticity theory, the plastic multiplier is determined from the solution of the consistency condition:

0f (2-3)

The above condition together with the yield criterion (2-1) imply that during continuous plastic deformation the stress stays on the yield surface (yield function is zero) and the increments of the yield function are zero too. In the numerical solution the latter property can be used in addition to the consistency condition.

2.2 Internal variables and hardening

If the yield stress Y and other yield criterion parameters depend on plastic strain, one can describe the hardening of material. The kinematic hardening translates the yield surface whereas the isotropic hardening changes the size of the yield surface. Models that include the effect of developing anisotropy will also change the shape of the surface.

Modelling of the isotropic hardening is the main focus of this study, and a review of existing models is presented in the subsequent chapters of the thesis. In the appended PAPER “B”, also kinematic hardening model is adopted for the one-dimensional stress-strain tests. In the literature there exist various formulations to generalise such models to three dimensions.

The plastic multiplier can be treated as an internal variable controlling plastic deformation. It is simply equal to the equivalent plastic strain in the case of von Mises yield criterion. Using the concept of internal variables, it is also possible to describe more complex hardening behaviour, when the amount of hardening can not be directly determined by the equivalent plastic strain alone. For example, the microstructure characteristics of the material can be defined as internal variables which influence the plastic deformation. In this study, a dislocation density

8

and related parameters are treated as internal variables, thus providing the bridge between the micro-level phenomena (discussed in the Chapters 4, 5 and 6) and macro-level continuum theory.

2.3 Rate-dependent plasticity (viscoplasticity)

The basic idea in rate-dependent plasticity is similar to rate-independent theory. A yield surface is used as a reference. The stress state can be outside this surface, and the plastic strain rate is a function of the distance to the yield surface, the overstress. Creep can be considered as a special case of viscoplasticity without elastic domain. The magnitude of the effective plastic strain rate is obtained from the flow strength equation:

fgp (2-4)

where fg is a general notation of the increase in rate when the stress state is outside the yield surface and its value is zero when the stresss state is inside the yield surface. There exists a large range of the choices of functions in different viscoplasticity models. Similar expressions are also used in so called unified models (Miller 1987).

3 Stress-update algorithms in FEM

In this chapter the implementation of material models in a finite element program is discussed. The three main tasks of a constitutive subroutine for plasticity in an FEM-code:

Yield criterion calculation (elastic-to-plastic and plastic-to-elastic transition). Updating the stress state and internal variables from given total strain increment. Calculation of a consistent constitutive matrix.

If the full implicit Newton-Raphson procedure is executed for equilibrium iterations, the consistent constitutive matrix is required in the computation of the tangent stiffness matrix to achieve overall 2nd order rate of convergence. Otherwise various approximations of the tangent matrix can be used. Furthermore, the consistency of the constitutive matrix is not important in the explicit FE solvers where no Newton-Raphson iterations are used.

The choice of either rate-independent plasticity or rate-dependent viscoplasticity framework dictates the choice of a corresponding stress-update algorithm: with the consistency condition for the former model, or with a flow-strength equation for the latter model (e.g. Belytschko et al.2000, and see PAPER “B” for more references). In the case of rate-dependent models, a unified-viscoplasticity form of stress-update algorithm could be used (Bammann 1984, 1990). However, it is also possible to formulate the rate-dependent and rate-independent plasticity in parallel, and in the actual implementation to use just one generalised formulation where the former model is a sub-case of the latter (Ponthot 2002). Similar approach is used in the present study.

Essentially, the rate-independent plasticity framework is chosen, and the rate-effects are introduced by means of dependency of the hardening parameters on the plastic strain rate (PAPERS “B” to “D”). Thus we may have a yield limit that is rate-dependent but applied in the context of rate-independent plasticity.

9

3.1 Large deformation aspects of constitutive routines

Simulation of typical manufacturing processes, e.g. sheet metal forming or extrusion, requires a large deformation analysis. The stress-strain relations can be either of hyperelastic type or hypoelastic type. Hypoelastic approach is based on the choice of an objective stress rate measure defining the increment in stress. The Cauchy stress and the rate of deformation tensor (velocity strain) are commonly used as a conjugate stress-strain pair.

In the current study, implicit solvers of the two commercial finite element codes are employed: ABAQUS and MSC.MARC. Both of these FE programs provide a set of “user subroutines” for implementation of the user-defined constitutive stress-update algorithms. The internal workings of the above-mentioned software are somewhat different. However they share certain common theoretical background for the solution of the large-deformation problems.

An additive decomposition of strain rate is assumed. The Green-Naghdi objective stress rate is proportional, via Hooke´s law, to the elastic strain rate tensor (Simo & Pister 1984). Alternatively, Jaumann objective stress rate can be used depending on the FE software, choice of element type and other program options. The hypo-elastic relations are the basis for applying the constitutive model, and the stress computation is performed in a given reference configuration. The stress computation algorithm will then be form-identical to the small strain case (Hughes 1983). PAPER “E” contains a detailed review of this approach and references to the original articles with rigorous proofs on this matter and for more in-depth explanations.

Accordingly, in the constitutive routines of these implicit FE programs the incremental form of constitutive equations is used with an objective co-rotational stress rate. The numerical algorithm ensures objectivity (frame-indifference) of the constitutive relation by means of a co-rotational transformation:

Tnn RR*1 (3-1)

where R is an incremental rotation matrix, appropriately computed. Hence, the large-rotation related issues of the constitutive routine are taken care of by the ABAQUS or MARC solvers. The user-defined material subroutine will only perform calculations which are form-identical to the small strain case: update stresses, plastic strains and internal variables, and compute consistent tangent matrix:

** CCT (3-2)

These calculations are made in a reference configuration and the rotation to the final configuration at the end of an increment is made outside the stress-strain algorithm. Therefore, in the subsequent section, the discussion of specific algorithms will use stress and strain relations without distinction of large or small strains case.

3.2 Stress-update algorithms for plasticity – general remarks

The choice of a yield criterion and a flow rule usually demands a certain stress updating algorithm, which in turn defines the derivation of a tangent moduli matrix consistent with the solution procedure (Crisfield 1991, 1997). There exist a number of algorithms and techniques to update the stresses and plastic strains, such as operator splitting, the standard predictor, “predictor + corrector”, sub-incrementation, pure incremental or forward Euler scheme, generalised trapezoidal or mid-point algorithms, a backward Euler (return-map), etc.

10

In this work the implicit, backward Euler algorithm was chosen. The method has some desirable properties and has a good performance for large strain increments. Enforcing the consistency condition at the end of the time step will lead to a symmetric consistent constitutive matrix (Simo & Taylor 1986) for many material models. The radial-return algorithm is a special case of backward Euler return-map, and it is especially effective for von Mises criterion: it is indeed a simple ‘radial return’ in deviatoric stress space. It is used in the PAPERS “B” and “C”.

User-defined material subroutine(ABAQUS - UMAT)

Update internal variables

Calculate yield stress

Calculate hardening modulus

nnp

n 1111

nnn

pn

1nY

1n

p

Yn d

dH 1

Return-map algorithmINPUT OUTPUT

User-defined material subroutine(ABAQUS - UMAT)

Update internal variables

Calculate yield stress

Calculate hardening modulus

nnp

n 1111

nnn

pn

1nY

1n

p

Yn d

dH 1

Return-map algorithmINPUT OUTPUT

Figure 3-1 Dataflow and tasks of a user-defined material subroutine.

The dataflow and the task of a user-defined subroutine is schematically shown on the Figure 3-1 (for the case of isotropic hardening, rate-independent plasticity). The generalisation of the return-map algorithm to the case of anisotropic yield criteria is straightforward and is used in the PAPER “A”.

The computation of the so-called consistent tangent moduli matrix (which corresponds to the implicit iterative equations of full Newton–Raphson method) instead of standard elasto-plastic moduli matrix (which corresponds to the original stress-strain relation in the rate form) is favourable, because the full Newton-Raphson’s solution procedure for the non-linear system of coupled equations in finite element formulation exhibits second order convergence only if the used stiffness matrix is the consistent tangent matrix (Simo 1988; Simo & Hughes 1997).

The importance of the efficiency of constitutive update algorithms within FE analysis workflow is apparent, considering the number of times the routine is called during the solution: every cycle of external load increments, every cycle of equilibrium iterations, in every finite element, in every integration point of an element.

3.3 Return-map algorithm. Special cases

The return-map stress-update algorithm is constructed in two steps. First, the so called trial state is defined as the state of stress when the given total strain increment is taken as elastic and zero plastic strain increment is assumed (elastic predictor step). Second step is the plastic corrector step: the trial stress is relaxed back onto the yield stress in the direction of the closest projection, which is also the direction of the plastic strain increment in the radial-return method. According to the implicit backward Euler approach the plastic strain increment is computed in the updated configuration, which depends on the updated stresses to be determined.

11

In the heart of the return-map algorithm is the calculation of the equivalent plastic strain increment (plastic multiplier) using the condition that the stress remains on the yield surface:

)()(11pp

nYp

nnf (3-3)

In the case of von Mises plasticity with radial-return, the expression of effective stress is a linear function of the plastic strain increment, which is used e.g. in the PAPER “E”. Otherwise it may be a non-linear function, e.g. for anisotropic yield criteria used in the PAPER “A”. In either case, the expression of the yield stress is a non-linear function of plastic strain (the hardening curve). Hence, the whole condition, Eq. (3-3), becomes a non-linear equation, which should be solved for the plastic strain increment. In other words, in the case of non-linear hardening, the solution of the above equation requires scalar iterations.

Thus one may consider possible simplifications or complications of the algorithm depending on the particular hardening law of the material, especially when additional internal variables are used.

(1) The most trivial case, when an analytical expression is available for the yield stress as a function of the plastic strain. If the yield stress is expressed via internal variables,. then the evolution equations for internal variables w.r.t. plastic strain must have an analytical solution over the increment of strain. This kind of model is used in the PAPER “A” to create the user-defined material subroutine UMAT for the ABAQUS implicit solver. The same formulation is used in the custom optimisation toolbox for Matlab in the PAPER “B” to carry out the parameter identification.

(2a) A less trivial case, when there is no analytical expression of the yield stress available, and the evolution equations for the internal variables have to be integrated numerically. In such case, one might still choose the first simplified approach and perform numerical integration for the yield stress in every iteration during the solution of the Eq. (3-3). Such approach was used in the PAPERS “C” and “D”. The advantage is that for uniaxial and proportional loadings it is possible to compute the hardening curve with higher accuracy e.g. using sub-incrementation. However, such method is obviously not the most efficient computationally .

(2b) A numerically more efficient approach to the previous case is simultaneous solution of both the consistency condition and the evolution equations. The effect of the used numerical integration method for the internal variables on the hardening modulus must then be accounted for. The hardening curve is then computed with a reduced accuracy when a lower order update formulas are applied to the evolution equations.

(3) In the PAPER “E”, an even more complicated situation is encountered. The hardening equations are of the form “generalised hardening law”, i.e. a combination of differential and algebraic equations, involving several internal variables and implicitly coupled with the current stress tensor. In addition, elastic moduli and thermal dilatation are coupled with plastic strain. To achieve better numerical efficiency, an extended radial return method had to be used. The formulation is derived similar to the one found in the book by Belytschko et al. (2000) with certain modifications. The details are given in the PAPER “E”.

As it was already pointed out in the Chapter 2, the material models used in the appended PAPERS “A” to “D” are physically based models, involving dislocation densities and related parameters. In the logic of the return-map algorithm these dislocation related variables are introduced as internal variables to define the hardening behaviour and yield stress. The fundamentals and derivations of the dislocation models are discussed in the subsequent chapters.

12

4 Dislocation density models. Background

In this chapter, the three mainstream dislocation density based models (Bergström 1983, Gottstein et al 1987, and Estrin et al 1984) are discussed. They all have similar basic assumptions and ideas about dislocation density evolution processes, which basically consist of storage and recovery. But the particular forms of the equations are often different. In the subsequent chapter it will be shown also that such phenomena as dislocation cell substructure, strain-rate and temperature effects, cyclic loading conditions can be accounted for by this kind of dislocation models. The accuracy of one or another model should be validated by further experimental observations. There are also numerous papers in which similar semi-empirical and dislocation models are discussed, and some suggestions made concerning temperature and strain-rate dependence of the flow stress, and some other micro-structural aspects of the deformation. Those papers are not reviewed here, as all the key issues are already addressed in the presented models.

4.1 Deformation mechanisms. Dislocation density

Dislocations provide the mechanism of plastic deformation of metals. Despite the fact that there exist many elementary geometrical types of dislocations in crystals (e.g. jogs, loops, dipoles etc), it is possible to establish a fundamental relation between macroscopic plasticity effects and the properties of dislocations. The plastic strain is directly related to the motion of dislocations, and hardening/softening of metals is attributed to the interaction of dislocations with each other and with surrounding crystal microstructure.

The microstructure is evolving during the deformation. This is also affected by the strain rate and temperature. The active mechanisms depend on the current structure of the material and applied stress or strain rate at the given temperature. An overview of the dominating deformation mechanisms can be outlined in a deformation map (Frost. & Ashby 1982). The models presented in this thesis concern the following deformation mechanisms: Low-temperature plasticity by dislocation glide (limited by a lattice resistance or Peierl’s stress, by discrete obstacles etc) and Power-law creep by dislocation glide or glide plus climb (limited by glide processes; by lattice-diffusion controlled climb; etc). PAPER “D” contains a brief review of these mechanisms.

Furthermore, dislocations in the crystal can form loops, can pile up on the grain boundaries and precipitate particles, and arrange themselves in various types of cells or substructures called dislocation networks or low energy dislocation structures. These arrangements act as obstacles to the motion of other dislocations, thus providing an important mechanism of hardening. The point, surface and volume defects interact with dislocations and also play important role in hardening mechanisms.

The dislocation density concept links the macroscopic stresses and strains to the underlying micro-structural processes of plastic deformation. A generally accepted assumption for the influence of dislocation density on hardening behaviour is

aGbfy (4-1) where is the so called “forest” dislocation density in cm of dislocations per unit volume, G is the shear modulus, b is the magnitude of the Burger’s vector, and a is a material constant related to the crystal and grain structure. The first term, f is the extrapolated stress to zero dislocation density (strain-independent “friction stress”), corresponds to the short-range interactions. It is a friction stress needed to move dislocations through the lattice and to pass short-range obstacles.

13

Thermal vibrations can assist the stress to overcome these obstacles. The second term is due to long-range interactions with the dislocation substructure. It is an athermal stress contribution. This model does not account for different elementary types of dislocations and dislocation arrangement structures.

4.2 Mobile and immobile dislocations. Evolution equations

The essential property of a dislocation is its mobility, and the simplest dislocation model should distinguish at least two types of dislocations: mobile and immobile. Motion of mobile dislocation carries the plastic strain, and immobile dislocations contribute to the plastic hardening. The direct simulation of the process, in the spirit of discrete dislocation dynamics, is costly to compute. Hence, an average treatment of dislocation processes is favourable, and the concept of dislocation density is found useful (Estrin 1999).

The dislocation processes include generation of new dislocations, annihilation, immobilisation and re-mobilisation (recovery) of dislocations. It is necessary to have a model for strain hardening of a material due to these processes. Most commonly a kind of evolution equation is derived for each type of dislocation density, which may generally be expressed in the form of “hardening term – recovery term”:

)()( ddd (4-2) The increments in the above equation are w.r.t. either time or plastic strain.

Often the following assumptions are also accepted (Bergström 1969, 1983): (a) mobile dislocation density is strain independent and much smaller than the immobile one. (b) mobile dislocations move, on average, a distance (mean free path) before they are immobilised or annihilated. The basis of this modelling is related to the statistical theory of work-hardening by Kocks (1966).

4.3 Further advances in dislocation density modelling

The dislocation density based models can account for static and dynamic strain-aging effects, if extra internal variables are introduced: partially immobilized and locked dislocation densities (Roberts & Bergström 1973). Their evolution laws are similar to presented above.

The dislocation model by Estrin & Mecking (1984) is based on the developments of Kocks (1966, 1976), and is very similar to the Bergström’s model. A one-internal-variable model is usually sufficient for describing monotonic deformation, without abrupt changes of the deformation rate or the deformation path. More general two-internal-variables models were developed by (Estrin 1998) that can account for transients associated with such changes. Then a second, ‘rapid response’ internal variable is added. The model distinguishes between the mobile dislocation density and the density of less mobile (forest) dislocations. The one-internal-variablevariant of the model is obtained as a special case of the two-internal-variable one. The need for a second internal variable also arises when a complex strain hardening behaviour at large strains is to be accounted for. The one-internalvariable model predicts a linear decrease of the strain-hardening coefficient with stress, the stress asymptotically approaching saturation. This is referred to as stage III hardening. In reality, a new stage of hardening intervenes, stage IV, when strain-hardening rate stays at a nearly constant low level until it drops again in stage V. The emergence of a dislocation cell structure may be responsible for these hardening features. A

14

single-phased material then effectively becomes a two-phase one, with soft cell interior regions separated by hard cell walls with a much higher dislocation density.

This situation is also assumed in a more sophisticated three-internal-variable model described in Roters, Raabe & Gottstein 2000, which is en extension of a previous basic concept by Gottstein & Argon 1987. The details of the proposed dislocation model are also discussed in Aretz et al.2000, and Luce et al. 2001. The model distinguishes three dislocation populations (Figure 4-4):

mobile dislocations travelling through cell structure, mthe immobile dislocation density inside cells, ithe density of immobile dislocations in the cell walls, w.

Figure 4-4. Schematic drawing of the arrangement of the three dislocation classes (after Roters, Raabe & Gottstein 2000).

The mean free path of mobile dislocations is determined by the effective grain size and three obstacle spacing: the forest dislocation spacing in the cell walls, in the cell interior, and the spacing of the precipitates. The effective shear stresses are calculated separately for the inside of the cell and inside of the cell walls. The macroscopic flow stress is an average of the effective stress in cells and cell walls.

5 Strain-rate and temperature effects on hardening

The current study (appended PAPERS “A” to “E”) uses the concept of rate-independent plasticity and a yield surface. It is an approximation where the plastic strains are assumed to develop instantaneously in such a way that the stress state stays on the yield surface during a plastic process. The applied stress is then equal to the yield stress during a plastic deformation. On the other hand, the physical processes that determine this yield surface are rate-dependent. Therefore, it leads to a class of models where a rate-dependent yield limit is introduced in the context of rate-independent plasticity.

The thermal effects are introduced by means of the temperature-dependence of the yield limit and other hardening parameters. (In one of the appended papers, PAPER “E”, elastic moduli and thermal dilatation are temperature-dependent properties too.) Furthermore, it is assumed that the deformation process does not affect the temperature and also the thermal conductivity phenomena are left outside the scope of the material models. This will be the case e.g. in a mechanical step of a staggered solution of coupled thermo-mechanical problems, with a thermal

15

step followed by a mechanical step. Then the temperature is simply prescribed in the mechanical stage. However, the models themselves can be generalised to include any form of coupling between temperature and deformation, e.g. by means of appropriate equation for heat generated due to plastic dissipation.

The physical basis of the strain-rate and temperature dependence in the dislocation models is discussed in this chapter. There exists a fundamental relation between plastic strain rate and average dislocation velocity, the Orowan equation:

,...,,,vv,v piymmmm

p (5-1)

where mv is a function of stresses, dislocation densities, strain rate, temperature etc. Most dislocation models accept this relation to introduce rate-dependency. However, there is no agreement in particular choices of the velocity function.

5.1 Simplified and semi-empirical models

First, an extension of the basic dislocation model to account for rate-dependent yield limit (Bergström & Hallen 1982) is described. See also the PAPERS “A” and “B” for further explanations of the symbols and parameters. The dislocation multiplication process is assumed to be rate-independent. The recovery process is strongly dependent on temperature and strain rate. The rocevery term of the equaton (4-2) is assumed to be proportional to the dislocation density,

p)( (5-2) The expression for recovery parameter, , consists of an athermal component 0 and a thermal component:

31

0 )(,ref

p

refp TT (5-3)

The formula is derived on the basis of the following assumption: the re-mobilisation of immobile dislocations takes place predominantly in the cell walls by means of diffusion controlled vacancy climb. Dislocations in motion experience resistance in their glide plane, and certain stress should be applied in order for a dislocation to start moving. The expression for the friction stress relating these effects is assumed to be

)(

0,Tm

ref

p

refp

f TT , gpsT0 (5-4)

where s , p , g are the solution, precipitation and grain-size hardening components (Bergström 1983). As temperature rises, the strain rate dependent part of the friction stress quickly drops to zero.The above two equations are used in the material model in the PAPER “B”.

The relation in Eq. (5-4) is equivalent to the kinetic equation of viscoplasticity (Hasegawa et al.2000):

niy

mp vv

0

)(ˆ, (5-5)

which could be used in a unified-viscoplastic form of stress-update algorithm.

16

Estrin (1998) uses unified viscoplasticity approach, but with a different formula:

mpmp

1

00 (5-6)

Thus, a multiplicative split of strain and strain-rate terms is assumed, which is often replaced by an additive split in other dislocation models in order to fit experimental data better, and based on some physical reasoning.

According to Gottstein (1987), their dislocation model (see section 4.3) can be further enhanced to include rate- and temperature-dependency. The effective shear stresses in cells and walls are derived from the kinetic equation of state for this model – the Orowan equation bvM m ,where v is the glide velocity according to the following formula,

TkV

TkQfv

B

eff

B

~sinhexp (5-7)

where is the jump width, i.e. the mean spacing of obstacles (the immobile forest dislocation in this case), f the attack frequency (assumed to be constant), Q activation energy for dislocation glide (or “forest cutting”), and V is the activation volume.

5.2 Dislocation models with vacancy concentration

A different and somewhat more systematic approach is used in the PAPER “D” to introduce various effects, including strain-rate and temperature. The ideas are collected from the literature devoted to physical modeling of the plastic deformation, and equations are organised in sub-models responsible for particular effect or process. An abridged and modified version of this model is also used in the PAPER “C”.

First, in the Orowan equation, Eq. (5-1), the velocity is related to the time it takes for a dislocation to pass an obstacle, the waiting time tw, as the time of flight to the next obstacle is negligible. The velocity is written as

kTGa

kTGa eebv (5-7)

where b is the mean free path between two successful events. a is the attempt frequency and depends on the obstacles. G is the activation energy, k is the Boltzmann constant and T is the temperature in Kelvin. Considering possible mechanisms controlling the dislocation glide in a crystal structure and using a general formula for the activation energy, the expression of the friction stress as a function of the effective plastic strain rate is obtained,

pq

pref

FkT

11

ln1ˆ* (5-8)

where F is the free energy required to overcome the lattice resistance or obstacles without aid from external stress. The quantity ˆ is the athermal flow strength that must be exceeded in order to move the dislocation across the barrier without aid of thermal energy. It reflects not only the strength but also the density and arrangement of the obstacles. Thus, the explicit rate- and temperature-dependence of the yield stress is introduced in this short-range stress component.

17

The “hidden” rate- and temperature-dependence of the yield stress is also introduced: in the long-range stress component via dislocation density evolution equation. Following the logic of Militzer et al. (1994) with some modifications, an equation for the dislocation recovery is obtained,

223

2 eqieqv

vvi kT

GbccDc (5-9)

where cv is the vacancy concentration, and the meaning of other parameters is explained in the PAPER “D”. Furthermore, an evolution equation for the vacancy concentration is used, coupled with dislocation density and other parameters. Together, they render the process of dislocation recovery as being controlled by temperature and strain-rate. See the PAPER “D” for detailed derivation and an overview of the literature on the subject.

6 Other aspects of material modelling

In this chapter modelling of certain aspects of material behaviour is addressed, which typically may be encountered in the process of sheet metal forming such as deep drawing, and in cyclic tests.

6.1 Strain-path dependent hardening

The formability of sheet metals (characterised by forming limit diagrams, FLD) is observed to depend strongly on the strain path during the deformation. Besides, experiments with rolled steel sheets show (Laukonis & Ghosh 1978) that the effective stress–equivalent strain curves are different in uniaxial and in balanced biaxial tension. Another noteworthy feature of the FLD is that the decrease of failure strains after a pre-strain is material dependent. All this suggests that the material model should account for such behaviour.

A plausible explanation was suggested by Bergström & Ölund (1982): the hardening of the material is dependent on the strain path. Their theoretical treatment is adopted in the PAPER “A” to compute forming limit diagrams. However, the physical interpretation of some of the resulting equation parameters is not straightforward.

In the numerical implementation, the strain path is defined by the ratio of principal strains, and all hardening parameters are assumed to depend on this ratio. It is quite similar to the logic of the rate-dependent hardening discussed in the Chapter 3. Effectively, material model has isotropic hardening, but the hardening parameters are evaluated based on a strain path parameter. Thus, a kind of strain path dependent isotropic hardening is obtained.

6.2 Back-stress and kinematic hardening

The most prominent factor in the cyclic tension-compression tests is the presence of kinematic hardening associated with the back-stress. In the appended PAPER “B” an advanced dislocation model by Estrin et al. (1996) is used to accommodate the evolution of the back-stress internal variable. The back-stress is attributed to the formation of channel-like dislocation structures upon cyclic deformation, with the immobile dislocation density in channel walls consisting of ‘trapped’ dislocations and of ‘recoverable’ dislocations. The changing curvature of the dislocation segments in the channels is believed to be the driving force of the process. Accordingly, an evolution equation for this quantity is chosen in the following form:

18

backp

ipback AsignAAA 2210 )( , (6-1)

The first term, which is associated with a decrease of the radius of curvature of the mobile segments as they bow out under the action of applied stress, is governed by the wall spacing and is thus proportional to the square root of forest dislocation density. This bowing out is counter-acted by the build-up of the back-stress, which is represented by the second term. Once the radius of curvature reaches its critical value, a steady state is achieved, and the subsequent motion of the screw segment is translational. In the original Estrin et al. (1996) model an additional internal variable is introduced: a density of immobile dislocations that are trapped in the walls but partially recoverable upon stress reversals. This process is not taken into account in the present study.

6.3 Anisotropy and texture evolution

This aspect of material behaviour is not included in any of the dislocation models in this study. However, texture and anisotropy are prominent factors in many material processes, especially in sheet metal forming. The subject is considered important and worth more in-depth research in the future studies.

Texture is generally associated with the crystal and grain microstructure. The crystal and poly-crystal plasticity approach is believed to be an appropriate method to account for these properties. A brief review of the basics of this approach was given in the introduction section 1.5.2. Various phenomenological models are often employed to describe self- and latent-hardening of the slip systems in a crystal grain. Attempts have been made to incorporate dislocation models instead, such as in Busso & McClintock (1996).

7 Parameter optimisation

The parameter identification is carried out with the help of the custom toolbox for Matlab. The optimisation algorithms and their implementation are discussed in detail in Lindgren et al.(2003).Essentially, an error measure is minimised with respect to the varying values of the material parameters (Fletcher 1980, 1981). The minimisation process may be subject to the linear or non-linear constraints and may also include user defined weight functions, when necessary for particular material model or test data. The problem is considered as strain driven, and the strain-stress calculations are performed according to the general algorithm for stress-update described in the preceding section.

A direct parameter fitting can be applied for tests where it is possible to measure over a homogenous deformed volume in order to obtain strain and stress, Figure 7-1. This is the case in the current study. pfinal is the material parameters that minimize the difference between computed

c and measured stress e.

In the PAPER “B”, a kind of gradient method of optimisation is chosen. It is based on the Matlab function for solving constrained minimisation problem, fmincon. The gradient methods can be quite efficient but find only local minima. Two error functions are used: the maximal discrepancy between calculated and measured stress normalised by a maximum measured stress magnitude; and the average discrepancy between calculated and measured stresses weighted by corresponding strain increments.

19

Constitutive model

Measureddata from

uniaxial test

Read test data, initial guess and

constraints

T(t), (t)

c(t)

e(t) T(t), (t)

Error function

Minimise

pinit

pfinal

T(t), (t), e(t)

ptrial Constitutive model

Measureddata from

uniaxial test


constraints

T(t), (t)

c(t)

e(t) T(t), (t)

Error function

Minimise

pinit

pfinal

T(t), (t), e(t)

ptrial

Figure 7-1: Parameter fitting with homogenous test giving stress, strain and temperature data

The results of the parameter optimisation for a dislocation model are found to be non-unique, very sensitive to the starting guess-value, and also sensitive to the choice of the error function (PAPER “B”). A step-wise reduction of parameter space (“divide and conquer”) had to be used in order to circumvent the numerical difficulties. The numerical instabilities due to the non-uniqueness of the solution appear to be a common problem with advanced material models. It is discussed, for instance, by Mahnken and Stein (1996). Similar numerical difficulties during parameter optimisation have been encountered in the PAPERS “C” and “D” too.

8 Summary of appended papers and author’s contribution

PAPER “A”

Domkin, K., L.-E. Lindgren & L. Troive (2001) “Physically based material model in sheet metal forming”, in "Simulation of Materials Processing: Theory, Methods and Applications" (ed. K. Mori), Balkema. Proc. 7th Int. Conf. on Num. Meth. Industr. Forming Processes NUMIFORM 2001, Toyohashi, Japan, June 2001, pp.221-226.

A simple dislocation density model is used to describe the isotropic non-linear hardening. The effect of strain-path dependency of hardening is included. The model is implemented in the ABAQUS user-defined material subroutine UMAT, using implicit return-map algorithm for anisotropic rate-independent plasticity in the large deformation context. The model is applied in the finite element simulations of several cases of a circular cup forming, and the forming limit diagrams (FLD) are also computed.

Author’s contribution: selection of the material model and parameters from literature, programming the user-subroutine, simulations of cup forming, derivation of forming limit criteria and estimation of FLD; writing the greater part of the paper.

20

PAPER “B”

Domkin, K., L.-E. Lindgren & P. Segle. (2003) “Dislocation density based models for plastic hardening and parameter identification”, in “Computational Plasticity - Fundamentals and Applications” (eds. D.R.J. Owen & E. Oñate). Proc. VII Int. Conf. on Comput. Plast. COMPLAS VII, CIMNE, Barcelona, Spain, April 2003, on CD-ROM.

The dislocation model introduced in PAPER “A” is further elaborated to include non-linear kinematic hardening and the strain-rate dependency. The model is implemented in a custom optimisation toolbox for Matlab. Using this toolbox, the parameter optimisation procedure is carried out to fit the dislocation model to the experimental data for a high-strength steel: the constant amplitude fully reversed strain controlled cyclic test curves. Parameter sensitivity and other numerical issues are briefly discussed.

Author’s contribution: selection of the material model from literature; programming the Matlab scripts for the toolbox according to the model; carrying out the parameter optimisation procedure; writing the greater part of the paper.

PAPER “C”

Hansson, S. & K. Domkin (2005). “Physically based material model in finite element simulation of extrusion of stainless steel tubes.” accepted for publication in conf. proc. of the 8th ICTP to be held in Verona (Italy), October 2005

A dislocation model is set up similar to those in PAPERS “A” and “B” in its basic assumptions. However, the temperature and strain-rate dependency is introduced in a different way: based on the concepts of short-range and long-range interactions of dislocations and obstacles. The evolution equation for the dislocation density is also modified to account for the static and dynamic recovery controlled by diffusional climb and interactions with vacancies. The material model is implemented in a user subroutine for MSC.Marc and in a custom optimisation toolbox for Matlab (see also PAPER “B”). Using the toolbox, the model is calibrated for the AISI type 316L stainless steel by results from compression tests at different temperatures and strain rates. A thermo-mechanically coupled axisymmetric finite element model is used to simulate the extrusion process. Model predictions of extrusion force and exit surface temperature are compared with measured values.

Author’s contribution: selection of the material models from literature and ideas how to modify or simplify them; programming the user-subroutine for MSC.MARC and the Matlab scripts for the toolbox according to the model; carrying out the parameter optimisation procedure; writing relevant parts of the paper.

PAPER “D”

Lindgren, L.-E., K. Domkin & S. Hansson, “Dislocations, vacancies and solute diffusion in physically based plasticity model for AISI 316L”, manuscript to be submitted to an international journal

Dislocation processes mentioned in the PAPER “C”, such as short-range and long-range interactions of dislocations and obstacles, interactions of dislocations and vacancies, are further explored in a comprehensive literature study. An advanced dislocation model is constructed in a

21

systematic way from the “sub-models,” each for particular dislocation mechanism. The resulting model consists of a coupled set of evolution equations for dislocation density and vacancy concentration. Furthermore, it includes the effect of diffusing solutes in order to describe dynamic strain-aging. The model parameters are calibrated by comparison with a set of compression tests for an AISI 316L steel using custom optimisation toolbox in Matlab. The paper also describes the numerical algorithm used to solve these strongly nonlinear relations applicable for user routines in finite element codes. The model has been formulated in a way that alleviates the replacements of different sub-models of the material model. Thus the proposed formulation is a platform for further development.

Author’s contribution: part of the initial literature survey, discussions of possible options in the dislocation modelling; derivation of the stress-update algorithm and ideas how to circumvent numerical issues with the system of non-linear evolution equations, programming the Matlab scripts for the toolbox according to the derived algorithm; writing relevant parts of the paper.

PAPER “E”

Domkin, K. & L.-E. Lindgren, “Stress update algorithm extended to strain induced martensite formation”, manuscript to be submitted to an international journal

A model for the strain-induced martensite transformation (SIMT) in metastable austenitic stainless steel is combined with a simple mixture rule for the properties of austenite-martensite phase mixture to deliver the yield stress, elastic moduli and thermal dilation as functions of the martensite fraction, accumulated plastic strain of the two phases, and temperature. The model is laid out as a set of coupled differential equations for martensite fraction, fraction of the shear bands, plastic strain in austenite and plastic strain in martensite. These four variables are identified as internal state variables in the framework of the rate-independent deviatoric plasticity.The implicit return-map algorithm of standard plasticity is extended to accommodate the above model of strain-induced martensite formation. This formation affects the thermo-elastoplastic properties of the material. Thereby, the phase change also indirectly makes all material properties dependent on the plastic strain which manifests itself in a major physical unsymmetry of the material behaviour: deviatoric strains affect the volumetric stress. This is further complicated by the evolution equations of internal variables being coupled with the current stress tensor via triaxiality factor. The stress-strain algorithm has been derived to accommodate these complications and a consistent algorithmic constitutive matrix has also been derived. The algorithm and the model are implemented (both in its general 3-D form and in a simplified 1-D form) in a custom optimisation toolbox in Matlab. Some examples of convergence properties of the stress-strain algorithm as well as its consistent constitutive matrix are shown for a uniaxial case. The formulation of the algorithm is based on quite general assumptions about the form of the hardening and evolution equations, and thus it is believed to be useful not only for the presented SIMT model but also perhaps for the other future models, such as advanced dislocation models.

Author’s contribution: derivation of the stress-update algorithm and consistent tangent; programming it in the Matlab scripts for the optimisation toolbox; running the test load cases to demonstrate convergence properties; writing half of the paper.

22

9 Conclusions and future work

The aim of the present work is to find and implement physically based material models of metal plasticity applicable for finite element simulations of metal forming with the research questions given in Chapter 1.1.The questions of their advantages compared with more commonly used empirical models may be discussed with respect to the following issues:

Do physically based models describe material behaviour with higher accuracy? Do they have a larger range of validity? Is determination of material parameters tractable in terms of number of parameters and necessary tests as well as their robustness? Is accurate and efficient implementation of the models into finite element codes possible?

The literature review of the advances in material modelling of metal plasticity has revealed that in principle it is possible to find material models that (a) are formulated on the macro-level in the framework of standard continuum plasticity, and therefore suitable for simulations of metal forming processes; and (b) are based on considerations of the underlying physical mechanisms of deformation. These are the dislocation density models.

Here are the conclusions which follow from the study of this kind of models.

Empirical models and table interpolations can be fitted to the mechanical test data with high accuracy. There is no guarantee that a dislocation density model would be automatically more accurate. It depends very much on finding the correct models, equations, for the physical processes dominating the deformation behaviour.

There are strong reasons to believe that physically based models can be extrapolated to a wider range of strains, strain rates, temperatures and other process conditions than engineering models, provided that the physical processes described by the model are still dominating and no new phenomena occur.

Physically based models of properties dependent on material microstructure can be more suitable for simulations of processes with complex histories of microstructure changes, where empirical models and table interpolations may be too cumbersome to handle.

Once an adequate and robust physically based material model is developed, the effect of particular material parameters can be, to a certain extent, evaluated numerically without further mechanical testing. This may be useful in design of materials.

Parameter optimisation procedures are algorithmically simple and well established. However, numerical difficulties such as non-uniqueness of the solution, high sensitivity to the starting guess-value and to the choice of the error function, appear to be a common problem with advanced material models. To describe more complex material behaviour a more advanced dislocation density model is needed, which in turn requires greater number of parameters to be determined from experiments. Simultaneous curve-fitting of multiple experimental curves of different mechanical testing types is advised – to trigger all desired effects in the model and to achieve more robust optimisation results.

Parameters of dislocation density models usually have clear physical interpretation, and therefore it is possible to obtain the values of some of the parameters from sources other than mechanical testing.

23

Numerical implementation of dislocation density models into finite element codes is straightforward. With numerically accurate and computationally efficient standard plasticity stress-update algorithms it is possible to apply these models in large-scale simulations of metal forming with little computer-time overhead for the material model.

As noted above, performance of dislocation density models very much depends on which physical mechanisms of deformation are included in the model, and if these mechanisms are well-understood and adequate equations are found to describe them. This appears to be the central problem of this kind of modelling, with many unresolved issues, which opens up quite a number of possibilities for the future study.

For instance, the simple model of kinematic hardening introduced in the PAPER “B” to describe the cyclic tests could be improved. Further study of the mechanism of dynamic strain aging together with the effects of twinning could improve the model in PAPER “D”.

10 References

Aretz, H. et al. (2000). Integration of physically based models into FEM and application in simulation of metal forming processes. Model. Simul. Mater. Sci. Eng. 8(6): 881-891.

Asaro, R.J., Rice, J.R. (1977). Strain localization in ductile single crystals. J. Mech. Phys. Solids, 25, 309-338

Bammann, D.J. (1984). Internal variable model of viscoplasticity. Int. J. Eng. Sci. 22(8-10): 1041-1053.

Bammann, D.J. (1990). Modeling temperature and strain rate dependent large deformations of metals. Appl. Mech. Rev. 43(5, part 2): 312-319.

Belytschko, T. , W.K. Liu and B. Moran, (2000). Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons.

Bergström, Y. (1969/70). “Dislocation model for the stress-strain behaviour of polycrystalline alpha-iron with special emphasis on the variation of the densities of mobile and immobile dislocations”, Mater. Sci. Eng., Vol. 5, pp.193-200.

Bergström, Y. (1983). The plastic deformation of metals - A dislocation model and its applicability. Reviews on powder metallurgy and physical ceramics 2(2,3): 79-265.

Bergström, Y. & Roberts, W. (1971). Dislocation model for dynamical strain ageing of alpha-fe in the jerky-flow region. Acta Metall. 19(11): 1243-1251.

Bergström, Y. & Hallen, H. (1982). An improved dislocation model for the stress--strain behavior of polycrystalline alpha-iron. Mater. Sci. Eng. 55(1): 49-61.

Bergström, Y. & Olund, S. (1982). The forming limit diagram of sheet metals and effects of strain path changes on formability: a dislocation treatment. Mater. Sci. Eng. 56(1): 47-61.

Busso, E.P. (1998). A continuum theory for dynamic recrystallization with microstructure-related length scales. Int. J. Plast. 14(4-5): 319-353.

Busso, E.P. & McClintock, F.A. (1996). A dislocation mechanics-based crystallographic model of a B2- type intermetallic alloy. Int. J. Plast. 12(1): 1-28.

24

Crisfield, M.A. (1991). Non-linear Finite Element Analysis of Solids and Structures, Volume 1: Essentials, John Wiley & Sons.

Crisfield, M.A. (1997). Non-Linear Finite Element Analysis of Solids and Structures, Vol. 2 Advanced topics, J Wiley & Sons.

Devincre, B. & Kubin, L.P. (1994). Simulations of forest interactions and strain-hardening in fcc crystals. Model. Simul. Mater. Sci. Eng. 2(3A): 559-570.

Estrin, Y. (1998). Dislocation theory based constitutive modelling: foundations and applications. J. Mater. Process. Technol. 80-81: 33-39.

Estrin, Y. (1999). Syntheses: Playing scales - a brief summary. Model. Simul. Mater. Sci. Eng. 7(5): 747-751.

Estrin, Y., Braasch, H., & Brechet, Y. (1996). A dislocation density based constitutive model for cyclic deformation. J. Eng. Mater. Technol. 118(4): 441-447.

Estrin, Y. & Mecking, H. (1984) A unified phenomenological description of work-hardening and creep based on one-parameter models. Acta Metall. 32: 57-70.

Fletcher, R. (1980). Practical Methods of Optimization, Vol. 1 Unconstrained Optimization, John Wiley & Sons.

Fletcher, R. (1981). Practical Methods of Optimization, Vol. 2 Constrained Optimization, John Wiley & Sons.

Frost, H.J. and M.F. Ashby. (1982). Deformation-Mechanism Maps - The Plasticity and Creep of Metals and Ceramics, Pergamon Press.

Gottstein, G. & Argon, A.S. (1987). Dislocation theory of steady-state deformation and its approach in creep and dynamic tests. Acta Metall. 35(6): 1261-1271.

Hasegawa, T., Takahashi, T., & Okazaki, K. (2000). Deformation parameters governing tensile elongation for a mechanically milled Al-1.1at.%Mg-1.2at.%Cu alloy tested in tension at constant true strain rates. Acta Mater. 48(8): 1789-1796.

Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford University Press.

Hughes, T.R.J. (1983). Numerical Implementation of Constitutive Models: Rate-independent Deviatoric Plasticity, Workshop on the Theoretical Foundation for Large-Scale Computations of North Western University, Evanston, Illinois.

Kocks, U.F. (1966). A statistical theory of flow stress and work hardening. Philos. Mag. 13: 541.

Kocks, U.F. (1976). Laws for work-hardening and low temperature creep. J. Eng. Mat. Tech. 98: 76-85.

Laukonis, J.V. & Ghosh, A.K. 1978. Met. Trans. 9A: 1849-56.

Lemaitre, J. & J.-L. Chaboche. (1990). Mechanics of Solid Materials, Cambridge University Press.

Lindgren, L.-E., H. Alberg & K. Domkin. (2003). “Constitutive modelling and parameter optimisation” in “Computational Plasticity - Fundamentals and Applications” (eds. D.R.J. Owen & E. Oñate). Proc. VII Int. Conf. on Comput. Plast. COMPLAS VII, CIMNE, Barcelona, Spain, April 2003, on CD-ROM.

Luce, R. et al. (2001). Application of a dislocation model for FE-process simulation. Comput. Mater. Sci. 21(1): 1-8.

Mahnken, R. and E. Stein, (1996). "Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigation", Int. J. Plasticity, 12 No.4, pp. 451-479

25

Marshall, P., (1984). Austenitic Stainless Steels - Microstructure and mechanical properties, Essex, England: Elsevier Applied Science Publishers Ltd.

McDowell, D.L. (2000). “Modeling and experiments in plasticity”. International Journal of Solids and Structures. 37(1-2): p. 293-309.

Meissonnier, F. T., Busso, E. P., & O'Dowd, N. P. (2001). Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains. Int. J. Plast. 17(4): 601-640.

Militzer, M., W. Sun, and J. Jonas (1994), Modelling the effect of deformation-induced vacancies on segregation and precipitation. Acta Metall. Mater., 42(1): p. 133-144

Miller, A.K. (ed.), (1987). Unified Constitutive Equations for Creep and Plasticity, Elsevier Applied Science Publishers Ltd.

Ponthot, J.P. (2002). "Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes", International Journal of Plasticity, 18(1), 91-126.

Reed-Hill, R.E. and R. Abbaschian, (1992). Physical Metallurgy Principles, PWS-KENT Publishing Company

Rhee, M. et al. (1998). Models for long-/short-range interactions and cross slip in 3D dislocation simulation of BCC single crystals. Model. Simul. Mater. Sci. Eng. 6(4): 467-492.

Roberts, W. & Bergström, Y. (1973). The stress-strain behaviour of single crystals and polycrystals of f.c.c. metals. A new dislocation treatment. Acta Metall. 21: 457-469.

Roters, F., Raabe, D., & Gottstein, G. (2000). Work hardening in heterogeneous alloys - A microstructural approach based on three internal state variables. Acta Mater. 48(17): 4181-4189.

Simo, J.C. (1988). "A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part II. Computational aspects", Computer Methods in Applied Mechanics and Engineering, 68, 1-31

Simo, J.C. & T.J.R. Hughes. (1997). Computational Inelasticity, Springer-Verlag.

Simo, J. C. & K. Pister, (1985). "Remarks on rate constitutive constitutive equations for finite deformation problems: Computational implications", Computer Methods in Applied Mechanics and Engineering, 46, 201-215

Simo, J.C. & Taylor, R.L. (1986). A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Methods Eng. 22: 649-670.

Stouffer, D.C. & L.T. Dame. (1996). Inelastic Deformations of Metals - Models, mechanical properties and metallurgy, John Wiley & Sons, Inc.

Zbib, H.M. et al. (2000). 3D dislocation dynamics: stress-strain behaviour and hardening mechanisms in fcc and bcc metals. J. Nuclear Mater. 276: 154-165.

Paper A

1 INTRODUCTION

A large number of constitutive models of the plastic behavior of metals are available for finite element modeling. However, their applicability is usually limited in terms of varying strain, strain rate and temperature. Material models based on consideration of the underlying physical processes are expected to have a larger range of usability in this respect. This work focus on the use of dislocation density as a way to model plastic deformation. During the 1970´s, in a series of investigations by Swedish re-searchers, several basic dislocation material models were developed (Roberts & Bergström 1973, Bergström 1983). Similar ideas were independently exploited by Kocks (1976), and later developed by Estrin & Mecking (1984) and Estrin (1998). Many other researchers incorporated dislocations either in-directly (Bammann 1990) or directly (Busso 1998) into material models. In this paper, one of these models for anisotropic plasticity (Bergström 1983) has been implemented in a user material routine for the implicit FE-code ABAQUS v5.8. Different cup-forming cases have been investigated, and the form-ing limit diagram has been computed.

2 MATERIAL MODEL

2.1 Large deformation analysis Simulation of sheet metal forming requires a large deformation analysis. We will focus on the constitu-tive equations in the following and will therefore only comment shortly how the described stress up-dating is implemented in a large deformation con-text. An additive decomposition of strain rate is as-sumed. The Green-Naghdi objective stress rate is

proportional, via Hooke´s law, to the elastic velocity strain tensor (Simo & Pister 1984). The hypo-elastic relations are the basis for applying the constitutive model and the stress computation is performed at a given reference configuration. The algorithm, for computing stresses will then be form-identical to the small strain case (Hughes 1983). The following presentation will therefore only describe stress and strain relations without any reference to large or small strains. The hypoelastic stress-strain relation is

pe CC (1)

whereC is the constitutive matrix for the elastic ma-terial in this reference configuration.

2.2 Theory of plasticity The theory of rate-independent plasticity consists of three parts: a yield criterion, a flow rule and a hard-ening rule. These are used together with the consis-tency condition, which states that the stress must remain on the yield surface during a plastic process, to compute the plastic strains. We will describe each one of them below as a background to their numeri-cal implementation, which is described thereafter. Crisfield (1997) has a thorough discussion of finite element implementation of constitutive models.

The Hill’s yield criterion for anisotropic metal plasticity (Hill 1947, 1950) can be written as

yT

yf P21 (2)

where y is the yield limit, is the effective stress, is the stress tensor and P is a symmetric matrix,

which describes the shape of the yield surface (Cris-field 1997).

Physically based material model in sheet metal forming

K. Domkin Dalarna University, SE-781 88 Borlänge, Sweden

L.-E. Lindgren Luleå University of Technology, SE-971 87 Luleå, Sweden

L. Troive Dalarna University, SE-781 88 Borlänge, Sweden

ABSTRACT: A model for dislocation density is incorporated into a material model for anisotropic plasticity. The change in dislocation density is computed from the plastic straining so that the hardening becomes nonlinear. The material model has isotropic hardening, but the hardening parameters are evaluated based on uniaxial and biaxial material tests. A parameter is used to interpolate between these two tests when computing the dislocation density at every increment. Thus, some kind of strain path dependent isotropic hardening is obtained. The model is applied to a circular cup forming simulation. The forming limit diagram is also pre-dicted.

1

The associated flow rule is

aaP222

pp f (3)

where is the plastic multiplier, and Pa is the flow vector. The last equality above comes from the definition of effective plastic strain, p . It is as-sumed that the hardening is isotropic so the initial anisotropy of the yield surface, described by P is not changed during the plastic yielding. The stress-updating algorithm needs to accommodate variable hardening.

2.3 Tasks for the stress-strain algorithm The tasks for a constitutive routine are i. Update stress from given strain increment. ii. Compute consistent constitutive matrix. We have chosen the implicit, backward Euler al-

gorithm. The method has some desirable properties. It is unconditionally stable for non-smooth surfaces and has a good performance for large strain incre-ments (Ortiz & Popov 1985). Enforcing the consis-tency condition at the end of the time step will lead to a symmetric, consistent constitutive matrix (Simo & Taylor 1986). The latter will also be appropriate, as we also prefer to set up the equation of motion at the end of the time step when formulating the global finite element equations. Then the stresses and dis-placements presented at the end of each time step fulfill the equation of motion and also the consis-tency condition of the plastic flow.

2.3.1 Stress updating The stress updating is based on a given initial state, A, and a given increment in strain, . The algo-rithm must find the final stress at state C that re-quires the knowledge of the elastic and plastic parts of the total strain increment. The hypo-elastic rela-tion between increment in (objective) stress and elastic strain, Equation 1, is combined with Equation 3 to give

CC

BCC

AC CPCPC22

(4)

where the plastic multiplier is denoted in this in-cremental context, B is the so-called trial stress and is computed based on assuming an elastic increment in Equation 1. The consistency condition is evalu-ated at the end of the time step in the Euler back-ward algorithm in order to find the amount of plastic flow during the increment. We rewrite Equation 4 as

BBC

C BCPI 1-1

)(2

(5)

where I is the unit matrix.

This is inserted into the consistency condition for a plastic process, 0f , in Equation 2.

021

B1

yCTT

BCf PBB (6)

This is a non-linear scalar equation in as we have B and yC ( ). It is possible to solve it by using a truncated Taylor series for iterative im-provements of a given approximate . Denoting the current value of a variable with a left superscript, i[], and the iterative change in the plastic multiplier,

ii 1 , after some algebra, we obtain

CCyCC

Ci

HK

Kf

4

(7)

where

BTT

BK CPBPBB 11 (8)

The solution of is then used to obtain the in-crement in yield stress, the plastic strain components and the stresses.

2.3.2 Consistent tangent The Newton-Raphson’s solution procedure for the non-linear system of coupled equations in finite element formulation gives second order convergence if the used stiffness matrix is a so-called tangent stiffness matrix. The constitutive matrix needed for when computing the tangent stiffness matrix is a consistent constitutive matrix defined as

ctc with ctc (9)

where is the change in stress due to a varia-tion in the strain increment, ii 1 .The superscript i is now the iterative counter in the Newton-Raphson iterations. The consistent constitu-tive matrix, ctc , should be evaluated at the instant in the time step for which the equilibrium equations or equations of motion are to be solved, which is at state C.

The evaluation of Equation 9 for the specific nu-merical solution procedure given in the previous section gives

RaaRaRaRc T

yCC

T

ct H 24 (10)

where R = B–1C, and CCH1 . It should be noted that the stresses used should

refer to the configuration at the end of the time step. The stress updating that gives C, in a large defor-mation algorithm, is performed in that reference configuration. Then the consistent constitutive ma-trix should be set up using the updated stresses after

2

they have been rotated to the configuration at the end of the time step.

3 DISLOCATION MODEL AND YIELD STRESS

3.1 The yield stress It is assumed that the yield stress, y, of crystalline materials is related to the total dislocation density, ,as

Gboy (11)

where o is a strain independent friction stress, is a nondimensional constant, G the shear modulus, and b the magnitude of dislocation Burgers vector. The essential correctness of this relation has been verified both experimentally (Bailey & Hirsch 1962) and theoretically (Hirsch 1960). The main problem in formulating a dislocation theory, for the behav-iour of the material, is to derive the relationship be-tween and the deformation. The effective plastic strain, p , is assumed to characterize the latter.

3.2 The dislocation density evolution The theoretical model for bcc metals proposed by Bergström (1983) states that the mobile dislocation density is (a) strain independent, and (b) much smaller than the immobile dislocation density, i.Thus, i , d = d i, and the evolution equation is expressed as

AUdd p

p )( (12)

where U is the rate of dislocation immobilization; —a measure of dislocation re-mobilization, or re-

covery; and A—total rate of dislocation annihilation. All these parameters may depend on temperature and strain rate.

The parameter U is expressed as:

)()( p

p

sbmU (13)

where m is an orientation factor, b is the magnitude of the Burgers vector, and s is the mean free path of a mobile dislocation.

It was found by Bergström (1983) that the annihilation process could be ignored, especially when temperatures are below 0.5Tm (homologous temperature). Additionally, he pointed out, based on several TEM studies of ferritic steel, that the mean free path, s, is strain independent in the whole interval of strains. Then, Equation 11 becomes:

pp

eeUGboP

y 01 (14)

where 0 is the initial dislocation density.

For bcc metal, this model provides a very good description of experimental stress-strain data, and also a good understanding of the physical mecha-nisms and parameters involved (Chao 1999).

3.3 Strain path dependent hardening In the following, the theories and material data used are taken from Bergström & Öhlund (1981). The values of the basic parameters for the investigated material (AK-steel) are shown in Table 1.

Table 1. Basic material parameters, AK-steel. ______________________________________________Variable Value Unit ______________________________________________Shear modulus, G 7.8 .104 MPa Burgers vector, b 2.5 .10–12 m Initial dislocation density, 1.0 . m–2

Constant, 0.8 - ______________________________________________

The experimental stress-strain data was obtained in an earlier study (Laukonis & Ghosh 1978). The sheet AK-steel specimens were tested in two ways: in uniaxial, and in balanced biaxial tension. The theoretical expression, Equation 14, was then fitted to the two sets of data using the least square method, to obtain the parameters U1, 1, 01 and U2, 2, 02for the two curves, respectively (Fig. 1).

0 0.05 0.1 0.15 0.2 0.25100

200

300

400

500

Figure 1. True effective stress (MPa) vs. true effective strain. Theoretical curves for AK-steel.

Finally, to evaluate the hardening parameters un-der arbitrary conditions of in-plane straining, a vari-able is introduced, which defines the strain path:

1

2

dd (15)

where d 1 and d 2 are the major and minor princi-pal plastic strain increments in the plane of the sheet. To combine the results received from the two cases of straining, a linear dependence of the parame-ters U and is assumed (Bergström & Öhlund 1981), which gives the following relations:

21510)73.127.0( mU (16a)

Uniaxial straining ( = –0.5)U1 = 1.6 . 1015 m–2

1 = 4.0 01 125 MPa

Balanced biaxial straining ( = 1) U2 = 2.0 . 1015 m–2

2 = 0.0 02 125 MPa

y

3

67.267.2 (16b)

The friction stress parameter, , was found to be approximately unaffected by variations in and was therefore set to the mean value of 01 and 02, equal to 125 MPa.

For proportional straining, Equation 14 can be used directly. In a multistep deformation, is no longer a constant, so the equation should be used stepwise, with 0 now meaning the dislocation den-sity from previous steps, and p the effective plastic strain increment.

4 THE FORMING LIMIT DIAGRAM

4.1 Experimental FLD The formability of sheet metals is traditionally char-acterized by the forming limit diagram or FLD, first introduced by Keeler & Backofen (1964) and Good-win (1968). The FLD of the material is represented as a curve of limiting strains in a plot of major strain

1 vs. minor strain 2. Experimental FLD are ob-served to depend strongly on the strain path.

According to Laukonis & Ghosh (1978), the formability test procedure consists of clamping rec-tangular sheet blanks of various widths over a lock-bead and stretching to failure over a hemi-spherical punch. The FLD is constructed by measuring princi-pal in-plane strains from regions just outside visible necks or fractures. A noteworthy feature of the FLD is that the decrease of failure strains after a prestrain is material dependent. So the material model should account for this behavior.

4.2 Plastic instability and necking Numerous attempts have been made to formulate the theoretical relationship for the FLD. The approach proposed by Marciniak and Kuczynski (1967) is based on the assumption that an initial inhomogene-ous region exists in the sheet plane where a localized neck can develop. In the present work, the classical conditions of plastic instability by Hill (1952) and Swift (1952) are used. It should be noted that these theories are exact only for a rigid-plastic material and for uniform fields. In the general case of elasto-plasticity and arbitrary loading, these criteria pro-vide a reasonably good approximation.

According to Hill’s approach, the localized neck is expected along the line of zero extension. This condition is realized only if one of the strain incre-ments is negative (i.e. < 0). The instability crite-rion by Swift is appropriate for the case of biaxial tension ( > 0). It indicates the beginning of diffuse necking, i.e. plastic straining under constant loading.

It is possible to express both criteria in the form of condition for the hardening rate and yield stress:

zy

py (17)

The parameter z is a function of stress and normal anisotropy coefficient R and has different values for localized (Hill) and diffuse necking (Swift):

xxRRxRRzHill 1

12123/2 21221

(18)

322

23221

112112123/2

xRxxRRRxRRxRRzSwift (19)

where x is the ratio of principal stresses,

RRRRx

11

1

2 . (20)

Substitution of Equation 14 into Equation 17 leads to the analytical solution for effective strain to necking. Thus, given the value of , principal strains at plastic instability are obtained, and the theoretical FLD is constructed.

One should note that FLD obtained in this way is valid for the in-plane deformation, as the in-plane material test data was used to evaluate the parame-ters. To account for the out-of-plane forming limit, which usually falls above that for in-plane, the re-covery parameter, , is reduced by the factor k = 3.0 (Bergström & Öhlund 1981). Although such reduc-tion does not emerge naturally for any physical rea-son and therefore seems questionable, it is the only way, in terms of the present theory, to explain the higher out-of-plane strain limit. The resulting theo-retical limit curves for different amount of balanced biaxial prestrain (Fig. 2) are in good agreement with experimental FLD by Laukonis & Ghosh (1978).

-30 -20 -10 0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

Figure 2. Theoretical out-of-plane FLD (engineering strains) as a function of in-plane balanced biaxial prestrain.

1 (%)

2 (%)

in-plane biaxial prestrain 0.0 0.070 0.152 0.271

4

5 COMPUTATIONAL MODEL

5.1 Implementation of the dislocation model In order to test the dislocation model, it has been implemented in an ABAQUS user routine UMAT according to the anisotropic plasticity algorithm dis-cussed in section 2. The incremental form of Equa-tions 11 and 14 were used:

ACyAyC Gb (21)

)exp()exp(1 ACU (22)

CC

C UGbH2

(23)

where variables with subscript A and C refer to the beginning and to the end of an increment. The pa-rameters U and depend on , which is updated every increment based on principal values of p in the sheet plane. The rotation of the principal axes is ignored. If the –value falls outside the range [–0.5; 1.0], the hardening is assumed to be the same as in uniaxial straining.

5.2 Finite element model The circular cup forming case as described in sec-tion 4.1 (the punch test) was chosen for simulation. The holding force = 12,000 N. Table 2 contains the model geometry data.

Table 2. Dimensions of the cup-forming model. ______________________________________________Punch pole radius 50.80 mm Circular punch diameter 101.60 mm Circular die diameter 105.66 mm Die lubricant radius of curvature 6.735 mm ______________________________________________

Several blank width cases were investigated: (I) 53 mm; (II) 76 mm; and (III) 152 mm. The blank length = 152 mm in all cases. For the models I and II, additional simulations were performed with a given balanced biaxial in-plane prestrain = 0.2. The material parameters presented in section 3.3 were used. For the sake of simplicity, isotropic (von Mises) plasticity was adopted, although the formula-tion can deal with anisotropic yielding as well.

True curvature 3D quadrilateral shell (S4) ele-ments were used for the blank. Only a quarter of the structure was modeled due to the symmetry.

5.3 Evaluation of the limit strains According to the formulation in section 4.2, a post-processing variable ‘risk of necking’ is defined:

y

Hz1 (24)

where z = zHill if < 0, and z = zSwift otherwise. Dur-ing elastic or stable plastic deformation is nega-tive. When, at an integration point, the condition

0 first satisfied, the principal total strains in the sheet plane are computed, and these are assumed to be the limiting strains at this point.

6 RESULTS AND DISCUSSION

For convenience, the –value was modified:

:= exp( ) – 1, if < 0; else := 1 – exp(– ) The field for the case III (square blank) is pre-sented in Figure 3. The regions with positive are considered unsafe with respect to necking.

Figure 3. The field of the ‘risk of necking’ variable on a de-formed cup, case III. Positive values indicate plastic instability.

The left side FLD was studied numerically in the cases I and II (Fig. 4). A good agreement with the theoretical curves is achieved, as the strain paths during the deformation in these cases were nearly proportional (after initial prestrain, if prestrained).

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

Figure 4. Limit strains for the cases I and II, with and without prestrain. The strain paths for certain points are also shown.

The right side FLD can be computed for the model case III, where biaxial tension dominates

+ case I . case II

case I, prestrained case II, prestrained

analytical curves actual strain paths

2

1

5

(Fig. 5). The scattering in the limit strain points is due to significant non-proportionality of strain path, which varies from point to point. Moreover, the FE-solution gives limit strains lower than theoretical FLD, which was derived for exactly proportional paths. Obviously, some improvements to the treat-ment of diffuse necking are required, as well as to the hardening behavior in the range of positive in out-of-plane deformation.

-0.2 -0.1 0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

Figure 5. Limit strains for the case III (square blank). The ac-tual strain paths and theoretical curve are also shown.

7. CONCLUSIONS

The present dislocation model, based on the average behavior of large numbers of dislocations, has been shown (Chao 1999) to give a very good description of the conventional stress-strain curves. The varia-tion of the parameters involved is in good agreement with their physical meaning.

The model is incorporated into an FE-code, al-lowing the direct computation of dislocation density and related variables. Several cases of cup forming process are solved in this work. The strain path ef-fects are found to be significant.

Application of the model to the forming limit analysis shows its capability to capture the main trends in the FLD shape and strain path dependence. However, there remains some questions about the validity of using the parameter approach for non-proportional loading, which is believed to be the cause of the discrepancy between the theoretical FLD diagram and the actual simulations of cup forming for case III.

In the future work, it is planned to study the tem-perature and strain rate effects within the dislocation model, as well as the texture evolution and its rela-tion to the strain path dependence of FLD, as it is expected that texture development plays a signifi-cant role during forming process.

ACKNOWLEDGEMENT The financial support by the Swedish Council for Technical Research, TFR, is gratefully acknowl-edged.

REFERENCES

Bailey, J.E. & Hirsch, P.B. 1962, Phil. Mag. 7: 69. Bammann, D.J. 1990. Modeling temperature and strain rate

dependent large deformations of metals. Appl. Mech. Rev.43(5, part 2): 312-319.

Bergström, Y. 1983. The plastic deformation of metals - A dis-location model and its applicability. Reviews on powder metallurgy and physical ceramics 2(2,3): 79-265.

Bergström, Y. & Öhlund S. 1981. Material Center, Royal Inst. of Technology, Stockholm. TRITA-MAC-0187.

Busso, E.P. 1998. A continuum theory for dynamic recrystalli-zation with microstructure-related length scales. Int. J. Plast. 14(4-5): 319-353.

Chao, Ò. 1999. Plastic deformation mechanisms in soft an-nealed SAE 52100 ball bearing steel. Degree project II. Ovako Steel AB, Sweden.

Crisfield, M.A. 1997. Non-linear Finite Element Analysis of Solids and Structures Vol 2 Advanced Topics. J Wiley.

Estrin, Y. 1998. Dislocation theory based constitutive model-ling: foundations and applications. J. Mater. Process. Tech-nol. 80-81: 33-39.

Estrin, Y. & Mecking, H. 1984 A unified phenomenological description of work-hardening and creep based on one-parameter models. Acta Metall. 32: 57-70.

Goodwin, G.M. 1968. Application of strain analysis to sheet metal forming problems in the press shop. SAE Technical paper No.680093.

Hill, R. 1947. A theory of the yielding and plastic flow of ani-sotropic materials. Proc. Roy. Soc. A193: 281-297.

Hill, R. 1950. The Mathematical Theory of Plasticity. Oxford University Press.

Hill, R. 1952. On discontinuous plastic states, with special ref-erence to localized necking in thin sheets. J. Mech. Phys. Solids 1: 19-30.

Hirsch, P.B. 1960, Phil. Mag. 5: 485. Hughes, T.R.J. 1983. Numerical Implementation of Constitu-

tive Models: Rate-independent Deviatoric Plasticity, Work-shop on the Theoretical Foundation for Large-Scale Com-putations of North Western University, Evanston, Illinois.

Keeler, S.P. & Backofen, W.A. 1964. Plastic instability and fracture in sheets stretched over rigid punches. ASM Trans.56: 25-48.

Kocks, U.F. 1976. Laws for work-hardening and low tempera-ture creep. J. Eng. Mat. Tech. 98: 76-85.

Laukonis, J.V. & Ghosh, A.K. 1978. Met. Trans. 9A: 1849-56. Marciniak, Z. & Kuczynski, K. 1967. Limit strains in the

processes of stretch forming sheet metal. Int. J. Mech. Sci.9: 609.

Ortiz, M. & Popov, E.P. 1985. Accuracy and stability of inte-gration algorithms for elastoplastic constitutive relations. Int. J. Numer. Methods Eng. 21: 1561-76.

Roberts, W. & Bergström, Y. 1973. The stress-strain behaviour of single crystals and polycrystals of f.c.c. metals. A new dislocation treatment. Acta Metall. 21: 457-469.

Simo, J.C. & Pister, K.S. 1984. Remarks on Rate constitutive equations for finite deformation problems: Computational implications. Comp. Met. Appl. Mech. Engnr. 46: 201.

Simo, J.C. & Taylor, R.L. 1986. A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Methods Eng. 22: 649-670.

Swift, H.W. 1952. Plastic instability under plane stress. J. Mech. Phys. Solids 1: 1-18.

2

1

case III

analytical curve actual strain paths

6

Paper B

VII International Conference on Computational Plasticity COMPLAS 2003

E. Oñate and D. R. J. Owen (Eds) CIMNE, Barcelona, 2003

DISLOCATION DENSITY BASED MODELS FOR PLASTIC

HARDENING AND PARAMETER IDENTIFICATION

Konstantin Domkin†, Lars-Erik Lindgren

*,†and Peter Segle

‡

†Dalarna University, 781 88 Borlänge, Sweden e-mail: [email protected], web page: http://www.du.se

*Luleå University of Technology, 971 87 Luleå, Sweden e-mail: [email protected], web page: http://www.cad.luth.se

‡Swedish Institute for Metals Research, Drottning Kristinas Väg 48, 114 28 Stockholm, Swedene-mail: [email protected], web page: http://www.simr.se

Key words: Identification, constitutive, cyclic tests, modelling, dislocation density.

Abstract. In this paper the parameter identification using dislocation density based materialmodel is studied. The model is rate-dependent and includes isotropic strain-hardening/softening as well as kinematic hardening. The model is implemented as a part of the custom toolbox for parameter identification (described in the accompanying paper) using Matlab®. A general stress-strain algorithm is used in the calculations, so the same logic can also be used when implementing the material models into a finite element code. The stress-update algorithm of rate-dependent plasticity is chosen in the form that has the yield surface for which a so-called consistency condition exists. The amount of plasticity in a strain increment is determined by the consistency condition, whereas the internal variables historyand yield stress depend on the plastic strain-rate. The paper focuses on the use of physically based material models. The dislocation density concept links the macroscopic stresses and strains to the underlying micro-structural processes of plastic deformation. The material models define evolution equations for the densities of mobile, immobile locked and immobile recoverable dislocations. The physical significance of the model parameters is highlighted. The developed toolbox is used to determine material parameters of a high-strength steel for achosen dislocation density model fitted to the constant amplitude fully reversed strain controlled cyclic test curves. Parameter sensitivity is briefly discussed.

1

mailto:[email protected]

http://www.du.se


http://www.cad.luth.se


http://www.simr.se

Konstantin Domkin, Lars-Erik Lindgren and Peter Segle

1 INTRODUCTION

The material modelling is crucial for obtaining accurate results in simulations. It is possible in general to try out different spatial and temporal discretisation in order to reduce the error due to these factors. However, errors in the material model or in the pertaining parameterscannot be reduced by the numerical procedures. Modelling the material behaviour poses anextra challenge in simulation of manufacturing processes. The model must not only account for varying strain, strain rate and temperature but sometimes also changing microstructure ortexture evolution. The desired features of such models also include the possibility to obtain model parameters from conventional test data and to implement the model in a finite elementcode via user-defined material subroutines. This paper focuses on the physically based material models of metal plasticity, their numerical implementation and parameteridentification.

Two classical types of deviatoric plasticity models are outlined in this paper. One is based on the concept of a yield surface and the other uses a flow strength equation. The radial returnapproach is used to compute the stress from given strain in the parameter determinationprocess. It is a logic that can also be used in finite element codes. The way the algorithm handles both types of models is described. Material models are formulated to fit into thisframework. Although many empirical models for plastic hardening are widely used in FE simulations, their domain of validity and overall accuracy are usually limited, and thereforethe explicit physically based models are preferred. These models include evolution equations for some aspects of the underlying physical process, e.g. the generation of dislocations, and therefore are expected to have a larger range of validity than empirical models. Parameterdetermination for a rate-dependent dislocation model is described. It is then applied to cyclictests of high-strength steel.

2 CONSTITUTIVE MODELS

2.1 Terminology

Physically based material models are models where knowledge about the underlying physical process, dislocation processes etc, is used to formulate the constitutive equations.Engineering or empirical models, on the other hand, are determined by means of fitting modeland parameters with experimental data without considering the physical processes causing the observed behaviour. These empirical models are also named engineering models as they aremore common in engineering applications than the physically based material models.Otherwise, both types of models can be considered “engineering”.

Furthermore, two different types of physically based models exist. One option is to explicitly include the physical model as an evolution equation in the constitutive model:explicit physically based models. The other possibility is to determine the form of theconstitutive equation based on knowledge about the physical process causing the deformation.The latter is a so-called “model-based-phenomenology”, according to Frost and Ashby1. Wewill call it an implicit physically based model.

2


2.2 Viscoplasticity and rate-independent plasticity

The general framework of the deviatoric plasticity models are given in the accompanyingpaper2. We define a yield surface, f, for which a stress state inside, or and

, corresponds to an elastic deformation process. The surface is written as 0f 0f

0f

yf (1)

where y is called the yield limit and is the effective von Mises stress. The magnitude of the effective plastic strain rate is obtained from the so-called consistency condition for thistype of models with a yield surface

0 (2)f

as we can never have a stress state outside the yield surface in this model. In the followingsections we will refer to this type of models as “models with consistency condition”. Such models are usually rate-independent plasticity models but it is possible to include the effect of strain rate by having a rate-dependent yield limit.

The magnitude of the effective plastic strain rate is obtained from the flow strengthequation in the other type of models. These models are always rate-dependent plasticity(viscoplasticity) models. We will use relations where the rate is a monotonous increasingfunction3 of the excess stress f.

fgp (3)

where the bracket denotes that the rate is zero if 0f . We can in this case have .The introduction of g, and facilitates the identification with some models like those inRefs. 3 and 4. These notations will be used in the discussions later.

0f

Finally, maximum plastic dissipation is postulated leading to associated plasticity. Thus the yield or the flow potential surfaces are plastic potentials for the plastic strains. Then the components of the rate of the plastic strain is1

LL23pp f (4)

The hardening/softening behaviour of the material can have an isotropic and a kinematicpart. The first is accounted for by the change of the yield limit of the material and the latter by translating the yield surface by changing the back-stress .

The evolution equations for isotropic and kinematic hardening behaviour are discussed below and these relations can be used both for rate-dependent and rate-independent models.Thereafter, the stress-strain algorithm and dislocation models are discussed.

A broad class of deviatoric plasticity models with a consistency condition or a flow strength equation will be covered within the framework given above. We will not make any

1 We use the so called Voigt notation with engineering strain definition. See the Appendix.

3


difference between plasticity, viscoplasticity or creep strains but only use the term plastic strains in our notations. Temperature effects, phase changes, high strain rates and anisotropy are not addressed in the discussions below. See the accompanying paper2 for some examplesfor isotropic and kinematic hardening models.

2.3 Isotropic hardening

The isotropic hardening includes both hardening and recovery processes. They may be accommodated in the yield limit in yield function but also in viscosity, , of the flow strengthequation. We will describe some evolution equation for the yield limit below. The initial yieldlimit is given as

00 yy (5)

Its evolution equation can be written via the evolution of internal variables vector, :i

)(),(),(

,)()()( ts

ipp

ipp

ii

piyy (6)

)(i denotes hardening and dynamic recovery. Both are active during plastic straining.

Furthermore, we may have a static recovery term . The dislocation density will be used as internal variable in the Section 4.

)(i

)(si

2.4 Kinematic hardening

The so-called Bauschinger effect is accommodated by the use of back-stresses, . Theycorrespond to internal micro-stresses from pile up of dislocations and their interactions 5. The amount of observed kinematic hardening also depends on whether a small or large offsetdefinition of yield is used 6,7. Kocks argues (page 67 in Ref. 8) that it should be evaluated so it only includes “permanent” softening. However, modelling using a small offset definition can also be useful for some applications 6,7. The evolution equation below for kinematic hardening also includes hardening and recovery processes. The initial back stress is given as

00 (7)

and its evolution equation can be written as

L kinp

kinp

kin SDH 1

32 (8)

where the first term denotes the hardening process and the second term denotes dynamicrecovery. The hardening term corresponds to the Prager model. It does not work directly in plane stress subspace but can be reformulated for this case, see Ch 15 in Ref. 9. Furthermore,we may have a static recovery term Skin(t). It should be noted that the kinematic hardeningcould be very important for cyclic loading. It has been found that it maybe necessary toinclude nonlinear kinematic hardening for many cases, see for example References 11-16.

4


3 STRESS-STRAIN ALGORITHMS

The numerical solution of the constitutive relations for use in finite element simulationsmust be strain driven. This means that the strain is given and the stress should be computed.Furthermore, a consistent constitutive matrix is needed in order to retain the convergenceproperties of the Newton-Raphson method. The book by Simo and Hughes describes these aspects17 and Crisfield gives some basic numerical methods in Ch. 6 in Ref.18 and extensions in Ref. 9. Belytschko et al. also devote some chapters to constitutive models and stress-strainalgorithms19.

We use an operator-split approach where an elastic predictor is followed by a plastic corrector. We will describe it in small-deformation context as the algorithm is form-identicalfor large deformations if the stress updating is done in an unrotated configuration like when using the Green-Naghdi stress rate (Box 3 in Ref. 20, Ch. 19.3 in Ref. 9). We will formulatethe rate-dependent and rate-independent plasticity in parallel. A formulation can be used where the first is a sub case of the latter21,22. We will use another approach here as it wasfound to be better for models with high exponents in the power law. We follow the ideas in Ch. 15.12 in Ref. 9 and Ch 5.9.8 in Ref.19 where the flow strength equation is solved to obtain the effective plastic strain increment. This approach is also used in Refs.1, 23, 24.

We will specialise to isotropic elastic behaviour and will not include the static recoveryterm of Eq (8) in the derivation below. The back-stress recovery terms require additionalconsiderations, as the back-stress will not be coaxial with the increment in plastic strains. Thiscan be solved by the simultaneous solution of all stresses and back stresses using a fullyimplicit backward Euler, described in Ch 5.9.2 in Ref.19, or a semi-implicit, described in Ch 5.9.6 in Ref.19 and in Ref.24, with the back stress fixed to the value at the beginning of the increment. The latter leads to an un-symmetric consistent tangent matrix.

A generalized mid-point scheme is used by Chaboche and Cailletaud21, who discuss different simplifications. Saleeb et al. 25 use implicit integration of system of the unknowns, which they found was stable and effective for large increments. They uncouple the back stress and stress during iterations as an alternative to fixed-point iterations. They combine theNewton solution procedure used in the stress updating with a line search technique. Kirchner and Kollmann26 discuss the problem of integrating stiff constitutive equations using the modelby Hart27 as a test case.The fully implicit Euler scheme is presented in Boxes 1-4. It is identical to the radial returnalgorithm as the trial deviatoric stress is coaxial with the final deviatoric stress. Ortiz andPopov28 show that the fully implicit backward Euler scheme, closed point projection scheme,has a good stability as the strain increments become large and is the only unconditionalmidpoint scheme for non-smooth yield surfaces. Furthermore, it guarantees that the consistency condition or the flow strength equation is fulfilled at the end of the time step.

The stress-strain updating consists of the steps shown in Box 1 and the determination ofthe amount of plasticity for the plastic corrector phase is shown in Box 2. The derivative needed for the Newton iterations in Box 2 are given in Box 3 with some details in Box 4. Itcan be noted that this derivative is also part of the derivation for the consistent tangent. Thelatter is not included here.

5


Box 1. Overall logic for stress updating. 1.1 Compute ela tic predictor, trial states

CCCCCnnthnnenthnntr 111 (9)

ntrtr (10)

P trM

trtr

23 (11)

C is elastic stiffness matrix with temperature-dependent properties.1.2. Check for plastic or elastic increment

ytrtrtr f (12)

wherepnn

yytr T ,1 (13)

1.3. If 0 then plastic corrector is appliedftr

See Box 2 1.4. Update variables.

Box 2. Computation of increment in effective plastic strain Use

equationstrengthflowfor the.conditionyconsistencfor the.

1

1

pn

np

tfgf

(14)

Find the effective plastic strain increment that gives 0p

2.1. Initialise i=000

p (15)p

00 (16)2.2. Next estimate i=i+1

ppi

pi (17)

where a Taylor expansion gives

pi

p

ip

dd

1

1 (18)

See Box 3 for pdd and Box 4 for the evaluation of f needed for .

2.3. Check convergenceIf the function and/or p are small enough then stop the iterations else go back to step 2.2Note that is not non-dimensional for the case of consistency condition and therefore an appropriate scaling should be applied when setting the convergence tolerance.

6


Box 3. Derivative of with respect to effective plastic strain.The case with consistency conditionWe use the definition of the yield function with effective stress and yield limit. Thederivation in Box 4 gives

kinn

isonn

py

n

p

n

p

n

p

p

HHGd

dd

dd

fdd

d 111111

3 (19)

where py

py

ison

tH 11 is the isotropic hardening modulus,

and is the kinematic hardening modulus.kinn H1

The case with flow strength equation01 pn fgt

11

pp

n

pf

fgt

where the last derivative above is obtained from Eq. (19)

131

isokinp

n

p HHGf

gt

hHHGf

gt isokinp

n

p 31

(20)

11

fgth

n

(21)

Box 4. Evaluation of yield function and its derivative The yield function at the end of the increment is needed for the evaluation of f and inBox 2. Its derivative is also needed in the Newton procedures (Boxes 2 and 3).

Von Mises effective stress and its derivative with respect to plastic strainWe have incremental versions of Eqns (4) and (8).

L1

123 n

npp (22)

LL 11

111 132

32 n

np

kinnp

kinn HH (23)

Updating the stress by including the increment in back-stress and the plastic relaxationinto the trial stress gives

C kinnpntr

kinnpntrn HGH 11111

322

32 (24)

7


where the last term is obtained for isotropic elastic behaviour. Combining these gives

1

1111 3 n

n

kinnnptrn HG (25)

This shows that the final deviatoric stress direction will be coaxial with the trialdeviatoric stress for the class of models under consideration. Thus we can write this in the form that shows that this is the same as the radial return method.We utilise that

tr

tr

n

n

1

1

(26)

that gives

kinnn

tr

ptrn HG 111 31 (27)

and thereby

kinnn

tr

ptrn HG 111 31 (28)

The derivative is

kinnn

p

n

HGd

d 111

3 (29)

where the kinematic hardening, Hkin, is evaluated at the current estimate of the effective plastic strain at the end of the increment.The yield limit and its derivative with respect to plastic strain increment

pppnny

ny

n T ,,111 (30)and its derivative is

py

n

py

n

ison

tH

111 1 (31)

where the hardening, Hiso, is evaluated at the current estimate of the effective plasticstrain at the end of the increment.

4 DISLOCATION DENSITY BASED MODELLING

To conform to the above stress-strain computation procedures, the model should provide macroscopic description of plastic hardening. This is achieved by means of the dislocation density concept that links the macroscopic stresses and strains to the underlying micro-structural processes of plastic deformation. In general, such models for the evolution of theyield limit can be implemented into both types of constitutive models described earlier. Wewill only apply it to the concept of a yield surface. However, the yield limit in this yieldsurface may become rate-dependent.

8


A generally accepted assumption for the influence of dislocation density on hardeningbehaviour is, Ch 5.9 in Ref. 29,

ify aGb (32)

where i is the immobile (“forest”) dislocation density in cm of dislocations per unit volume,G is the shear modulus, b is the magnitude of the Burger’s vector, and a is a material constantrelated to the crystal and grain structure. f is the extrapolated stress to zero dislocationdensity (strain-independent “friction stress”). This model does not account for different elementary types of dislocations and dislocation arrangement structures. The formula can bemotivated as follows24. The stress field of a dislocation is

rGb (33)

where r is the distance from the dislocation. The average distance between dislocations is proportional to . Combining this with the previous relation leads to Eq (32)2/1

i

4.1 Basic dislocation processes

It is necessary to have model for strain hardening of a material due to dislocation generation, motion, immobilisation, recovery and annihilation, see Figure 1.

creationDensity ofimmobile

dislocationsi

Density ofmobile

dislocationsm

recovery, R

immobilisation

A annihilation

U

creationDensity ofimmobile

dislocationsi

Density ofmobile

dislocationsm

recovery, R

immobilisation

A annihilation

U

Figure 1. Basic dislocation processes: generation, immobilisation, recovery and annihilation.

The following assumptions are accepted according to Bergström’s theoretical model30 forbcc and fcc metals:(a) average behaviour of large numbers of dislocations is considered; (b) the mobile dislocation density is strain independent and is less than the immobiledislocation density; (c) after generation, the mobile dislocations move, on average, a distance, s, before they are immobilised or annihilated; this distance (a mean free path) is ultimately related to averagedislocation cell size and grain size;(d) annihilation term can be neglected, especially when temperatures do not rise above half the melting point. Thus, the evolution equation is expressed as

9


0)0(i

ip

pi U

(34)

The dislocation storage parameter, U, is linked to the mean free path:

pp

sbU 1 (35)

For ferritic steels it is found that U const( p ) under certain metallurgical conditions31.The remobilisation or recovery parameter, is a measure of probability by which immobile dislocations remobilise. The basis of this model is related to the statistical theory of work-hardening by Kocks32.

4.2 Extension to rate-dependent model

First, we describe the extension of the basic Bergström’s model (Eqns. 32 and 34) to account for rate-dependent yield limit.

The parameter U is assumed to be rate-independent31. The recovery parameter isstrongly dependent on temperature and strain rate33. The expression for consists of an athermal component 0 and a thermal component:

31

0 )(,ref

p

refp TT (36)

The formula is derived on the basis of the following assumption: the re-mobilisation of immobile dislocations takes place predominantly in the cell walls by means of diffusioncontrolled vacancy climb.

Dislocations in motion experience resistance in their glide plane, and certain stress should be applied in order for a dislocation to start moving. The expression for the friction stressrelating these effects is found to be33

)(

0,Tm

ref

p

refp

f TT (37)

As temperature rises, the strain rate dependent part of the friction stress quickly drops to zero. As shown by Hasegawa et al. 34, such relation is equivalent to the kinetic equation of viscoplasticity:

niy

mp vv

0

)(ˆ, (38)

which could be used in a unified-viscoplastic form of stress-update algorithm22,35,36. However,in this work the former equation is used to account for rate-dependency of the yield limit.

10


4.3 Extension to kinematic hardening model

The most prominent factor in the cyclic tension-compression tests is the presence of kinematic hardening associated with the back-stress. An advanced dislocation model by Y.Estrin37 is used to accommodate the evolution of the back-stress internal variable. The back-stress is attributed to the formation of channel-like dislocation structures upon cyclic deformation, with the immobile dislocation density in channel walls consisting of ‘trapped’ dislocations and of ‘recoverable’ dislocations. The changing curvature of the dislocation segments in the channels is believed to be the driving force of the process.

Accordingly, an evolution equation for this quantity is chosen in the following form:

backp

ipback AsignAAA 2210 )( , (39)

where, in terms of Eq.8, 210 AAAH ikin , Dkin = A2 and Skin = 0. The first term, which is associated with a decrease of the radius of curvature of the mobile

segments as they bow out under the action of applied stress, is governed by the wall spacing and is thus proportional to the square root of forest dislocation density. This bowing out is counter-acted by the build-up of the back-stress, which is represented by the second term.Once the radius of curvature reaches its critical value, a steady state is achieved, and the subsequent motion of the screw segment is translational.

In the original Estrin’s model37 an additional internal variable is introduced: a density of immobile dislocations that are trapped in the walls but partially recoverable upon stress reversals. This process is not taken into account in the present study.

The derivation in section 3 does not include these recovery terms. However, the currentimplementation is only for 1D-relations and then the stress-strain logic is simple to extend toinclude this term.

5 APPLICATION OF THE MODEL

5.1 Cyclic deformation tests. Dislocation model simplification

The dislocation model has been applied to a high-strength steel in cyclic deformation,strain-controlled tests. The strain cycles are fully reversed, with the strain-rate history being a kind of sine wave, or an approximation thereof. The irregularities are due to the naturallimitations of the precision-control in the testing machine. The cycle frequencies are 4.20,1.72 and 1.08 Hz; the nominal strain amplitudes, max, are 0.2%, 0.6% and 0.9%, respectively; and the average magnitude of the total strain rate is 0.03, 0.04 and 0.04 sec–1, respectively.The tests are carried out at room temperature.

The measured curves (Figures 2, 3) show significant Bauschinger effect that suggests the usage of the kinematic hardening term in the dislocation model. During the first few cycles anincrease of the maximum yield stress is observed. This suggests that isotropic hardening mechanism is active as well. In the subsequent cycles the maximum yield stress starts todecrease slightly that could be accounted for by the thermal softening due to plastic friction-heat generation. However, in the present study this temperature effect is neglected and the

11


model is fitted to the test data of only the first two cycles so that the strain-hardening processis dominant.

(a) (b)

Figure 2: Experimental hysteresis loops and model results for the high strength steel tested with a nominal strainamplitude of 0.9%: (a) rate-independent model (Table 1, set 1); (b) rate-dependent (Table 1, set 2).

(a) (b)

Figure 3: Experimental hysteresis loops and rate-dependent model (Table 1, parameter set 2) results for the highstrength steel tested with a nominal strain amplitude of (a) 0.2% and (b) 0.6%.

As there is no apparent transient effect associated with isotropic hardening, only one “isotropic” internal variable is used – the immobile dislocation density. The mobiledislocation density is assumed to be constant. The immobile dislocations are not differentiated into locked and partially recoverable, but instead are treated collectively as one single population of dislocations. So, effectively the Bergström’s dislocation model is used to describe isotropic hardening (Equations 32, 34, 36 and 37).

The kinematic hardening is taken according to Equation 39, with an assumption that the effect of forest dislocation density change on the back-stress hardening term can be neglected(A1 = 0). It is motivated by the observed near-saturation state of isotropic hardening.

12


5.2 Optimisation procedure

The parameter identification is carried out using the custom optimisation toolbox for Matlab®. The toolbox is described in details in Ref. 2. A gradient method of optimisation is chosen. It is based on the Matlab function for solving constrained minimisation problem – fmincon. The error function to be minimised is: for the parameter set 1 and 2, the maximaldiscrepancy between calculated and measured stress normalised by a maximum measuredstress magnitude; and for the parameter set 3, the average discrepancy between calculated and measured stresses weighted by corresponding strain increments. The model parameters are identified for the cyclic test curve of the max = 0.9% case. Due to the complexity of the dislocation model with a relatively large number of the parameters to be determined, theoptimisation is undertaken in a step-by-step manner: the results of fitting the simplified modelwith fewer parameters are used as a starting guess-values in the following steps, and only in the last step all model parameters are included.

First, the Young’s modulus is estimated by fitting those parts of the curve that apparentlycorrespond to the elastic loading-unloading regime (Figure 4). It appears that elastic modulusvalue is slightly decreasing every cycle: E 206; 188; 186; 183; 184 (GPa), so a representative average value is selected and fixed, E 1.9E+05 (MPa).

Figure 4: Determination of the Young’s modulus, E, by linearly fitting the assumed elastic unloading-reloadingranges of the measured dataset. An average value of E is selected.

Next, the number of points in the dataset is reduce by half to speed-up the intermediateoptimisations, and the rate-independent version of the model (Equations 32, 34 and 39) isfitted to the reduced dataset. The initial dislocation density, 0, and recovery parameter, , are assigned to their physically motivated values, and these parameters are fixed. So there arefour free parameters: f , U, A0, A2. The resulting values are in the Table 1, parameter set 1, and the calculated curve is shown on the Fig. 2a.

Next, the abridged ( ref = 0, 0 = 0) rate-dependent model is fitted to the reduced dataset,with the following parameters fixed: 0, U, 0, ref = 0.04 sec–1, and these four parameters

13


free: ref, m, A0, A2. Then, the rate-dependency of recovery parameter is included (free ref,0), but with the fixed kinematic hardening (fixed values of A0, A2).Finally, all parameters are allowed to vary unconstrained, and the model is fitted to the full

dataset. The resulting parameter values are in the Table 1, parameter set 2 and 3, and the calculated curves are shown on the Fig. 2b and Fig. 6a, respectively.

Rate-independent model Rate-dependent modelParameter set 1 Parameter set 2 Parameter set 3

0 (m–2) 1.00e+12 1.06e+12 1.27e+12U (m–2) 2.19e+14 1.90e+14 1.95e+14

0 (-) - 10.2 9.42ref (-) 50.0 43.2 44.9

ref (sec–1) - 0.04 0.040 (MPa) - 0 0

f ref (MPa) 302.72 302.22 275.69m (-) - 0.0457 0.0467

A0 (MPa) 182 221 232A2 (MPa) 309 207 207

a (-) 1.0 1.0 1.0b (m) 2.5e-10 2.5e-10 2.5e-10

E (MPa) 1.9e+05 1.9e+05 1.9e+05

Error-function 0.107284 0.101661 0.051243Table 1: Results of the parameter optimisation for the case of max = 0.9%. Parameter set 1 and 2 on one

hand and set 3 on the other hand are obtained with two different error-functions (details in the text).

5.3 Discussion. Model accuracy and parameter sensitivity

The optimised parameters of rate-independent version of the dislocation model (Table 1) appear to have the values quite similar to those of the corresponding parameters of rate-dependent model. Also the resulting error-function values are very close, and the shape of the curves is similar too (Figures 2 and 3). This suggests that rate-dependent model does not provide a dramatic improvement over the basic model in this test case. It could be explained by the periodic strain-rate variation during this cyclic test: the strain rate, although not a constant, does not change significantly, oscillating about the average value. For the rate effects to become more apparent, the strain rate should vary on the scale of several orders of magnitude. During this test it happens only for a very short periods of time, as the strain-ratepasses its zero points. Comparison of the two calculated curves reveals that at these points theshape of the curves does differ: the rate-independent model produces distinct sharp corners in the points of unloading-reloading, while the rate-dependent model “rounds” them up (Fig. 5

14


and Fig. 6b). The latter “rounded corners”-shape is characteristic to the measured curve as well, so the rate-dependent model does indeed represent the real rate-effect correctly.

(a) (b)

Figure 5: Effect of the strain-rate on the shape of the curves, comparison with experiment – “rounded corners”near the points of zero strain-rate: (a) rate-independent model; (b) rate-dependent model.

(a) (b)

(c) (d)

Figure 6: Experimental hysteresis loops and rate-dependent model results (Table 1, parameter set 3) for the highstrength steel tested with a nominal strain amplitude of (a),(b) 0.9%; (c) 0.2% and (d) 0.6%.

The calculated curves of the other test cases – with strain amplitude, max = 0.2% and 0.6%– are shown on the Figure 3. The model parameters there are set to those from the Table 1,set 2, i.e. fitted to the test case of the max = 0.9%.

15


Apparently, with the dislocation model used in this study it is possible to achieve only moderate, not very high accuracy in predicting the exact shape of the strain-stress variation –for this particular cyclic tests of the chosen high-strength steel. Evidently no other parameterset results in a significantly better approximation (i.e. with the lesser value of the selectederror-function) than the one presented in Table 1. Moreover, those parameter sets are not unique (both set 2 and 3), in the sense that there are other sets of parameters with the error-function values very close to the one above. The results of the parameter identification are found to be highly sensitive to the starting guess-values. The numerical instabilities due to the non-uniqueness of the solution appear to be a common problem with advanced materialmodels. It is discussed, for instance, by Mahnken and Stein38. They also discuss the use of the eigenvalues of the Hessian matrix in order to evaluate the sensitivity of the solution.

Furthermore, the optimised parameter values and the shape of the curves depend on the kind of the error-function used in the optimisation. The curves calculated with the modelparameters from the set 3 (Table 1) for all three test cases are shown on the Figure 6.Comparing them with the curves from the Figures 2 and 3, one can notice that in this case theoverall curve-fit has somewhat improved, at the expense, however, of a greater discrepancy during the initial loading stage.

6 CONCLUSIONS

The formulation of a rate-dependent dislocation model with isotropic and kinematichardening has been presented. The model has been implemented in a custom toolbox for Matlab according to the general framework of stress-update algorithm for rate-dependent plasticity. The model parameters have been identified for the cyclic deformation of a high-strength steel. The following observations have been made:

- A step-wise optimisation approach had to be used in order to circumvent the numerical difficulties due to large number of parameters involved.

- The accuracy of the curve-fit depends on the choice of the error-measure function. A satisfactory agreement between experiment and the model is achieved (Fig.6). However the overall accuracy of the model is quite moderate.

- For these particular test data, the strain-rate effect in the model is not very strong. Still, certain minor rate effect is found to be in agreement with experiment (Fig.5).

- The solution of optimisation problem is found to be non-unique, as the model in its present form cannot predict the measured stress-strain curve with very high accuracy,and different parameter sets result in the values of error-function close to the optimalone. Some additional experimental curves (other than cyclic test) are required toeliminate this non-uniqueness.

ACKNOWLEDGEMENT

The financial support from KK-Stiftelsen (http://www.kks.se/) is gratefully acknowledged.

16

http://www.kks.se

http://www.kks.se


REFERENCES

[1] H.J. Frost and M.F. Ashby, Deformation-Mechanism Maps - The Plasticity and Creep of Metals and Ceramics, Pergamon Press, (1982).

[2] L.-E. Lindgren, H. Alberg and K. Domkin, "Constitutive modelling and parameteroptimisation", in VII International Conference on Computational PlasticityCOMPLAS 2003, Barcelona, (2003).

[3] P. Perzyna and R. Pecherski, "Modified theory of viscoplasticity Physical foundations and identification of material functions for advanced strains", Arch. Mech., 35(3), 423-436 (1983).

[4] A.K. Miller, "The MATMOD equations", in Unified Constitutive Equations for Creep and Plasticity, A.K. Miller ed., Elsevier Applied Science Publishers Ltd, 139-219, (1987).

[5] D.C. Stouffer and L.T. Dame, Inelastic Deformations of Metals - Models, mechanical properties and metallurgy, John Wiley & Sons, Inc., (1996).

[6] D.J. Bammann and A.R. Ortega, "The influence of the Bauschinger effect and yield definition on the modeling of welding processes", in Modeling of Casting, Welding and Advanced Solidification Processes VI, Warrendale, The Minerals, Metals & Materials Society, 543-551, (1993).

[7] M.P. Miller, E.J. Harley and D.J. Bammann, "Reverse yield experiments and internalvariable evolution in polycrystalline metals", International Journal of Plasticity,15(1), 93-117 (1999).

[8] U.F. Kocks, "Constitutive behaviour based on crystal plasticity", in UnifiedConstitutive Equations for Creep and Plasticity, A.K. Miller ed., Elsevier Applied Science Publishers Ltd, 1-88, (1987).

[9] M.A. Crisfield, Non-Linear Finite Element Analysis of Solids and Structures, Vol. 2 Advanced topics, J Wiley & Sons, (1997).

[10] D.J. Bammann, "Modeling temperature and strain rate dependent large deformationsof metals", Applied Mechanics Review, 43(5), 312-319 (1990).

[11] J.L. Chaboche and G. Rousselier, "On the plastic and viscoplastic constitutive equations - Part I: Rules developed with internal variable concept", ASME Journal of Pressure Vessel Technology, 105(May), 153-158 (1983).

[12] J.L. Chaboche and G. Rousselier, "On the plastic and viscoplastic constitutive equations - Part II: Application of internal variable concept to the 316 stainless steel",ASME Journal of Pressure Vessel Technology, 105(May), 159-164 (1983).

[13] B. Chun, J. Jinn and J. Lee, "Modeling the Bauschinger effect for sheet metals, part I: theory", International Journal of Plasticity, 18, 571-595 (2002).

[14] B. Chun, J. Jinn and J. Lee, "Modeling the Bauschinger effect for sheet metals, part II: applications", International Journal of Plasticity, 18, 597-616 (2002).

[15] J.L. Chaboche, "Cyclic plasticity modeling and ratchetting effects", in ConstitutiveLaws for Engineering Materials: Theory and Applications, C.S. Desai ed., Elsevier Science Publishing, 47-58, (1987).

17


[16] J. Chaboche, "Constitutive equations for cyclic plasticity and cyclic viscoplasticity",International Journal of Plasticity, 5, 247-302 (1989).

[17] J.C. Simo and T.J.R. Hughes, Computational Inelasticity, Springer-Verlag, (1997). [18] M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, Volume

1: Essentials, John Wiley & Sons, (1991). [19] T. Belytschko, W.K. Liu and B. Moran, Nonlinear Finite Elements for Continua and

Structures, John Wiley & Sons, (2000). [20] T. Hughes, "Numerical implementation of constitutive models: rate-independent

deviatoric plasticity", in Theoretical Foundation for Large Scale Computation of Nonlinear Material Behaviour, Northwestern University, Evanston, Illinois, US, Nijhoff, (1984).

[21] J.L. Chaboche and G. Cailletaud, "Integration methods for complex plasticconstitutive equations", Computer Methods in Applied Mechanics and Engineering,133(1-2), 125-155 (1996).

[22] J.P. Ponthot, "Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes", International Journal of Plasticity, 18(1), 91-126 (2002).

[23] P. Raboin, A deformation-mechanism material model for Nike3D, Lawrence Livermore National Laboratory, 47, (1993).

[24] E. Marin and D. McDowell, "A semi-implicit integration scheme for rate-dependentand rate-independent plasticity", Computers & Structures, 63(3), 579-600 (1997).

[25] A. Saleb, T. Wilt and W. Li, "Robust integration schemes for generalizedviscoplasticity with internal-state variables", Computers and Structures, 74, 601-628 (2000).

[26] E. Kirchner and F. Kollmann, "Application of modern time integrators to Hart's inelastic model", International Journal of Plasticity, 15, 647-666 (1999).

[27] E. Hart, "Constitutive relations for non-elastic deformation of metals", ASME Journal of Engineering Materials and Technology, 98, 193-202 (1976).

[28] M. Ortiz and E. Popov, "Accuracy and stability of integration algorithms forelastoplatic constitutive relations", International Journal for Numerical Methods inEngineering, 21, 1561-1576 (1985).

[29] R.E. Reed-Hill and R. Abbaschian, Physical Metallurgy Principles, PWS-KENTPublishing Company, (1992).

[30] Y. Bergström, “Dislocation model for the stress-strain behaviour of polycrystalline alpha-iron with special emphasis on the variation of the densities of mobile and immobile dislocations”, Mater. Sci. Eng., Vol. 5, pp.193-200 (1969/70)

[31] Y. Bergström, “The plastic deformation of metals - A dislocation model and its applicability”, Reviews on powder metallurgy and physical ceramics, 2 No. 2 & 3, pp. 79-265 (1983).

[32] Kocks, U.F. "A statistical theory of flow stress and work hardening". Philos. Mag., 13,p.541 (1966).

[33] Y. Bergstrom and Hallen, H. "An improved dislocation model for the stress--strain behavior of polycrystalline alpha-iron". Mater. Sci. Eng., 55(1), pp. 49-61 (1982).

18


[34] T. Hasegawa, Takahashi, T., & Okazaki, K. "Deformation parameters governing tensile elongation for a mechanically milled Al-1.1at.%Mg-1.2at.%Cu alloy tested in tension at constant true strain rates". Acta Mater., 48(8), pp.1789-1796 (2000).

[35] Y. Estrin and Mecking, H., "A unified phenomenological description of work-hardening and creep based on one-parameter models", Acta Metall., 32, 57-70 (1984).

[36] Y. Estrin, "Dislocation theory based constitutive modelling: foundations and applications", J. Mater. Process. Technol., 80-81, pp. 33-39 (1998).

[37] Y.Estrin, Braasch, H., and Brechet, Y., “A dislocation density based constitutivemodel for cyclic deformation”, J. Eng. Mat. Technol., Vol. 118, pp.441-447 (1996).

[38] R. Mahnken and E. Stein, "Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigation", Int. J. Plasticity, 12 No.4, pp. 451-479 (1996).

[39] E. Ramm and A. Matzenmiller, "Consistent linearization in elasto-plastic shellanalysis", Engineering Computations, 5(Dec), 289-299 (1988).

7 APPENDIX

7.1 Explanation on the use of some matrices

The matrix LUse of Voigt notation instead of tensor notations needs a correction in the scalar product

between the stress or strain vector as for example in the case of the computing the second invariant of the deviatoric stress

(A1)

23

13

12

33

22

11

2313123322112

223

213

212

233

222

2112

200000020000002000000100000010000001

222

ssssss

ssssssJ

ssssssssJ

T

ijij

Lss

The matrix is also used when translating tensor formulas for strains to vector notation, as the engineering shear definition is two times the tensor definition. For example, computing the effective plastic strain rate using tensor and vector notation

19


p

p

p

p

p

p

pppppppTpp

pppppppij

pij

p

23

13

12

33

22

11

2313123322111

223

213

212

233

222

211

2/10000002/10000002/1000000100000010000001

22232

32

L (A2)

The matrix PM and the projection matrix PProjection of the stresses to deviatoric stresses is done by

23

13

12

33

22

11

23

13

12

33

22

11

100000010000001000000211000121000112

31

ssssss

Ps (A3)

The effective von Mises stress for isotropic material is computed by the use of L as shown

in Eq. (A1) as we have 223 J . This gives together with Eq. (A3)

PLPPLPPLss MTTTTT

23

23

23

23 (A4)

This can be modified to accommodate anisotropic plasticity, see page 123 in Ref. 9. Ramm and Matzenmiller39 use the same notation for shells with a zero normal stress constraint.

The projection matrix has the properties 2PP (A5)

and

sPs (A6)

20

Paper C

PHYSICALLY BASED MATERIAL MODEL IN FINITE ELEMENT SIMULATION OF EXTRUSION OF STAINLESS STEEL TUBES

S. Hansson1,2, K. Domkin2

1 Sandvik Materials Technology, Sweden; 2 Dalarna University, Sweden

Summary

Stainless steel tube extrusion is a metal-forming process associated with large deformations, high strain rates and high temperatures. The finite element method provides a powerful tool for analysis of phenomena that occur during extrusion. In the present study, a dislocation density-based material model for the AISI type 316L stainless steel is implemented in a user subroutine for the commercial finite element code MSC.Marc. The model is calibrated using results from compression tests at different temperatures and strain rates. A thermo-mechanically coupled axisymmetric FE model is utilized to simulate the extrusion process. Model predictions of extrusion force and exit surface temperature are in good agreement with experimental values.

Keywords: physically based material model, extrusion, stainless steel, finite element

1 Introduction

Process simulation has become an important tool in design and development of extrusion and other manufacturing processes. A key obstacle is that the simulation of hot extrusion processes is quite difficult due to the significant deformations of the workpiece. To model the metal flow during extrusion, continuous remeshing is needed. Another main challenge when simulating manufacturing processes is to accurately model the material behavior. A good constitutive model, which correctly reflects the changes in material properties, is crucial to the reliability of the final simulation result. For processes associated with large deformations, high strain rates and high temperatures, such as extrusion, extrapolation from existing material data is often necessary. Physically based models [1,2] are of particular interest as it is expected that these models can more accurately describe the material behavior over a larger range of strains, strain rates and temperatures. Material parameter fitting can be simplified since data from other tests than mechanical testing can supply information.

In the present study, a dislocation density-based material model for the AISI type 316L stainless steel was implemented into a finite element code and applied to the stainless steel tube extrusion process. In order to validate the simulation results, exit surface temperature and extrusion pressure were measured during extrusion in a 14.5 MN extrusion press.

1

2 Stainless steel tube extrusion

The basic principle of extrusion is very straightforward. A billet is placed in a closed container and squeezed through a die to reduce the cross-section area and increase the length of the billet. The design of the die opening determines the cross-section of the extruded product. When extruding tubes, a mandrel is inserted in the middle of the die. Today, the most important method for steel extrusion is the Ugine-Séjournet process, where molten glass is used as lubrication [3]. The principles of this process are shown in Figure 1.

Figure 1: Glass-lubricated tube extrusion.

In glass-lubricated extrusion, there is a layer of glass between the billet and the container, between the billet and the mandrel, and between the billet and the die. Each heated billet is coated with powdered glass during transportation to the extrusion chamber. Glass powder is also applied inside the billet to assure good lubrication between billet and mandrel. Lubrication through the die is provided by a disc of compacted glass, the glass pad, which is placed between the billet and the die, see Figure 1. During extrusion, the glass pad is pressed against the die by the hot metal. The glass pad will deform with the billet and melt progressively to surround the extrusion with a lubricant glass film. The glass-lubricated extrusion process has been studied by Baqué, Pantin and Jacob [4] among others. Generally, it is quite difficult to predict the shape of the interface between the glass pad and the die. However, in FE analysis of extrusion, an assumption of the shape of the glass pad has to be done since it forms the die profile with the metal. One solution is to examine the butt of the extrudate to determine the path of the metal flow [5].

3 Methods and models

3.1 Mechanical testing

In order to calibrate the material model, compression tests were conducted over a temperature range of 1100-1300 ºC at strain rates of 0.01, 1 and 10 s-1. The experimental steel was of AISI type 316L, i.e. an austenitic stainless steel. The nominal chemical composition of the test material is given in Table 1. Cylindrical specimens of 10 mm diameter and 12 mm height were used for the hot compression tests.

2

The temperature was recorded during the test to see whether there was a temperature rise due to plastic dissipated energy.

Table 1: Chemical composition of experimental steel (wt %).

C Si Mn P S Cr Ni Mo Cu N 0.009 0.27 1.74 0.030 0.024 16.82 10.26 2.08 0.31 0.029

3.2 Material model

The yield limit is assumed to consist of two components:

Gy σσσ += ∗ , (1)

where ∗σ is the friction stress and Gσ is an athermal component. The friction stress

accounts for the short-range interactions between dislocations and discrete obstacles. The process is thermally activated and characterized by the free energy, FΔ , required to overcome the lattice resistance or obstacles without aid from external stress. Relation of the friction stress to the effective plastic strain rate, pε& , and temperature, T, is derived from the expression of the activation energy [2]. The classical Orowan equation is used for the dislocation velocity and the strain rate. The resulting expression can be written as

pq

F

kT

11

prefln1ˆ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

−=∗

ε

ετσ

&

&, (2)

where τ̂ is the athermal flow strength and k is the Boltzmann’s constant. The parameters p and q are chosen as 1. The reference strain rate, refε& , depends on the

density of mobile dislocations and is assumed be constant. In the equation above, the non-dimensional parameters 0τ and 0fΔ are introduced as G0ˆ ττ = and 3

0GbfF Δ=Δ ,

where G is the shear modulus, and b is the magnitude of the Burger’s vector.

The athermal component in Equation 1, Gσ , is a result of the long-range interactions

with the dislocation substructure. It is expressed via the density of immobile dislocations, ρ, as

ρασ Gbm=G , (3)

where m is the average Taylor orientation factor and α ≈ 1 is a proportionality factor. To establish a relationship between dislocation density and plastic deformation, an evolution equation is utilized,

3

2p ρερ Ω−= && U , (4)

where U is the dislocation multiplication parameter, which can be related to the mean free path, Λ, by [6]

Λ=

b

mU . (5)

The remobilization parameter, Ω, accounts for the static and dynamic recovery controlled by diffusional climb and interactions with vacancies [7],

kT

Gbe

c

cD

kT

GbD kT

Q 3

eqv

v0

3

v

v

22−

∗ ==Ω . (6)

Here, the self-diffusivity coefficient, D0, is related to the vacancy migration and vacancy self-diffusion, and vQ is the combined activation energy of vacancy formation and

vacancy migration. In the present formulation of the model, the vacancy concentration,

vc , is assumed not to deviate from that of the thermal equilibrium at the current

temperature, i.e. ( )Tcc eqvv = .

The parameters of the model were obtained by fitting it to the compression tests introduced in the previous section. The values of the parameters are summarized in Table 2. The experimental yield stress curves and corresponding curves predicted by the model are presented in Figure 2. Predicted curves for the strain rate of 100 s-1 are also shown for comparison since such high strain rates are quite common in extrusion processes.

Table 2: Model parameters for AISI 316L stainless steel.

Burger’s vector, b 2.58·10-10 m

Athermal strength coefficient, 0τ 0.003

Free energy coefficient, 0fΔ 0.6

Reference strain rate, refε& 106 s-1

Average Taylor factor, m 3.06

Self-diffusivity coefficient, D0 4.15·10-4 m2/s

Vacancy activation energy, Qv 5.07·10-19 J

Mean free path, Λ 53·10-6 m at T = 1100 °C, 167·10-6 m at T = 1300 °C

4

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

180

200

Strain

Stre

ss [M

Pa]

0.01 s-1

1 s-1

10 s-1

100 s-1

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

180

200

Strain

Stre

ss [M

Pa]

0.01 s-1

1 s-1

10 s-1

100 s-1

(a) (b)Figure 2: Experimental (solid line) and predicted (circle) flow stress at (a) T=1100 °C

and (b) T=1300 °C.

The material model was implemented in the commercial finite element code MSC.Marc using the YIEL user subroutine, which computes the yield stress as a function of the current plastic strain, strain rate and temperature. In this context, the dislocation density is treated as an internal state variable and computed by numerical integration of the evolution equation.

3.3 FE model

The extrusion process was modeled with an axisymmetric FE model due to rotational symmetry in loading and workpiece. The implicit FE code MSC.Marc was used for the simulations. The behavior of the metal during extrusion was simulated in a thermo-mechanically coupled analysis. The extrusion process parameters and thermal initial conditions are summarized in Table 3. Since steady-state condition is the dominating phase in tube extrusion, only part of the total billet length was considered. The billet length considered was long enough to correctly simulate the start-up of extrusion until steady-state was obtained. The billet consisted of four-node quadratic elements together with a few three-node triangular elements. Due to the large deformations of the billet, the elements became heavily distorted and remeshing was required frequently during analysis. Automatic remeshing schemes, based on given remeshing criteria, are available in MSC.Marc and were used successfully.

The glass pad was modeled as a rigid surface with a constant temperature of 1100 °C. The shape of the glass pad is an approximation based on an earlier study where the amount of glass used during extrusion was weighed. In accordance with those measurements an assumed profile of the glass pad during extrusion was constructed. The other tools, i.e. die, container, mandrel and pressure pad, were modeled as rigid bodies with heat transfer properties. When glass lubrication is used, the metal flow is almost frictionless and a constant Coulomb friction coefficient of 0.03 was assumed at the above contact areas. Between the billet and the ram the friction factor was set to 0.35.

5

Table 3: Extrusion process parameters and initial conditions.

Billet material AISI 316L

Container, die, mandrel, ram material AISI H13

Ram speed 93 mm/s

Container diameter 125.3 mm

Billet inner diameter 33 mm

Billet outer diameter 121 mm

Billet length 421 mm

Tube outer diameter 33.2 mm

Tube wall thickness 3.5 mm

Billet initial temperature 1150 °C

Die initial temperature 20 °C

Container initial temperature 200 °C

Mandrel initial temperature 100 °C

Ram initial temperature 20 °C

The surface and contact heat transfer coefficients were assumed to be 0.66 kW/m2/°C and 9 kW/m2/°C, respectively [8]. Thermal conductivity and specific heat capacity data for different temperatures were obtained from literature [9].

4 Results and discussion

The material model with the chosen parameter set somewhat underestimates the stress in the case of T = 1100 ºC and rate = 10 s-1, see Figure 2a. The agreement with the other measured curves is satisfactory. The reason for the variation of the mean free path with temperature does not have an apparent physical foundation, but in terms of the presented model it accounts for a weaker hardening rate at higher temperatures. The values of the parameters found from curve-fitting are physically reasonable and in the same order of magnitude as those found in literature.

Figure 3: Plastic strain rate distribution in extrusion simulation.

6

After the start-up of the extrusion process, the metal flow conditions soon stabilized. In the FE simulation, steady state was reached within a ram travel distance of only 13 mm. During steady state, the maximum effective plastic strain rate in the deformation zone was approximately 240 s-1. The plastic strain rate distribution in the extrusion model is shown in Figure 3.

The billet temperature in the deformation zone increased markedly due to heat generation by the plastic deformation. A maximum temperature of 1290 °C was recorded. The back end of the billet, on the other hand, was heavily chilled due to contact with the tooling. Temperatures as low as 760 °C were observed. There are some uncertainties with this result. Heat transfer between the billet and the tooling was difficult to model due to large temperature gradients. These gradients resulted in discretization errors of temperature at the interface nodes. The heat transfer coefficients may also be outside the margin of error.

The exit surface temperatures, evaluated from the FE simulation, are shown in Figure 4together with values obtained from experiments in a production extrusion press. 425 mm of extruded tube was considered in the numerical model. Temperatures from the FE model seem to agree quite well with the temperatures measured in the production extrusion press at the beginning of extrusion. The maximum difference in temperature between model and experiment was measured to 40 °C, which corresponds to an error of 3.2%. After 0.15 s, the difference between predicted and measured exit temperature was 22 °C. One source of this difference in temperature stems from the experimental measurements of the temperature profile since the sensitive nature of installing the optical pyrometer makes it susceptible to error. However, this exit surface temperature is the only temperature available for measurements during extrusion. There are also uncertainties in the model due to errors in the thermal initial conditions. The billet initial temperature, for example, is modeled as uniform, while in reality, temperature gradients exist.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.351120

1140

1160

1180

1200

1220

1240

1260

1280

Time [s]

Exi

t sur

face

tem

pera

ture

[°C

]

ExperimentalFE model

Figure 4: Exit surface temperature during extrusion.

7

The extrusion force predicted by the FE model was 11.0 MN compared to the experimentally obtained value of 11.5 MN; a difference of approximately 4.4%, which must be considered acceptable. As mentioned earlier, the shape of the glass pad against the billet and the die needs further investigations and analyses. The underestimation of the force may also be due to the extremely low friction factor used.

5 Conclusions

The stress-strain curves of AISI 316L, obtained from compression tests at temperatures 1100 ºC and 1300 ºC, were well described by the dislocation density-based material model. Application to an axisymmetric FE analysis of glass-lubricated tube extrusion showed that the model performs well for these simulations. The extrusion pressure and the exit surface temperatures predicted by the FE model were in good agreement with the experimental measurements.

6 Acknowledgements

The financial support from the Swedish Knowledge Foundation (KK-stiftelsen) and the Swedish Steel Producers’ Association (Jernkontoret) is gratefully acknowledged.

7 References

[1] Domkin, K., Physically Based Models of Metal Plasticity, (2003), Licentiate Thesis, Luleå University of Technology.

[2] Frost, H.J. and Ashby, M.F.: Deformation-Mechanism Maps - The Plasticity and Creep of Metals and Ceramics, (1982), Pergamon Press.

[3] Laue, K., Stenger, H.: Extrusion, (1976), Aluminium-Verlag GmbH Düsseldorf. [4] Baqué, P., Pantin, J. and Jacob, G.: Theoretical and Experimental Study of the Glass

Lubricated Extrusion Process, Trans. ASME, Journal of Lubrication Technology, No.74 (1975), 18-24.

[5] Damodaran, D., and Shivpuri, R.: Prediction and Control of Part Distortion During the Hot Extrusion of Titanium Alloys, Journal of Materials Processing Technology, Vol 150 (2004), 70-75.

[6] Bergström, Y.: The Plastic Deformation of Metals - A Dislocation Model and its Applicability. Reviews on Powder Metallurgy and Physical Ceramics, Vol. 2/3 (1983), 79-265.

[7] Militzer, M., Sun, W. and Jonas, J.: Modelling the Effect of Deformation-induced Vacancies on Segregation and Precipitation, Acta Metall. Mater., Vol. 42, No.1 (1994), 133-144.

[8] Sellars, C.M., and Whiteman: Temperature Changes Arising from Radial Heat Flow During Hot Extrusion of Cylindrical Steel Billets, Metal Technology, (1981), 10-21.

[9] Material Database of STEELTEMP 2D: (2003), MEFOS, the Foundation for Metallurgical Research.

8

Paper D

1 of 49

Dislocations, vacancies and solute diffusion in a physically based plasticity model for AISI 316L

Lars-Erik Lindgren1,2, Konstantin Domkin2 and Sofia Hansson2, 3

1Luleå University of Technology, 971 87 Luleå, Sweden 2Dalarna University, 781 88 Borlänge, Sweden 3Sandvik Materials Technology, 811 81 Sandviken, Sweden

1 Introduction The description of material behaviour is crucial for all simulations of material deformation processes. The use of simulations to study manufacturing, [1, 2], includes additional complications due to the changing microstructure of the material. Therefore, it is preferable to use physically based models that can catch the essential phenomena dominating the deformation based on the underlying physics of the deformation, Kocks & Mecking [3], coupled to microstructure evolution. However, the models must still be tractable for large scale computations. Additional advantages of using physically based models is expected to be a larger domain of validity as well as the possibility to utilise other information from other sources than mechanical testing to determine the model parameters.

This study is part of a project aiming at developing a material model for describing the plastic behaviour of AISI 316L for a large range of temperatures, strains and strain rates. The material has fcc structure and a low stacking fault energy (SFE). The current scope is limited to temperatures from room temperature (RT) up to 1300°C, strains up to 0.6 and strain rates from 0.0005 up to 10 s-1. This also includes the phenomenon of dynamic strain ageing (DSA). The paper gives an overview of the formulation and relevant publications. The model shares its basic features with the model in Cheng, Nemat-Nasser and Guo [4]. Some submodels are different and the effect of excess vacancies on diffusion processes is included. The chosen model is based on a coupled set of evolution equations for dislocation density and (mono) vacancy concentration. Furthermore, it includes the effect of diffusing solutes in order to describe DSA. The model parameters are calibrated by comparison with a set of compression tests. It has been decided to avoid adjusting physical quantities like diffusivities etc in the parameter fitting procedure.

The paper also describes the numerical algorithm used to solve these strongly nonlinear relations in an efficient manner applicable for user routines in finite element codes. The model has been formulated in a way that alleviates the replacements of different sub-models of the material model. Thus the proposed formulation is a platform for further development.

2 Overview of deformation behaviour The inelastic deformation depends on the interaction between dislocations and the microstructure of the material. This structure is evolving during the deformation. Hardening as well as recovery processes may occur. This is also affected by the strain rate and temperature. The active mechanisms depend on the current structure of the material and applied stress or strain rate at the given temperature. An overview of the dominating deformation mechanisms can be outlined in a deformation map [5, 6]. The inelastic deformations that are of concern when constructing these maps [6] are shown in Table 1. The model presented in this paper is based on the dislocation mechanism, #2 and 4 in the table. The tests that are used to calibrate the model are compressive tests and the time scale of creep processes is not evaluated. It should be noted that twinning is not included in the current model.

2 of 49

Table 1. Mechanisms for inelastic deformations.

# Deformation mechanism

1 Collapse at the ideal strength when the ideal shear strength is exceeded.

2 Low-temperature plasticity by dislocation glide a) limited by a lattice resistance or Peierl’s stress; b) limited by discrete obstacles; c) limited by phonon or other drags; and d) influenced by adiabatic heating. This plasticity is mainly obstacle-limited in the case of austenitic stainless steels [5].

3 Low-temperature plasticity by twinning

4 Power-law creep by dislocation glide or glide plus climb a) limited by glide processes; b) limited by lattice-diffusion controlled climb (high-temperature creep); c) limited by core-diffusion controlled climb (low-temperature creep); d) power-law break down creep at transition from dislocation glide plus climb to glide alone at higher stresses; e) Harper-Dorn creep; and f) creep accompanied by dynamic recrystallisation.

5 Diffusional flow a) limited by lattice diffusion (Nabarro-Herring creep) at higher temperatures; b) limited by boundary diffusion (Coble creep) at lower temperatures; and c) interface-reaction controlled diffusional flow.

3 Dislocation density model

3.1 Physically based models Physically based models are formulated on the underlying physical processes causing the deformation in contrast to empirical models of a more curve-fitting nature. Due to the need for averaging and also limited knowledge some relations for these processes may be phenomenological. Two different types of physically based models exist. One option is to explicitly include the physical model as an evolution equation in the constitutive model. This is the approach in the current paper. The other possibility is to determine the form of the constitutive equation based on knowledge about the physical process causing the deformation. The latter is a so-called “model-based-phenomenology”, Frost and Ashby [6]. The same hardening model can be obtained from empirical models and models based on dislocation mechanisms as in the case of power law creep. However, in general it is expected that the physically based models can have a larger range of validity than purely empirical equations that do not consider the underlying physical process.

The focus in this study is on dislocation density models that can be used to describe the hardening behaviour of the material. The evolution of the flow stress during the deformation determines the isotropic hardening of the material. We will not account for back-stresses, kinematic hardening, in this study.

3.2 Plasticity and dislocations The current study uses the concept of rate-independent plasticity and a yield surface. It is an approximation where the plastic strains are assumed to develop instantaneously in such a way that the stress state stays on the yield surface during a plastic process. The applied stress is then equal to the yield stress during a plastic deformation. The processes that determine this yield surface are rate-dependent. Therefore, we will have a rate-dependent yield limit applied in the context of rate-independent plasticity.

The yield limit is assumed to consist of two components [7].

*Gy (1)

3 of 49

where G is due to long-range interactions with the dislocation substructure. It is an athermal stress contribution. The second term, * , is the short-range interactions. It is a friction stress needed to move dislocations through the lattice and to pass short-range obstacles. Thermal vibrations can assist the stress to overcome these obstacles.

The long-range term in Eq. (1) is written as

GbmG (2)

where m is the Taylor orientation factor translating the effect of the resolved shear stress in different slip systems into effective stress and strain quantities. It depends on the crystal structure and is affected by texture. It is assumed to be constant and is 3.06 for fcc crystals [8]. is a proportionality factor, is the dislocation density and G is the temperature dependent shear modulus. We will assume to be constant although it is known to depend on the dislocation substructure [9]. This dependency is not so important [3]. We will ignore the temperature dependency of Burger’s vector b. Variants of Eq. (2) have been used when different dislocations densities are accounted for [10-14]. The model for the evolution of the dislocation density needed for Eq. (2) is formulated in the next section.

The second term in Eq. (1), the short-range stress component, is obtained as described below. The dislocation velocity is related to the plastic strain rate via the Orowan equation

mvbmp , (3)

where v is the average velocity of mobile dislocations, m . This velocity is related to the time it takes for a dislocation to pass an obstacle, the waiting time tw, as the time of flight to the next obstacle is negligible. The velocity is written as [6, 15-17]

kTGa

kTGa eebv (4)

where b is the mean free path between two successful events. a is the attempt frequency [12, 15] and depends on the obstacles. G is the activation energy, k is the Boltzmann constant and T is the temperature in Kelvin. Combining Eq.s (3-4) gives

kTGamp em

b (5)

where 2bb is sometimes called the activation area.

Eq. (5) is rewritten as [6] kTG

Gp ef (6)

where is the effective stress due to the applied load, G is long-range stresses. The form of the function f and G depend on the stresses, type of obstacle etc. The stress available to move the dislocation is thus the applied stress minus the long-range flow stress component. The effective stress is assumed, in the theory of plasticity, to be equal to the flow stress during the deformation

y (7)

and then Eq. (1) gives

Gy* (8)

Combining Eq.s (6-8) gives

4 of 49

kTGp ef * (9)

The short-range stress component of the yield stress requires a model that determines f and G.This is discussed in chapter 3.4.

The flow stress variation with the control variables temperature, strain and strain-rate is further complicated by dynamic ageing. It is due to diffusion of solute atoms that affects the plastic deformation. Several mechanisms are proposed based on drag effects on moving dislocations and thereby the short-range stress component or the pinning of dislocations and thereby the long-range stress component [10, 18-20]. Probably, both phenomena are present simultaneously. The latter effect is chosen for the model in this study and the phenomenon is discussed in chapter 3.4.

Models for the dislocation evolution are discussed next and thereafter models for the short-range component are described.

3.3 Evolution of density of immobile dislocations The direct simulation of the process, in the spirit of discrete dislocation dynamics, is costly to compute. Hence, an average treatment of dislocation processes is favourable, and the concept of dislocation density is found useful. The dislocation density based models are formulated at the macro level, i.e. all the quantities (like dislocation densities, flow stress etc) are calculated for a representative material volume that can be considered homogeneous. Every dislocation model considers at least two different densities, mobile and immobile dislocations. That is the case for the model by Bergström [8, 21-23] and this is also the case in the current study. Zikry [24] traces evolution of the dislocation densities in different slip systems and their interactions and effect on the long-range stress component, Eq. (2). Mughrabi [13, 14] and Estrin [25] use two types of densities for immobile dislocations, channels and walls. They can also accommodate cell size and width as variables to describe hardening behaviour or, as in [26], cell area density and cell fraction. The channels are low density volumes whereas the walls are dense dislocation walls in the substructure created by the dislocations. They have separate, and coupled, evolution equations. A similar approach was also used by Nes [10]. This split into two different densities for immobile dislocations corresponds to the use of separate densities for statistically stored dislocations (SSD) and geometrically necessary dislocations (GND) that have separate evolution equations [27, 28]. SSD is the minimum density of dislocations needed to accommodate a given strain gradient and GND represents dislocations that are needed to preserve lattice compatibility [29]. They accumulate near grain boundaries whereas the SSD are stored in grain interiors. Evers et al. [30, 31] used an model with GND and SSD for each slip direction and a complex model for their interaction. Mugrabi [32] discusses the association with SSD and GND densities with gradient plasticity models. Seefeld and Klimanek [33] set up a coupled set of differential equations for mobile and immobile dislocations combined with disinclination densities.

The use of two different densities for immobile dislocations is used to model back-stress and pertaining kinematic hardening in [34, 35]. Peeters et al. [36, 37] introduced a third type of directional sensitive dislocation density in order to include the effect of deformation path changes.

The basic evolution equations for different dislocation densities do all have the same characteristics, storage and recovery processes. This kind of evolution equation is derived for each type of dislocation density. They may be expressed in the form of “hardening term – recovery term”

)()(xxx (10)

where x denotes a specific dislocation density. In the following we have x=m for mobile dislocations and x=i for immobile dislocations.

5 of 49

The model used in the current approach [8, 21, 22, 38] is based on the assumption that themobile dislocation density is stress and strain independent and much smaller than the immobile one [4, 12, 39]. This is expressed as

0,0 pmm

im

mi

(11)

Thus transients when changing stress [15, 17] that are associated with changes in the mobile dislocation density, like Lüder’s band formation, can not be modelled with the assumptions above.

The immobile dislocation density is used in Eq. (2) for the long-range stress contribution to the flow stress.

iG Gbm (12)

We will in the following describe the evolution equations for the density of immobile dislocations, i . The initial dislocation density is denoted as 00 ii .

3.3.1 Hardening

It is assumed that mobile dislocations move, on average, a distance (mean free path), , before they are immobilised or annihilated. The density of mobile dislocations and their average velocity are related to the plastic strain rate according to Eq. (3). The increase in the immobile dislocation density is also assumed to be proportional to the plastic strain rate [8, 11, 40-42]

pi b

m 1 (13)

where m is the Taylor orientation factor, 3.06 for fcc crystals [8]. This factor is evolving with the deformation but is assumed to be constant in the current model. The mean free path, , is assumed to be a combination of the distance between grain boundaries, g, and dislocation subcell or subgrain diameter, s,

others111sg

(14)

where “others” denotes contributions from varying types of obstacles like precipitates [43] or the distance between martensite lathes [44]. The initial grain size is g0 and it is assumed to be constant in the current model. Models for recrystallisation and grain growth can be included and used to modify the grain size term in Eq. (14). The effect of grain size on flow stress, the Petch-Hall effect, is accounted for via this equation and its effect depends on whether it will be masked by the other terms or not [45, 46]. Karaman et al. [47] consider twins as permeable obstacles to dislocations although they included this effect in a separate term and not via Eq. (14).

The formation and decrease in size of subcells is one sequence of dislocation patterns that are formed during straining [48]. Holt [49] derives a relation between dislocation subcell diameter and dislocation density

icKs 1 (15)

The analysis by Holt also indicates the approximate value of the parameter Kc by relating the dislocation density to the long-range stress component, Eq. (2), and plotting this for a number of

6 of 49

materials. Holt [49] states that the Eq. (15) is not valid when the dislocation mobility is low or when it is high.

The subcell formation is a complex process in which dislocations form different low energy structures [34, 35, 50, 51]. Despite these complications the relation proposed by Holt has been found to be useful, e.q. [52, 53]. Michel et al. [54] found Kc to be 16 for AISI 316 from room temperature up to 816ºC. Raj and Pharr [55] analysed a large amount of data and noticed that there is a support for a linear relation between flow stress and subcell size in many cases. They evaluated a somewhat more general relation

2/

1ni

cKs (16)

The best fit to data gave the exponent n the value of 0.84. A statistical correlation was found between Kc and n

1log37.40.3log nnc bmnK (17)

However, their analysis of the uncertainties in data lead Raj and Pharr to recommend the use ofn=1. Thus, they obtained Eq. (15) and a Kc in the same order as Holt [49]. Raj and Pharr [55] noticed that the case of austenitic stainless steel AISI 316 from [56] was a clear exception with n=1.52. Kuhlmann-Wilsdorf [45] discusses the microstructure evolution and, of particular interest in this study, how material with fcc structure first forms a Taylor lattice that becomes a carpet structure and then a cell structure. The Taylor lattice has long glide paths, i.e. a large Kc in Eq. (15). One discussed example is -brass with Kc=250. Hansen [57] draws the conclusion that, due to the complex geometry of formed cells or subgrains, Kc should not be considered as a true constant. The review by Gil Sevillano et al. [9] states that larger strains are required to develop a cellular substructure and that a decrease in stacking fault energy (SFE) or temperature or a higher strain rate gives a more uniform dislocation distribution in fcc materials compared to bcc materials. Higher SFE and higher temperatures give larger cells [58]. Furthermore, they observed that the complexity in the substructure formation in low SFE metals [59] accounts for the lack in quantitative data to be used to formulate models. The temperature influence is small until it reaches about half homologous temperature. The deformation changes character from planar slip to cross-slip as the stacking fault increases with increasing temperature [60]. It was also noted in [60] that cell formation is difficult for strain below 0.25 and also that planar slip was found at 823 K due to dynamic strain ageing (DSA).

We, like in [61], consider s to be a characteristic length in general applicable for other dislocation substructures than cells. It describes the effect of the dislocation structure on the mean free path of a dislocation and may therefore not directly be related to a geometric measure. This can also be motivated from the observations that a glide dislocation may pass several cells before it is trapped [3, 62]. Seefeld and Klimanek [33] also includes a factor for the probability that a dislocation is trapped in a cell wall. The parameter s is taken as a combination of a geometric measure between obstacles to dislocation motion and their strength. One possibility is to use Eq. (15) and allow the value of Kc to depend on temperature and plastic strain rate in an ad-hoc manner in lack of models for substructure formation. Thus strain rate and temperature will affect the dislocation storage via this part of the model. This was noted by Gil Sevillano etal. [9] as a deficit of the original proposal by Bergström. In this study we will assume that Kc is dependent on temperature only. We also introduce a lower limit for the subcell size, s , and then Eq. (15) is written as

sKsi

c1 (18)

7 of 49

3.3.2 RecoveryDifferent processes may contribute to the reduction in the dislocation density. The formulations by Bergström [8] and Estrin [41], for example, lead to a remobilisation of dislocations proportional to the current immobile dislocation density

pii (19)

where is a recovery function. This kind of models can only accommodate dynamic recovery due to the strain-rate term in Eq. (19). Engberg [63] reviews different models for recovery that also can be used for static recovery. and Nes [10] formulates a recovery proportional to x

i with cross-slip controlled recovery having x=1 and climb controlled having x=2. Li [64] assumes diffusional climb controlled recovery leading to a formulation in the format

22eqii (20)

where eq is an equilibrium value towards which the density decreases.

Siwecki and Engberg [43] use a model for the recovery by climb accounting for the interaction with vacancies [65]. If the inclusion of the effect of precipitates is excluded from the model in Siwecki and Engberg [43], then it can be written as

23

ivmvri kTGbDcc (21)

where vmv Dc is a product of the fraction of vacancies, vc , and the diffusivity of migration, Dvm.

rc is a material parameter.

Militzer et al. [66] use a quite similar formulation based on Sandström and Lagneborg [40] and Mecking and Estrin [67] with a modification of the diffusivity that is consistent with the formulation in chapter 3.5,

23

23

* 22 ieqv

vvivi kT

Gbcc

DkT

GbD (22)

where eqvc is the thermal equilibrium vacancy concentration. The model in Lúkac and Balik [68]

for climb recovery can be reformulated to a similar expression and with a multiplying parameter, like rc in Eq. (21), that must be fitted with experiments.

The model in Eq. (22) is combined with Eq. (20) to give

223

2 eqieqv

vvi kT

Gbcc

D (23)

Solid-solution for steels like 316L may reduce the possibility for partial dislocations to unite in order to climb. This is believed to be, in co-operation with formation of fine subgrains, a reason for the high creep strength of 18-8 austenite stainless steel [62, 69, 70]. The creep rates are reduced [62] with a factor of

3.39.1Gb (24)

where is the stacking-fault energy. The stacking fault energy for 316 is 23 mJ/m-2 [71]. Therefore, we introduce a material parameter into Eq. (23) in the same way as in Eq. (21) giving

223

2 eqieqv

vvi kT

Gbcc

Dc (25)

8 of 49

The difference between this expression and Eq. (21) is the temperature dependent equilibrium vacancy concentration, eq

vc , discussed later.

3.4 Mobile dislocations The short-range stress is the component of the flow stress that moves the mobile dislocations through the lattice with its obstacles etc. There is the resistance of the lattice itself, the Peierl’s stress [72], which is not believed to be the controlling factor for fcc metals [12, 15]. Schoeck [15] notices that the different models used for thermal activation all lead to a reaction rate of the form of Eq. (9),

kTGp ef * [ Eq. (9) ]

Frost and Ashby [6] discuss different mechanisms controlling the dislocation glide in a crystal structure. A general formula for the activation energy, also in [73, 74], is

qp

FGˆ*1 (26)

F is the free energy required to overcome the lattice resistance or obstacles without aid from external stress. The quantity ˆ is the athermal flow strength that must be exceeded in order to move the dislocation across the barrier without aid of thermal energy. It reflects not only the strength but also the density and arrangement of the obstacles. The conditions for the exponents in Eq. (26) are

2110

qp

(27)

The stress dependency in the pre-exponential term f in Eq. (9) can be ignored when the activation energy is large. It arises from the variation of the mobile dislocation density [6, 17, 42] and thus assuming it is constant is consistent with the assumption of stress independent mobile dislocation density in Eq. (5). It is assumed in this study, like in [16, 42, 73, 75], that this is the case for the deformation process that has the largest contribution to the friction stress. The function f is then treated as a constant and Eq. (9) can be combined with Eq. (26) to

qp

kTF

refp e ˆ

*1

(28)

where the reference strain rate, ref , is constant as motivated above. Comparing with Eq. (5) gives that

mb am

ref (29)

Eq. (28) can be rewritten to give the friction stress as a function of the effective plastic strain rate

pq

pref

FkT

11

ln1ˆ* (30)

Nemat-Nasser and Guo [73] used the values p=2/3 and q=2. Uenishi and Teodosiu [16] used p=1/2 and q=1 and Follansbee [75] used p=1/2 and q=3/2. The influence of the coefficients p and q is small when the activation energy F is large [6] as in the case of obstacle limited dislocation glide. Therefore Frost and Ashby [6] propose the values p=q=1, which is also used in

9 of 49

this study. Table 2 lists approximate ranges for the activation energy and athermal strength for different obstacles. This study utilises only one obstacle activation energy although there may exist a range of obstacles of various types [76].

Table 2. Activation energy for different obstacles (from [6]).

Obstacle strength F ˆ Example

Strong 32Gbl

Gb Dispersions; large or strong precipitates (spacing l)

Medium 30.12.0 Gbl

Gb Small or weak precipitates (spacing l)

Weak 32.0 Gbl

Gb Lattice resistance; solution hardening (solute spacing l)

The stress needed for moving the dislocation in the lattice with different short-range obstacles is further influenced by dynamic strain ageing (DSA) or the Portevin-Le Chatelier effect (PLC). Then [4, 7, 77-80] presence of solute atoms can influence the stress needed to move the dislocations. The solute atoms can also affect the immobilisation and remobilisation of dislocations [41]. This was utilised by Bergström [38] and van Liempt et al. [23] to model strain-ageing by changing the evolution if immobile dislocation density. Their approach is not used in the current study.

If the dislocation velocity is low, i.e. the plastic strain rate is small, then the solute atoms can move with the dislocation without any problem. When the strain rate is higher, then there is an additional stress that must be overcome to move the dislocation. When the dislocation moves even faster, then the solute atoms are passed like fixed obstacles and the drag stress is reduced. DSA can be manifested as peaks or plateau in the variation of flow stress with temperature and negative strain rate sensitivity (SRS). It exhibits itself as serrations on the stress-strain curve. The aim in the current study is only to be able to represent the global response and not these serrations.

The DSA effect is believed to be caused by diffusion of C and N atoms but also by the solutes Cr and Ni at higher temperatures [78, 80, 81]. Cheng et al. [4] modified the ˆ and F in Eq. (30) for the activation energy whereas Rizzi and Hähner [77] added a transient term to the activation energy in order to model DSA. Both approaches give an increase in flow stress when the strain rate is low, in relation to the diffusion of solute atoms. None of the models do account for the observed influence of accumulated strain [80, 82] for triggering DSA. The model by Cheng et al.[4] modified the ˆ and F in Eq. (30) by

00ˆˆ

iyref

y

cc (31)

and

yref

y

ccFF 0 (32)

where yc is the solute concentration, y denotes the solute atom, of the dislocation and yrefc is the

bulk solute concentration corresponding to the evaluated 0F . Eq. (31) is further simplified by

10 of 49

removing the effect of bowing out of the dislocations accounted for by the dislocation density term 0i . Then the expression will be identical to the original equation when y

refy cc . We

introduce the following scaling of the obstacle parameters

yref

y

ccG0ˆ (33)

and

yref

y

ccGbfF 3

00 (34)

Thus we will use

pq

pref

yref

yyref

y

ccGbf

kTccG

11

30

0 ln1* (35)

Diffusion of solutes for obtaining yc is discussed in chapter 3.6.There is a lower limit of the plastic strain rate in this model giving zero friction stress. It is

yref

y

cc

kTGbf

refp e

30

min (36)

3.5 Vacancy migration and generation The existence and motion of vacancies is coupled with the recovery of dislocations as well as the diffusion of solute atoms. Creation of vacancies increases the entropy but requires energy. This leads to an equilibrium level of vacancies when a crystal is retained a sufficient time at a given temperature. Changing the temperature or deforming the material generates excess vacancies. The models considered here is only concerned with mono-vacancies. Haasen [12] gives a coupled set of evolution equations for vacancies and divacancies. However, that formulation does not account for generation of vacancies by plastic deformation.

The relation for the equilibrium concentration at a given temperature is [7, 12]

kTQ

vkTQ

kS

eqv

vfvfvf

eceec 0 (37)

where vfQ is the activation energy for forming a vacancy and vfS is the increase in entropy when creating a vacancy.

The self-diffusion coefficient can be written as [7]

kTQ

vkT

QQ

k

SS

v

vvfvmvfvm

eDeeaD 02 (38)

where a is the lattice constant, is the lattice vibration frequency, vmS is the entropy increase and vmQ is the energy barrier for the jumping of an atom into a vacancy. The parameters given in the last equality in Eq. (38) are those obtained from measurements. Using Eq. (37) the self-diffusivity can be written as

11 of 49

kTQ

kS

eqvv

vmvm

eeacD 2 (39)

Thus the self-diffusivity is proportional to the equilibrium concentration of vacancies. We assume that the diffusivity at non-equilibrium vacancy concentrations can be written as

veqv

vv D

ccD* (40)

The vacancy migration is used in the model below for vacancy annihilation. It is the part of the self-diffusion associated with the motion of existing vacancies. Thus it is written as

kTQ

vmkT

QkS

vm

vmvmvm

eDeeaD 02 (41)

The model in Siwecki and Engberg [43] for vacancy generation and recovery is written as

ieqvvvv

pv ccDccbcc 21 (42)

Militzer et al. [66] use a similar model

eqvv

ivm

j

vf

eqvv

exv cc

gD

bbc

Qbccc 22

02

14

(43)

where 0 is the atomic volume and cj is the concentration of thermal jogs. The term depends on the subcell formation of the dislocations and can from [67] be found to be

cK (44)

We therefore rewrite the equation by introducing the subcell size, s, from Holt’s relation discussed earlier. This gives

eqvvvm

j

vf

eqvv

exv cc

gsD

bbc

Qbccc 22

02

114

(45)

The factor in the mechanical production term, , is the fraction of the mechanical work spent for the vacancy formation [67]. Militzer et al. assume

1.0 (46)

The concentration of thermal jogs is given by

kTQ

j

fj

ec (47)

where the formation energy is approximated as

14

3GbQ fj (48)

and is Poisson’s ratio.

The parameter describes the neutralisation effect by vacancy emitting and absorbing jogs. It is given as

0

00

/5.0if0/5.0if5.0

j

jj

ccc

(49)

12 of 49

Militzer et al. [66] use 100 .

The rate of change in equilibrium concentration in Eq. (43) is only due to change in temperature

TTQ

cc vfeqv

eqv 2 (50)

We assume that the stress in Eq. (43) is equal to the effective stress, which is equal to the flow stress during a plastic deformation. Then we have

eqvvvm

j

vf

yexv cc

gsD

bbc

Qb

c 220

2

114

(51)

This equation is chosen as the model for vacancy generation and recovery. Further simplification makes it possible to relate to the model in Eq. (42). Excluding the effect of thermal jogs, taking Eq. (12) and ignoring the short-range component of the flow stress y leads to

eqvvvmi

vf

exv cc

gsDb

QG

c 220 11 (52)

The first part in the first term above can be directly identified with c1 in Eq. (42). The second term has a similar structure as the second term in Eq. (42).

An additional remark can be made regarding Eq. (52). Militzer et al. [66] refer to Mecking and Estrin [67]. The latter reference also supports the possibility to relate the recovery term of the vacancy concentration with the mean free path of a moving dislocation, Eq. (14). Then one could also write

eqvvvmi

vf

exv cc

gsDb

QG

c2

0 11 (53)

3.6 Diffusion of solutes The diffusion of solute atoms affects the motion of dislocations as discussed earlier. In the studied material the largest amount of solute atoms are Cr and Ni. The diffusivity is affected by the deformation due to creation of vacancies and dislocations [83-87]. The latter is called pipe diffusion. Diffusion is also easy along grain boundaries. The relative effects of the different contributions from lattice, pipe and grain boundary diffusion vary with temperature as they have different activation energies. However, the effect of solutes on mobile dislocations is only dependent on lattice diffusion as they only move within a grain or subcell before they are immobilised.

The equation for the temperature dependency of the diffusion is

kTQ

yl

yl

yx

eDD 0 (54)

where l denotes lattice diffusion and y is a symbol for the diffusing atom. Diffusivities are measured at an equilibrium value of vacancies and we assume similar to the Eq. (40) that it increases linearly with the vacancy concentration as

yleq

v

vyl D

cc

D * (55)

The diffusion of the solute atoms into the dislocations is dependent on two time scales. The characteristic time of the diffusion process is denoted tD. The dislocation motion is characterised

13 of 49

by the waiting time at obstacles, tw. Assuming the time of flight between obstacles is negligible and using Eq.s (3) and (29) give

apref

w vt (56)

The characteristic time for diffusion of solutes is assumed to be [4]

*

1y

lD KD

t (57)

where *ylD is proportional to the vacancy concentration, Eq. (55). K is a parameter to be

determined. The influence of the interaction energy between the solute and the dislocation is only accounted for via K. The evolution equation for the solute concentration is taken from [4]

D

wyys

yy

tthcccc 000 exp1 (58)

where ysc is a saturation value, yc0 is the bulk concentration of the solute in [4] and h0 is a

parameter to be determined. is 1/3 for pipe diffusion along dislocation line and 2/3 for volume diffusion in the lattice, which is the case in the DSA model. The assumption in Eq. (57) makes it possible to write

2

*

00 exp1b

tDhcccc wy

lyys

yy (59)

where h is a combination of h0 and K. The Burgers vector is introduced in order to make h non-dimensional. The model has been formulated for the case of several solutes in [4].

4 Stress update algorithm

The model for the yield stress will be used in a strain-stress algorithm in finite element codes. The radial return method and other similar algorithms are commonly used [88-91] for this purpose. They all need to update the stress and internal state variables from the control variables of a thermo-mechanical problem, i.e. from given increments in total strains, temperature and time, Figure 1. The lower box in Figure 1 illustrates the role of the logic described below for updating the flow stress. The plastic strain is an internal variable common to most plasticity models. The current model requires two additional internal state variables, immobile dislocation density and vacancy concentration. They are determined by the control variables and by the effective plastic strain increment. In additon, further dependent internal variables are introduced in order to create a framework where it is straightforward to replace different submodels. Collectively all these internal variables are denoted by the vector q in the formulas below.

Essential part of the stress-update algorithms is the solution of a non-linear equation known as the consistency condition [88-91] to compute the increment of effective plastic strain. It requires calculation of the yield stress and hardening modulus for the current iterative estimation of the plastic strain and internal variables. Moreover, the hardening modulus is also needed in the computation of the consistent tangent matrix of a stress-strain algorithm. This matrix is used in the implicit finite element codes in order to preserve the 2nd order rate of convergence of the full Newton-Raphson iterations for equilibrium [92]. The hardening modulus is defined as

14 of 49

py

py

dd

dd

H qq

q)( (60)

It is derived analytically in Appendix 1. Alternatively it can be computed by a numerical perturbation, which can be straight-forward and efficient as there is only one independent variable involved, the effective plastic strain.

Figure 1. Internal variables and yield stress in the context of finite element analysis.

The basic, essential internal variables needed for the model are the immobile dislocation density and the vacancy concentration, scaled by their initial or reference values as appropriate,

eqv

v

i

iTei c

c

0

q (61)

These are the only necessary state variables for the current model. We also introduce dependent internal variables; subcell size s , mobile dislocation density m , velocity of mobile dislocations v , solute concentration crc and grain size g . We write

000 ggc

vv

ssss Cr

refm

mTdiq (62)

This makes it simple to use other relations for subcell formation, stress dependent mobile dislocation density etc. than those in this study. The current model uses constant grain size but it is possible to include grain growth and recrystallisation into the numerical scheme. Finally, we include the two components of the flow stress

0,0

*

G

G

ref

Ty G

q (63)

into the vector of internal variables. It is then possible to have more elaborate relations for these variables as well. This is of particular interest for the short-range stress component * where several other options are common.

Strain-stress

algorithm tTTn

npnn

,,,,, q q111 ,, npnn

ddn 1

Determine yield stress, hardening modulus

and evolution ofinternal variables

py

Gy

dd

H

*TTt npn ,,,,q q1n

Determination of the plastic strain

increment

15 of 49

We use the notation Ty

Tdi

Tei

T qqqq (64)

The evolution of the essential internal state variables is governed by a coupled set of differntial equations,

tp

pi PP (65)

tp

pv XXc (66)

where Eq.s (13, 14) and Eq (25) give

bmPp (67)

sg111 (68)

223

2 eqieqv

vvt kT

Gbcc

DcP (69)

where eq is set zero. Eqs (50-51) together with Eq. (1) give

bbc

Qj

Gvf

p0

20

4* (70)

TTQ

cccsg

D vfeqv

eqvvvmt 222

11 (71)

These evolution equations are solved incrementally by a fully implicit method. We assume constant plastic strain rate during the increment

t

pp (72)

Thus we can write the system of evolution equations we have to solve as

ttnp

pn

i11 (73)

tc tnp

pn

v11 (74)

where the left superscript denotes that all variables are evaluated at time n+1t in the incremental procedure. Equations (73) and (74) is a set of nonlinear equations for ei

n q1 that will be solved simultaneously with the submodel equations for di

n q1 and yn q1 . The left superscript n+1 is

omitted in all the following equations in this section to simplify the writing. Equations (73) and (74) give

01

001 tPtPH t

pp

ii

i (75)

012 tXtX

ccc

H tp

peqv

eqv

v (76)

The model for formation of subcells by immobile forest dislocations, Eq. (18), is written as

16 of 49

0100

3i

i

ssssH (77)

where we scaled with

0

0

i

cKss (78)

The mobile dislocation density is assumed to be independent of stress in the current model and the relation H4 below is just used to compute this density. It can be used to evaluate the basic assumption of small mobile dislocation density compared with the immobile dislocation density or replaced with a more advanced model. Eq. (29) embodies the assumption of stress and strain independent mobile dislocation density corresponds to

014ref

am

mbH (79)

The Orowan equation, Eq. (3), is rewritten in non-dimensional form as

011 mvb m

p

(80)

where we arbitrarily choose strain rate of 1 s-1 as a reference. Furthermore, we define the reference dislocation velocity at this strain rate via the initial mobile dislocation density,

1-1

0 s1m

vb refm (81)

Combining the last two equations gives

001

5refm

mp

vvH (82)

The Cr and Ni solute atoms are combined in the diffusion model into one solute. This means that cCr is the sum of Cr and Ni concentrations. The diffusivity of Cr is used in the model. Then Eq. (59) gives

0exp1 2

*

006 btDhccccH w

CrlCrCr

sCrCr (83)

where

kTQ

Crleq

v

vCrleq

v

vCrl

Crl

eDcc

Dcc

D 0* (84)

and

pa

refwt [ Eq. (56) ]

The current model uses a constant grain size and thus the trivial relation is

010

7 ggH (85)

17 of 49

The flow stress is a sum of the long-range and the short-range components, Eq. (1). They are directly included in the system of equations to be solved. The model for short range obstacles, Eq. (35), can be written as

0ln*1

30

21

22

08

q

pref

qp

Crref

Crp

Crref

Crp

GbfkT

cc

cc

GH , (86)

where the bulk solute concentration, Crc0 , is used as a reference concentration.

The model for the long-range stress component, Eq. (12), is

iGG

G GbmH0,0,

9 , (87)

where initial value of this stress component is used for scaling.

We combine the complete set of equations of the present model with 9 submodels in a compact vector form,

0qH Ttp ,,, (88)

where the left superscript n+1 for the variables at the end of the increment is omitted. It is also left out in the equations below.

Variation of internal variables with the effective plastic strain is defined implicitly through this system of equations, Eq. (88), where the time step, t, and change in temperature are assumed to be given and fixed. For the present material model it is not feasible to derive analytical expression of such function, pqq . An iterative solution of the system of equations in Eq. (88) is required. Newton iterative procedure is used, with the given initial data for the beginning of the time step and change in control variables (Figure 1). The linearised system of equations for an iteration (i) is written as

0qq

HH )(

)(i

i (89)

where the subscript (i) is an iteration counter and

Ttpii ,,,qHH . (90)

From Eq. (89) the iterative change in internal variables is computed as

)(

1)(

ii H

qH

q (91)

Then the variables are updated,

qqq )()1( ii (92)

The iterations are terminated when the iterative changes for the independent variables are small,

01 *TOLi (93)

andeqvv cc *TOL2 . (94)

18 of 49

The matrix q

H )(i is given in Appendix 1.

The iterations need a predictor for the first iteration and usually this can be taken as

qq n)0( (95)

However, in the authors experience with the present model, the Newton iterations in such case tend to diverge, especially when increments are not very small. To prevent divergence during early iterations and to facilitate asymptotically quadratic convergence rate of the Newton method, a rough approximate solution is obtained first, which is then used to start the Newton iterations.

5 Experiments

A number of experiments were performed in order to characterise AISI 316L and to calibrate the dislocation-density model. Compression tests are the basis for the model development in the current study and the model parameters were calibrated by results from tests performed at different temperatures and strain rates. Tensile tests have been carried out for comparison. The chemical composition of the experimental steel is shown in Table 3.

Table 3. Chemical composition of AISI 316L [%].

C Si Mn P S Cr Ni Mo Cu N

0.009 0.27 1.74 0.030 0.024 16.82 10.26 2.08 0.31 0.029

5.1 Experimental procedure

Compression tests were performed at several temperatures and strain rates up to a total strain of 0.6, as shown in Table 4. Furthermore, four tests were performed with a holding time of 300 s at a strain of 0.2 before the compression was continued to 0.6. These tests were performed with a strain rate of 0.01s-1 at the temperatures given in Table 4. Four strain rate jump tests were also carried out. The initial strain rate of 0.01 s-1 was then increased to 1 s-1 at a total strain of 0.1. After compression to a total strain of 0.3 the rate was reduced to 0.01 s-1 and the test continued until 0.6. The nominal temperatures for all tests can be seen in Table 4. The actual temperatures were recorded and used in the optimisation together with the computed true stress and strain data. The true strain was obtained from measurements of the diameter change during the test. At room temperature (RT), this was calibrated with a measurement of the change in length. This calibration was also used for the high temperature tests. Tests at RT, 800ºC and 1100ºC were performed both in tensile and compression mode.

19 of 49

Table 4. Compression tests up to a total strain of 0.6.

[s-1] 20ºC 200ºC 400ºC 600ºC 700ºC 800ºC 900ºC 1100ºC 1300ºC

10 x x x x x x x x x

1 x x x x x x x x x

0.01 x x x x x x x x x

0.0005 x x

0.01

held for 300 s

x x x x

0.01 - 1 - 0.01 strain rate jump

x x x x

The mechanical test specimens were machined from a hot rolled, round bar of diameter 127 mm, with the rolling direction of the bar corresponding to the loading axis of the test specimens. Cylindrical specimens of diameter 10 mm and height 12 mm were used for the compression tests performed at temperatures above 800ºC. For tests conducted in the temperature range of RT-800ºC, cylindrical specimens of diameter 6 mm and height 9 mm were used. The tensile test specimens were rods, 10 mm in diameter and 115 mm long, with threaded ends. To ensure a uniform deformation zone, the specimens were machined with a thinner central part of approximately 25 mm. The diameter of the deformation zone was 5 mm for the RT tests and 9.5 mm for the tests performed at elevated temperature. The tests were performed in a Gleeble 1500 thermomechanical simulator. Between the compression-test specimen and the tungsten carbide anvils, a graphite foil and a tantalum foil were inserted in order to reduce the friction and to prevent sticking, respectively. For the tests with high reheating temperature, above 900ºC, the end surfaces of the specimens were coated with an electroplated nickel coating or a nickel foil in order to prevent carbon diffusion.

Each test specimen was first heated to 1100°C at a rate of 20°C/s where it was held for five minutes before it was cooled to the test temperature at a rate of 5°C/s. This procedure was carried out to ensure that all mechanical tests were performed with specimens of the same grain size. For comparison, one test at room temperature was performed both with and without the heating procedure. The resulting stress-strain curves were, however, very similar.

A grain size investigation was performed on the test specimens compressed at RT, 400ºC, 800ºC and 1300ºC at a strain rate of 1 s-1. A polished and etched cross-section of each specimen was studied in a light optical microscope (LOM) and the grain size was measured according to the mean linear intercept method [93]. To determine if any martensite had been induced during the mechanical tests, magnetic balance measurements were carried out. The tensile and compression test specimens tested at RT and the compression-test specimens tested at 400ºC were analysed.

5.2 Experimental results Some tests were duplicated and performed both in tensile and compression mode. The true stress-strain curves from these tests are shown in Figure 2. The tests with a strain rate of 10 s-1

are not shown in the figure but they follow the same trend. Many tests have been duplicated as the results have been reproducible. Variations are less than 5%. The difference between tension and compression can also be seen in Figure 2.

20 of 49

Figure 2. Tensile and compression tests at 20°C, 800°C and 1100°C at strain rates of 0.01 s-1.

Figure 3 shows the results from compression tests in the temperature range RT-1100°C at a strain rate of 0.01 s-1. Four strain rate jump tests are also included.

Figure 3. Compressions test for different temperatures at strain rate 0.01 s-1 including four strain rate jump tests up to 1 s-1.

21 of 49

Figure 4 shows the results from compression tests in the temperature range RT-1100°C at a strain rate of 1 s-1. Four strain rate jump tests, also shown in Figure 3, are included.

Figure 4. Compression tests at different temperatures at strain rate 1 s-1 including four strain rate jump tests from 0.01 to 1 s-1.

Compression-tests results at different temperatures at strain rate 10 s-1 are given in Figure 5.

Figure 5. Compression tests at different temperatures at strain rate 10 s-1.

22 of 49

The results from the compression tests, strain rate jump tests and holding tests at 900ºC and 1100ºC can be seen in Figure 6.

Figure 6. Compression tests at strain rate 0.01 s-1, strain rate jump and holding tests for 900ºC and 1100ºC.

Figures 7-10 show compression tests at different strain rates at 400ºC, 600ºC, 700ºC and 800ºC, respectively. The serrated flow, characteristic for DSA, is clearly seen at low strain rates at these temperatures.

Figure 7. Compression tests at different strain rates at 400ºC.

23 of 49


Figure 9. Compression tests at different strain rates at 700ºC. The flow stress curves at the strain rates of 0.01 and 1 s-1 are near each other. The flow stress for the lower rate exhibits serrations whereas the curve for 1 s-1 is smooth.

24 of 49


The microstructural investigation revealed that a small amount of martensite had been induced in the test conducted at RT. Deformation twins were observed at 1300ºC. As expected, the grain sizes were found to be in the same order of magnitude independent of testing temperature. The average grain size was 90 μm.

The results from the magnetic balance investigation are shown in Table 5. The compression test carried out at RT and strain rate 0.01 s-1 was found to contain 8 % of a magnetic phase. In this case, the magnetic phase most certain is deformation induced martensite.

Table 5. Measurements of magnetic phase

Test temperature [ºC]

Mechanical test type Strain Rate [s-1]

Amount of magnetic phase [%]

20 Compression 0.01 8.0

20 Compression 1 3.7


20 Tensile 0.01 1.3

20 Tensile 1 1.5

20 Tensile 10 1.0

400 Compression 0.01 1.0



25 of 49

6 Parameter fitting procedure The parameter determination is done by an error minimisation procedure using a developed Matlab toolbox also used in [2, 94, 95]. An error measure is minimised with respect to the material parameters. We minimise

nRppe (96)

subject to constraints

UpL (97)

where p is a vector with the n unknown parameters and e is an error measure. L and U are lower and upper limits for the parameters.

We consider the problem as strain driven. The strain history is given as ninc number of strain increments. Bruhns and Anding [96] write the error measure as

Grrp Te21 (98)

where G is a diagonal matrix with individual standard deviation errors for the measured data. The vector r has the elements

iecir (99)

where ininci ...1 (100)

denotes the different sampling times of the strain during the tests. The matrix G can also be used to make the error non-dimensional [97] and include user defined weight functions, w( ). We use

iiii wG (no summation on i) (101)

The weights wi make it possible to place more emphasis on more important parts of the measured data, see chapter 6.2. The overall logic of the parameter fitting is shown in Figure 11. pfinal is the material parameters that minimize the difference between computed c and measured stress e.

Constitutive model

Measureddata from

uniaxial test


constraints

T(t), (t)

c(t)

e(t) T(t), (t)

Error function

Minimise

pinit

pfinal

T(t), (t), e(t)

ptrial Constitutive model

Measureddata from

uniaxial test


constraints

T(t), (t)

c(t)

e(t) T(t), (t)

Error function

Minimise

pinit

pfinal

T(t), (t), e(t)

ptrial

Figure 11. Parameter fitting procedure for minimisation of difference between measured and computed stress.

26 of 49

6.1 Parameter space for optimisation The parameters to be determined from the mechanical tests are given in Table 6. Ranges are given when the parameters is unknown. These ranges are used as upper and lower limit in the parameter optimisation procedure described above.

It is assumed that Poisson’s ratio is constant 3.0 and we have

12EG (102)

Young’s modulus is assumed to be 2·105 MPa at room temperature and decreases linearly to 1.16·105 MPa at 1200 °C. This is consistent with the values in the report by Burgan [98] and also used by Frost and Ashby [6] for AISI 316. refG is the shear modulus at 27 °C and its value is 81 GPa. These data are extrapolated down to 1.09·105 MPa at 1300 °C. The thermal expansion is also needed when computing the stress. It is given in Table 7.

Some physical constants that are used are given in Table 8. It also contains measured or approximated or assumed values for some parameters. Tm is the melting temperature, 1440ºC. The entropy increase due to the formation of a vacancy, Eq. (37), can be estimated from a simplified analysis for temperature higher than the Debye temperature to be in the order of 0.5k-2k. We follow Militzer et al. [66] and assume kSvf . Approximate relations [81] indicates that

vvm QQ 4.0 (103)

and then

vvf QQ 6.0 (104)

This is in fair agreement with the data in Table 8.

We assume in this study that Cr and Ni are the important solutes causing DSA in the temperature range around 600ºC. Their concentrations are added into one concentration denoted Crc . We assume lattice diffusion is dominating giving =2/3 and use the diffusivity for Cr in the model.

27 of 49

Table 6. Parameters to be determined.

Notation Used in Eq. (#) Value Dimension Comment

12, 87 0.4 - [34, 35]

Kc 18, 78 - See chapter 3.3.1.

s 18, 78 [m] Moteff [99] give examples around 0.5 m.

0i Initial and scaling value

120

10 101101 i[m/m3] The lower limit was used

in [34].

c 25, 69 <1 -

0 35, 86 1001.0 0 - [6] used 6.5·10-3 for AISI 316

0f 35, 86 21.0 0f - [6] used 0.5 for AISI 316

p 35, 86 10 p -

q 35, 86 21 q -Crsc 59, 83 CrCr

s cc 01 -

h 59, 83 -

Table 7. Thermal dilatation

Temperature [°C] 0 50 150 250 350 450 550 650 750 1500

Thermal strain [-]·103

0 0.58 1.73 3.18 4.75 6.43 8.12 9.88 11.6 2.49

28 of 49

Table 8. Known or assumed parameters.

Notation Used in Eq. (#)

Value Dimension Comment

k Boltzmann’s constant

1.38·10-23 [J/K]

m Taylor factor 3.06 - For fcc [6]

b Burgersvector

2.58·10-10 [m] For AISI 316 from [6]

a 29, 56, 79 1. 1012 - [15]

ref29, 35, 56,

79, 86 1. 106 [s-1] For AISI 316 from [6]

Crc059, 83 0.27 - Combined bulk

fraction of Ni and Cr

vfS 37 (gives eqvc ) k [J/K] From [66]

vfQ 37, 50-51, 70-71

2.46·10-19 [J] From [66] and gives eqvc

0vmD 41, 51, 71 1.37·10-5 [m2/s] Frost and Ashby [6]

vmQ 41, 51, 71 2.38·10-19 [J] From [6].

kS

vmv

vf

eDD 00

38, 25, 69 3.7·10-5 [m2/s] Relation is derived from Eqns (38) and

(41). Data is consistent with [100] and [6].

vmvfv QQQ 38, 25, 69 4.65·10-19 [J] From [6], Kim [101], Perkins [100] give a similar value for the 18-8 steel

0 49, 51, 70 10 - From [66]

fjQ 47, 51, 70 [J] From [66]

51, 70 0.1 [-]

0 51, 70 1.21·10-29 [m3] From [6]

CrlD0

(1) 54,59, 83-84 1.3·10-5 [m2/s] Perkins [100]

CrlQ (1) 54,59, 83-84 4.37·10-19 [J] Perkins [100]

59, 83 2/3 [-]

0g 85 90·10-6 [m] Measured, see chapter 5.2.

(1) The lattice diffusivity of Cr is used both for Cr and Ni.

29 of 49

6.2 Strategy for optimisation and found value of parameters The basic approach is to use the physical constants in Table 8 without modifications. The divide and conquer strategy described below was used to determine the parameters in Table 6. The weighting function in Eq. (101) was set up so that the part of the strain-stress curve up to 0.2 was given five times larger weights than the last part of the curve. Thus more emphasis is placed on fitting this part of the curves when performing optimisations for the compression tests. The optimisation strategy can be summarised as

1. Determination of p, q, and 0i

An evaluation of the virgin yield limits, Tpy ,0 , from all the compression tests at the

different temperatures and strain rates was made. The virgin yield limit in the model depends on the parameters p, q, 0 , 0f , h, Cr

sc , and 0i . The tests at 200 and 400ºC show that the recovery, chapter 3.3.2, and the rate effects in the friction stress term, chapter 3.4, are negligible. These tests were used to evaluate the parameters for the long-range stress component and the pertaining hardening model. These parameters are Kc, s and 0i . The last parameter does not affect the results much within the range that is possible 10101001 . The higher value was chosen. The focus was on matching the initial slope of the curve as the model could not reproduce the complete curve. The the values for and 0i were fixed together with p and q.

2. Determination of 0 , 0f and Crsc and excluding DSA model.

Preliminary evaluation for all tests at each on of the test temperatures was done. This showed that the parameters 0 and 0f become quite stable and they were then taken as constant. It was found at this stage that the model for DSA did not work well. The value for the solute fraction in the mobile dislocation was assumed to be the bulk value (0.27) up to 400ºC and quite arbitrarily fixed to 8.0Cr

sc at higher temperatures.

3. Determination of Kc, s and c versus temperature.

Evaluation of the remaining temperature dependent parameters Kc, s , c separately for each test temperature. c obtained a quite constant value for temperatures below 400ºC and is of no importance in the low temperature range as the recovery processes are negligible in these tests.

4. Demonstration of DSA model, h-term.

A test optimisation for determining h using 8.0Crsc was done at 800ºC.

Table 9. Obtained parameters. Kc

(1) s (1)0i c (1)

0 0f p q Crsc h (2)

0.5 1·1012 0.0045 0.8 0.27 1.34 0.8 160

- - [ m] - - - - - - - - (1) See Table 10. (2) Only evaluated at 800 ºC.

30 of 49

Table 10. Obtained parameters that are taken as temperature dependent.

T [ºC] 20 200 400 600 700 800 800(1) 900 1100 1300

Kc [-] 24 20 20 16 14.7 10 29 59 140 200

s [ m] 0.25 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0

c [-] 0.001 0.001 0.001 0.003 0.007 0.01 0.01 0.018 0.01 0.04 (1) accounting for DSA with h=160

7 Comparison of model and experiments The computed results using the parameters given in previous chapter are shown together with measurements below. Figure 12 shows them for the temperatures 20ºC, 200ºC and 400ºC. Symbols denote experiments for the different strain rates (triangles 10s-1, stars 1s-1 and circles 0.01s-1). Solid line is computed results for 0.01s-1 and dashed line for dashed lined for 1s-1 anddotted line for 10s-1. The same for the tests in the temperature range 600 - 1300ºC is given in Figure 13-18, respectively. Figure 13 and Figure 14 have an additional strain rate of 0.0005s-1.and Figure 16 and Figure 17 includes rate jump tests and hold performed test as described earlier. These tests for lower temperatures were not included for comparison as these rate effects are small at the lower temperatures as can be seen in for example Figure 3. The DSA model is only utilised the stress-strain curves in Figure 19. This curve corresponds to the results in Figure 15 where the solute concentration was fixed to 0.8 for all rates. Further results for the DSA model at 800°C are shown in the later figures. Solute concentration is shown in Figure 20, vacancy concentration in Figure 21. Figure 22 shows the computed dislocation density.

All figures use symbols for smoothed experimental values for the different strain rates. The dots are used for the strain rate 0.0005 s-1, circles for 0.01 s-1, stars for 1 s-1 and triangles for 10 s-1.Fat solid line is denotes computed results for 0.0005 s-1, solid line for 0.01 s-1, dashed line for 1 s-1 and dotted line for 10 s-1.

Figure 12. Smoothed measured data (symbols) and computed flow stress (lines) for 20, 200 and 400°C, for strain rates 0.01 (blue), 1.0 (green) and 10 s–1 (red). It is assumed that 27.0Crc .

31 of 49

Figure 13. Smoothed measured data (symbols) and computed flow stress (lines) for 600°C. It is assumed that 8.0Crc .


32 of 49


Figure 16. Smoothed measured (symbols) and computed flow stress (lines) for 900°C. Strain rate jump test and hold test for this temperature are also shown. It is assumed that 8.0Crc .

33 of 49

Figure 17. Smoothed measured (symbols) and computed flow stress (lines) for 1100°C. Strain rate jump test and hold test for this temperature are also shown. It is assumed that 8.0Crc .

Figure 18. Smoothed measured (symbols) and computed flow stress (lines) for 1300°C. It is assumed that 8.0Crc .

34 of 49

Figure 19. Smoothed measured (symbols) and computed flow stress (lines) for 800°C including the DSA effect.

Figure 20. Computed solute concentration, cCr, for 800°C, including the DSA effect.

35 of 49

Figure 21. Computed vacancy concentration for 800°C including the DSA effect.

Figure 22. Computed dislocation density for 800°C including the DSA effect.

36 of 49

8 Discussions

8.1 Experiments

8.1.1 Tension-compression assymmetryThe difference between tensile and compressive tests in Figure 2 is yet to be explained. The difference seems to disappear at 1100°C. The same differences have been observed in earlier tests performed at Sandvik Materials Technology for Sanmac 316L. Then the difference disappeared at 800°C.

A difference in tensile and compressive behaviour was also found by Yapici et al. [102]. They produced test pieces by the equal channel angular extrusion (ECAE) process at 700°C and 800°C. These were quenched and then tested at room temperature. Asymmetry in tension and compression was also observed for test pieces produced at 700°C but not at 800°C. This asymmetry was not observed in as-received samples. They considered the assymmetry to be due to either texture or residual stresses. Twinning occurs more easily under tensile loading in the [1 1 1] direction of the crystal whereas the opposite holds for the [0 0 1] direction. However, their samples did not exhibit any texture and, based on further investigations, they concluded that the samples produced at 700°C had residual stresses causing the asymmetry.

Texture measurements have not been completed in the current study. The test pieces have not been produced in the same way as in [102] and tests in Figure 2 indicate about the same initial yield limit for compression and tension tests. The difference is mainly the hardening behaviour. Thus the difference can not be explained based on the current available information.

It should be noted that the model proposed in this paper has been calibrated using the compression tests. The model would have fitted the tensile tests better with their more linear hardening.

8.1.2 Rate dependency effectsThe difference between the different strain rates for the test from RT up to 400°C is considered to be small, see Figure 3 and Figure 4.

8.1.3 TwinningTwins have been observed even in the test performed at 1300°C. Moteff [99] observed twinning for cold-worked bar but not in cold-worked sheet of AISI 316. The first case had larger initial grain sizes. This is in agreement with Karaman et al. [47, 103]. El-Danaf et al. [104] performed compression tests for AISI 316L with a strain rate of 0.001 s-1 up to a total strain of 1.0. The samples had a grain size of 40 m. They noticed primary twinning for strains between 0.1-0.25 leading to a reduced hardening rate. No increase in hardening due to twin intersections was observed. Yapici et al. [102] observed twinning in their tests at 700 and 800°C.

8.1.4 Dynamic strain agingDSA can be seen in Figure 7-10. The temperature range is 400-800°C. Barnby [105] observed jerky flow for tensile tests of AISI 316 in the temperature range of 300-700°C for a strain rate of 0.00013 s-1. The material had a grain size of 180 m. The rate dependency of the flow stress was small below 600°C. He observed a smooth region before DSA occurred and his analysis supported that vacancy assisted diffusion of Cr could cause this. Hong and Lee [80] studied AISI 316L. The specimens had an initial 17 % cold work and were thereafter cyclically loaded. The strain rates were in the range of 0.0001 – 0.01s-1. They plotted the region of serrated flow versus strain rate and temperature. Lower rates gave a wider region shifted to lower temperatures for serrated flow. They also classified the serrations into different types. Mannan [106] investigated an AISI 316 steel and found that finer grains lowers the temperature interval for DSA. A grain

37 of 49

size of 0.04 mm gave a temperature range for DSA from 250 to 600°C whereas a grain size of 0.65 mm gave a range of 350-700°C. Yapici et al. [102] discuss the work of others. Interstitials and/or solutes cause DSA in the region of 300-900°C depending on strain rate.

8.2 Model versus experiments

8.2.1 HardeningComparisons between computed and measured stress-strain curves in chapter 7 show that they agree quite well at higher temperatures. The hardening evolution is more linear in the model, Figure 12, than in the measurements at lower temperatures. This is believed to be due to twinning that reduced the hardening [104]. The current model does not include twinning. The microstructure measurements also found twinning at higher temperatures. The agreement between tests and computations are probably due to a too large recovery in the model. The optimisation procedure could make the tests and computations agree by determining a large value for c . The recovery term at low temperatures is negligible and thus no large c could make up for the missing twinning effect in the model.

The subcell formation model, Eq. (18), was found to work quite well despite the complexities dissuced in chapter 3.3.1 when allowing Kc to be temperature dependent. The value for Kc may become different when extending the current model with a model for twinning.

No measurements of dislocation densities have been performed. Thus the computed dislocation densities and vacancy concentrations are not validated. Angella et al. [107] investigated the influence of grain size and shape by torsion tests of 316L around 1000ºC at a strain rate of 0.006 s-1. They found that i was 11·1014 m/m3 near grain boundaries and much lower in interior at

36.0p . Sauzay et al. [108] performed cyclic tests and measured a dislocation density of 3·1013 m/m3 in the interior of grains.

8.2.2 Dynamic strain ageingThe model for solute diffusion is a large simplification of the actual phenomenon as Kubin et alpoint out [20]. An attempt is made to model DSA in the same manner as in Cheng et al. [4]. This model was extended with the effect of vacancy concentration on the diffusion of solutes. This was made in order to enable the triggering of DSA occurring after some strain. For example, Figure 10 shows that the flow stress for the test with a strain rate of 1 s-1 becomes larger than the flow stress for the rate 10 s-1 around a strain of 0.2. It was hoped that the increase in vacancy concentration would increase the diffusivity of the solutes so that DSA occurs at that instant. However, it was found that it is not possible to adjust the parameter h in Eq. (59) so that the solute only diffuses into the mobile dislocations for the lower rates and not the higher rates. This is due to the fact that the difference in the evolution of vacancy concentration for different strain rates is not large enough, Figure 21. Cheng et al. [4] used numerical values for the diffusivities that did not correspond to any real atoms which is in contrast to our approach to not calibrate the physical constants in Table 8. There are several simplifications in the phenomenological model for solute diffusion, Eq. (59) as well as in the assumption on how it affects the friction stress, Eq. (35). This would need improvement or the alternative approach of letting the solute diffusion pin dislocations, also discussed in chapter 3.4 can be used. The latter would mean that their effect would contribute to Eq. (13).

38 of 49

9 Conclusions A physically based constitutive model has been proposed and calibrated versus compression tests from RT up to 1300°C and for strain rates up to10 s-1. The model requires 11 parameters to be determined from mechanical testing. The overall agreement is good provided the strain is less than 0.25. The model can not represent the DSA behaviour as hoped.

The inclusion of deformation due to twinning has been found to be an important extension in order to improve accuracy at higher strains. The model for dynamic strain aging is not sufficiently accurate. The numerical procedure has been adapted to what is needed in finite element simulation and is suitable for large scale computations as the computational overhead is negligible. The model with its submodels has been implemented in a flexible way that will alleviates future extensions/modifications.

10 Acknowledgements The financial support from the Swedish Knowledge Foundation and Sandvik Materials Technology is gratefully acknowledged. Furthermore, the authors also express their appreciation to PhD Mahesh Somani and the staff at Mechanical Engineering Dept, University at Oulu, Oulu, Finland for performing the mechanical testing in an excellent way.

11 References

[1] Lindgren, L.-E., Finite element modelling and simulation of welding, Part 2 Improved material modelling. Journal of Thermal Stresses, 2001. 24: p. 195-231.

[2] Lindgren, L.-E. Models for forming simulations of metastable austenitic stainless steel. in 8th International Conference on Numerical in Industrial Forming Processes, NUMIFORM '04. 2004. Ohio, USA.

[3] Kocks, U.F. and H. Mecking, Physics and phenomenology of strain hardening: the FCC case. Progress in Materials Science, 2003. 48(3): p. 171-273.

[4] Cheng, J., S. Nemat-Nasser, and W. Guo, A unified constitutive model for strain-rate and temperature dependent behavior of molybdenum. Mechanics of Materials, 2001. 33(11):p. 603-616.

[5] Frost, H.J. and M.F. Ashby, Deformation-mechanism maps for pure iron, two austenitic stainless steels and a low-alloy ferritic steel, in Fundamental Aspects of Structural Alloy Design, R.I. Jaffee and B.A. Wilcox, Editors. 1977, Plenum Press. p. 26-65.

[6] Frost, H.J. and M.F. Ashby, Deformation-Mechanism Maps - The Plasticity and Creep of Metals and Ceramics. 1982: Pergamon Press. 166.

[7] Reed-Hill, R.E. and R. Abbaschian, Physical Metallurgy Principles. Third ed. 1992: PWS-KENT Publishing Company.

[8] Bergström, Y., The plastic deformation of metals - A dislocation model and its applicability. Reviews on powder metallurgy and physical ceramics, 1983. 2/3: p. 79-265.

[9] Gil Sevillano, J., P. van Houtte, and E. Aernoudt. Large strain work hardening and textures. in Progress in Materials Science. 1981.

39 of 49

[10] Nes, E., Modelling of work hardening and stress saturation in FCC metals. Progress in Materials Science, 1997. 41(3): p. 129-193.

[11] Estrin, Y., Dislocation theory based constitutive modelling: foundations and applications. Journal of Materials Processing Technology, 1998. 80-81: p. 33-39.

[12] Haasen, P., Physical Metallurgy. Third ed. 1996: Cambridge University Press. 420.

[13] Mughrabi, H., Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metall., 1983. 31(9): p. 1367-1379.

[14] Mughrabi, H., A two-parameter description of heterogeneous dislocation distributions in deformed netal crystals. Materials Science and Engineering, 1987. 85: p. 15-31.

[15] Schoeck, G., Thermal activation in plastic deformation, in Multiscale Phenomena in Plasticity: From experiments to phenomenology, modelling and materials engineering,G. Saada, Editor. 2000, Kluwer Academic Publications. p. 33-55.

[16] Uenishi, A. and C. Teodosiu, Constitutive modelling of the high strain rate behaviour of interstitial-free steel. International Journal of Plasticity, 2004. 20(4-5): p. 915-936.

[17] Bonneville, J. and J.-L. Martin, Multiplication, mobility and exhaustion of dislocations,in Multiscale Phenomena in Plasticity: From experiments to phenomenology, modelling and materials engineering, J. Lépinoux, et al., Editors. 2000, Kluwer Academic Publications. p. 57-66.

[18] Keh, A., Y. Nakada, and W. Leslie. Dislocation Dynamics. in Batelle Institute Materials Science Colloquia. 1967. Seattle, USA: McGraw-Hill.

[19] Bressers, J. and M. Steen, Fatigue and microstructure in austenitic high temperature alloys. International Journal of Pressure Vessels and Piping, 1991. 47(2): p. 217-245.

[20] Kubin, L., Y. Estrin, and C. Perrier, On static strain ageing. Acta Metall. Mater., 1992. 40(5): p. 1037-1044.

[21] Bergström, Y., Dislocation model for the stress-strain behaviour of polycrystalline alpha-iron with special emphasis on the variation of the densities of mobile and immobile dislocations. Mat. Sci. Eng., 1969/70. 5: p. 193-200.

[22] Bergstrom, Y. and W. Roberts, The dynamical strain ageing of [alpha]-iron: Effects of strain rate and nitrogen content in the jerky-flow region. Acta Metallurgica, 1973. 21(6): p. 741-745.

[23] van Liempt, P., M. Onink, and A. Bodin, Modelling the influence of dynamic strain ageing on deformation behaviour. Advanced Engineering Materials, 2002. 4(4): p. 225-232.

[24] Zikry, M.A. and M. Kao, Inelastic microstructural failure mechanisms in crystalline materials with high angle grain boundaries. Journal of the Mechanics and Physics of Solids, 1996. 44(11): p. 1765-1798.

[25] Estrin, Y., L.S. Toth, A. Molinari, and Y. Brechet, A dislocation-based model for all hardening stages in large strain deformation. Acta Materialia, 1998. 46(15): p. 5509-5522.

[26] Schwink, C., Flow stress dependence on cell geometry in single crystals. Scripta Metallurgica et Materialia, 1992. 27: p. 963.968.

[27] Cheong, K.S., E.P. Busso, and A. Arsenlis, A study of microstructural length scale effects on the behaviour of FCC polycrystals using strain gradient concepts. International Journal of Plasticity. In Press, Corrected Proof.

40 of 49

[28] Arsenlis, A., D. M. Parks, R. Becker, and V. V. Bulatov, On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals. Journal of the Mechanics and Physics of Solids, 2004. 52(6): p. 1213-1246.

[29] Ashby, M., The deformation of plastically non-homogeneous materials. Philisophical Magazine, 1970. 21: p. 399-425.

[30] Evers, L.P., W.A.M. Brekelmans, and M.G.D. Geers, Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids, 2004. 52(10): p. 2379-2401.

[31] Evers, L.P., W.A.M. Brekelmans, and M.G.D. Geers, Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures. In Press, Corrected Proof.

[32] Mughrabi, H., On the current understanding of strain gradient plasticity. Materials Science and Engineering A, 2004. 387-389: p. 209-213.

[33] Seefeldt, M. and P. Klimanek, Modelling of flow behaviour of metals by means of a dislocation-disclination reaction kinetics. Modelling and Simulation in Materials Science and Engineering, 1998. 6(4): p. 349.

[34] Feaugas, X., On the origin of the tensile flow stress in the stainless steel AISI 316L at 300 K: back stress and effective stress. Acta Materialia, 1999. 47(13): p. 3617-3632.

[35] Feaugas, X. and C. Gaudin, Different levels of plastic strain incompatibility during cyclic loading: in terms of dislocation density and distribution. Materials Science and Engineering A, 2001. 309-310: p. 382-385.

[36] Peeters, B., et al., Work-hardening/softening behaviour of b.c.c. polycrystals during changing strain paths: I. An integrated model based on substructure and texture evolution, and its prediction of the stress-strain behaviour of an IF steel during two-stage strain paths. Acta Materialia, 2001. 49(9): p. 1607-1619.

[37] Peeters, B., et al., Work-hardening/softening behaviour of b.c.c. polycrystals during changing strain: : Part II. TEM observations of dislocation sheets in an IF steel during two-stage strain paths and their representation in terms of dislocation densities. Acta Materialia, 2001. 49(9): p. 1621-1632.

[38] Bergstrom, Y., A dislocation model for the strain-ageing behaviour of steel. Materials Science and Engineering, 1972. 9: p. 101-110.

[39] Michalak, J., The influence of temperature on the development of long-range internal stress during the plastic deformation of high-purit iron. Acta Metall., 1965. 13(3): p. 213-222.

[40] Sandström, R. and R. Lagneborg, A model for hot working occurring by recrystallization.Acta Metallurgica, 1975. 23(March): p. 387-398.

[41] Estrin, Y., High temperature plasticity of metallic materials, in Thermodynamics,Microstructures and Plasticity, A. Finel, D. Mazière, and M. Veron, Editors. 2003, Kluwer Academic Publications. p. 217-238.

[42] Estrin, Y., Constitutive modelling of plastic flow, in Multiscale Phenomena in Plasticity: From experiments to phenomenology, modelling and materials engineering, J. Lépinoux, et al., Editors. 2000, Kluwer Academic Publications. p. 339-348.

[43] Siwecki, T. and G. Engberg. Recrystallisation controlled rolling of steels. in Thermo-Mechanical Processing in Theory, Modelling & Practice [TMP]2. 1996. Stockholm, Sweden: the Swedish Society for Materials Technology.

41 of 49

[44] Perlade, A., O. Bouaziz, and Q. Furnemont, A physically based model for TRIP-aided carbon steels behaviour. Materials Science and Engineering A, 2003. 356(1-2 SU -): p. 145-152.

[45] Kuhlmann-Wilsdorf, D., Theory of plastic deformation: - properties of low energy dislocation structures. Materials Science and Engineering A, 1989. 113: p. 1-41.

[46] Thompson, A., Substructure strengthening mechanisms. Metall. Trans. A, 1977. 8: p. 833-842.

[47] Karaman, I., H. Sehitoglu, H.J. Maier, and Y.I. Chumlyakov, Competing mechanisms and modeling of deformation in austenitic stainless steel single crystals with and without nitrogen. Acta Materialia, 2001. 49(19): p. 3919-3933.

[48] Nabarro, F.R.N., Sequences of dislocation patterns. Materials Science and Engineering A, 2001. 317(1-2): p. 12-16.

[49] Holt, D., Dislocation cell formation in metals. Journal of Applied Physics, 1970. 41(8): p. 3197-3201.

[50] Bocher, L., P. Delobelle, P. Robinet, and X. Feaugas, Mechanical and microstructural investigations of an austenitic stainless steel under non-proportional loadings in tension-torsion-internal and external pressure. International Journal of Plasticity, 2001. 17(11):p. 1491-1530.

[51] Kuhlmann-Wilsdorf, D. and J.H. Van Der Merwe, Theory of dislocation cell sizes in deformed metals. Materials Science and Engineering, 1982. 55(1): p. 79-83.

[52] Staker, M.R. and D.L. Holt, The dislocation cell size and dislocation density in copper deformed at temperatures between 25 and 700 C. Acta Metallurgica, 1972. 20(4): p. 569-579.

[53] Hughes, D.A., D.B. Dawson, J.S. Korellis, and L.I. Weingarten, A microstructurally based method for stress estimates. Wear, 1995. 181-183(Part 2): p. 458-468.

[54] Michel, D.J., J. Moteff, and A.J. Lovell, Substructure of type 316 stainless steel deformed in slow tension at temperatures between 21[deg] and 816[deg]C. Acta Metallurgica, 1973. 21(9): p. 1269-1277.

[55] Raj, S.V. and G.M. Pharr, A compilation and analysis of data for the stress dependence of the subgrain size. Materials Science and Engineering, 1986. 81: p. 217-237.

[56] Challenger, K. and J. Moteff, ??++. Metall. Trans., 1973. 4: p. 749.

[57] Hansen, N., Microstructure and flow stress of cell forming metals. Scripta Metallurgica et Materialia, 1992. 27: p. 947-950.

[58] Hughes, D. and W. Nix, Strain hardening and substructural evolution in Ni-Co solid solutions at large strains. Materials Science and Engineering A, 1989. 122: p. 153-172.

[59] Owen, D., et al., Modeling of cutting for residual stresses. to appear in one of two monographs about residual stresses, Residual Stress and Mechanical Design.

[60] Sandhya, R., K. Bhanu Sankara Rao, S.L. Mannan, and R. Devanathan, Substructuralrecovery in a cold worked Ti-modified austenitic stainless steel during high temperature low cycle fatigue. International Journal of Fatigue, 2001. 23(9): p. 789-797.

[61] Molinari, A. and G. Ravichandran, Constitutive modeling of high-strain-rate deformation in metals based on the evolution of an effective microstructural length. Mechanics of Materials, 2005. 37(7): p. 737-752.

42 of 49

[62] Nabarro, F. and H. de Villiers, The Physics of Creep. 1995, London: Taylor & Francis. 413.

[63] Engberg, G., Recovery in a titanium stabilized 15% Cr - 15¤ Ni austenitic stainless steel.1976, Royal Institute of Technology: Stockholm. p. 50.

[64] Li, J., Recrystallisation, Grain Growth and Textures. 1966: ASM.

[65] Nabarro, F., Work hardening and dynamical recovery of fcc metals in multiple glide.Acta Metall., 1989. 37(6): p. 1521-1546.

[66] Militzer, M., W. Sun, and J. Jonas, Modelling the effect of deformation-induced vacancies on segregation and precipitation. Acta Metall. Mater., 1994. 42(1): p. 133-144.

[67] Mecking, H. and Y. Estrin, The effect of vacancy generation on plastic deformation.Scripta Metallurgica, 1980. 14: p. 815-819.

[68] Lukác, P. and J. Balik, Kinetics of plastic deformation. Key Engineering Materials, 1994. 97-98: p. 307-332.

[69] Sherby, O., Factors affecting the high temperature strength of polycrystalline solids.Acta Metallurgica, 1962. 10(Feb): p. 135-147.

[70] Sherby, O. and P. Burke, Mechanical behavior of crystalline solids at elevated temperature, in Progress in Materials Science, B. Chalmers and W. Hume-Rothery, Editors. 1967, Pergamon Press: Oxford. p. 325-390.

[71] Ferreira, P.J. and P. Mullner, A thermodynamic model for the stacking-fault energy. Acta Materialia, 1998. 46(13): p. 4479-4484.

[72] Schoeck, G., The Peierls energy revisited. Philosophical Magazine A, 1999. 79(11): p. 2629-2636.

[73] Nemat-Nasser, S. and W.-G. Guo, Thermomechanical response of HSLA-65 steel plates: experiments and modeling. Mechanics of Materials, 2005. 37(2-3): p. 379-405.

[74] Schulze, V. and O. Vöhringer, Influence of alloying elements on the strain rate and temperature dependency of the flow stress of steels. Metallurgical and Materials Transactions A, 2000. 31A(March): p. 825-830.

[75] Follansbee, P.S., High-strain-rate deformation of FCC metals and alloys, in Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena, L.E. Murr, K.P. Staudhammer, and M.A. Meyers, Editors. 1986, Marcel Dekker: New York. p. 451-479.

[76] Robertson, I.M., et al., Dislocation-obstacle interactions: Dynamic experiments to continuum modeling. Materials Science and Engineering A. In Press, Corrected Proof.

[77] Rizzi, E. and P. Hahner, On the Portevin-Le Chatelier effect*1: theoretical modeling and numerical results. International Journal of Plasticity, 2004. 20(1): p. 121-165.

[78] Cho, S.-H., Y.-C. Yoo, and J. Jonas, Static and dynamic strain aging in 304 austenitic stainless steel at elevated temperatures. Joruna of Materials Science Letters, 2000. 19(22): p. 2019-2022.

[79] Nemat-Nasser, S. and W.-G. Guo, Thermomechanical response of DH-36 structural steel over a wide range of strain rates and temperatures. Mechanics of Materials, 2003. 35(11): p. 1023-1047.

43 of 49

[80] Hong, S.-G. and S.-B. Lee, The tensile and low-cycle fatigue behavior of cold worked 316L stainless steel: influence of dynamic strain aging. International Journal of Fatigue, 2004. 26(8): p. 899-910.

[81] Jenkins, C. and G. Smith, Serrated plastic flow in austenitic stainless steel. Transactions of the Metallurgical Society of AIME, 1969. 245(October): p. 2149-2156.

[82] Chen, M., L. Chen, and T. Lui, Analysis of the critical strain associated with the onset pf Portevin-LeChatelier effect of substitutional fcc alloys. Acta Metall. Mater., 1992. 40(9): p. 2433-2438.

[83] Kothari, R. and R.W. Vook, The effect of cold work on surface segregation of sulfur on oxygen-free high conductivity copper. Wear, 1992. 157(1): p. 65-79.

[84] Tvrdy, M., R. Seidl, L. Hyspecka, and K. Mazaneo, Effect of cold work on the surface segregation of phosphorus in AISI 321 austenitic stainless steel. Scripta Metallurgica, 1985. 19(10): p. 1199-1202.

[85] Ruoff, A. and R. Balluffi, Strain-enhanced diffusion in metals II. Dislocation and grain-boundary short-circuiting models. Journal of Applied Physics, 1963. 34(1): p. 1848-1853.

[86] Balluffi, R. and J. Blakely, special aspects of diffusion in thin films. Thin Solid Films, 1975. 25: p. 363-392.

[87] Bower, A. and L. Freund, Analysis of stress-induced void growth mechanisms in passivated interconnect lines. Journal of Applied Physics, 1993. 74(6): p. 3855-3868.

[88] Simo, J. and R. Taylor, A return mapping algorithm for plane stress elastoplasticity.International Journal for Numerical Methods in Engineering, 1986. 22: p. 649-670.

[89] Simo, J.C. and T.J.R. Hughes, Computational Inelasticity, ed. J.E. Marsden, L. Sirovich, and S. Wiggins. 1997, New York: Springer-Verlag. 392.

[90] Belytschko, T., W.K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures. 2000: John Wiley & Sons. 650.

[91] Crisfield, M.A., Non-linear Finite Element Analysis of Solids and Structures, Volume 1: Essentials. 1991, Chichester: John Wiley & Sons. 345.

[92] Simo, J. and R. Taylor, Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering, 1985. 48: p. 101-118.

[93] Pickering, F., The basis of quantitative metallography. Vol. 1. 1976, London: Institute of Metallurgical Technicians. 38.

[94] Lindgren, L.-E., H. Alberg, and K. Domkin. Constitutive modelling and parameter optimisation. in VII International Conference on Computational Plasticity COMPLAS 2003. 2003. Barcelona.

[95] Domkin, K. and L.-E. Lindgren. Dislocation density based models for plastic hardening and parameter identification. in VII International Conference on Computational Plasticity COMPLAS 2003. 2003. Barcelona.

[96] Bruhns, O.T. and D.K. Anding, On the simultaneous estimation of model parameters used in constitutive laws for inelastic material behaviour. International Journal of Plasticity, 1999. 15(12): p. 1311-1340.

[97] Zentar, R., P. Hicher, and G. Moulin, Identification of soil parameters by inverse analysis. Computer and Geotechnics, 2001. 28: p. 129-144.

44 of 49

[98] Burgan, B., Elevated temperature and high strain rate properties of offshore steels. 2001, Offshore Technology Report: Norwich.

[99] Moteff, J., Correlation of substructure with mechanical properties of plastically deformed AISI 304 and 316 stainless steel. 1973, University of Cincinnati: Cincinnati. p. 44.

[100] Perkins, R., R. Padgett, and N. Tunali, Tracer diffusion of Fe and Cr in Fe-17 wt pct Cr-12 wt pct Ni austenitic alloy. Metalurgical Transactions, 1973. 4(Nov): p. 2535-2540.

[101] Kim, S., Vacancy formation energies in disordered alloys. Physical Review B, 1984. 30(8): p. 4829-4832.

[102] Yapici, G.G., et al., Microstructural refinement and deformation twinning during severe plastic deformation of 316L stainless steel at high temperatures. J Mater. Res., 2004. 19(8): p. 2268-2278.

[103] Karaman, I., et al., Modeling the deformation behavior of Hadfield steel single and polycrystals due to twinning and slip. Acta Materialia, 2000. 48(9): p. 2031-2047.

[104] El-Danaf, E., S. Kalindi, and R. Doherty, Influence of grain size and stacking-fault energy on deformation twinning in fcc metals. Metallurgical and Materials Transactions A, 1999. 30A(May): p. 1223-1233.

[105] Barnby, J.T., Effect of strain aging on the high-temperature tensile properties of an AISI 316 austenitic stainless steel. Journal of The Iron and Steel Institute, 1965. 203(April): p. 392-397.

[106] Mannan, S., K. Samuel, and P. Rodriquez, Dynamic strain ageing in type 316 stainless steel. Transactions of the Indian Institute of Metals, 1983. 36: p. 313-320.

[107] Angella, G., B.P. Wynne, W.M. Rainforth, and J.H. Beynon, Microstructure evolution of AISI 316L in torsion at high temperature. Acta Materialia. In Press, Corrected Proof.

[108] Sauzay, M., et al., Creep-fatigue behaviour of an AISI stainless steel at 550 [deg]C. Nuclear Engineering and Design, 2004. 232(3): p. 219-236.

45 of 49

Appendix 1. Hardening modulus and derivatives involved in the solution of the evolution equations for internal variables

The hardening modulus is defined as

py

py

dd

dd

H qq

q)( [ Eq. (60) ]

The linearisation of the hardening law, Eq. (88), in the converged state gives,

0pp

HqqH , 0)(iH (A-1)

Hence the derivative of internal variables w.r.t. effective plastic strain is expressed as,

ppdd H

qHq

1

(A-2)

The expression of the hardening modulus and also the iterative solution procedure for the internal variables in chapter 4 involves the derivatives which are given below. Derivatives not given are zero.

The internal variables are

0,00000

*

G

GCr

refm

meqv

v

i

iT

Gggc

vv

ssss

ccq [ Eq. (64) ]

Because the scaling of internal variables is applied only for the solution of the system of equations, arbitrary scaling factors can be chosen. To simplify some formulas here we choose the current value of the shear modulus, G, as one of the reference values.

The following derivative vector is trivially obtained from the Eq. (1) and Eq. (64),

0,00000000 Gy G

q (A-3)

Evolution equation for the immobile dislocation density leads to

02111 223

001 t

kTGb

cc

Dcsgb

mH eqieqv

vv

p

ii

i (A-4)

which gives

tkT

Gbcc

DcHqH

ieqv

vvi

i

3

01

1

1 41 (A-5)

tckT

GbcD

cccH

qH eq

vi

eqieqv

veqv

v 0

2231

2

1 2 (A-6)

46 of 49

p

i sss

bmss

sH

qH

20

00

1

3

1 )()( (A-7)

p

i gg

bmg

gH

qH

20

00

1

7

1 (A-8)

bmH

tH

ip

ipp

00

11 111 (A-9)

Evolution equation for the vacancy concentration leads to

011-

4*1

222

02

02

TTQ

ctccsg

D

bbc

Qccc

H

vfeqv

eqvvvm

pjpG

vfeqv

eqv

v

(A-10)

which gives

tsg

DccH

qH

vmeqv

v22

2

2

2 111 (A-11)

tccs

sscDss

sH

qH eq

vveqv

vm3

00

2

3

2 2 (A-12)

tccgg

cD

gg

HqH eq

vveqv

vm30

02

7

2 2 (A-13)

GQc

GHqH p

vfeqv

00

02

8

2

* (A-14)

0,0

0,2

9

2G

p

vfeqv

GG Qc

HqH (A-15)

bbcc

QccH

tH

eqv

jG

vfeqv

peqv

pp0

2022

4*11 (A-16)

Subcell size formation

012

1

003

i

i

ssssH [ Eq. (77) ]

which gives

21

000

3

1

3

21

i

ii

i ssssH

qH

(A-17)

21

00

3

3

3

i

isss

HqH

(A-18)

Mobile dislocation density relation

47 of 49

014ref

am

mbH [ Eq. (79) ]

which gives

ref

amm

m mbH

qH 0

04

4

4 (A-19)

sssm

bsss

HqH

ref

am02

2

04

3

4 (A-20)

02

2

04

7

4 ggm

bg

gH

qH

ref

am (A-21)

where 2

2

ss and 2

2

gg

Mobile dislocation velocity relation

001

5refm

mp

vvH [ Eq. (82) ]

with

1-1

0 s1m

vb refm [ Eq. (81) ]

which gives

refm

m vvH

qH

05

4

5 (A-22)

0

5

5

5

m

mrefv

vH

qH (A-23)

ttH

tH

ppp111

1

55 (A-24)

Solute fraction in mobile dislocation

0exp1006CrCr

sCrCr ccccH [ Eq. (83) ]

where

2

*

btDh w

Crl , (A-25)

kTQ

Crleq

v

vCrleq

v

vCrl

Crl

eDcc

Dcc

D 0* , [ Eq. (84) ]

apref

wt . [ Eq. (56) ]

We will use

48 of 49

10

6 )exp(CrCrs ccH (A-26)

which gives

Crl

weqveq

v

Crlw

eqv

v

Crl

Crl

eqv

v

DbhtHc

cD

bhtH

cc

DD

HccH

qH

26

26

*

*66

2

6

(A-27)

where we used

2* bht

Dw

Crl

(A-28)

and

eqv

Crl

v

Crl

cD

cD *

. (A-29)

Furthermore,

16

6

6Crc

HqH

, (A-30)

tt

bDhH

tHH

tH

pw

Crl

ppp111

2

*6666 (A-31)

Grain size model

010

7 ggH [ Eq. (85) ]

where all derivatives are zero except 107

7

7 gg

HqH , as the grain size is constant.

Short range flow stress contribution

0ln*1

30

21

2

0

2

008

q

pref

qp

Cr

Crp

Cr

Crp

GbfkT

cc

cc

GH [ Eq. (86) ]

If the above expression yields 0* , a cut-off is introduced,

0*

08 G

H , (A-32)

and the only non-zero derivative in this case is

1* 08

8

8 GHqH . (A-33)

Otherwise the first expression of H8 is used, which gives remaining derivatives:

49 of 49

q

pref

qp

Cr

Cr

Cr

p

Cr

Cr

CrCr GbfkT

cc

qp

ccc

cp

cH

qH

1

30

121

2

00

12

00

8

6

8 ln21

21

2 (A-34)

1

00

8

8

8 **

p

GpGH

qH (A-35)

tGbfkT

GbfkT

cc

qH

tH

p

q

pref

qp

Cr

Cr

pp1ln11

30

11

30

21

2

0

88 (A-36)

The equation for the long-range stress contribution is written in non-dimensional form as

iGG

G GbmH0,0,

9 [ Eq. (87) ]

which gives

iGii

i

GbmHqH 1

2 0,00

9

1

9 (A-37)

10,9

9

9G

G

HqH (A-38)

Paper E

1 of 35

Stress update algorithm extended to strain induced martensite formation Konstantin Domkin1 and Lars-Erik Lindgren1,2

1Dalarna University, 781 88 Borlänge, Sweden 2Luleå University of Technology, 971 87 Luleå, Sweden AbstractThe implicit Euler backward algorithm is extended to constitutive models for material exhibiting strain induced martensite formation. This formation affects the thermo-elastoplastic properties of the material. Thereby, the phase change also indirectly makes all material properties dependent on the plastic strains. The stress-strain algorithm has been modified to accommodate this and a consistent constitutive matrix has also been derived. The paper includes some numerical examples for a uniaxial stress case to demonstrate its convergence properties.

1. Introduction

Metastable austenitic stainless steels are of interest for automotive applications. They have good ductility during the initial forming and can reach high strength after the forming as a result of strain induced martensite formation. However in the process both strain and temperature must be controlled carefully in order to utilise these properties fully. Designing the forming process in order to achieve this is most efficiently done by simulations. Thereby, it is of interest to use appropriate material models, determine pertaining parameters and implement stress-strain algorithms into simulations tools for forming processes. The current paper is focussing on the latter part. The implicit Euler backward or, in case of von Mises plasticity, the radial return algorithm is an efficient and common stress-strain algorithm. It has been adapted to the special features of the model for strain induced martensite formations; the dependence of all material properties on the plastic strain. The paper starts with the theory of the model for strain induced martensite formation combined with mixture rules for the mechanical properties of the two-phase material. These equations are based on a set of four internal variables for the microstructure evolution. Thereafter the incorporation of the model into the radial return algorithm is described with details of the derivations in appendices. Finally, some examples of convergence properties of the stress-strain algorithm as well as its consistent constitutive matrix are shown for a uniaxial case.

2. Martensite formation

Metastable austenitic stainless steel can form martensite under certain conditions. Spontaneous martensite formation occurs when the temperature sinks below the Ms-temperature. The Koistinen-Marburger [1] equation is usually applied for this case. The transformation is called stress-assisted for temperatures above Ms and starts on the same sites responsible for spontaneous transformation during cooling. This is sometimes modelled by including the effect of stress on the Ms-temperature [2-4]. The strain-induced martensite transformation is active at even higher temperatures. The martensite then nucleates at new sites at intersection of shear bands created by the plastic deformation [5] but also mechanical twins can be nucleation sites [6]. It is believed [7, 8] that -martensite forms first at stacking faults of phase. Thereafter ´-martensite is formed at shear band intersections. It is also likely that -martensite transforms into

´-martensite at larger strains. The latter is magnetic and is measured during the tests used in the current study. No martensite can form at all if the temperature is above Md. The three ranges of martensite formation [9] are summarised in Figure 1.

2 of 35

TdM

y

sMsM

Spon

tano

usm

arte

nsite

th

erm

alfo

rmat

ion

Plas

tic d

efor

mat

ion

of

aust

enite

, no

mar

tens

ite

Stre

ss-a

ssist

ed

Strain-inducednucleation

Yield limit of austenite

TdM

y

sMsM

Spon

tano

usm

arte

nsite

th

erm

alfo

rmat

ion

Plas

tic d

efor

mat

ion

of

aust

enite

, no

mar

tens

ite

Stre

ss-a

ssist

ed

Strain-inducednucleation

Yield limit of austenite

Figure 1. Martensite formation in different stress and temperature regimes.

3. Strain induced martensite formation

The current work focuses on strain induced martensite formation (SIMT). Olson and Cohen [10] proposed a physical based model for SIMT where shear band intersections has a probability to be nuclei for martensite formation. Stringfellow et al. [11] extended this model by including stress into the probability function. The model was further developed by Tanaka and Iwamoto [12-19].

It is assumed that only two phases coexist in the material. They are austenite ( ) and ´-martensite (M). It is only necessary to compute the martensite fraction and use

MXX 1 . (1)

The amount of shear band formation is related to plastic strain in austenite by

sbsbsb XAX 1 , (2)

where Xsb is the volume fraction of shear bands and is the effective plastic strain in austenite. The factor sb is written as

M

refsb KAATATAA 432

21 , (3)

where the temperature, T, is given in Kelvin, ref is a reference rate and the triaxiality factor Kis defined as

3kkK , (4)

where kk is the sum of the normal stresses and is the von Mises effective stress. The strain rate effect has not been included in the current work due to lack of experimental data. Thus the rate-dependency factor, M, is set to zero. The fraction of shear bands is related to the number of shear band intersections per unit volume,NI, by a power law,

nsbI XN , (5)

where is a geometric constant and the exponent n typically has a value around 4-5. The number of operational martensite nucleation sites, NM, are assumed to be equal to NI multiplied by a probability, , [11] as

3 of 35

IM NN . (6)

The increase in martensite fraction is assumed to be proportional to MN giving

nsbMMMM Xg

dtdXNXX 11 . (7)

This together with Eq. (2) gives

nsbsbsb

nsbMM Xg

dgdgHXAnXXX 11 1 . (8)

The Heaviside function, gH , is included to reflect the irreversibility of the martensite formation process. The probability function is defined as

g gg

g

gdeg g

20

21

21 , (9)

where the driving force is

KgTg 1 . (10)

Thus the evolution equation for the martensite formation becomes

gBAXX MMMM 1 , (11)

with

sbnsbsbM XnXAA 11 , (12)

gHdgdXB n

sbM , (13)

KgTg 1 . (14)

The last term in Eq. (11), gBM , is the effect of the change in probability in Eq. (7) and is ignored in [14, 15]. The parameters were determined as described in [20] and are given in Table 1 for HyTensX, a metastable austenitic stainless steel. However, the rate dependency factor could not be determined from those experiments and is thus excluded from the table.

Table 1. Martensite formation parameters for HyTensX.

A1 A 2 A 3 A 4 g g0 g1 n

-3.9 10-4 -2.7 10-2 56.9 7.15 65.8 -182 67.7 11.1 4.5

4. Mechanical properties of two-phase material

The macroscopic properties of the austenite-martensite mixture is obtained from simple mixture rules and properties assigned to the individual phases. These mixture rules imply that the properties are dependent on the plastic strain via the strain induced martensite formation described earlier. The mechanical properties of interest are elastic modulus, thermal expansion and yield limit. The Young’s modulus is computed as

4 of 35

MMM EXEXE 1 , (15)

with data from Tomita and Shibutani [21], in MPa,

10027325700215700 TE (16)

and

10027325800237300 TEM . (17)

It is assumed that Poisson’s ratio is constant, = 0.3, for both martensite and austenite. The split into deviatoric and volumetric stress states in the incremental stress-strain algorithm described later needs shear modulus,

MMM GXGXG 1 , (18)

and bulk modulus,

MMM KXKXK 1 . (19)

Shear and bulk moduli are determined by Young’s modulus and, in this case, the fixed value for Poisson’s ratio. The thermal dilatation is computed as

thMM

thM

th XX1 , (20)

with data from Iwamoto [22],

10002730173.0015.0 Tth (21)

and

1000273011.0 Tth

M . (22)

The shift between the curves in Eqns (21) and (22) corresponds to the difference in specific volume between the two phases. The macro yield limit is computed as

YMM

YM

Y XX1 , (23)

with the model for the yield limit of each phase taken from Tomita and Iwamoto [12, 19] and other studies by the same group. It is written as

325 114

CCTCY eCeC (24)

for austenite, andM

MMM CCMTCMY

M eCeC 325 114 (25)

for martensite. The parameters in Eqns (24-25) for the material studied in the current work are determined as described in [20] and shown in Table 2. The distribution of the plastic strain between the two phases is obtained, as in [9], by assuming

MYM

Y (26)

5 of 35

and

MMMp XX1 . (27)

The higher strength of the martensite can also be used to motivate that there should be no plastic straining of the martensite. Iwamoto [22] use a multiscale model of austenite-martensite and found that the plastic strain in the martensite was considerably smaller than in the austenite. Furthermore, it is assumed in the derivation above that the formed martensite has the same effective plastic strain as the existing martensite. Other options would be to assume it inherits the value from the existing austenite phase.

Table 2. Yield limit parameters for HyTensX from [20]

1C 2C 3C 4C 5C MC1MC2

MC3MC4

MC5

998 0.82 0.66 1673 0.0058 1701 15.74 4.25 7733 0.0148

5. The return mapping algorithm

The context of the backward Euler or implicit return mapping algorithm for computation of stress from given strain is presented. General continuum mechanics and the theory of plasticity are not given but can be found in [23, 24] as well as in [25] where also numerical procedures are described. The numerical solution of the microstructure evolution equations is described together with the extension of the return mapping algorithm. Details of the derivations are given in the appendices.

5.1. Large deformation context

The Updated Lagrangian (UL) formulation is common in metal forming analyses and other large deformation problems. The basic idea is to update the mesh so that the initial geometry of an increment is the reference geometry and then iterate for the solution at the end of the time step. This iterative, incremental approach provides the stress-strain algorithm with a strain increment that should be used to update the stress in order to iterate for equilibrium. The stress-strain relations can be based on a hyperelastic model or hypoelastic model. The latter is used in this study where the increment in stress is computed from increment in strain. The Cauchy (true) stress and the rate of deformation tensor (velocity strain) are commonly used conjugate stress and strain measures. Hughes and Winget [26] describe a simple finite element implementation to compute an increment in strain that is a 2nd order accurate approximation of the rate of deformation tensor. Differences between the stresses computed by the hypoelastic and hyperelastic models become negligible when the elastic strains are small, such as in metal plasticity [25, 27]. Hypoelastic relations are based on the choice of an objective stress rate measure defining the increment in stress. The Green-Naghdi stress rate has become popular since the Jaumann rate was found to produce non-physical oscillatory stresses for large shear deformation [28, 29]. The Green-Naghdi stress rate can be conveniently implemented by the use of an unrotated stress [30], which is sometimes called co-rotated stress [31]. It is denoted here by a right superscript R.The variables at time nt are unrotated and the strain increment associated with the midpoint geometry is unrotated by

ljn

klnT

iknR

ijn RR , (28)

ljnp

klnT

iknRp

ijn RR , (29)

6 of 35

ljn

klTik

nRij RR 2

12

1, (30)

where R is the rotation tensor obtained from the polar decomposition of the deformation gradient at the state n or n+1/2. Thereafter the stress update is performed in the unrotated configuration. The stress-strain algorithm described in the next chapter can be written symbolically as

,...,,,algorithm-stress, TT, nRpij

nRij

nRij

Rpij

Rij .

These computed increments are added to Rij

n and Rp

ijn giving R

ijn 1 and Rp

ijn 1 . Finally, these

variables are rotated forward to the current geometry at the end of the increment by Tlj

nRkl

nik

nij

n RR 1111 , (31)

Tlj

nRpkl

nik

npij

n RR 1111 . (32)

See [29-33] for further details about finite element formulations for large deformation analysis. The previous summary shows one possibility of stress updating in a large deformation analysis using hypoelastic stress-strain relations. Crisfield shows that the consistency condition, which plays an essential role in the radial-return method, is unchanged in the case of the Green-Naghdi stress rate [29]. Therefore, the algorithm is form-identical [30] with the small deformation version and is described in terms of stress and strain without further consideration of possible large deformation context.

5.2. Strain-stress algorithms

The return mapping method is an operator-split approach where an elastic predictor is followed by a plastic corrector. The elastic predictor step is the computation of a trial state assuming no plastic strain increment occurs. If the effective trial stress is larger than the yield limit, then a plastic corrector step must be performed. There are different options for the return direction of the stress state towards the yield surface. The backward Euler or implicit method uses the return direction at the end of the increment. Ortiz and Martin [34] shows that only the implicit approach for the return mapping gives a symmetric consistent tangent for von Mises plasticity and the associated flow rule. They also note [35] that this method may also be a better choice when large strain increments are expected. The algorithm presented in this paper is limited to the rate-independent, deviatoric von Mises plasticity with the associated flow rule. Then the plastic strain increment is normal to the yield surface, the radial direction. It is a closest-point projection of the stress state onto the yield surface. The stress-strain algorithm has two tasks. They are the computation of stress from given strain and computing the consistent constitutive matrix. The latter is required when the 2nd order convergence of the Newton-Raphson method should be preserved. The book by Simo and Hughes describes plasticity and viscoplasticity models and their numerical implementation [25]. The book by Crisfield describes basic numerical methods in chapter 6 in [36] and extensions in [29]. Belytschko et al. also devote some chapters to constitutive models and stress-strain algorithms [31]. Application to viscoplasticity can be found in chapter 15.12 in [29] and chapter 5.9.8 in [31] where a flow strength equation is solved to obtain the effective plastic strain increment. The flow strength equation is an equation that defines the viscoplastic strain rate in terms of other variables. This approach is also used in [37-39]. Ponthot [40] describes also the application of the radial-return method to viscoplasticity.

5.3. Extended radial return algorithm for SIMT

The previous sections give general background of the radial return algorithm for stress updating. It is described in detail in the following with extensions in order to accommodate strain induced

7 of 35

martensite formation (SIMT). These are necessary because both the thermo-elastic and plastic material properties are dependent on the plastic strain as shown in the discussion of SIMT earlier and the way the properties of the austenite-martensite mixture are computed. The numerical approach is based on [31]. It should be noted that here we deal with the stresses and strains in the unrotated configuration, and the superscript R is omitted.

5.3.1 Basic equations

An additive decomposition of the strain rates and thereby strain increments is assumed. This can be motivated in different ways starting from the multiplicative decomposition of the deformation gradient, see for example [25, 29, 41-43]. Some of the differences in these derivations disappear when elastic strains are small, see chapter 19 in [29] or [27], as in metal plasticity. Thus it is assumed that

ijthp

ijeijij , (33)

where ij is the total strain rate tensor, eij is the elastic strain rate tensor, p

ij is the plastic strain

rate tensor and th is the rate of the thermal dilatation.

In order to simplify the derivation, it is advantageous to treat the deviatoric and volumetric stresses and strains separately, as the plastic strains are assumed to be purely deviatoric, and the thermal strains are purely volumetric. The stress tensor is split into the volumetric part, or hydrostatic stress, p, and the deviatoric part, ijs ,

ijijij sp , (34)

3kkp , (35)

ijkk

ijijs3

. (36)

Furthermore, with the hypo-elastic approach in the unrotated configuration the following incremental update formula is assumed,

ekk

nkkn

n

nkk

n

KKK 1

11

33, (37)

eij

nij

nn

n

ijn eGs

GGs 1

11 2 , (38)

where eije is the deviatoric part of the elastic strain tensor, e

ij .

The evolution equations for the microstructure presented in chapter 3 define a set of additional internal state variables beside the (macroscopic) effective plastic strain, p . Those are effective

plastic strain in austenite and in martensite, and M respectively, martensite fraction, MX ,and fraction of the shear bands, sbX . Therefore, a vector of the internal variables due to the strain induced transformation is introduced as

sbMMT XXq . (39)

Following the model of the SIMT plasticity and mixture rules for a two-phase material discussed in chapter 4, the thermo-mechanical properties can now be expressed in a more general form as functions of the above internal variables and temperature. Shear modulus and bulk modulus, thermal dilatation, and yield stress are expressed as

8 of 35

TGG ,q , (40)

TKK ,q , (41)

Tthth ,q , (42)

TYY ,q , (43)

and the specific functions employed in this work are identified with equations (18), (19), (20) and (23), respectively.The Eqns (26) and (27), which determine the distribution of the plastic strain in martensite and austenite phases, together with the evolution equations for martensite fraction, Eq. (11), and the fraction of the shear bands, Eq. (2), can be written in the compact vector form,

0qqH Tpp ,,,,, , (44)

where

0,,1 TTH MYMM

Y , (45)

012p

MMM XXH , (46)

013 MMM XAXH , (47)

014 sbsbsb XAXH . (48)

Appearance of the hydrostatic stress, p, and effective stress, , in the Eq. (44) is due to the

parameters Asb and AM dependency on the triaxiality factor, pK , Eqns (3) and (12),

TKAA sbsb , (49)

TKXAA sbMM ,, (50)

Note also that the rate of the driving force, g , is ignored in the evolution equation for the martensite fraction, Eq. (47). Finally, the classical von Mises effective stress, , the yield surface and associative flow rule are employed for determination of plastic strain rate,

Yf , (51)

ij

ppij

f , (52)

with the loading-unloading conditions,

0,0,0 ff pp . (53)

Thus, the analytical model of rate-independent plasticity for SIMT is defined with the evolution equations for internal variables combined with the classical von Mises plasticity and the associated flow rule. Furthermore, it is assumed that the temperature is given to the stress-strain algorithm.

5.3.2 Incremental strain-stress algorithm – overall logic

A numerical procedure for the above plasticity model is summarised in Box 1 and is described below. Rate terms will be replaced by incremental terms.

9 of 35

The deviatoric stress at the start of an increment is

ijkk

n

ijn

ijns

3, (54)

where ijn is the stress tensor at state n. Thus the trial deviatoric stress is computed first in step

1.1 in Box 1 by

ijnij

ntr

ijeij

ntrij

tr eGs

GeeGs2

22 , (55)

where ije is the increment in deviatoric total strains, the deviatoric elastic strain at the start of the increment is

Gs

e nij

neij

n

2, (56)

and the trial shear modulus is

TGG nntr 1,q . (57)

The effective trial stress is thereafter compared with the trial yield limit, step 1.2 in Box 1, which accounts for temperature change but no increment in internal variables. If the trial stress state is inside the yield surface, then the process is elastic and trial state is the true state for the end of the increment. Otherwise, the logic in Box 2 is executed in order to determine the updated stresses, increment in plastic strains and internal variables. That logic is described in chapter 5.3.3.The change in shear modulus due to plastic strain is introduced as

TGTGGGG nnnntrn 1111 ,, qq . (58)

The final deviatoric stress can be written as

pij

ntr

ijtr

trpijijn

ijn

nij

n eGG

sGGee

Gs

Gs 111 22

22

2 , (59)

where Eqns (55) and (58) were used in the last step. The implicit return mapping method assumes that the plastic flow increment is in the final deviatoric stress direction. This is the same as the deviatoric trial stress direction in the case of von Mises plasticity.

ijtr

tr

pp

ijpij se

23 , (60)

Introducing the radial-return assumption, Eq. (60), into Eq. (59) gives

ijtr

tr

pn

ijtr

trijtr

tr

pn

ijtr

trijn sGs

GGsGs

GGs 111 31

2321 , (61)

and a simple relation between trial and final deviatoric stresses is obtained as

ijtrn

ijn ss 11 , (62)

where the deviatoric stress scaling factor, 1n , is

10 of 35

tr

pn

trn G

GG 11 31 . (63)

Eqns (62) and (58) also lead to the expression of the final effective stress as

trnpnpntrtr

ntrnn TG

GG ,,,3 111

111 q (64)

Thus the scaling parameter and effective stress are functions of the increment in plastic strain and internal variables (via shear modulus, n+1G).

Box 1. Summary of the stress-update algorithm.

1.1 Elastic predictor: Compute deviatoric and effective stress in the trial state.

TGG nntr 1,q (1.1)

ijnij

ntr

ijtr e

Gs

Gs2

2 (1.2)

ijtr

ijtrtr ss

23 (1.3)

1.2. Check for plastic or elastic process

Tf nnYtrYtrtrtr 1,q (1.4)

IF 0ftr THEN elastic process:

0p , qq nn 1 , trn 1 (1.5)

TKK nnn 111 ,q (1.6)

TT nnthnnthth ,, 11 qq (1.7)

thkkn

kkn

nkkn

n

KKp 3

331

11 (1.8)

ELSE ( 0ftr ) Plastic corrector step.

See Box 2. Compute plastic strain increment, p , internal variables, q1n , hydrostatic and

effective stress, pn 1 and 1n , and thermo-elastic properties, Kn 1 , Gn 1 and th .

1.3. Update stress tensor and other variables.

ijn

ijtr

tr

n

ijn ps 1

11 (1.9)

pij

pij

npij

n 1 (1.10)

ppnpn 1 (1.11)

11 of 35

The volumetric part of the stress can be computed as

kknnth

kkekk

nnkkn

n TpKp ,,33

1111

1 q , (65)

where the volumetric elastic strain is

Knkk

nekk

n

3. (66)

The volumetric stress dependency on the internal variables and temperature stems from Eqns (41) and (42) for the bulk modulus and thermal dilatation. Thereafter, the deviatoric stress obtained from Eq. (62) is added to the volumetric stress in Eq. (65) to give the total stress at the end of the increment,

ijkk

n

ijn

ijn s

3

111 . (67)

One should note that, in addition to the dependencies in Eqns (64-65), the volumetric and deviatoric stresses are also coupled with internal variables and effective plastic strain increment through the generalised hardening evolution equation (44), and therefore all these variables have to be iteratively updated in the numerical procedure of the Box 2. This is described in the next section.

5.3.3 Incremental strain-stress algorithm – plastic corrector step

The remaining step in the procedure summarised above is the updating of the internal variables for the SIMT model, q, the increment in effective plastic strain, p , and the stress tensor, ij.Here, in the context of the plastic corrector step, the total strain increment, ij , is fixed, and

hence the “trial”-values of variables are fixed too. On the other hand, the variables q and p at the end of an increment i.e. their “n+1”-values are treated as independent unknowns. One should also note that temperature is known (it is a prescribed explicit function of time). The unknowns are obtained as a solution to the system of equations which includes the generalized hardening law, Eq. (44), and the consistency condition, 01 fn ,

0qqH Tpp ,,,,, , (68)

0111 Ynnn f . (69)

The latter one, the consistency condition, is a non-linear algebraic equation with the unknowns qand p . It is formulated for the “n+1” state at the end of the increment according to the fully implicit backward Euler procedure of the return-map algorithm. The generalized hardening law, Eq. (68), is a system of non-linear equations involving the unknowns themselves and also their rates, and thus it defines a set of differential equations. A simple single-increment update formula according to the mid-point rule, or “ -method”, is chosen here for the integration of these equations,

0qqH Tpp ,,,,, (70)

where the rate-variables are substituted with

qqqq nn

tt11

, (71)

12 of 35

t

pp , (72)

and the variables in the “ -state” are defined as

qqqqq 11 nnn , (73)11 nnn , (74)

and the same for p and T .One should note that for the specific SIMT model used in this study the rate-variables enter the equations of the hardening law, Eqns (45-48) and hence Eq. (70), merely as linear coefficients. Thus the time step, t, is a common multiplier in all four equations and can be omitted, as in Eqns (79-82) below. Hence, these rate equations are not truly rate-dependent, but rather define evolution of internal variables with plastic strain, in the spirit of the rate-independent plasticity. Essentially, the approach described in chapter 5.9 in [31], is followed with extension specific to the SIMT model, i.e. with the thermo-elastic properties dependent on plastic strain and pertaining internal variables, and with the parameter for mid-point method of integration of the hardening equations. Also the notation of [31] is preserved for the most part, even though the particular expressions behind some of the symbols are different due to the modifications of the algorithm. With regards to the computation of the stresses, a variant of the algorithm is chosen here, in which the stress tensor is not considered independent unknown, but rather it is expressed as a function of the other unknowns with the help of the developments of the last section 5.3.2. The radial-return assumption allows one to express the effective stress as a function of the unknown variables, q and p , via the scalar scaling parameter, n+1 , according to the Eq. (64). Also the hydrostatic stress, p, is expressed as a function of the unknowns according to the Eq. (65). Thus the equation for the stress tensor is reduced to these two scalar functions,

thkk

ekk

nnn Kp 311 , (75)

pntrtr

nn G

GG 1

11 3 (76)

These are substituted directly into the system of equations (69-70).

Thus, the following system of equations nonlinear in q1n and p has to be solved,

0qqHH Tpp ,,,,, , (77)

0,,, 11111 TTf nnYnnpn qq , (78)

where the hydrostatic stress, n+1p, and effective stress, 1n , are given by Eqns (75-76), yield stress given by Eqns (23-25). Hardening law is based on Eqns (45-48), and hence the vector equation for H, Eq. (77), denotes the four scalar equations,

01 MYM

YH , (79)

012p

MMM XXH , (80)

01),,(3 MsbMM XTKXAXH , (81)

01),(4 sbsbsb XTKAXH , (82)

where “ -variables” are defined in Eqns (73-74), and increments are nn 1 etc.

13 of 35

The task is approached with a standard Newton iterations method. First, the equations are linearised using the current updated values denoted below with an iteration counter, subscript (i),

0)(1)(

1)(

1)(

1)(

ip

pni

ni

ni

ni p

pH

HHHq

qH

, (83)

0)(1

1)(

1

1)(

1

inp

pni

Yn

ni

Yn

fqq

, (84)

where a vector derivative and a vector derivative of another vector are defined as

4321 qqqqq and

m

k

km qH

qH , (85)

and the partial derivatives of H are computed using the definitions of the “ -variables” and the trivial chain-rule, such as, for instance,

qH

qH

qH1n . (86)

The variations of hydrostatic and effective stress are obtained from Eqns (75-76),

01)(

1

qqn

in p

p , (87)

ppni

n

ni

n

1)(

1

1)(

1

qq

. (88)

It is of interest to write a simple expression (see also Appendix 1, Eq. A1-44) of the last derivative in Eq. (88), which often appears in formulations of the classical radial-return method,

Gnpn

n1

1

1

3 . (89)

Note also that the hydrostatic stress, Eqns (75) and (87) does not depend on the plastic strain explicitly.Next, the linearised system of equations (83) and (84) is re-written in the compact matrix form,

0arqA )()(1

)(~~

ip

ii , (90)

0)(1

)()( inp

ipiq ffqf , (91)

where the following symbols have been introduced,

qH

qH

qHA 1

1

11

1

111

n

n

nn

n

nnp

p, (92)

11

11

1

11 3~n

npnpn

n

npn G HHHHr , (93)

Ha~ , (94)

qqf 1

1

1

1

n

Yn

n

n

q , (95)

14 of 35

GGTf nnpn

Yn

pn

n

p11

1

1

1

1

303),(q . (96)

Finally, the system of linear equations defined by Eqns (90-91) has the following structure,

0aq

frA

)(1

)(

)()(

)(1)(

~~

in

ip

ipiq

ii

ff. (97)

It is solved for the iterative change in internal variables and effective plastic strain, and the new estimates of their current values are computed. The iteration cycle is repeated until convergence is reached, i.e. the residual vector is close to zero within chosen tolerance. Note that residuals and unknowns are not necessarily non-dimensional, and therefore an appropriate scaling should be applied when setting the convergence tolerance. Box 2 summarises the solution procedure. Further details of the computation of the derivatives involved can be found in the Appendix 1.

5.3.4 Consistent constitutive matrix

To achieve quadratic rate of convergence of the full implicit Newton-Raphson procedure within the finite element solution, the consistent constitutive tensor, or algorithmic moduli, a

ijklC , is needed for the computation of the tangent stiffness matrix. Its definition is

klaijklij dCd , (98)

kl

ijaijkl d

dC , (99)

where all derivatives are computed at state n+1, at the end of the increment. For convenience of implementation (in the context of the finite element method), the tensor notation is traditionally replaced with the vector-matrix notation. The stress and strain vectors are defined as

T312312332211 , (100)

T312312332211 . (101)

In the strain vector the “engineering shear strain” components are used, 1212 2 etc, which necessitates appearance of the diagonal matrix )222111Diag( ,,,,,L in the Eq. (103) below.

Thus, the definition of consistent matrix, Eq. (98), in vector-matrix form becomes

C dd a . (102)

Algorithmic constitutive matrix for the radial-return stress update algorithm and the constitutive model of SIMT is derived in Appendix 3. The final result is

1111

1IL11C

KDGDGDKD

GZGGZGGK

SPPSSSPP

P

tr

Sdevtra

33

332 1

(103)

Note that here, for the sake of brevity, the “n+1” superscripts of the quantities at the “n+1” state, i.e. at the end of the increment, are omitted. Notice also that all this derivation has deliberately omitted the superscript R denoting the unrotated configuration. Thus this obtained matrix must be rotated to the “true n+1” state in the same way as discussed for strains and stresses in chapter 5.1.

15 of 35

The following variables and symbols have been introduced:

”Unit vector” T0001111 . Operation is the product of two vectors, which results in a matrix (e.g. T1111 ). Symmetric deviatoric projection matrix (where I is the unit matrix),

11II31dev . (104)

The trial direction of the plastic flow is defined by a vector

tr

tr s . (105)

In the radial-return algorithm it is also the final flow direction vector. Symbol s denotes the deviatoric stress vector, analogous to Eq. (100). The following coefficients represent the numerical coupling terms between volumetric and deviatoric stresses and strains due to triaxiality factor in the SIMT hardening law, H:

pD PPP

HAd , (106)

HAdSSSD , (107)

HAdPPSD , (108)

pD SSP

HAd , (109)

qfd pZ qPP , (110)

qfd qSS Z1 . (111)

Coefficient ZP stands for the coupling between volumetric stress and deviatoric plastic strain due to the thermal dilatation and bulk modulus being dependent on internal variables:

pqP f

p

ZrAf

rAq

~

~

(112)

Coefficient ZS is related to the change in yield stress, i.e. hardening:

pqS f

GZ

rAf

rAq

~

~31 (113)

The symbols qf , pf , [A] and r~ have been introduced in the section 5.3.3, Eqns (92-96). Further explanations and a detailed derivation can be found in Appendix 3.

16 of 35

Box 2. Plastic corrector step.

Find internal variables, q1n , hydrostatic and effective stress, n+1p and 1n , and effective plastic

strain increment p , fulfilling the generalized hardening law and the consistency condition, according to the radial-return method for SIMT plasticity

i.e. Solve the following system of nonlinear equations:

0qqHH Tpp ,,,,, (2.1)

0, 1111 Tf nnYnn q (2.2)

2.1. Initialize i = 0.

Set the values of unknowns “near” the desired solution: pn)0()0(

1 ,q (see Appendix 2)

2.2. Iteration. (The subscript (i) is an iteration counter.)

Update thermo-elastic properties (bulk and shear moduli, and thermal dilatation):

TKK ni

ni

n 1)(

1)(

1 ,q and TGG ni

ni

n 1)(

1)(

1 ,q (2.3)

thnni

nththi T1

)(1

)( ,q (2.4)

Update hydrostatic and effective stress: thikk

ekk

ni

ni

n Kp )()(1

)(1 3 (2.5)

pii

ntrtr

in

in G

GG

)()(1)(

1

)(1 3 (2.6)

Compute new residual vector and yield function:

Tp iiipiii ,,,,,~

)()()()()()( qqHa (2.7)

Tf ni

nYi

ni

n 1)(

1)(

1)(

1 ,q (2.8)

2.3. Check convergence.

IF residuals ai TOL)(~a , fi

n TOLf )(1 and/or increments TOL , qTOLq

THEN stop the iterations and return to Box 1;

ELSE proceed to step 2.4

2.4. Update estimates of the unknown variables.

Compute derivatives (see Appendix 1) and solve the linearised system of equations for q and according to Eqns (92-97):

0aq

frA

)(1

)(

)()(

)(1)(

~~

in

ip

ipiq

ii

ff (2.9)

Update internal variables and increment in effective plastic strain:

qqq )(1

)1(1

in

in (2.10)

ppi

pi )()1( (2.11)

Set i = i + 1 and repeat the cycle from step 2.2.

17 of 35

6. Convergence studies of the algorithm

The presented radial return algorithm with extension to accommodate strain induced martensite formation has been implemented in a custom toolbox for Matlab™ [44]. The same tool was also used in the parameter fitting for HyTensX to obtain the data for the martensite formation model given in Table 1 and for the flow stress given in Table 2, [20].

6.1. Prescribed strain

This case demonstrates the 2nd order convergence of the plastic corrector algorithm in Box 2. Results from tensile tests have been used to evaluate the different choices of in previous derivation and to demonstrate their convergence properties. The choice = 0 is labelled explicit/implicit, = 0.5 is labelled midpoint/implicit and = 1 is called fully implicit method. The second word “implicit” in these labels refers to the implicit nature of the return-map algorithm fulfilling the consistency condition at the end of the time step, Eq. (78). The tensile tests were computed by prescribing strain increments up to the final strain of 40% and the temperature is prescribed to increase linearly from 20 to 40°C during the same time. Test with cooling from 40 to 20°C were also done and compressive tests with heating from 20 to 40°C. They are not shown in this paper. They do also support the conclusions concerning the convergence in the plastic corrector step. Initial martensite fraction = 0.008, and initial values of other internal variables are zero. The computed stresses and martensite fractions for the tensile tests with heating are shown in Figure 2 for the explicit/implicit approach, Figure 3 for the midpoint/implicit case and Figure 4 for the fully implicit method. The needed average number of iterations per increment for the logic in Box 2 when solving the cases in Figures 2-4 is shown in Table 3. Convergence is assumed if the change in plastic strain is less than 1·10-15, the changes in the other internal variables are less than 1·10-12 and the stress state is less than 1·10-9 MPa from the yield surface. The quadratic convergence of the algorithm is shown in Tables 4-6. The accuracy for the different choices of and the different number of increments can be estimated from the final stress and martensite fractions shown in Table 7.

0 0.1 0.2 0.3 0.40

200

400

600

800

1000

1200

1400 [MPa]

10 20 40 80 160 320 6401280

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6 XM

10 20 40 80 160 320 6401280

Figure 2 Computed stress and martensite versus strain for explicit/implicit method and different number of increments.

18 of 35

0 0.1 0.2 0.3 0.40

200

400

600

800

1000

1200

1400, MPa

10 20 40 80 160 320 6401280

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6 XM

10 20 40 80 160 320 6401280

Figure 3 Computed stress and martensite versus strain for midpoint/implicit method and different number of increments.

0 0.1 0.2 0.3 0.40

200

400

600

800

1000

1200

1400 [MPa]

10 20 40 80 160 320 6401280

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6 XM

10 20 40 80 160 320 6401280

Figure 4 Computed stress and martensite versus strain for fully implicit method and different number of increments.

Table 3. Average number of iterations to determine plastic strain increment.

No of increments Explicit/explicit Midpoint/implicit Fully implicit

1280 3.0 3.6 3.8

640 3.3 3.8 4.0

320 3.5 4.0 4.0

160 3.7 4.0 4.0

80 3.8 4.0 4.1

40 3.9 4.1 4.6

20 3.8 4.5 4.8

10 3.8 4.4 5.0

19 of 35

Table 4. Convergence of iterations using 40 strain increments with explicit/implicit approach.

Iteration no Increment 1 ( =1%) Increment 2 ( =2%) Increment 10 ( =10%)

1 0.0093 0.0102 0.0098

2 9.1e-4 -3.0e-7 -3.5e-5

3 1.6e-9 --3.2e-16 -4.7e-10

4 -1.8e-19 Convergence 4.3e-20

5 convergence Convergence

Table 5. Convergence of iterations using 40 strain increments with midpoint/implicit approach.


1 0.0093 0.0102 0.0098

2 9.1e-4 -1.1e-5 -4.5e-5

3 6.6e-9 -1.4e-8 1.2e-8

4 -2.9 e-13 -2.6e-13 3.9e-016

5 -6.9e-19 8.8e-19 Convergence

6 Convergence convergence

Table 6. Convergence of iterations using 40 strain increments with fully implicit approach.


1 0.0093 0.0102 0.0098

2 9.0e-4 -2.7e-5 -5.9e-5

3 -2.6e-9 -2.4e-7 1.2e-7

4 -1.2e-12 -1.4e-11 1.1e-13

5 1.6e-19 1.6e-15 -1.8e-18

6 Convergence Convergence Convergence

Table 7. Final stress and martensite fraction for different number of increments.

Num.of incremnts

Explicit/implicit Midpoint/implicit Fully implicit

XM XM XM

1280 1265.55 0.53840 1265.80 0.53869 1266.07 0.53899

640 1265.27 0.53810 1265.80 0.53869 1266.33 0.53928

320 1264.72 0.53750 1265.79 0.53869 1266.85 0.53986

160 1263.60 0.53627 1265.79 0.53867 1267.87 0.54099

20 of 35

80 1261.24 0.53367 1265.73 0.53860 1269.85 0.54322

40 1256.05 0.52798 1265.55 0.53832 1273.49 0.54739

20 1242.55 0.51399 1264.88 0.53713 1279.57 0.55471

10 1210.01 0.47225 1259.15 0.53048 1288.04 0.56597

6.2. Prescribed stress

The prescribed stress case demonstrates the 2nd order convergence the consistent tangent matrix can give when used to set up the tangent stiffness matrix in a full Newton-Raphson procedure in a finite element program. This is the iterative process for solving the nonlinear system of coupled equations of increment in nodal displacements/rotations.

The consistent tangent, aC , is evaluated by applying a prescribed stress and using the algorithm to determine the needed strain. Then an iterative process, outside the stress-strain algorithm, is applied where the change in estimated total strain increment to achieve a wanted stress increment is determined by

ipaa dd CC 11 (114)

where i is the stress obtained in the previous estimate for the strain and p is the prescribed (wanted) stress. A tensile loading was tested where the uniaxial stress is prescribed along arbitrary fixed direction in increments up to 1250 MPa. Temperature is constant = 20°C. Three cases with different -values were tested using 50 increments in each case giving a load increase of 25 MPa per increment. The cases were explicit/implicit, midpoint-implicit and fully implicit variants of the return mapping algorithm. Initial martensite fraction = 0.008, and initial values of other internal variables are zero. Stress-strain and martensite formation curves similar to those of the prescribed strain case above were thereby obtained. A full Newton-Rapshon procedure with the derived consistent tangent was used to determine the needed strain to balance the wanted stress. Convergence was assumed when the deviation between obtained and wanted stress was less than 1·10-15 · Young’s modulus. The evolution of the error for two of the increments of each of the cases is shown in Table 8. Table 8. Convergence of equilibrium iterations using prescribed stress increments.Iteration Explicit/implicit method Midpoint/implicit method Fully implicit method

Increment 10 Increment 50 Increment 10 Increment 50 Increment 10 Increment 50

1 +2.6E+001 +2.6E+001 +2.6E+001 +2.6E+001 +2.6E+001 +2.6E+001

2 +3.7E+002 +7.7E+002 +3.7E+002 +1.1E+002 +3.7E+002 +4.3E+001

3 +5.7E-014 +5.8E-002 +5.7E-014 +1.0E+001 +5.7E-014 +1.5E+000

4 Convergence +4.4E-006 Convergence +1.4E-002 Convergence +5.4E-004

5 +9.4E-011 +1.5E-008 +4.3E-011

6 Convergence +2.4E-010 Convergence

Convergence

21 of 35

7. Remarks on the constitutive model for SIMT and on the computational algorithm

The coupling of the model for strain induced martensite formation with a mixture rule for material properties within the context of a standard model for von Mises plasticity is a heuristic approach. The formulation has not been based on thermo-dynamics from the outset and we have not endeavored to derive limits for properties set by Clausius-Duhems theorem. Furthermore, it is assumed that the deformation process does not affect the temperature. This will be the case e.g. in a staggered approach to coupled thermo-mechanical problems, with a thermal step followed by a mechanical step. Then the temperature is prescribed in the mechanical stage. However, the model itself can be generalised to include any form of coupling between temperature and deformation, e.g. by means of appropriate equation for thermal energy due to plastic dissipation. It should be noted also that the change in internal variables and in stress is effectively driven by the change in the equivalent plastic strain. In case of a more trivial hardening law it might be possible to resolve the equations in terms of plastic strain alone. However, for the present model such analytical solution is not readily available, and therefore a numerical integration of a system of ordinary differential equations (ODE) has to be performed. One alternative would be to use any standard ODE-solver based on sub-increments to obtain a numerical “exact” solution. However, a simple single-increment update formula is chosen here, because it is expected to deliver adequate accuracy in the context of the return-map algorithm. One more note concerns the step 2.1 in Box 2. It would seem straightforward to start the plastic corrector iterations with the initial values of the unknowns set equal to their trial values. In terms of the formulation of the Box 2, these would be: qq n

)0( and 0)0(p . It often works well

with many hardening models which are not so complex. However, in the authors experience with the present SIMT model, the Newton iterations in such case tend to diverge, especially when increments are not very small. To prevent divergence during early iterations and to facilitate asymptotically quadratic convergence rate of the Newton method, it is advantageous first to obtain the rough approximate solution by means of some other numerical procedure to obtain a predictor, and only after that proceed with the Newton method. Here we choose simply to initialize the unknown variables so that their values are “near” the desired solution, which is defined in Appendix 2. The unsymmetrical terms in the expression of the consistent tangent matrix, Eq. (103), correspond to these two different kinds of unsymmetry: constitutive unsymmetry of the model, and “algorithmic” one. The ZP-term represents the essential unsymmetry of the constitutive model: deviatoric strains affect volumetric stress. Deviatoric strains produce plastic strains, which triggers the evolution of the martensite fraction, which in turn changes the thermal dilatation and bulk modulus, and hence changes the volumetric stress. Elastic properties in the model vary only slightly w.r.t. plastic strain, but variation of the dilatation is significant. On the other hand, changes in the volumetric strain do not affect the deviatoric stress, which leads to unsymmetry. Because of this, even the tangent matrix of the original continuous problem (classical “inconsistent” elasto-plastic continuum tangent moduli) is unsymmetrical too. The other non-symmetrical terms are due to the coupling between the parameters in the hardening evolution equation and the stress tensor (via triaxiality factor). These terms do not appear in the continuum elasto-plastic tangent moduli matrix. In the original continuous problem there is no unsymmetry related to this coupling. The nature of these non-symmetrical terms is numerical, algorithmical. It is a side effect of using implicit formulas for the stress-dependent hardening equations in the return-map algorithm. In the explicit-implicit variant of the algorithm, i.e. when = 0, these unsymmetrical terms vanish. Ignoring either of these unsymmetrical terms of the tangent matrix results in a slower, first order convergence rate of the equilibrium iterations.

22 of 35

8. Conclusions

An extension of the radial return algorithm to a model where the elastic properties and the thermal dilatation also depends on the plastic deformation has been presented and demonstrated. The need for this extension is due to the presence of strain induced martensite formation that together with mixture rules for thermo-elastoplastic properties caused this dependency. The convergence properties of the algorithm have been demonstrated for the uniaxial load case. The mixed midpoint-implicit variant of the algorithm, as expected, provides better accuracy of the integration of the SIMT evolution equations and hence better accuracy of the stress-strain curves for the uniaxial proportional loading tests. The implicit-implicit variant may be preferred if stability of the analysis is of concern, for very large increments in particular. Also the second order convergence of the equilibrium iterations is achieved owing to the derived consistent constitutive tangent, as demonstrated for the case of prescribed uniaxial stress. The algorithm can be adapted to different types of models for strain induced martensite formation as well as mixture rules and assumptions about the distribution of the plastic strain between the two phases.

References1. Koistinen, D. and R. Marburger, A general equation prescribing the extent of the

austenite-martensite transformation in pure-carbon alloys amd plain carbon steels. Acta Metallica, 1959. 7: p. 59-60.

2. Inoue, T. and Z. Wang, Coupling between stress, temperature, and metallic structures during processes involving phase transformations. Material Science Technology, 1985. 1: p. 845-850.

3. Fischer, F., A micromechanical model for transformation plasticity in steels. Acta Metallurgical et Materialia, 1990. 38(8): p. 1535-1546.

4. Tanaka, K. and Y. Sato, A mechanical view of transformation plasticity. Ingeniur-Archiv, 1985. 55: p. 147-155.

5. Olson, G. and M. Cohen, Kinetics of strain-induced martensitic nucleation. Metallurgical Transacations A, 1982. 6A(April): p. 791-795.

6. Tsakiris, V. and D.V. Edmonds, Martensite and deformation twinning in austenitic steels. Materials Science and Engineering A, 1999. 273-275: p. 430-436.

7. Lebedev, A.A. and V.V. Kosarchuk, Influence of phase transformations on the mechanical properties of austenitic stainless steels, in International Journal of Plasticity.2000. p. 749-767.

8. Ganesh Sundara Raman, S. and K. Padmanabhan, Tensile deformation-induced martensitic transformation in AISI 304LN austenitici stainless steel. Journal of Materials Science Letters, 1994. 13(5): p. 389-392.

9. Perlade, A., O. Bouaziz, and Q. Furnemont, A physically based model for TRIP-aided carbon steels behaviour. Materials Science and Engineering A, 2003. 356(1-2 SU -): p. 145-152.

10. Olson, G. and M. Cohen, Kinetics of strain-induced martensitic nucleation. Metallurgical Transactions A, 1975. 6A(April): p. 791-795.

11. Stringfellow, R., D. Parks, and G. Olson, A constitutive model for transformation plasticity accompanying strain-induced martensitic transformation in metastable austenitic steels. Acta Metallurgical, 1992. 40(7): p. 1703-1716.

23 of 35

12. Tomita, Y. and T. Iwamoto, Constitutive modeling of trip steel and its application to the improvement of mechanical properties. International Journal of Mechanical Sciences, 1995. 37(12): p. 1295-1305.

13. Iwamoto, T. and T. Tsuta, Computational simulation of the dependence of the austenitic grain size on the deformation of TRIP steels. International Journal of Plasticity, 2000. 16:p. 791-804.

14. Iwamoto, T., Y. Kawagishi, and S. Morita, Identification of constitutive equation for TRIP steel and its application to improve mechanical properties. JSME International Journal. Series A., 2001. 44(4): p. 443-452.

15. Iwamoto, T., T. Tsuta, and Y. Tomita, Investigation on deformation mode dependence of strain-induced martensitic transformation in TRIP steels and modelling of transformation kinetics. International Journal of Mechanical Sciences, 1998. 40(2-3): p. 173-182.

16. Tomita, Y. and T. Iwamoto, Computational prediction of deformation behavior of TRIP steels under cyclic loading. International Journal of Mechanical Sciences, 2001. 43(9 SU -): p. 2017-2034.

17. Tsuta, T. and J. Cortes, Flow stress and phase transformations analyses in austenitic stainless steel under cold working, Part 2, Incremental theory under multiaxial stress state by the Finite-Element method. JSME International Journal. Series A., 1993. 36(1):p. 63-72.

18. Iwamoto, T. and T. Tsuta, Computational simulation on deformation behaviour of CT specimen of TRIP steel under mode I loading for evaluation of fracture toughness.International Journal of Plasticity, 2002. 18(11 SU -): p. 1583-1606.

19. Iwamoto, T. and T. Tsuta, Computational simulation of the dependence of the austenitic grain size on the deformation behavior of TRIP steels. International Journal of Plasticity, 2000. 16(7-8 SU -): p. 791-804.

20. Lindgren, L.-E. Models for forming simulations of metastable austenitic stainless steel.in 8th International Conference on Numerical in Industrial Forming Processes, NUMIFORM '03. 2004. Ohio, USA.

21. Tomita, Y. and Y. Shibutani, Estimation of deformation behavior of TRIP steels -- smooth/ringed-notched specimens under monotonic and cyclic loading. International Journal of Plasticity, 2000. 16(7-8 SU -): p. 769-789.

22. Iwamoto, T., Multiscale computational simulation of deformation behavior of TRIP steel with growth of martensitic particles in unit cell by asymptotic homogenization method.International Journal of Plasticity, 2004. 20(4-5): p. 841-869.

23. Malvern, L.E., Introduction to the Mechanics of a Continuous Medium. Engineering of the Physical Sciences, ed. J. Reswick and W. Rohsenow. 1969, Englewood Cliffs: Prentice-Hall.

24. Khan, A. and S. Huang, Continuum Theory of Plasticity. 1995: John Wiley & Sons. 420. 25. Simo, J.C. and T.J.R. Hughes, Computational Inelasticity, ed. J.E. Marsden, L. Sirovich,

and S. Wiggins. 1997, New York: Springer-Verlag. 392. 26. Hughes, T. and J. Winget, Finite rotation effects in numerical integration of rate

constitutive equations arising in large-deformation analysis. International Journal for Numerical Methods in Engineering, 1980. 15(9): p. 1862-1867.

27. Simo, J. and K. Pister, Remarks on rate constitutive constitutive equations for finite deformation problems: Computational implications. Computer Methods in Applied Mechanics and Engineering, 1985. 46: p. 201-215.

24 of 35

28. Johnson, G. and D. Bammann, A discussion of stress rates in finite deformation problems. International Journal of Solids and Structures, 1984. 20(8): p. 725-737.

29. Crisfield, M.A., Non-Linear Finite Element Analysis of Solids and Structures, Vol. 2 Advanced topics. 1997: J Wiley & Sons.

30. Hughes, T. Numerical implementation of constitutive models: rate-independent deviatoric plasticity. in Theoretical Foundation for Large Scale Computation of Nonlinear Material Behaviour. 1984. Northwestern University, Evanston, Illinois, US: Nijhoff.

31. Belytschko, T., W.K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures. 2000: John Wiley & Sons. 650.

32. Bathe, K.-J., Finite Element Procedures. 1996: Prentice-Hall. 1037. 33. Bonet, J. and R.D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis.

1997: Cambridge University Press. 34. Ortiz, M. and J. Martin, Symmetry-preserving return mapping algorithm and

incrementally extremal paths: A unification of concepts. International Journal for Numerical Methods in Engineering, 1989. 28: p. 1839-1853.

35. Ortiz, M. and E. Popov, Accuracy and stability of integration algorithms for elastoplastic constitutive relations. International Journal for Numerical Methods in Engineering, 1985. 21: p. 1561-1576.

36. Crisfield, M.A., Non-linear Finite Element Analysis of Solids and Structures, Volume 1: Essentials. 1991, Chichester: John Wiley & Sons. 345.

37. Raboin, P., A deformation-mechanism material model for Nike3D. 1993, Lawrence Livermore National Laboratory. p. 47.

38. Frost, H.J. and M.F. Ashby, Deformation-Mechanism Maps - The Plasticity and Creep of Metals and Ceramics. 1982: Pergamon Press. 166.

39. Marin, E. and D. McDowell, A semi-implicit integration scheme for rate-dependent and rate-independent plasticity. Computers & Structures, 1997. 63(3): p. 579-600.

40. Ponthot, J.P., Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes. International Journal of Plasticity, 2002. 18(1): p. 91-126.

41. Arif, A., T. Pervez, and M. Mughal, Performance of a finite element procedure for hyperelastic-viscoplastic large deformation problems. Finite elements in analysis and design, 2000. 34: p. 89-112.

42. Simo, J.C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation.Computer Methods in Applied Mechanics and Engineering, 1988. 66: p. 199-219.

43. Simo, J.C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part II. Computational aspects.Computer Methods in Applied Mechanics and Engineering, 1988. 68: p. 1-31.

44. Lindgren, L.-E., H. Alberg, and K. Domkin. Constitutive modelling and parameter optimisation. in VII International Conference on Computational Plasticity COMPLAS 2003. 2003. Barcelona.

25 of 35

Appendix 1. Derivatives needed for iterative solution in the plastic corrector step The formulation of the radial return algorithm for SIMT, in particular step 2.4 in Box2 in section 5.3.3, is applied to the specific SIMT model which has been introduced in chapter 3. First we recall the definition of the internal state variable vectors,

sbMMT XXqqqq 4321q , (A1-1)

and the definition of “ -values” of the variables,

qqq 11 nn , (A1-2)

and the same for other “ -variables”.In the plastic corrector step, the residual vector corresponding to the generalized hardening law is based on Eq. (77) and for the present SIMT model it takes the form,

01 MYM

YH , (A1-3)

012p

MMM XXH , (A1-4)

01),,(3 MsbMM XTKXAXH , (A1-5)

01),(4 sbsbsb XTKAXH . (A1-6)

The yield stress of austenite and martensite are functions of corresponding “ -values” of strains and temperature, Eqns (24) and (25),

TYY , , (A1-7)

TMYM

YM , , (A1-8)

and the derivatives p

Y

and pM

YM are trivially obtained from these functions.

The increments are defined for all variables, such as, for instance, nn 1 . Also we use the following relation between partial derivatives,

qH

qH

qH1n . (A1-9)

Hence, we obtain first for H1 ,Y

Yn

HHq

H 11

11

1 , (A1-10)

MM

YMY

MMM

nHH

qH 11

21

1 , (A1-11)

and all other partial derivatives are zero. Next, non-zero derivatives of H2:

Mn XHq

H 102

11

2 , MM

n XHq

H 02

21

2 , (A1-12)

MM

n XH

qH 2

31

2 0 , 1021

2ppn

HH , (A1-13)

26 of 35

Next, all non-zero derivatives of H3:

MMn XAHq

H 103

11

3 , (A1-14)

MMM

n AXH

XH

qH 133

31

3 , (A1-15)

Msb

M

sbn X

XA

XH

qH 10 3

41

3 , (A1-16)

MM

n Xp

Ap

Hp

H 10 31

3 , (A1-17)

MM

n XAHH 10 31

3 . (A1-18)

Finally, all non-zero derivatives of H4:

sbsbn XAHq

H 104

11

4 , (A1-19)

sbsbsb

n AXH

XH

qH 144

41

4 , (A1-20)

sbsb

n Xp

Ap

Hp

H 10 41

4 , (A1-21)

sbsb

n XAHH 10 41

4 . (A1-22)

Derivatives of AM are obtained trivially from Eq. (12), which we re-write here,

sbnsbsbM XXnAA 11 , (A1-23)

sbnsbsb

sb

M XnnXnAXA 12 , (A1-24)

pK

KA

pA MM and K

KAA MM , (A1-25)

KA

KAXXn

KA

KA

AA

KA

sbsb

sbnsb

Msb

sb

M 11 (A1-26)

Derivative of Asb is obtained trivially from Eq. (3), which we re-write here too,

KAATATAAsb 4322

1 , (A1-27)

where the rate-dependency is ignored. Its derivative w.r.t the tri-axiality factor becomes

4AKAsb . (A1-28)

Probability, , has been defined as a function of the driving force, g, Eqns (9) and (10), as

27 of 35

g gg

g

gdeg g

20

21

21 , (A1-29)

KgTg 1 . (A1-30)

Thus its derivative

121

20

21 ge

Kg

dgd

Kg

gg

g

. (A1-31)

Finally, triaxiality factor and its partial derivatives are computed as

pK 1pK and 2

pK . (A1-32)

The expressions listed above complete the matrices q

H1n , pn 1

H ,pn 1

H and 1nH . The

remaining quantities needed are the derivatives of hydrostatic stress, effective stress, and the yield stress. First, we recall the constitutive law for the hydrostatic stress,

thkk

ekk

nnn Kp 311 . (A1-33)

Thus its derivative can be written as

qqqqq 1

11

1

1

1

1

1

1

1

1

1

1

33 n

thnn

n

nth

kkekk

nn

th

th

n

n

n

n

n

n

n

KKpKKpp .(A1-34)

The bulk modulus and thermal dilatation in the SIMT model, Eqns (19) and (20), are computed as

)()(1 11111 TKXTKXK nMM

nnM

nn , (A1-35)

)()(1 11111 TXTX nthMM

nnthM

nthn , (A1-36)

and their only non-zero partial derivatives (w.r.t. internal variables q) are

thnthM

n

Mn

thn

X11

1

1

and KKXK n

Mn

Mn

n11

1

1

, (A1-37)

where the bulk moduli and thermal dilatation of austenite and martensite are functions of temperature only. Hence the only non-zero partial derivative of the hydrostatic stress is

thnthM

nnnM

nthkk

ekk

nn

n

KKKqp 11111

31

1

33 (A1-38)

Next, we compute the derivatives of the effective stress, using the definition of the deviatoric stress scaling parameter, ,

trnn 11 , (A1-39)

tr

pn

tr

trnn G

GGG 1

11 31 . (A1-40)

28 of 35

The shear modulus is computed according to the Eq. (18),

)()(1),( 1111111 TGXTGXTXGG nMM

nnM

nnM

nn , (A1-41)

where the shear moduli of austenite and martensite are functions of temperature only. Accordingly, the derivative w.r.t. martensite fraction in the end of the increment is

GGXG

qG n

Mn

Mn

n

n

n11

1

1

31

1

. (A1-42)

The scaling parameter, and hence effective stress, is a function of only XM, p and T, with trial effective stress and trial shear modulus being fixed. The non-zero derivatives are

GGGq

GGqq

nM

nptr

trn

tr

p

trtr

n

ntr

n

n11

3

1

31

1

31

1

331 (A1-43)

GG ntr

ntrp

ntr

pn

n11

1

1

1

313 (A1-44)

Finally, we recall the definition of the (macro) yield stress,

Mn

Mnn

MnYn XX 11111 1 . (A1-45)

The yield stress of austenite and martensite are functions of corresponding “n+1”-values of strains and temperature, defined in Eqns (24) and (25),

Tnnn 111 , and TnM

nMM

n 111 , , (A1-46)

and their derivatives, 1

1

n

n

and M

nM

n

1

1

, are trivially obtained.

Hence, the non-zero derivatives of the yield stress are as follows,

1

11

11

1

1 n

n

Mn

n

Yn

Xq

,M

nM

n

Mn

n

Yn

Xq 1

11

21

1

, (A1-47)

11

31

1n

Mn

n

Yn

q. (A1-48)

Appendix 2. Initial values of the unknowns in the plastic corrector step This is the details for step 2.1 in Box 2. First, using the Eq.(20) for thermal dilatation, an estimate of the martensite fraction, )0(

1M

n X , is found with the assumption that the hydrostatic stresses in the end and in the beginning of the increment are equal,

)0(1

Mn X pTXK nn

Mnth

kkekk

ntr ),(3 1)0(

1 . (A2-1)

Here we use trial values of bulk and shear moduli, trG and trK, see Box 1. Effective plastic strain increment is calculated with the assumption that the effective stresses in the end and in the beginning of the increment are equal,

p)0(

nptrtrtr

tr

GGG

)0(3 . (A2-2)

29 of 35

Next, estimates of the plastic strain in austenite and martensite, )0(1n and )0(

1M

n , are

calculated to satisfy the Eqns (79) and (80), where we use the above estimates of )0(1

Mn X and

p)0( , and also the trial values of austenite and martensite yield stress.

)0(1n 0)0()0( M

YM

trYtr (A2-3)

)0(1

Mn 01 )0()0()0()0()0(

pMMM XX (A2-4)

Finally, the parameter Asb is calculated using the value of triaxiality factor at the beginning of the increment. It is substituted together with the estimate of )0(

1n into the first-order update

formula for the shear-bands fraction, Eq. (82), to find an estimate of )0(1

sbn X ,

)0(1

sbn X 01),( )0()0()0( sb

nsbsb XTKAX . (A2-5)

Thus we have a predictor for all terms in q.

Appendix 3. Derivation of the consistent constitutive matrix

According to the definition of the consistent (or algorithmic) constitutive tensor, aijklC ,

klaijklij dCd , (A3-1)

where the variation of stresses and strains in the “n+1” state is considered, that is in the end of the increment. In the following derivation, we shall omit the “n+1” superscript.

A3.1 Variation of the total stress tensorThe variation in stress is split into volumetric and deviatoric parts,

ijijij dsdpd . (A3-2)

From Eq. (62), the variation in deviatoric stress is found as

)()( ijtr

ijtr

ijtr

ij sdsdsdds . (A3-3)

The variation in the trial deviatoric stress in the last term above is

kldevijkl

trijkl

klij

trij

trij

tr dGIddGGdesd 23

22 , (A3-4)

where the relation between total and deviatoric strain has been used, introducing the symmetric deviatoric projection tensor,

klijjkiljlikdevijklI

31

21 . (A3-5)

Then the variation of the effective trial stress becomes

kltrkl

trtr

trkl

trkl

tr

trkl

trkl

trtr dsGGdessdsd 3

223

2)(3 . (A3-6)

The last equality is possible due to the property of the deviatoric stress tensor,

klkltr

klkltr dsdes . (A3-7)

30 of 35

Next, the variation in scaling parameter is expressed in terms of effective stress,

dddd trtr

trtr1

2 . (A3-8)

Finally, all the above variations are substituted into Eq. (A3-2) to give variation in total stress,

dsdpdIGdssGd ijtr

trijkldevijkl

trklkl

trij

trtr

trij

1213 2 . (A3-9)

We re-write it in vector notation,

ddpdGdGd devtrtr 1IL 123 , (A3-10)

where stress and strain vectors are defined as

312312332211T , (A3-11)

312312332211T . (A3-12)

In the strain vector we use the “engineering shear strain” components, 1212 2 etc, which necessitates appearance of the diagonal matrix )222111Diag( ,,,,,L in the formulas. Additionally the following symbols have been used. ”Unit vector” T0001111 .Operation is the product of two vectors, which results in a matrix (e.g. T1111 ).Symmetric deviatoric projection matrix (where I is the unit matrix) is

11II31dev . (A3-13)

The trial direction of the plastic flow is defined by a vector,

tr

tr s . (A3-14)

In the Eq. (A3-10) all quantities are defined, except for variations in hydrostatic and effective stress in the last two terms. These are derived below.

A3.2 Variations of the hydrostatic and effective stressVariations of hydrostatic and effective stress, as well as variation of internal variables, are obtained as a result of the linearisation with respect to the total strain increment applied to the plastic corrector procedure of the Box 2. In the context of such linearisation, all variables are considered to depend directly or indirectly on the total strain increment, ij.First, we recall that hydrostatic and effective stresses depend on the total strain increment directly, or indirectly via trial effective stress,

thkk

ekk

nKp 3 , (A3-15)

ptr

tr GG

G 3 . (A3-16)

Hence the expressions of their variations, Eqns (87) and (88), should be modified to account for the change in total strain,

pddpdp n 01 qq

, (A3-17)

31 of 35

dddd ppn q

q1 , (A3-18)

where the additional symbols denote variations due to change in total strain alone and are derived using either tensor or vector notation,

1 dKdpdKKddppd klklkkklkl

, (A3-19)

s dGdGddsGddd trtrklkl

trtr

trtrkl

kl

333 .(A3-20)

The second term in Eq. (A3-18) is taken from Eq (A1-44) from Appendix 1,

Gp 3 . (A3-21)

For the studied SIMT model, not only the yield function, f, but also the hardening law, H, both depend on total strain via the hydrostatic and effective stress. Thus the linearisation of the non-linear system of equations (77) and (78) has to account for the change in total strain too. On the other hand, the yield function and residuals here are assumed to be zero, as a result of the solution in Box 2. Hence we obtain,

0~~1

fdd

dd

fdfd

ppq

aqf

rAH, (A3-22)

where the new symbols again denote the variations due to change in total strain alone,

dddTddffdY

0),(q , (A3-23)

dpdp

dTpdd nn

p

11,,,,,~~ HHqqHaa . (A3-24)

Inserting Eqns (A3-19) and (A3-20) into the last equation above gives

H1HHHa1111 3

~nnnn G

pKp

p. (A3-25)

Next, the first vector equation of the system (Eq. A3-22) is inverted to give the variation in internal variables,

aArAq ~~ ddd p . (A3-26)

It is inserted into the last equation of the system, which is then solved for the variation in effective plastic strain,

pq

qp

fdfd

drAf

aAf~

~. (A3-27)

The latter is inserted back into eq (A3-26), which with some manipulation gives

aArAf

aAfrAq d

f

ddf

dpq

q ~~

~

~ . (A3-28)

32 of 35

Variation in internal variables, Eq. (A3-28), and variation in effective plastic strain, Eq. (A3-27), together with the Eqns (A3-19) and (A3-20) are inserted into the Eqns (A3-17) and (A3-18), and hence the complete expressions of the variations in hydrostatic and effective stress are obtained. For the hydrostatic stress, we derive

1aAqrAf

aAfrA

qdKdp

f

ddfpdp n

pq

q

n

~~

~

~11

Introducing a symbol,

pq

n

P f

p

ZrAf

rAq

~

~1

, (A3-29)

we arrive at

aAq

f1 dpZdfZdKdp nPP

~1 . (A3-30)

For the effective stress we derive

rAf

aAf

aAqrAf

aAfrA

q

df

ddf

df

ddf

d

pq

q

p

npq

q

n

~

~

~~

~

~11

(A3-31)

Introducing another symbol,

pq

np

S fZ

rAf

rAq

~

~

11

, (A3-32)

and with some algebraic manipulations, we obtain

aAq

f dfZdZdZd SnqSS 1~

1 1 (A3-33)

The last term in the above equation is zero, according to the Eq. (A3-23).

A3.3 Expression of the algorithmic constitutive matrixInserting the variations of hydrostatic and effective stress into the last two terms of the Eq. (A3-10), and also using Eqns. (A3-20) and (A3-23) gives

aAq

fq

1f1

1111

dZpZ

dGZdGZdKddp

nqSnqP

SP

~1

33

11

(A3-34)

33 of 35

Hence the complete tangent matrix is expressed as

aAq

fq

1f1

1IL11C~

1

3332

11

1

nqSnqP

PStrdevtra

ZpZ

GZGZGGK (A3-35)

Using Eq. (A3-25) and by means of tedious algebraic transformations, the last term is expanded and re-arranged, and the tangent matrix finally takes the form presented in the section 5.3.4,

1111

1IL11C

KDGDGDKD

GZGGZGGK

SPPSSSPP

P

tr

Sdevtra

33

332 1

(A3-36)

The definitions of the symbols DPP, DSS, DPS, and DSP have been also introduced there in the section 5.3.4.

Appendix 4. Uniaxial stress case version Stress updating for the uniaxial case is conveniently done without split into deviatoric and volumetric parts. Furthermore, the direction of plastic flow is 1 and is determined by the sign of the trial stress trsgn .

The trial stress is

TE

TEEE nnthn

nnnthtrentretrtrtr 11 ,, qq . (A4-1)

and its absolute value is compared with the trial yield limit as in Box 1 earlier. If the stress state is outside the yield surface, then a plastic corrector step is executed. Modification of the Box 2 algorithm for the uniaxial stress case is described below.

First simplification concerns the triaxiality factor: it is constant in the uniaxial case, 3/1K .Hence the stress-dependency of the hardening law, which in this model is only due to triaxiality factor, can be dropped and the corresponding derivatives are equal zero,

011 nn pHH . (A4-2)

As a consequence, computation of the hydrostatic stress, n+1p, in the Box 2 is unnecessary too. Secondly, the ”radial return” assumption in the uniaxial case is reduced to

ptrp sgn . (A4-3)

The effective stress becomes 11 sgn ntrn , (A4-4)

where the uniaxial stress in the end of the increment depends on the increment in plastic strain and on the corresponding changes in Young’s modulus and thermal dilatation,

pnnthn

nnnennn T

ETEE 1111111 ,, qq . (A4-5)

34 of 35

Thus, instead of using Eq. (2.6) in the 3-D algorithm of the Box 2, the effective stress in the uniaxial case is defined by Eqns. (A4-4) and (A4-5). Accordingly, the derivatives of the yield function needed for the logic in Box 2 become

qq

qqqf ),()sgn(

111 TYntr

Ynn

q (A4-6)

and

p

n

p

ntr

p

Yn

p

n

pTf

1111

0)sgn(),(q . (A4-7)

Derivatives of the current yield stress, q

Yn 1

and p

Yn 1

, are identical to those in the general

3-D case and are given in Eqns. (A1-47, A1-48) in the Appendix 1. Non-zero derivatives of the current uniaxial stress, Eq. (A4-5), are computed as

M

thnnthtrp

n

n

M

n

M

nn

XE

EXE

Xq

11

11

3

1

sgn , (A4-8)

Enp

n1

1

. (A4-9)

The Young’s modulus and thermal dilatation are computed according to the model, Eqns (15) and (20). Corresponding derivatives are computed similar to the Eq. (A1-37), Appendix 1. With the above simplifications of the algorithm, with the definitions of the uniaxial stress and partial derivatives of the yield function, the logic of the Box 2 and Appendix 1 can be applied in the uniaxial case to compute the increments in effective plastic strain and internal variables.

Next, the logic of Appendix 3 is followed to derive the consistent tangent for the uniaxial case. The consistent constitutive “matrix” is defined as

11 nan dCd . (A4-10)

From the Eq. (A4-4) it follows trivially that 11 sgn ntrn dd . (A4-11)

Next, the variation of effective stress is expressed in the form analogous to the Eq. (A3-18),

ddddn

pp

nnn

1111 q

q, (A4-12)

where partial derivatives are obtained from the Eq. (A4-5) according to Eqns (A4-6) to (A4-9), and additionally,

Entrn

trn

111

sgnsgn . (A4-13)

Introducing the “modified Young’s modulus”,

Edd

ddE n

p

n

p

n

p

n1

111

modq

qq

q, (A4-14)

the variation of effective stress, Eq. (A4-12), is re-written as

EddEd ntrpn 1mod

1 sgn . (A4-15)

35 of 35

In the uniaxial case the employed hardening law is independent of the current stress, Eq.(A4-2), hence the term a~d in the Eq. (A3-26) in Appendix 3 is equal zero, and the full derivative of internal variables w.r.t. effective plastic strain can be obtained simply as

rAq ~pd

d . (A4-16)

Finally, variation of the effective plastic strain is obtained from the consistency condition,

HdddHdfd

nppnn

111 0 , (A4-17)

where the hardening modulus is defined as

p

Yn

p

Yn

ddT

ddH q

qq ),(11

. (A4-18)

Combining Eqns (A4-15) and (A4-17) and using Eq. (A4-11) with some manipulation gives variation of the uniaxial stress w.r.t. the total uniaxial strain,

1

mod

11 n

nn d

HEHEd . (A4-19)

constitutive models based on dislocation density. formulation and

Documents