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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2016; 107:733–785Published online 26 January 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.5186

The particle finite element method (PFEM) inthermo-mechanical problems

J. M. Rodriguez1, J. M. Carbonell2,*,† , J. C. Cante3 and J. Oliver2

1Universitat Politècnica de Catalunya (UPC), Campus Terrassa, C/Colom 11, 08222 Terrassa, Spain2International Center for Numerical Methods in Engineering (CIMNE), Campus Nord UPC, Gran Capitán, s/n., 08034

Barcelona, Spain3Escola Tècnica Superior d’Enginyeries Industrial i Aeronàutica de Terrassa, Rambla de Sant Nebridi 22, 08222

Terrassa, Spain

SUMMARY

The aim of this work is to develop a numerical framework for accurately and robustly simulating the differentconditions exhibited by thermo-mechanical problems. In particular, the work will focus on the analysisof problems involving large strains, rotations, multiple contacts, large boundary surface changes, andthermal effects.

The framework of the numerical scheme is based on the particle finite element method (PFEM) in whichthe spatial domain is continuously redefined by a distinct nodal reconnection, generated by a Delaunaytriangulation. In contrast to classical PFEM calculations, in which the free boundary is obtained by a geo-metrical procedure (˛�shape method), in this work, the boundary is considered as a material surface, andthe boundary nodes are removed or inserted by means of an error function.

The description of the thermo-mechanical constitutive model is based on the concepts of large strainsplasticity. The plastic flow condition is assumed nearly incompressible, so a u-p mixed formulation, with astabilization of the pressure term via the polynomial pressure projection, is proposed.

One of the novelties of this work is the use of a combination between the isothermal split and the so-calledIMPL-EX hybrid integration technique to enhance the robustness and reduce the typical iteration number ofthe fully implicit Newton–Raphson solution algorithm.

The new set of numerical tools implemented in the PFEM algorithm, including new discretization tech-niques, the use of a projection of the variables between meshes, and the insertion and removal of pointsallows us to eliminate the negative Jacobians present during large deformation problems, which is one of thedrawbacks in the simulation of coupled thermo-mechanical problems.

Finally, two sets of numerical results in 2D are stated. In the first one, the behavior of the proposedlocking-free element type and different time integration schemes for thermo-mechanical problems is ana-lyzed. The potential of the method for modeling more complex coupled problems as metal cutting and metalforming processes is explored in the last example. Copyright © 2015 John Wiley & Sons, Ltd.

Received 10 July 2015; Revised 1 December 2015; Accepted 2 December 2015

KEY WORDS: particle finite element method (PFEM); thermo-elastoplasticity; IMPL-EX integration;remeshing and geometry update

1. INTRODUCTION

1.1. Motivation

Thermo-mechanical processes, which involve large strains and rotations, multiple contacts, andgeneration of new surfaces, are a major challenge of the numerical simulation and thus requireadvanced numerical techniques. Numerical modeling has been applied to simulate complex thermo-mechanical processes, in order to predict incoming difficulties such as defects in formed parts,improper tool profile, and low lifetime.

*Correspondence to: J. M. Carbonell, International Center for Numerical Methods in Engineering (CIMNE), CampusNord UPC, Gran Capitán, s/n., 08034 Barcelona, Spain.

†E-mail: [email protected]

Copyright © 2015 John Wiley & Sons, Ltd.

734 J. M. RODRIGUEZ ET AL.

First, we lay out a summary of the advantages and drawbacks of the different numerical strate-gies capable to represent the complex phenomena taking place. The principal problem in the useof a conventional finite element model with a Lagrangian description is the mesh distortion inhigh deformation processes. Traditional Langrangian approaches such as the finite element method(FEM) cannot resolve the large deformations very well. Element distortion has been always a matterof concern that that turns the modeling into an incipient analysis. Instead, the FEM with a Eule-rian formulation requires the knowledge of the deformed geometry in advance, which, undoubtedly,restricts the range of problem conditions capable of being analyzed.

A Lagrangian formulation was used to simulate complex thermo-mechanical processes in [1–3]and [4]. The Eulerian formulation is used in [5]. In order to avoid the disadvantages of finite ele-ment Lagrangian and Eulerian formulations, other computational techniques have been investigated.One of them is the finite element arbitrary Lagrangian Eulerian (ALE) formulation in conjunctionwith adaptive mesh techniques [6–8]. The ALE formulation combines the best features of pureLagrangian analysis (in which the mesh follows the material) and Eulerian analysis (in which themesh is fixed and the material flows through the mesh). In ALE framework, mesh motion is indepen-dent of material motion, for that reason, high-quality finite element meshes are preserved during thenumerical simulation. In addition to the numerical strategies based on the finite element, there existsome recently developed meshless methods. Among them, we can mention the smoothed-particlehydrodynamics (SPH) [9], the finite pointset method (FPM) [10], the constrained natural elementmethod (C-NEM) [11, 12], the discrete element method (DEM) [13], a meshless technique based ona reproducing kernel particle method (RKPM) [14–16], and a meshless method based on maximumentropy meshfree approximants (MEM) [17]. The main advantage of a meshless method is that itdoes not need a finite element mesh to calculate derivatives. Material properties and state variablesare available at a set of points, called particles. This avoids severe problems associated with meshtangling and distortion that usually occur in finite element Lagrangian formulations involving largedeformation and strain rates. The main disadvantage of meshless methods in comparison with theFEM is the neighbors search, because updating the database of neighbor particles takes usually along time in comparison with other calculations needed during each time step. Another disadvan-tage of meshless methods is that due to a fix number of particles used in the numerical simulation,the particles usually concentrate in some region and disperse in other areas. Also, imposing essentialboundary conditions in meshless methods is not as straightforward as for the FEM. Furthermore,nearly incompressible conditions are a big challenge to meshfree methods. The main objective ofthis work is precisely to contribute to solve some of the problems described earlier through theextension of the particle finite element method (PFEM) to thermo-mechanical problems in solidmechanics that involves large strains and rotations, multiple contacts, and generation of new sur-faces. In this work, we exploit the particle and Lagrangian nature of the PFEM and the advantagesof finite element discretization. The new ingredients of PFEM are focused on the insertion andremoval of particles, the use of constrained Delaunay triangulation and a novel transfer operatorof the internal variables, which minimizes the numerical diffusion of internal variables, one of themain disadvantages of the standard finite elements. As a consequence of PFEM basic features, theelement distortion is minimized during all the numerical simulation.

1.2. Contents

Next, we present the mathematical and numerical ingredients necessary to simulate a classicalthermo-mechanical problem, including the balance of momentum and its finite element discretiza-tion, and the balance of energy and its finite element discretization. The constitutive equation forthe treatment of metals and the used time discretization will be detailed. Particular solutions relatedto the treatment of the incompressibility constraint and for the improvement of the time integrationscheme are explained. All developments are build within the PFEM.

The paper starts with the definition of the PFEM in Section 2, the basic general steps and thecustom characteristics used in the present formulation are explained. In Section 3, the coupledthermo-mechanical problem is described with a summarized form of the balance equations of theinitial boundary value problem (IBVP). The equations are stated using a thermo-elastoplastic split of

Copyright © 2015 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2016; 107:733–785DOI: 10.1002/nme

PFEM IN THERMO-MECHANICAL PROBLEMS 735

the problem and particularized with the mixed displacement-pressure formulation used for the finiteelement discretization. A review of the most used techniques for the treatment of the incompressibil-ity constraint is made. The pressure stabilization method used in this work is explained in detail inthis section. The main expressions of the finite element numerical integration of the IBVP are devel-oped in Section 4. In Section 5, we present an overview of the thermo-elastoplastic model at finitestrains proposed by Simo et al. in [18–20]. This model will be used in the examples of Section 7that test the capabilities of the present formulation. Also in Section 5, the IMPL-EX scheme for theconstitutive law integration is presented. The flow rules, the algorithmic constitutive tensor expres-sion and the expression of the linearization of the algorithmic dissipation are developed and givenfor the current scheme. The expressions for the dissipation will be used in the thermal solution andalso as error estimates for the mesh update within the PFEM. Three possible time integrations of theIBVP are explained in this work; in Section 6, the proposed isothermal IMPL-EX scheme for thetime integration of the thermo-mechanical problem is described.

The paper finishes analyzing three classical benchmark problems found in the literature. Withthese examples, we validate the formulation in front of the current state of the art. In Section 7,we also include an example of analysis of challenging thermo-mechanical problems. First, a clas-sical steel cutting test-type is presented to show the thermo-mechanical modeling capabilitiesof the PFEM. In second term, we make a comparison with experimental results and the resultsfrom commercial codes to prove the competitiveness of the PFEM in the field of the model-ing of thermo-mechanical processes. All problems analyzed represent qualitative validations of2D models.

The theory, presented in synthesized way in each one of the sections, is described in more detailin the appendices.

2. THE PARTICLE FINITE ELEMENT METHOD

The PFEM is founded on the Lagrangian description of particles and motion, and it combines ameshless definition of the continuum containing a cloud of particles with standard mesh-based finiteelement techniques.

The initial developments of the PFEM took place in the field of fluid mechanics [21, 22], becauseof the PFEM feasible features of tracking and modeling of free surfaces. Later on, the particle finiteelement (PFEM) was applied in a variety of simulation problems: fluid structure interaction withrigid bodies, erosion processes, mixing processes, coupled thermo-viscous processes, and thermaldiffusion problems [23–26].

The continuum, representing a solid or a fluid, is described by a collection of particles in space.The particles contain enough information to generate the correct boundaries of the analysis domain.Meshing techniques like the Delaunay tessellation and the ˛�shape concept [27] are used todiscretize the continuum with finite elements starting from the particle distribution. The meshingprocess creates continuum sub-domains and identifies the geometrical contacts between differentsub-domains.

First applications of PFEM to solid mechanics were carried out in problems involving largestrains and rotations, multiple body contacts, and creation of new surfaces (riveting, powder filling,ground excavation, and machining) [28–31]. In this work, we extended the PFEM to the numericalsimulation of process involving thermo-mechanical problems.

2.1. Basic steps of the particle finite element method

In the PFEM, the continuum is modeled using an updated Lagrangian formulation. That is, allvariables are assumed to be known in the current configuration at time t . The new set of variables issought for in the next or updated configuration at time t C�t (Figure 1). The FEM is used to solvethe continuum equations. Hence, a mesh discretizing the domain must be generated in order to solvethe governing equations in the standard FEM fashion. Recall that the nodes discretizing the analysisdomain are treated as material particles, which motion is tracked during the transient solution. Thisis useful to model the separation of particles from the main domain, in groups of particles such as a



Figure 1. Sequence of steps to update in time a ‘cloud’ of nodes representing a solid mass that is progres-sively fragmented the action of an external object using the particle finite element method. In the boundaries

the particles are fixed.

metal chips in metal cutting problems, or as single particles such as water drops in fluid problems.In the last case, it is possible to follow the motion of the domain as individual particles with aknown density, an initial acceleration and velocity, and subject to gravity forces. The mass of a givendomain is obtained by integrating the density at the different material points over the domain.

The quality of the numerical solution depends on the discretization chosen as in the standardFEM. Adaptive mesh refinement techniques can be used to improve the solution.

For clarity purposes, we will define the collection or cloud of nodes (C) belonging to the analysisdomain, the volume (V) defining the analysis domain, and the mesh (M) discretizing the domain.

A typical solution with the PFEM involves the following steps.

(1) The starting point at each time step is the cloud of points in the analysis domains. For instanceCn denotes the cloud at time t D tn (Figure 1).

(2) Identify the boundaries defining the analysis domain Vn. This is an essential step as someboundaries may be severely distorted during the solution, including separation and re-enteringof nodes. The ˛�shape method [27] is used for the boundary definition.

(3) Discretize the continuum domains with a finite element mesh Mn.(4) Solve the Lagrangian equations of motion in the domain. Compute the state variables at the

next (updated) configuration for t C �t : displacements, pressure, temperature, stresses andstrains, etc.

(5) Move the mesh nodes to a new position CnC1 where nC 1 denotes the time tnC�t , in termsof the time increment size. This step is typically a consequence of the solution process ofstep 4.



Figure 2. Remeshing steps in an standard particle finite element method numerical simulation.

(6) Go back to step 1 and repeat the solution process for the next time step to obtain CnC2. Theprocess is shown in Figure 1.

Figure 1 shows a conceptual example of application of the PFEM to model the progressivefragmentation of a solid mass under the action of an external object.

2.2. Meshing procedure and variables transfer in the PFEM

The original idea of the PFEM was to improve the mesh quality by performing a re-triangulation ofthe domain only when is needed, which allows to capture large changes in the continuum domainand avoid global remeshing and interpolation from mesh to mesh. Usually, that is performed accord-ing to some criteria associated to element distortion. This re-triangulation consists in re-computingthe element connectivity using a Delaunay triangulation [32–34] where the current position of theparticles (i.e., of the mesh nodes) is kept fixed. Mesh distortion is corrected and improved naturallywith the PFEM, because the Delaunay triangulations maximize the minimum angle of all the anglesof the triangles in the triangulation. Therefore, they tend to avoid skinny triangles.

This strategy has some important implications, the Delaunay triangulation generates the convexfigure of minimum area that encloses all the points and that may be not conformal with the externalboundaries. A possibility to overcome this problem is to couple the Delaunay triangulation with theso-called ˛�shape method. An example of the remeshing scheme using PFEM is shown in Figure 2.

In the Lagrangian approach, the particles move because of the material flow and it may happenthat particles concentrate in same regions of the domain and, on the contrary, in other regions thenumber of particles becomes too low to obtain an accurate solution. To overcome these difficulties,PFEM adds and removes particles comparing with a certain characteristic distance h. If the distancebetween two nodes dnodes is dnodes < h , one of the nodes is removed. If the radius of an elementcircumsphere rec is rec < h , a new node is added at the center of the circumsphere. The flowvariables in the new node are linearly interpolated from that of the element nodes, and the assignedmaterial properties are the ones of the elements.

The solution scheme described by the PFEM applied to fluid mechanics problems can besummarized by the following steps:

(1) The domain is filled with a set of points referred to as ‘particles’ that are endowed with initialvelocity v0, pressure p0, and position X0. The accuracy of the numerical solution is clearlydependent on the considered number of particles.

(2) A starting finite element mesh is generated using the particles as nodes through a Delaunaytriangulation and external boundaries are identified by means of the ˛�shape technique.



(3) As long as mesh distortion is acceptable, the nonlinear Lagrangian form of the governingequations is solved determining the velocity vn and the pressure pn at every node of the mesh.

(4) The position of the ‘particles’ is updated, and if the mesh distortion is not acceptable,a new finite element mesh is generated again using the particles as nodes through aDelaunay triangulation.

In the PFEM, the size of each time step is assumed small enough to avoid remeshing duringthe iterations for the solution of the nonlinear equations in the time step itself. Mesh distortion ischecked only at convergence.

The usual PFEM presents some weaknesses when applied in solid mechanics problems. Forexample, the external surface generated using ˛�shapes may affect the mass conservation of thedomain analyzed. To deal with this problem, in this work, we propose the use of a constrainedDelaunay algorithm [32–34]. In order to deal with complex material flows where the material canbe merged, the use of ˛�shapes will be essential. In these cases, a non-constrained Delaunay tes-sellation can be employed together with proper techniques to preserve the contour surface and forthe mass conservation [35]. That also allows the extension to 3D models where, in many cases, theconstrained Delaunay tessellation is not guaranteed. There are also some other works focussed pos-sible constructions of the constrained Delaunay tessellation in 3D [36–38]. For a certain non-convexdomains, where the constrained tessellation is not allowed, one possible solution consists in addingsome nodes on the border of the domain [37, 39]. Furthermore, addition and removal of particles arethe principal tools, which we employ for sidestepping the difficulties, associated with deformation-induced element distortion, and for resolving the different scales of the solution. The insertion ofparticles is based on the equidistribution of the plastic power, such that, elements exceeding theprescribed tolerance "tol are targeted for refinement.Z

�eDmech d.�/

e > "tol (1)

where Dmech is the mechanical power that will be introduced later on by the Equation (C.34) and�e is the domain of the element. When the condition is fulfilled, a particle is inserted in the Gausspoint of the finite element.

The removal of particles is based on a Zienkiewicz and Zhu [40, 41] error estimator defined bythe expression (2).

Error .e/ D

ˇˇ Ne� � NeNemax

ˇˇ (2)

where Ne� is the recovered equivalent plastic strain and Nemax is the maximum equivalent plasticstrain. A particle is removed if and only if the error in all the elements belonging to the particle isless than a given tolerance.

All the information necessary in subsequent time steps has now to be transferred to the new mesh,it includes the nodal information like displacements, temperatures, pressure in the new insertedparticles, and the Gauss point information like internal variables in the new element. This is achievedusing the procedure described in Box 1.

3. THE COUPLED THERMO-MECHANICAL PROBLEM

We start with the description of the system of partial differential equations governing the evolution ofthe thermo-mechanical IBVP. This is a solid mechanics problem that uses a Lagrangian descriptionof the continuum medium. Thus, the material and spatial Lagrangian descriptions of the governingequations can be used and will be presented in the Appendix A. These descriptions are equivalentin continuum mechanics. However, in this work, we will focus on the spatial one because it is moresuitable for the features of the PFEM. The IBVP described by the thermo-mechanical equations canbe written in a simpler way using an operator split. This is



(1) Isothermal elastoplastic problem

PZ D

24 P'''� PvP�

35 D

24 v.x; t /

div .��.'''; �; �.'''; �///0

35C

24 0b0

35 (3)

(2) Thermoplastic problem at a fixed configuration

PZ D

24 P'''� PvP�

35 D

24 0

0

�div .q.'''; �; �.'''; �///CDint

35C

24 00r

35 (4)

In Equations (3) and (4), � is the density, v is the velocity field, and Pv it the acceleration, where.P/ expresses the time derivative. The motion P' and the absolute temperature � are regarded as theprimary variables in the problem while b the body force per unit of spatial volume and r the heatsource per unit of spatial volume are prescribed data. In Equation (4), Dint is the internal dissipationper unit of spatial volume. In addition, the heat flux q and the Cauchy stress tensor � are defined viaconstitutive equations, where �.'; �/ expresses the set of internal constitutive variables.

Further details of the coupled thermo-mechanical IBVP are presented in the Appendix A.

PROCEDURE FOR MESH REFINEMENTAND

INFORMATION TRANSFER USED IN THE PFEM

1. Update the particle positions due to motion.

2. Refine elements. Criterion based on plastic dissipation values.

3. Refine boundary that is too distorted. Criterion based on curvature information and plasticdissipation values.

4. Remove particles if error estimators are less or equal than a given value.Error estimators based on plastic strain values or on the norm of the isochoric stresses. A particleis removed if all previous finite elements joined to that particle have an error value less or equalto a given tolerance.

5. Perform a constrained Delaunay triangulation in the refined mesh boundary.The triangulation must include remaining and new particles and must delete the triangles outsidethe boundary

6. Estimate the mesh quality. If mesh quality is less or equal than a given tolerance, develop aLaplacian smoothing [42] of the updated particle positions.

1. Find smoothed particles in the new mesh2. Transfer particle information (displacement, pressure, temperature) to the new particle

positions using the shape functions

7. Calculate the global coordinates of the gauss points of the new triangulation.

8. Using the information of the previous mesh, update the internal variables of the newtriangulation.This step states that the Gauss point information of finite element in the new mesh is the Gausspoint information of the closest finite element in the previous mesh.

It is important to remark that step 4 and 6 are optional.The main advantage of the proposed strategy is that:

It is not necessary to create a complete new mesh; we only adapt the mesh with the addition andremoval of particles and the mesh quality is improved using a Delaunay triangulation.

Box 1: Flowchart of the refining scheme and information transfer process.



3.1. Mixed displacement-pressure formulation for the IBVP

It is well known that pure displacement formulations are not suitable for problems in which theconstitutive behavior exhibits incompressibility because they tend to experience locking. Lockingmeans, in this context, that the constraint conditions due to incompressibility cannot be satisfied.These constraint conditions are related to the pure volumetric mode (in the elastic case, the conditionis det.Fe/ D 1, see Equation (B.4) and for the plastic flow the condition is det.Fp/ D det.Cp/ D 1,see Equation (B.5)). Thus, this behavior is also called volume locking. As locking is present in themodeling of metal plasticity, we adopt a mixed formulation in the momentum balance equation ofthe workpiece [44]. Introducing a pressure/deviatoric decomposition of the Cauchy stress tensor, thestandard expression of the weak form of the equilibrium equation becomes

Gu;dyn D hdev.� /C p1;rs�i � h�; bi � ht;�i�� C h�; � Pvi D 0G�;dyn D �hr�;qi � h�;Dint i � h�;q � ni�q

C h�; Pei D 0

G� DD ln.J / � 3 ˛ .1�ln.J //

J.� � �0/; q

EC hp; qi D 0

9>=>;8� 2 V;8� 2 T;8q 2 Q

(5)

where > 0 and ˛ can be interpreted as the bulk modulus and the thermal expansion coefficient,respectively. J is the determinant of the deformation gradient (Equation (B.4)). The pressure isdenoted by p, t is the surface traction, e is the internal energy per unit of spatial volume, and n isthe surface normal. The initial temperature is set as �0. Being � 2 V, � 2 T and q 2 Q valuedfunctions in the space of virtual displacements V, virtual temperatures T, and virtual pressures Q,respectively. The L2 inner product is represented as h�; �i, and with a slight abuse in notation h�; �i��and h�; �i�q is denoting the L2 inner product on the boundaries � and q, respectively.

The weak form of the IBVP and the details of the mixed displacement-pressure formulation aredeveloped in Appendix A.4.

3.2. Numerical treatment of the incompressibility constraint

The most common finite elements used in the numerical simulations involving plasticity at finitestrains are in 2D: the plane strain isoparametric quadrilateral element used in [3, 43, 45], the six-noded isoparametric triangle element used in [43] and [1], and the enhanced four-node quadrilateralwith one-point quadrature used in [2]. When finite elements have linear order of interpolation, theperformance for the treatment of the incompressibility is very poor. Usually the problems appearwith the use of linear triangles and linear tetrahedra under incompressible and nearly incompressibleconditions. This is exactly the case we encounter when the PFEM is employed. In order to surpassthis inconvenience, different type of finite elements have been developed. They can be classified infour groups mainly

(1) Mixed enhanced element. The enhanced strain technique, essentially consists in augmentingthe space of discrete strains with local functions [46].

(2) Composite pressure fields. The most representative finite elements with composite pressurefields are F-bar [47, 48] and composite triangles [49, 50].

(3) Average nodal pressure (ANP). The ANP was presented in [51] and [52] in the framework ofexplicit dynamics and by [53] in the framework of implicit dynamics. Another references forthe ANP are [54, 55], and [54]. There are alternative formulations based in node average ofthe variables. The node-based uniform strain elements [56, 57], the average nodal deforma-tion gradient [58] and the mixed discretization technique [59], improved in [60] creating theirformulation called nodal mixed discretization.

(4) Pressure stabilized finite elements.

In this work, we have chosen the pressure stabilized option for the treatment of the incompress-ibility constraint. The details set out subsequently.

3.2.1. Pressure stabilization. This element technology is applied to linear finite elements formu-lated in a mixed displacement-pressure or velocity-pressure field. When the order of interpolation



of the pressure field of the finite element is the same as the order of interpolation of the displace-ment field, the solution presents nonphysical oscillations. Mathematically, it means that equal-orderinterpolation for displacement and pressure does not satisfy Babuska–Brezzy condition. In order toremove these undesirable oscillations, a literature overview shows different strategies. Among them:the characteristic-based split (CBS)[61], the finite calculus (FIC) [62], the orthogonal subgrid scales(OSS) [63–65], and the polynomial pressure projection (PPP) [66, 67].

After looking at the advantages and disadvantages of the cited pressure stabilization techniques,we have chosen the use of the PPP for the development of our finite element. The theoreticalexplanation for this technique is summarized next.

3.2.2. Polynomial pressure projection. Mixed formulations have to fulfill additional mathematicalconditions, which guarantee its stability. Linear displacement-pressure triangles and tetrahedra finiteelements do not satisfy Babuska–Brezzy condition; consequently, a stabilization of the pressure fieldis needed.

In our approach the stabilization method utilized is the so-called PPP presented and applied tostabilize Stokes equations in [66, 67]. The PPP is based on two ingredients:

(1) The use a mixed equal-order interpolation of the pressure and displacement/velocity fields(2) The use of a L2 pressure projection

The method is obtained by modification of the mixed variational equation by using local L2PPPs. The application of the pressure projections in conjunction with minimization of the pressure-displacement mismatch eliminates the inconsistency of equal-order approximations and leads to astable variational formulation. Unlike other stabilization methods, the PPP does not require speci-fication of a mesh-dependent stabilization parameter or calculation of higher-order derivatives. ThePPP uses a projection on a discontinuous space and as a consequence can be implemented in anelementary level surpassing the need of mesh-dependent and problem-dependent parameters. Theimplementation of the PPP reduces to a simple modification of the weak continuity equation (incom-pressibility constraint). In this work, we extend the PPP to solid mechanics problems involvinglarge strains.

Given a function p 2 L2, the L2 projection operator Mp W L2 ! Q0 is defined by

G�;p DZVt

Mq .p � Mp/dVt D 0 8Mq 2 Q0 (6)

where Mp is the best approximation of the pressure p in the space of polynomials of order O.Q0/.Vt is the volume of the domain at the current configuration.

To stabilize the mixed form given by equations (5), we add the projection operator to the thirdequation in (5)

Gstab;p D

ZVt

.q � Mq/˛s

�.p � Mp/dVt D 0 (7)

where ˛s is the stabilization parameter and � is the material shear modulus.The use of the projection operator to the pressure test and trial functions removes the approxima-

tion inconsistency present for equal-order displacement and pressure spaces.The role of the form Gstab;p is to further penalize pressure variation away from the range of

the divergence operator. Taking into account the mixed formulation and the polynomial pressurestabilization terms to deal with the incompressibility phenomena, the momentum and energy balanceequations take the form:

Gu;dyn D 0G�;dyn D 0G� CGstab;p D 0

9=;8� 2 V;8� 2 T;8q 2 Q (8)

where

G� D G�;p C hp; qi (9)



G�;p D� ln.J / � 3 ˛

.1 � ln.J //

J.� � �0/; q

�8q 2 Q (10)

and

Gstab;p D

�˛s

�.p � Mp/; q � Mq

�8 Mp; Mq 2 Q0 (11)

The set of governing equations for the displacement, pressure, and temperature variablesis completed by adding the proper initial conditions and constraint equations related to theproblem variables.

4. FINITE ELEMENT NUMERICAL INTEGRATION OF THE IBVP

Consider a spatial discretization � DSneeD1�

.e/ into a disjoint collection of non-overlappingelements �.e/ with characteristic size h.e/, being ne the total number of elements. The FEM fornumerical solution of problem (8) consists on replacing the functional sets ¹u; V º, ¹�; T º, and¹p;P º with discrete subsets ¹uh; V hº, ¹�h; T hº, and ¹ph; P hº generated by a finite element dis-cretization h of the domain�. Let !.X; t / be a generic scalar or vector field defined over the domain�e of the element. The finite element interpolation of the field ! within element .e/ is obtained as

!h .e/.X; t / DnnXjD1

!jNj.e/.X; t / (12)

where nn is the number of nodes of an element, !j is the value of ! at node j , and Nj .e/ is theshape function such that its values are 1 at the node j and 0 at any other node of the element.

The interpolated function now defined over the approximated domain is given by

!h.X; t / DnpXjD1

!jNj .X; t / (13)

where Nj is a piecewise polynomial function (the global shape function) associated with the globalnode j and np is the total number of nodal points in the finite element mesh.

With the introduction of the earlier interpolation procedure, we generate the finite dimensionalsets. The discrete counterpart of (8) is given then by the equations:Z

Vt

BuT � dVt �ZVt

NT b dVt �ZVt

NT � Pv dVt �Z��

Nt d� D 0 (14)

ZVt

c NNT P� dVt �ZVt

B�T q dVt �

ZVt

NTDint dVt CZ�q

NT .q � n/ dq D 0 (15)

ZVt

1

NNTp dVt �

ZVt

NT�

ln.J / � 3 ˛.1 � ln.J //

J.� � �0/

�dVt C F.e/

p;stabD 0 (16)

where c is the specific heat of the material, Bu and B� are the strain-displacement matrix and theglobal gradient-temperature matrix, respectively. That matrices contain the derivatives of the shapefunctions used in the interpolation of the problem variables.

F.e/p;stab expresses the discrete counterpart of the projection operator, accounting that the pressureprojection is constant and discontinuous among finite elements.

F.e/p;stab

D

ZVt.e/

˛s

�p.e/

�N.e/NT .e/ � QN.e/ QNT .e/

�dVt

.e/ (17)

If N.e/ contains the set of polynomials of order k, QN.e/ contains the set of polynomials oforder k � 1.



The equilibrium incompressible equations (8) can be expressed alternatively as follows. Startingwith the balance of the linear momentum (14):

Fu;dyn. Ru/ � Fu;int .u;p/C Fu;ext D 0 (18)

where

Fu;int .u;p/ DZVt

BuT � dVt (19)

Fu;ext D

ZVt

NT b dVt �Z��

Nt d� (20)

Fu;dyn. Ru/ DZVt

NT � Ru dVt (21)

The equations for the energy balance (15) are written as

F�;dyn. R�/ � F�;int .�/C F�;ext D 0 (22)

where

F�;int .�/ DZVt

B�T q dVt �

ZVt

NTDint dVt (23)

F�;int DZ�q

NT .q � n/ dq (24)

F�;dyn. R�/ DZVt

c NNT P� dVt (25)

and the incompressibility balance equations (16) are written as

Fp;pres.p/ � Fp;vol.u/C Fp;stab.p/ D 0 (26)

where

Fp;pres.p/ DZVt

1

NNTp dVt (27)

Fp;vol.u/ DZVt

NT�

ln.J / � 3 ˛.1 � ln.J //

J.� � �0/

�dVt (28)

Fp;stab.p/ DZVt.e/

˛s

�p.e/

�N.e/NT .e/ � QN.e/ QNT .e/

�dVt

.e/ (29)



In finite element computations, the earlier force vectors are obtained as the assemblies of elementvectors. Given a nodal point, each component of the global force associated with a particular globalnode is obtained as the sum of the corresponding contributions from the element force vectorsof all elements that share the node. In this work, the element force vectors are evaluated usingGaussian quadratures.

5. THERMO-ELASTOPLASTICITY MODEL AT FINITE STRAINS

In this section, the formulation of the constitutive thermo-plasticity model at finite strains will besummarized. Some models are proposed in the literature to deal with thermo-plasticity accountingfor the characteristics of the material behavior. If the plastic behavior experiences isotropic harden-ing, the approach proposed by Simo et al. [18–20] is usually followed. When the strain and strainrate hardening and the thermal softening are considered, other models can be used: (i) Voce [68] andSimo et al. [69]; (ii) Johnson and Cook [70]; and (iii) Bäker [71]. In Box 2, we present the mainequations of the thermo-mechanical J2 flow model for rate-independent plasticity that will be usedin this work. Details of the theory of thermo-plasticity and the definition of the variables that appearin Box 2 are explained in Appendix B.

The purpose of presenting here the main equations for the constitutive model is to introduce latera new integration scheme for thermo-hyperelastoplasticity called IMPL-EX scheme [72].

5.1. Time integration of the constitutive law

The problem of integrating numerically the initial-value ODE equations configured by the evolutionequations and the Kuhn–Tucker conditions (see Box 2) is the main objective of this section.

In Box 3, the integration flowchard for the backward Euler method is presented. The implicitbackward Euler method is the most commonly used integration scheme for plasticity. The detailsof the constitutive law integration are given in the Appendix C. The equation g.��nC1/ related tothe obtention of the consistency parameter ��nC1 (Box 2) is effectively solved by a local Newtoniterative procedure. The convergence of the Newton–Raphson is guaranteed if g.��nC1/ is a convexfunction. In this work, we use isotropic hardening functions that make g.��nC1/ convex. Althoughthe convergence of the integration is guaranteed, the fully implicit integration of the constitutivelaw requires some relevant computational effort and can experience some numerical problems ofrobustness when the material failure appears.

To improve the performance in the integration of the constitutive law, we introduce the integra-tion of the constitutive law by means of the IMPL-EX scheme. Next, we develop the IMPL-EXintegration for the thermo-hyperelastoplastic constitutive model used in this work.

5.2. IMPL-EX integration scheme

The IMPL-EX (IMPLicit-EXplicit) adopted herein is the one pioneered by Oliver et al. [72], orig-inally conceived for addressing the problem of robustness and stability arising in the numericalsimulation of material failure. The essence of the method is to solve explicitly for some variables,in the sense that the values at the beginning of the increment are presumed known, and implicitlyfor other variables, with the primary motivation to enhance the spectral properties of the algorithmictangent moduli. However, our primary motivation of using IMPL-EX is to reduce the equation solv-ing effort associated to the solution of the fully implicit scheme. The explicit integration of somevariables in the coupled thermo-mechanical J2 flow theory and therefore, the use of extrapolatedvalues in the balance of momentum and energy, allow us to solve a coupled thermo-mechanicalproblem as a sequence of three uncoupled problems. First, an elastic problem with shear moduluschanging from element to element; second, a thermal problem with a temperature-dependent plasticheat source; and finally, a relaxation process affecting the stress and the internal variables at the inte-gration points. It is important, to remark, that the mechanical and thermal problems are solved usingthe IMPL-EX integration scheme for the J2 plasticity model, while in the relaxation steps, stressesand internal variables, are calculated using the implicit Back–Euler time integration presented in



COUPLED THERMO-MECHANICAL J2 FLOW THEORYRATE INDEPENDENT PLASTICITY

1. Free energy function.

O D OT .�/C OM.�; J e/C OU .J e/C OW . Nbe/C OK. Ne p; �/

2. kirchhoff stress.

� D J e p 1C s

p WD

��3 ˛

.1 � ln.J e//

J e.� � �0/C ln.J e/

s WD � dev. Nbe/

and the entropy

� D �p � �e C �t

�e WD �@� OT .�/

�t WD �@� OM.�; J e/ � @� OK. Nep; �/

3. Von Mises yield criterion.

ˆ. ; Ne p; �/ D kdev. /k �

r2

3

�y C ˇ

�6 0

4. Evolution equations � > 0; ˆ 6 0; �ˆ D 0

Lvbe D �2 � J�231

3tr. Nbe/n

PNepD �� @ˇˆ. ; Ne

p; �/

P�p D � @�ˆ. ; Nep; �/

The definition of the variables that appear in this box are explained in appendix B.

Box 2: Coupled thermo-mechanical J2 flow theory. Rate independent plasticity.

Box 3 and explained in detail in Appendix C.1. The arguments in support of the IMPL-EX integra-tion scheme for the numerical simulation of metal thermo-mechanical processes were already putforward earlier. Here, we simply choose the variable to be treated explicitly and derive the stressupdate algorithm arising from this choice.

By definition, the equivalent plastic strain is a monotonically increasing function of time, PNe p > 0.For this reason, it is a logical candidate to be treated explicitly, because its evolution can be predictedmore accurately than other variables exhibiting non-monotonic behavior. The following analysispursues to develop an expression for explicitly updating the equivalent plastic strain at tnC1 usingvalues obtained in previous time steps by an implicit backward Euler integration procedure.

Let us consider, the Taylor expansion of the equivalent plastic strain at tn�1 around tn:

Nepn�1 D Ne

pn �

@ Ne p

@t

ˇˇtn

.tn � tn�1/CO.�2tn/ (30)

Next, the Taylor expansion is carried out at tnC1 around tn, yielding

NepnC1 D Ne

pn C

@ Ne p

@t

ˇˇtn

.tnC1 � tn/CO.�2tnC1/ (31)



BACKWARD-EULER INTEGRATION FLOWCHART

1. Thermoelastic trial state:Initial data: Nben; Ne

pn ; �

pn

Current values of Fn;nC1; �nC1, where NFn;nC1 D J�13Fn;nC1

Let f trialnC1 D

��strialnC1

�� q23

�y;nC1 C ˇnC1. Ne

pn /�

IF f trialnC1 6 0 : Set . NbenC1; Ne

pnC1; �

pnC1/ D .

Nbe;trialn ; Nepn ; �

pn / and EXIT

ELSE:

2. Consistency parameter:Set N� D �

3tr. Nbe;trialnC1 /

Compute ��nC1 by solving:

g.��nC1/ D ftrialnC1 � 2��nC1 �

1

3tr. Nbe;trial

nC1 /

C

r2

3

�y;n C ˇn. Ne

pn /��

r2

3


pnC1/

�D 0

Return mapping:

Set nnC1 DstrialnC1

kstrialnC1k

and update

snC1 D strialnC1 � 2��nC1 �

1

3tr. Nbe;trial

nC1 /nnC1

NepnC1 D Ne

pn � �nC1�t

r2

3

�pnC1 D �

pn �

r2

3��nC1@�


pnC1/

�3. Update the intermediate configuration by the closed form formula:

NbenC1 D Nbe;trialnC1 � 2��nC1

1

3tr. Nbe;trialnC1 /nnC1

END

The definition of the variables that appear in this box are explained in appendices B and C.

Box 3: Implicit Backward-Euler integration flowchart for thermo-elastoplastic models.

The standard explicit difference scheme is obtained truncating the remainder terms O.�2tnC1/.The earlier explicit difference equation presents an inconvenience that ensures that the yield con-

dition is not enforced at tnC1, and as a result, it is possible for the solution, over many time steps,to drift away from the yield surface. In order to avoid that this drift from the yield surface growsunboundedly, Oliver et al. [72] propose to approximate the derivative in (31) using the derivativeappearing in (30).

Hence, truncating the terms O.�2tnC1/ in Equation (30), one obtains



Ne pn D Nepn�1 C

@ Ne p

@t

ˇˇtn

.�tn/ (32)

The earlier equation is a backward Euler integration of the equivalent plastic strain, in the sensethat the equivalent plastic strain at tn, Ne pn , is obtained by an expression that uses a derivative eval-uated at tn. As a result, Ne pn and Ne pn�1 are obtained at times tn and tn�1 using the implicit schemepresented in the previous section. From (32), we can deduce that

@ Ne p

@t

ˇˇtn

DNepn � Ne

pnC1

�tn(33)

Finally, inserting the expression (33) into (31), and truncating the remainder terms, yields

QNep

nC1 D Nepn C . Ne

pn � Ne

pn�1/

�tnC1

�tn(34)

Expression (34) constitutes an explicit extrapolation of the equivalent plastic strain at tnC1 interms of the implicit values computed at tn and tn�1. Note that the IMPL-EX algorithm is a multistepmethod, because two points are used to advance the solution in time to point tnC1.

The algorithmic plastic multiplier resulting from this extrapolation reads:

� Q�nC1 D

r3

2

�QNep

nC1 � Nepn

�

D

r3

2

Ne pn � Ne

pn�1

� �tnC1�tn

D

r3

2��n

�tnC1

�tn

(35)

Expression (35) reveals that the elastic or plastic nature of the response predicted by the IMPL-EX integration scheme at tnC1 is dictated by the response computed implicitly at tn. This may giverise to overshoots and oscillations in the transitions from elastic to inelastic and vice versa. Now,steps 3 and 4 in Box 3 can be pursued in terms of extrapolated plastic multiplier yielding the IMPL-EX integrated values of the remaining variables QSnC1; QNe

p

nC1 and Q� pnC1. Those IMPL-EX resultswill be replaced later in Box 6 to fulfill the momentum and energy equations. The IMPL-EX explicitstage for both cases is summarized in Box 4.

5.3. Algorithmic constitutive tensor and algorithmic dissipation

The ultimate goal in the numerical simulation of thermo-mechanical processes is to solve an IBVPfor the displacement and temperature fields. The numerical solution of this problem relies on thespatial discretization, via a Galerkin finite element, of the momentum and energy equations and atime discretization of the displacement, velocity, and temperature fields. In case of an implicit dis-cretization, the response is obtained by solving a sequence of linearized problems. The theoriesunderlying the spatial and temporal discretization are presented in the Sections 4 and 6. The lin-earization of the weak form of the momentum and energy equation is not addressed in this work.We refer the reader to [73, 74] for further details.

In the Appendix C.2, the expressions for the algorithmic tangent moduli for the implicit integra-tion scheme as well as the IMPL-EX scheme are provided. The algorithmic constitutive tensor is akey aspect in the linearization of the weak form of the momentum equation. In addition, in AppendixC.3, we provide a linearization of the plastic power relevant in the linearization of the weak form ofthe energy equation.



IMPL-EX INTEGRATION FLOWCHART

1. Explicit extrapolation stage:Initial data: Nben; Ne

pn ; �

pn

Current values of Fn;nC1; �nC1

� Q�nC1 D

r3

2��n

�tnC1

�tn

QNep

nC1 D QNep

n C

r2

3� Q�nC1

2. Let NFn;nC1 D J�13Fn;nC1 and set:

Nbe;trialnC1 D NFnC1 Nben NFTnC1

strialnC1 D � dev. Nbe;trialnC1 /

3. Compute stresses and plastic entropy:Set N� D �

3tr. Nbe;trialnC1 /

Set nnC1 DstrialnC1

kstrialnC1

kand update:

QsnC1 D strialnC1 � 2�Q�nC1 N�nnC1

Q�pnC1 D �

pn �

r2

3� Q�nC1@�

�Q�y;nC1 C ˇnC1. QNe

p

nC1/�

4. Compute plastic power:

QDnC1mech

D �

r2

3

�Q�y C Q

�nC1

� Q�nC1

�t

The definition of the variables that appear in this box are explained in appendices B and C.

Box 4: IMPL-EX explicit integration flowchart for thermo-elastoplastic models.

6. TIME INTEGRATION OF THE IBVP

The FEM allows different time discretization schemes. The most common are the implicit andexplicit time integration schemes. Each of them has its advantages or disadvantages (Appendix D).

The implicit time integration scheme using isothermal split will be used in this work. Based onthe global operator split for finite deformation plasticity presented in Equations (3) and (4), a formalsplit of the problem into a mechanical phase with the temperature held constant, followed by athermal phase at a fixed configuration is presented in the following lines.

The implicit coupled algorithm for a simultaneous solution of the thermo-mechanical equationsis presented in Appendix D.1.

6.1. Isothermal split

The following lines present a summary of the isothermal split, developed in [20]. Let tn ! tnC1 bethe initial and final time step. Let �t D tnC1 � tn be the time increment.

The algorithm in Box (5) is based on the application of an implicit backward Euler differencescheme to the momentum equation, for fixed initial temperature (temperature at previous time step)



COUPLED SYSTEM OF EQUATIONSISOTHERMAL SPLIT

1. Momentum equation for fixed initial temperature (3)

Fu;dyn. Ru�nC1/ DFu;int

� nC1.u�nC1;p

�nC1; �

�n I�nC1.u

�nC1; �

�n //�� Fu;ext .u�nC1/

2. Incompressibility �Mp CMstab

�p�nC1 D Fp;vol.J

�nC1.u

�nC1; �

�n //

where Mp and Mstab are the mass-type matrices of the linearized pressure and stabilizationforces respectively

3. Update nodal variables

vnC1 D vn C PvnC1�t

u�nC1 D u�n C vnC1�t

p�nC1 D p�n C�p�nC1———————————————————–

4. Energy equation at updated fixed configuration (4)

F�;dyn. P��nC1/ DF�;int .q.��nC1/ID�int .u�nC1; ��nC1/I�nC1.u�nC1; ��nC1// � F�;ext


��nC1 D ��n CP��nC1�t

The definition of the variables that appear in this boxare explained in sections 4 and 6.

Box 5: Implicit isothermal split scheme.

and the application of an implicit backward Euler difference scheme to the energy equation at afixed configuration (configuration obtained as a solution of the mechanical problem).

The solution of the balance of momentum equation for fixed initial temperature gives an updateof the primary variables u�nC1; p�nC1 and a first update of the internal variables (left Cauchy–Greentensor, internal energy, and entropy) of the form

ben; Nepn; �

pn ! .Box 3/ ! Qben; QNenC1; Q�

pn (36)

Along with an incremental value of the consistency parameter satisfying the Kuhn–Tuckerconditions and denoted by � Q�nC1

The solution of the balance of energy with initial conditions u�nC1;p�nC1; �

�n and initial internal

variables ben; Nepn ; �

pn gives an update of the primary variable ��nC1 and a second update of the internal

plastic variables (at fixed configuration) of the form

ben; Nepn; �

pn ! .Box 3/ ! QQbenC1;

QQNepnC1;

QQ�pnC1 (37)

Along with an incremental value of the consistency parameter satisfying the Kuhn–Tucker con-

ditions and denoted by � QQ�nC1. In general, � Q�nC1 ¤ �QQ�nC1 as a consequence Qben; QNe

pn ; Q�pn ¤

QQbenC1;QQNepnC1;

QQ�pnC1



In summary, the isothermal split solves the mechanical problem with a predicted value of temper-ature equal to the temperature of the last converged time step and, then, solves the thermal problemusing the configuration obtained as a solution of the mechanical problem. A full Newton–Raphsonscheme is used for the solution of the nonlinear system; the necessary linearization of the constitu-tive law has been presented in Appendix C.2. The details of the linearization of the weak form ofthe momentum and energy equation can be seen in [73, 74].

The well-known restriction to conditional stability is the crucial limitation of the isothermalapproach, which often becomes critical for strongly coupled problems. However, this restriction isnot significant for metal plasticity [20]. Armero and Simo [75] provide the sufficient conditions forstability of the isothermal split as

�t

h6 K2

p�c

˛,

�t

h6 K2

p2�c

˛

r�

.�C 2�/(38)

where �;� > 0 are the Lamé constants, ˛ the thermal expansion coefficient, �; c the density and thespecific heat, and h;�t;K are the minimum element size of the mesh, the maximum allowed timestep, and a given constant. In the cases where the mechanical inertia can be considered negligible,Armero and Simo [75] provide the sufficient conditions for stability of the isothermal split as

�t

h2> ˛

2 � 2E c

2E k$

�t

h2> c

2 k

�˛2

E c� 2

�(39)

Previous restrictions show that algorithms based on the isothermal split are not suitable forstrongly coupled problems, because the stability restriction phrased in terms of the Courant numberbecomes increasingly restrictive the higher the coupling (increase in the thermal expansion coeffi-cient). The numerical simulation of metal cutting and metal forming processes can be considered aweakly coupled problem (the thermal expansion coefficient of metals is usually small), as a result,the isothermal split will perform well in most of the numerical simulations of cutting and formingprocesses for metal presented in this work. The stability restriction of the isothermal split is cir-cumvented using an isentropic split, in which one must solve first a mechanical problem at constantentropy (estimates the temperature change in the mechanical problem), followed by a thermal heatconduction problem at constant (fixed) configuration [75].

6.2. Isothermal IMPL-EX split

The isothermal scheme presented in [20] decouples the thermo-mechanical problem in two moresimple problems, but yet, the mechanical problem is coupled with the evolution equations of theinternal variables and the thermal problem is also coupled with the evolution equations of the inter-nal variables, both of them are coupled through the plastic multiplier. The previous reason suggestsdecoupling the problem in the following three simple problems: (i) an elastic problem with shearmodulus changing from element to element; (ii) a thermal problem with a temperature-dependentplastic heat source; and (iii) a relaxation process affecting the stress and the internal variables at theintegration points.

In this work, we present a new staggered algorithm, which is based on the isothermal split pre-sented in [20] and the IMPL-EX integration scheme of the constitutive equations presented in [72].Using the ingredients presented earlier, the solution of the coupled system of ODE (16), (14), and(15) could be decoupled in the three simple problems mentioned previously. In addition, the elasticand the thermal problems update the internal variables according to a predicted plastic multiplier(explicit), while the constitutive equations keep the displacements, velocities, and temperaturesunchanged (implicit).

For simplicity, a partition of the time domain I WD Œ0; T � intoN time steps of the same length�tis considered. Let us focus on the time step tn ! tnC1, where�t D tnC1�tn. An implicit backwardEuler difference scheme is applied to the momentum equation and to the energy equation. In thefirst step, the extrapolation of the plastic multiplier ��nC1 D ��n is carried out. Consequently,the stresses � nC1 are computed via IMPL-EX integration scheme of the constitutive equation. Afterthat, the balance of momentum (A.1) is solved implicitly providing the nodal displacement and



COUPLED SYSTEM OF EQUATIONSIMPL-EX SPLIT

1. Momentum equation for fixed initial temperature (3)(elastic problem with shear modulus changing from element to element)

Fu;dyn. Ru��nC1/ DFu;int

� nC1.u��nC1;p

��nC1; �

��n I�nC1.u

��nC1; �

��n ///

�� Fu;ext .u��nC1/

2. Incompressibility�Mp CMstab

�p��nC1 D Fp;vol.J

��nC1.u

��nC1; �

��n //

where Mp and Mstab are the mass-type matrices of the linearized pressureand stabilization forces respectively



u��nC1 D u��n C vnC1�t

p��nC1 D p��n C�p��nC1———————————————————–

4. Energy equation at updated fixed configuration (4)(thermal problem with temperature dependent external heat source)

F�;dyn. P��/ DF�;int .q.��nC1/ID��int .u��nC1; ��nC1/I�nC1.u��nC1; ��nC1// � F�;ext


��nC1 D ��n C

P��nC1�t

6. Constitutive equation and update internal variables (Plastic algorithm)

QQQbenC1;QQQNenC1;

QQQ�pnC1 D f

.u��nC1; �

��n /; .benC1; NenC1; �nC1/

�The definition of the variables that appear in this box

are explained in sections 4 and 6.

Box 6: Isothermal IMPL-EX split.

pressure for fixed initial temperature. The balance of momentum equations, providing a fixed initialtemperature and an extrapolated value of the internal variables, constitutes a nonlinear system tosolve. In this case, the nonlinearity of the system appears because of the geometrical part of thelinearized equations. Therefore, they have to be iteratively solved until convergence is achieved.

The solution of the balance of momentum equation for a fixed initial temperature gives an updateof the primary variables u��nC1;p

��nC1 and a first update of the internal variables of the form

ben; Nepn ; �

pn ! .Box 4/ ! Qben; QNe

pnC1; Q�

pn (40)

Then, in the second step, the solution of the balance of energy with initial conditionsu��nC1;p

��nC1; �

��n , initial internal variables ben; Ne

pn ; �

pn , and the extrapolation of the plastic multiplier

��nC1 D ��n gives an update of the primary variable ��nC1 and a second update of the internalplastic variables (at fixed configuration) of the form

ben; Nepn ; �

pn ! .Box 4/ ! QQbenC1;

QQNepnC1;

QQ�pnC1 (41)



Finally, in the third step, the values of u��nC1;p��nC1; �

��n remain fixed, and an implicit backward

Euler integration of the constitutive model (B.47) is carried out using as initial internal variablesben; Ne

pn ; �

pn . Given, as a consequence, a finally update of the internal variables of the form

ben; Nepn ; �

pn ! .Box 3/ !

QQQbenC1;QQQNepnC1;

QQQ�pnC1 (42)

The set of internal variables obtained at the end of this time step will be the set of internal variablesused as the starting point in the next step of the fractional step method proposed in this work. Assummary about the isothermal IMPL-EX split is shown in Box 6.

It is interesting to note that the boundary values of the momentum equation are included in theelastic equations with shear modulus changing from element to element and the boundary values ofthe balance of energy are imposed on the thermal problem with temperature-dependent plastic heatsource. In addition, the plastic algorithm consists on a collection of systems of ordinary differen-tial equations, each one of which belongs to a different integration point. A full Newton–Raphsonscheme is used for the solution of the nonlinear system.

7. EXAMPLES

In this section, we present some 2D examples using the proposed formulation. First of all,two benchmarks, the Cook’s Membrane, and the Taylor impact test. With the solutions reportedin the literature, we validate qualitatively and quantitatively the pressure stabilization in quasi-incompressible elastic problems and in mechanical problems involving plasticity. Furthermore, aplane strain traction test is presented to validate the locking-free element type proposed for thermo-mechanical problems. In the traction test example, we also report the comparison of different timeintegration schemes, showing the advantages and disadvantages of the IMPL-EX solution scheme.Finally, the proposed formulation is used in the numerical simulation of a continuous steel cut-ting test in order to show the possibilities of the PFEM in the modeling of metal cutting and metalforming processes.

7.1. Plane strain Cook’s membrane problem

The Cook membrane problem is a bending dominated example that has been used by many authorsas a reference test to check their element formulation. Here, it will be used to validate the proposedformulation in incompressible elasticity and plasticity. The results of our formulation will be com-pared against Q1P0 finite element and a mixed finite element using OSS as a stabilization strategy.The problem consists in a tapered panel, clamped on one side, and subjected to a shearing load at thefree end, (Figure 3. In order to test the convergence behavior of different formulations, the problem

Figure 3. Cook’s membrane benchmark. Problem dimensions and the initial structured triangular mesh of16�16 elements.



has been discretized into 16 � 16, 24 � 24 and 40 � 40 elements per side. The following materialsproperties are assumed: Young’s modulus E D 70 UF

UL2, Poisson’s ratio � D 0:4999, and applied

force F D 1UF . Where UF means units of force and UL means units of length.Figure 4 shows the behavior of both quadrilateral and triangular finite elements in case of

nearly incompressible elasticity. The figure shows the poor performance of the Q1 and T1 stan-dard elements within the context of nearly incompressible elasticity, because of an extreme locking.Furthermore, the figure shows that the proposed formulation converges similarly to OSS but a lowcomputational cost. It is important to remark that in PPP strategy, the stabilization parameter ismesh size independent and that the stabilization terms added to the mixed formulation are elemen-tary depend. It shows that our proposal allows getting similar results to the OSS strategy but a lowcomputational cost. The stabilization parameter used in PPP and OSS was ˛ D D 1.

Figure 4. Plane strain Cook’s problem: convergence of different formulations for incompressible elasticity.T1 standard displacement for triangular elements, Q1 standard displacement for quadrilaterals elements,Q1P0 mixed mean dilatation/pressure approach for quadrilateral elements, T1P1 orthogonal subgrid scales(OSS) mixed formulation for linear triangles using OSS as a stabilization strategy, T1P1 polynomial pressure

projection (PPP) mixed formulation for linear triangles using PPP.

Figure 5. Plane strain Cook’s problem: convergence of different formulations for J2-plasticity. T1P1 orthog-onal subgrid scales (OSS) mixed formulation for linear triangles using OSS as a stabilization strategy, T1P1

polynomial pressure projection (PPP) mixed formulation for linear triangles using PPP.



Next, examples involves Cook’s membrane but J2-plasticity and the following assumed mate-rials properties: Young’s modulus E D 70 UF

UL2, Poisson’s ratio � D 0:4999, yield stress �y D

0:243 UF

UL2, hardening modulusH D 0:135 UF

UL2and kinematic hardening modulusK D 0:015 UF

UL2,

and an applied force of F D 1:8UF in 50 increments.Figure 5 shows a comparison of the top corner displacement for the mixed finite element using

OSS and PPP as stabilization strategies. It also shows that the convergence behavior of two formu-lations is really similar. As we say in case of elastic behavior, PPP is simple to implement and doesnot need an extra calculation like the projected pressure gradient in OSS.

Figure 6 presents pressure contour field at the end of the deformation process. A smooth contourfield can be identified in both mixed formulations. At the same time, the predicted results are verysimilar quantitatively.

Figure 6. Pressure field for mixed formulation using orthogonal subgrid scale and polynomial pressureprojection as stabilization strategies and J2-plasticity.



7.2. Taylor impact test

The problem consists of the impact of a cylindrical bar with initial velocity of 227m=s into a rigidwall. The bar has an initial length of 32:4mm and an initial radius of 3:2mm (Figure 7). Materialproperties of the bar are typical of copper: density � D 8930 kg=m3, Young’s modulus E D 1:17 �105MPa, Poisson’s ratio � D 0:35, initial yield stress �y D 400MPa, and hardening modulusH D 100MPa. A period of 80�s has been analyzed.

The problem is treated as a 2D axisymmetric model of the cylindrical bar shown in Figure 7. Wewill compare qualitatively and quantitatively the results obtained using the proposed formulationwith the results of the formulations based in the CBS [76], the ANP [51], and the De Micheliformulation [77]. In this problem, the effect of the temperature is not considered. The bar constitutivebehavior experiences plasticity but not thermo-plasticity.

First, we consider a finite element solution of the problem with the displacement-pressure stabi-lized element proposed in this work. The mesh is considered the same in the whole analysis, andthe PFEM features are not used. The final geometry of the bar is in good agreement with the resultsobtained in the literature, and locking is not present in the solution. However, some parts of the meshgets very deformed, the elements that received first the impact experience large plastic deforma-tions. That causes a pressure distribution somehow conditioned by the mesh shape. The final radius

Figure 7. Taylor impact test. Problem dimensions and the initial structured triangular mesh of 6�50elements.

Table I. Final radius of the rod after the Taylor impact test obtained with De Micheli formulation, CBSformulation, ANP formulation, and the proposed formulation of this work.

Formulation De Micheli [77] CBS [76] APN [51] PPP (FEM) PPP (PFEM)

Final radius 7.07 mm 7.07 mm 6.99 mm 7.24 mm 7.02 mm

CBS, characteristic-based split; ANP, average nodal pressure; PPP, polynomial pressure projection; FEM, finiteelement method; PFEM, particle finite element method.



Figure 8. Final mesh, equivalent plastic strain distribution and pressure field 80�s after the impact for theproposed formulation without geometry update.

Figure 9. Final mesh, equivalent plastic strain distribution and pressure field 80�s after the impact for theparticle finite element method formulation.



Figure 10. Plain strain nearly adiabatic shear banding traction test benchmark. Problem dimensions andinitial mesh.

Table II. Material properties.

Young modulus E 206:9 � 103 MPaDissipation factor � 0.9Thermal capacity c 0:46 � 109 mm2=s2K

Density � 7:8 � 10�9 Ns2=mm4

Expansion coefficient ˛ 1 � 10�5 K�1

in the base of the bar obtained using the proposed formulation (PPP with FEM) is of 7:24mm.Table I shows the comparison of the final radius obtained with present formulation with the resultspresented in the literature.

Figure 8 shows the final mesh and the numerical results of the pressure and effective plastic straindistribution using the formulation proposed in this work.

In order to improve the solution, the PFEM simulation with the intrinsic geometry update isused. In this case, the finite elements of the mesh have always a good shape. It avoids the artificialnumerical peaks that appear in the solution of the bad shaped linear triangles. The final geometry ofthe bar is in good agreement with the results obtained in the literature, without locking in the solutionand with a final radius in the base of the bar of 7:02mm (PPP with PFEM). Figure 9 shows thefinal mesh and the numerical results of the pressure and effective plastic strain distribution using thePFEM formulation proposed in this work. The values for the equivalent plastic strain and pressure



fields obtained with the PFEM coincide well with those given by the finite calculus and by the CBSformulation.

7.3. Thermo-mechanical traction test

We consider a rectangular specimen in plane strain submitted to uniform traction forces. The speci-men considered in the simulation has a width of 12:866mm and a length of 53:334mm (Figure 10.Figure 10 shows also the mesh in the initial configuration. The bar is assumed insulated alongits lateral face, while the temperature is held constant and equal to 293:15K on the upper andlower faces.

The total value of imposed displacement is increased to 5mm applied in 100 equal time steps,with a rate of increase of 1mm=s. The chosen values of thermo-mechanical properties of the spec-imen are given in Table A.1 and Table II, they correspond to steel. We consider the source term inthe energy equation defined as a fraction of the plastic work; in this example, we use a factor of 0.9.Because of the symmetry of the solution, only one quarter of the specimen is discretized, imposingthe corresponding symmetry boundary conditions. To solve the problem, we use the mixed lineardisplacement-linear pressure finite element presented in this work with the PPP as a stabilizationtechnique.

The simulations are performed under quasi-static conditions with the isothermal implicit split pro-posed by Simo and Miehe [20] presented in Section 6 and the isothermal IMPL-EX split proposedin this work. No specific features of the PFEM are used in this example. The purpose is to evalu-ate the thermo-hyperelastoplastic model and the stabilized element developed within the IMPL-EXintegration scheme. Next, some results are presented.

Figure 11 shows the temperature and von Mises field at the final configuration. Figure 12 showsthe load/displacement curve obtained with the proposed formulation. The same figure shows alsothe results presented by Ibrahimbehovic and Chorfi [78] using a four-node element with incom-patible modes and Beni and Movahhedy [79] using an ALE formulation. The predicted forces aresimilar during the strain hardening part of the force displacement curve, but in the softening branch

Figure 11. Plane strain nearly adiabatic shear banding. Temperature and von Mises stress field at 5 mm.



Figure 12. Plane strain nearly adiabatic shear banding. Load/displacement curve, from different authors.

Figure 13. Plane strain nearly adiabatic shear banding. Load/displacement curve.

of the force displacement curve, the predicted forces are different in the three formulations. Ourformulation predicts the force in the softening branch in a similar way as the results presented byIbrahimbehovic and Chorfi do. It means that the formulation does not lock in softening.

The load displacement curve obtained using the isothermal IMPL-EX split proposed in this workis presented in Figure 13. The total value of imposed displacement is increased to 5mm and appliedin 100-500-2000 equal time steps to analyze the overshoots and oscillations in the transitions fromelastic to inelastic state. The results presented in Figure 13 show that the overshoot decreases byincreasing the number of time steps used. Using 2000 time steps, the nonphysical overshot predictby the isothermal IMPL-EX split is negligible, although the results predicted with 500 time steps canbe considerable satisfactory, taking into account that we identify the overshoot as an nonphysicalprediction that comes from the integration scheme. On practice, the error on the norm of the stresscan be used to predict the suitable time step for the IMPL-EX integration scheme.

The computing time need to solve the thermo-mechanical traction test using the isothermal IMPL-EX split is slightly smaller compared with the computing time needed by the isothermal implicitsplit. Considering that in both cases, we are obtaining the same accuracy, the isothermal IMP-LEXsplit will be used in the numerical modeling of larger problems. In that problems, the IMPL-EX is



a substantially better choice, because it needs less computing time per time step in comparison withthe implicit split and because it introduces robustness in the integration of the constitutive law.

7.4. Challenging thermo-mechanical problems

7.4.1. Steel cutting test. Here, we introduce our first industrial application, it consists in the cuttingof a rectangular block of a common steel. The block (workpiece) has a length 3:7mm and a of width1:8mm. The cutting has an imposed velocity of 3333:3mm=s, a cutting depth of 0:10mm, a rakeangle of 0°, a clearance angle of 5°, and a tool radius of 0:025mm. The rigid tool is composed byto straight lines connected by a circular arch on the tool tip with the characteristics of the cuttingparameters described (Figure 14). The workpiece material behavior is given by a Simo law that takesinto account thermal softening (Table A.1 and Table II).

Conductivity and specific heat do not depend on temperature, we consider them constant. Thefollowing assumptions are made: First, the tool is supposed to be rigid and friction is neglected.Furthermore, the thermal exchange between the workpiece and the tool is also neglected. The inertiaof the workpiece is neglected. A classical penalty method is considered for the contact constraintgenerated by the action of the rigid tool.

An implicit quasi-static step with the isothermal IMP-LEX split is used. Time steps were of1:2 � 10�8 that take 2:5 � 104 steps for a tool to travel 1:0mm. The assumption that the tool is rigidis reasonable, because the deformation of the tool is negligible compared with the deformation ofthe workpiece.

Temperature, pressure, effective plastic strain rate, and von Mises contours are presented inFigures 15 and 16. Depicted pressure distribution shows the tension and compression zones. It iscompleted with the von Mises stress shown, which demonstrates that relatively high stresses arisein the primary shear zone and at the tool–chip interface. The localization of this zone agrees withsimplified models. It is also important to note the presence of residual stresses at and below the pro-duced new surface and in the upper part of the chip, especially near the tool–chip interface where

Figure 14. Steel cutting test. Workpiece, tool dimensions, and initial unstructured triangular mesh.



Figure 15. Continuous chip formation using a rate-independent yield function: von Mises (MPa) andtemperature (K).

unloading due to curling of the chip occurred. The effective strain rate in the primary and the sec-ondary shear zone is of the order of 105, and it has its highest value close to the tool tip. Finally,temperature distribution is also shown in the workpiece. Temperature reaches its peak on the tooltip zone located on the machined surface.

Figure 17 depicts the cutting and thrust forces applied on the tool, which are obtained from thesimulation. Although the predicted chip is continuous, the cutting and thrust forces do not reach asteady state because of the strong dependency of the yield hardening function on the linear hardeningmodulus. Figure 18 depicts the chip formation in different time step sequences and the contour fillof the temperature on the continuous chip.

The contact length between the tool and the workpiece, the deformed chip thickness and the shearangle are 0:16mm, 0:25mm, and 22°, respectively.

7.4.2. Analysis of the competitiveness of the PFEM. In order to validate PFEM as an effectivestrategy to deal with thermo-mechanical problems, we present here a small comparison with othercodes based in the FEM and with experimental results. The comparison is performed with a steel



Figure 16. Continuous chip formation using a rate-independent yield function: strain rate (1/s) andpressure(MPa).

cutting test example similar to the one presented in this article. In this case, the test is a model of theorthogonal cutting process proposed in [80].

The objective of this test is to predict the chip formation in the cutting process of 42CD4 steel at300 m/min, with a tool radius of 0.04 mm, rake angle of 6 and cutting depth 0.2 mm. Data aboutexperimental results have been obtained from data reported in the literature [80].

The validation was carried out comparing numerical results with experimental ones and numericalresults obtained from the commercial software ABAQUS, DEFORM, and ADVANTEDGE. It is impor-tant to mention that there are some differences and similarities between the formulations used inthose softwares. They use distinct time integration schemes, boundary conditions, material models,and contact laws. Main differences between the numerical models are listed next:

(1) PFEM, DEFORM, and ABAQUS use a Johnson’s Cook yield function to model the materialbehavior. However, in ADVANTEDGE, the material behavior is governed by Marusich law.

(2) ABAQUS, ADVANTEDGE, and DEFORM use a Coulomb friction law at the tool–chip interface,with a friction coefficient of 0.23. However, PFEM and DEFORM use a Norton–Hoff frictionlaw with a Norton Hoff constant value of 6e-5.



Figure 17. Cutting and thrust force versus simulation time applied on the tool for a rate-independent yieldfunction.

Figure 18. Continuous chip formation sequence: temperature (K).

(3) PFEM, DEFORM, and ADVANTEDGE consider the tool as a deformable while ABAQUS

consider the tool as a rigid body.

Furthermore, PFEM, ADVANTEDGE, and DEFORM use a Lagrangian description of motion whileABAQUS uses an ALE description of motion. Also, PFEM and DEFORM use implicit time integra-tion while ADVANTEDGE and ABAQUS use explicit time integration. More information about theconstitutive model, boundary conditions used in the numerical models developed in ABAQUS andADVANTEDGE are explained in detail in [80].

Taking in account mentioned differences, the results of the numerical simulation are presented inTable III. The obtained process variables of temperature, chip thickness, contact length, von Misesstress, and contact force are compared.

It is observed a good agreement between the experimental and numerical cutting forces pre-dicted by the PFEM, DEFORM, and ABAQUS. Instead, comparing experimental cutting forces withADVANTEDGE results, higher differences were found. Regarding to the chip thickness, a relativelyquite good agreement was found for all the results. However, the tool–chip contact length measuredin the experiments is about two times greater than the length predicted by the numerical simulations.



Table III. Experimental and numerical results (PFEM, ABAQUS, ADVANTEDGE, DEFORM).

Temperature Chip von Mises Contact Contact(K) thickness (mm) (MPa) force (N) length (mm)

Experimental — 0.32 — 395 0.49PFEM 1173 0.35 1400 383 0.26ABAQUS 1240 0.35 1348 412 0.32ADVANTEDGE 1442 0.27 2343 647 0.29DEFORM 1107 0.30 1210 365 0.24

PFEM, particle finite element method.

Comparing results for the maximum tool temperature, it is observed that the larger differenceoccurs between DEFORM–ADVANTEDGE, while the smaller takes place between the PFEM andDEFORM. The differences are due to material model and friction law used in each one of thesoftwares (stated before).

In the case of the von Mises stress, the results predicted by the PFEM, DEFORM, and ABAQUS

are really similar; however, the maximum von Mises stress predicted by ADVANTEDGE is 1000MPa greater than the average stress predicted by the other software. The differences and similari-ties among the predicted results are because of those existing ones between the PFEM, ABAQUS,DEFORM, and ADVANTEDGE (stated before).

The numerical model setup with PFEM is considered to be accurate enough to compete withcurrent commercial softwares based in the FEM method. Concerning with the computational cost, aMATLAB code with the PFEM implementation was used for this comparison. The calculation timein a serial execution was similar to the commercial ones. We guess that, with an implementation ina more optimized programming language, the PFEM would be clearly faster.

8. CONCLUSIONS

A numerical framework, based on the extension of the PFEM, for accurately and robustly simulatingthe different conditions exhibited by thermo-mechanical problems has been presented.

We show in examples 7.1 and 7.2 that the elastic and plastic incompressibility problem can besolved accurately with the proposed u-p formulation. This mixed nodal implementation adapts nat-urally to the core of PFEM because no additional transfer variables are required. It was observedthat the stabilization parameter is mesh independent and that the convergence rate is similar to OSStechnique but with lower computational cost.

In example 7.3, thermo-mechanical coupling, stabilization, and IMPL-EX integration schemewere analyzed. The main conclusion is that IMPL-EX scheme increases the robustness comparedwith a typical implicit integration scheme and the continuous particle discretization of the domainallows us to control the mesh homogeneity.

The potential of numerical methods for the modeling of cutting problems is widely accepted bythe industries of this field, the challenge being to promote effectively their industrial practice. InSection 7.4, we have attempted to make an exploratory step in this regard by extending the model tosimulate the main aspects of a typical continuous chip formation in 2D. The most relevant advantageof the formulation presented is the automatic update of the geometry and the natural generationof new boundary surfaces. It reduces the numerical diffusion due to remeshing because transientmesh additivity is used instead of remeshing, and usually needs less degrees of freedom and lesscomputing time than other methods to achieve the same accuracy. The results also show the goodagreement between our method and the results obtained with other commercial codes.

This paper is an exploratory work, intending to evaluate the capabilities and possibilities of thePFEM in modeling thermo-mechanical problems, rather than being an attempt of comparison withother well-established methods on the subject. In this sense, and for the sake of simplicity, allsimulations in the paper are in 2D.

The extension of the method to 3D cases is a future research field not in the scope of this paper. Itwill possibly bring some new challenges, for example, (i) facing the geometrical aspects to ensure



mass conservation; (ii) construction of a constrained Delaunay tessellations and advanced ˛�shapetechniques in 3D; and (iii) evaluation of competitiveness aspects of the PFEM-based methods withrespect to alternative techniques in 3D cases.

APPENDIX A: COUPLED THERMO-MECHANICAL IBVP

A.1. Balance equations

The coupled thermo-mechanical IBVP is governed by the momentum and energy balance equations,restricted by the second law of thermodynamics. The material form of the local governing equationsfor the body '.X; t / can be written as

P'.X; t / D V.X; t / (A.1)

DIV .P/C B D �0 PV (A.2)

PE C DIV .Q/ D NDint CR (A.3)

In the previous equations, �0 is the reference density, V is the velocity field, B are the prescribedforces per unit of reference volume, DIV .�/ is the reference divergence operator, and P is the firstPiola–Kirchhoff stress tensor. E is the internal energy per unit of material volume, Q the nominalheat flux, R is the prescribed reference heat source per unit of material volume, and NDint is theinternal dissipation per unit of reference volume.

The entropy N and first Piola–Kirchhoff stress tensor P are formulated in terms of the free energy‰ and subjected to the dissipation inequality often referred to as the Clausius–Plank form of thesecond law of thermodynamics.

NDint D P W PFC � PN � PE > 0 (A.4)

D P W PF � P�N � P‰ > 0 (A.5)

where the free-energy function per unit of material volume ‰ is obtained from the internal energyvia the Legendre transformation

‰ D E �N� (A.6)

The nominal heat flux Q is defined by Fourier’s law, subjected to the restriction on the dissipationby conduction NDcon

NDcon D �1

�GRAD.�/ �Q > 0 (A.7)

The spatial form of the local governing equations for the body '.x; t / can be writtenanalogously as

P'.X; t / D v.x; t / (A.8)

div .� /C b D � Pv (A.9)

Pe C div .q/ D Dint C r (A.10)

In these equations, the motion P' and the absolute temperature � are regarded as the pri-mary variables in the problem while b the body force per unit of spatial volume, e the internalenergy per unit of spatial volume, and r the heat source per unit of spatial volume are prescribed



data. In addition, the heat flux q, the entropy �; and the Cauchy stress tensor � are defined viaconstitutive equations.

These constitutive equations are subjected to the following restrictions on the internal dissipationand the dissipation arising from heat conduction per unite spatial volume

Dint D J� W dC � P� � Pe > 0 (A.11)

Dcon D �1

�grad.�/ � q > 0 (A.12)

where the free-energy function per unit of spatial volume is obtained from the internal energy viathe Legendre transformation

D e � � � (A.13)

A.2. Boundary conditions and initial conditions

The basic governing equations (A.1) and (A.8) and the constitutive constraints (A.4) and (A.11) arecompleted by the standard boundary conditions for the mechanical field

' D N' on �' (A.14)

t D P � N D Nt on �� (A.15)

where N' and Nt are the prescribed deformation and nominal traction.Together with the analogous essential and natural boundary conditions for the thermal

field, namely,

� D N� on �� (A.16)

Q � N D NQ on �Q (A.17)

where N� and NQ are the prescribed temperature and the normal heat flux maps.Additionally, we assume that the following initial data is specified for the mechanical and

thermal fields

'.X; t / jtD0 D N'0.X/V.X; t / jtD0 D NV0.X/�.X; t / jtD0 D N�0.X/

9=; in � (A.18)

A.3. Global operator split thermo-elastoplasticity

The IBVP described in Equation (A.8) can be written in a simpler way. Suppose that

PZ D

24 P'� PvP�

35 and Z D

24'v�

35 (A.19)

Then equations can be written in a generalized form as

PZ D A.Z/C f (A.20)

Where A is a nonlinear elliptic operator and f a prescribed function. The Cauchy stress tensor� , the heat flux vector q, the total � and the plastic �p entropies, and the mechanical dissipationDmech WD Dint will be regarded as dependent variables in the problem, defined in terms of theprimary variables Z and a set of internal strain-like variables � . The set of internal variables are



defined in terms of a constrained problem of evolution driven by the primary variables, with thefunctional form

P� D � ˘.� ;Z/ (A.21)

where � is an additional variable determined by means of the Kuhn–Tucker conditions, as follows

� > 0 ˆ.� ;Z/ 6 0 � ˆ.� ;Z/ D 0 (A.22)

and ˆ.� ;Z/ is the Mises yield function. The Kuhn–Tucker conditions are applied only for rate-independent plasticity models.

Generally, the nonlinear operator A can be decomposed in two simpler operators A1 and A2,where A D A1 C A2 [20]. The use of the additive operator split applied to the coupled system ofnonlinear ordinary differential equations leads to the following two simple problems:

(1) Isothermal elastoplastic problem

PZ D

24 P'� PvP�

35 D

24 v.x; t /

div .� .'; �; �.'; �///0

35C

24 0b0

35 (A.23)

(2) Thermoplastic problem at a fixed configuration

PZ D

24 P'� PvP�

35 D

24 0

0

�div .q.'; �; �.'; �///CDint

35C

24 00r

35 (A.24)

A.4. Weak form of the IBVP

We define the set of admissible displacements and admissible temperatures of the body domain �as the set of all sufficiently regular displacement and temperature functions that satisfy the essentialboundary condition, denoted here respectively as

U WD '.�/! R3 W det.F/ > 0 'ˇ�' D N' (A.25)

‚ WD �.�/! R W � > 0 �ˇ�� D

N� (A.26)

The spatial version of the virtual work principle states that the body � is in equilibrium if, andonly if, its Cauchy stress satisfies the equation. The weak form of the momentum balance equation

div .� /C b D � Pv (A.27)

can be justified by taking the L2 inner product of with any valued function � 2 V, being V thespace of virtual displacements

V WD®� 2 '.�/! R3 j �

ˇ�' D 0

¯(A.28)

and making use of the divergence theorem will lead to the following expression:ZVt

Œ� W rs� � � .b � � Pv/� dVt �Z��

t � � d� D 0 8� 2 V (A.29)

The dynamic weak form of the energy balance equations on the body � (A.8) in absence of aheat source (r D 0)

Pe C div .q/ D Dint (A.30)

can be obtained by taking the L2 inner product of with any valued function � 2 T, being T thespace of virtual temperatures

T WD®� 2 �.�/! R j �

ˇ�� D 0

¯(A.31)



making use of the divergence theorem, leading to the following expression:ZVt

�.Pe/ dVt �ZVt

r� � q dVt �ZVt

�Dint dVt CZ�q

�.q � n/ dq D 0 8� 2 T (A.32)

For simplicity, theL2 inner product will be represented as h�; �i, and with a slight abuse in notationh�; �i�� and h�; �i�q will denote the L2 inner product on the boundaries � and q, respectively.

As a consequence, Equations (A.28) and (A.32) can be written as

h� ;rs�i � h�;b � � Pvi � ht;�i�� D 0 (A.33)

h�; Pei � hr�; qi � h�;Dint i � h�;q � ni�qD 0 (A.34)

Denoting by Gu;dyn and Gu;stat the dynamic and quasi-static weak forms of the momentumbalance equations lead to

Gu;dyn D Gu;stat C h�; � Pvi (A.35)

Gu;stat D h� ;rs�i � h�;bi � ht;�i�� (A.36)

And denoting by G�;dyn and G�;stat , the dynamic and quasi-static weak forms of the energybalance equations lead to

G�;dyn D G�;stat C h�; Pei (A.37)

G�;stat D �hr�;qi � h�;Dint i � h�;q � ni�q(A.38)

The weak form of the momentum balance and energy equations for body � can be expressed inshort notation as

Gu;dyn D 0G�;dyn D 0

³8� 2 V;8� 2 T (A.39)

A.5. Mixed displacement-pressure formulation for the IBVP

It is well known that pure displacement formulations are not suitable for problems in which theconstitutive behavior exhibits incompressibility because they tend to experience locking. Lockingmeans, in this context, that the constraint conditions due to incompressibility cannot be satisfied.These constraint conditions are related to the pure volumetric mode (in the elastic case, the conditionis det.Fe/ D 1 see Equation (B.4) and for plastic flow the condition is det.Fp/ D det.Cp/ D 1,see Equation (B.5)). Thus, this behavior is also called volumetric locking. As locking is present inthe modeling of metal plasticity, we adopt a mixed formulation in the momentum balance equationof the workpiece. Introducing a pressure/deviatoric decomposition of the Cauchy stress tensor, thestandard expression of the equilibrium equations becomes

Gu;dyn D Gu;stat C h�; � Pvi (A.40)

Gu;stat D hdev.� /C p1;rs�i � h�; bi � ht;�i�� (A.41)

The pressure field p in the variational equation (A.40) is an additional variable determined by thevolumetric part of the material model. In our case a Neo–Hookean material [18, 74] is used. It willbe introduced in Appendix B.1. The resultant continuity equation is given by

p � ln.J /C 3 ˛ .1 � ln.J //

J.� � �0/ D 0 (A.42)

where > 0 and ˛ can be interpreted as the bulk modulus and the thermal expansion coefficient,respectively. J is the determinant of the deformation gradient (Equation (B.4)).

The weak form of the pressure constitutive equation can be obtained by taking the L2 innerproduct of with any valued function q 2 Q, being Q the space of virtual pressures



Q WD®q 2 p.�/! R j q

ˇ�p D 0

¯(A.43)

The variational equation that represents the weak form of the pressure constitutive equation canbe expressed asZ

Vt

q

�p � ln.J / � 3 ˛

.1 � ln.J //

J.� � �0/

dVt D 0 8q 2 Q (A.44)

or in an alternative form

G� D G�;p C hp; qi (A.45)

G�;p D� ln.J / � 3 ˛

.1 � ln.J //

J.� � �0/; q

�8q 2 Q (A.46)

Taking into account the mixed formulation for the momentum and energy balance equations takesthe form

Gu;dyn D 0G�;dyn D 0G� D 0

9=;8� 2 V;8� 2 T;8q 2 Q (A.47)

APPENDIX B: THERMO-ELASTOPLASTICITY MODEL AT FINITE STRAINS

In the treatment of the themo-mechanical coupling, the constitutive equations must account mate-rial and geometrical nonlinearities. In the mechanical part, a material model with the finite strainelastoplasticity and the multiplicative decomposition of the deformation gradient will be used.

The decomposition of the deformation gradient into elastic and plastic parts is defined by

F.X; t / D Fe.X; t /Fp.X; t / (B.1)

If we are taking in account finite strains, the deformation measures used are the Green–Lagrangeand the Almansi strain tensors that describe the strain in the material and in the spatial configuration,respectively.

E WD1

2.C � N1/ and e WD

1

2.1 � c/ (B.2)

where

C WD FTF and b WD FFT ; then c WD b�1 D .FFT /�1 D F�TF�1 (B.3)

and N1 and 1 denotes the symmetric unit tensor in the reference and the current configurations,respectively.

A Neo–Hookean model will be taken as a reference for the finite strains elastic behavior. Once thematerial reaches plasticity, the thermal behavior must be taken into account. In most materials, thestress–strain relationship is affected by the strain rate and temperature during plastic deformation.For a given value of strain, we can encounter: (i) the stress is higher for a higher strain rate and (ii)the stress is lower for higher temperatures.

The materials to be treated will be metals-type. In this case, the formulation of the constitutiveequations is based on two basic assumptions:

(1) The stress response is isotropic. Therefore, the free energy is independent of the orientation ofthe reference configuration.

(2) The plastic flow is isochoric (standard assumption in metal plasticity).

det.Fp/ D det.Cp/ D 1

det.F/ D det.Fe/ D J e D J(B.4)

where Cp is the plastic part of the Cauchy–Green tensor is defined as



Cp WD FpTFp (B.5)

With these two assumptions, we proceed to outline the governing equations of the modelbeginning with the thermo-hyperelastic model and continuing with the thermo-hyperelastoplasticone.

B.1. Constitutive thermo-hyperelastic model

The first model considered will be an hyperelastic model under temperature effects. The firstassumption is the material isotropy and the second one the thermal response. Volumetric changesin the constitutive response must be accounted because of the variation of the temperature in thematerial.

The Neo–Hookean material used to represent the phenomenology mentioned earlier is repre-sented with the following free-energy function ([18, 74]).

O .b/ D OU .J /C OW .b/C OM.�; J / (B.6)

The elastic part of the free energy is uncoupled into volumetric/deviatoric response describedby the functions OU .J / and OW .C/, respectively. The function OM.�; J / describes the thermo-mechanical coupling due to thermal expansion and provides the potential for the associated elasticstructural entropy.

OU.J / D1

2 ln2.J /

OW .b/ D1

2� t r. Nb/ � 3

�D1

2� t r. NC/ � 3

�OM.�; J e/ D �3 ˛

ln.J /

J.� � �0/

(B.7)

where � > 0, > 0, c > 0 and ˛ can be interpreted as the shear modulus, the bulk modulus, theheat capacity, and the thermal expansion coefficient, respectively. NC and Nb are the volume preservingright Cauchy–Green tensor and the volume preserving left Cauchy–Green tensor. If NF denote thevolume preserving part of the deformation gradient, then det. NF/ D 1. Recalling that J WD det.F/gives the volume change, then

NF WD J�13F) det. NF/ D 1 (B.8)

Associated with F and NF, we define the volumetric preserving part of the right Cauchy–Greentensor and the volume preserving left Cauchy–Green tensor as

NC D J�23C D J�

23FTF (B.9)

and

Nbe D J�23be where be WD FeFeT D F.Cp/�1FT (B.10)

The free-energy function O .b/ (B.6) satisfies two important properties:

� O .b/ is invariant when the current configuration undergoes a rigid body rotation. This isbecause O .b/ only depends on the stretching part U D

pC and is independent of the rotation

part R of F, F D UR (objectivity)� O .b/ on any translated and/or rotated reference configuration is the same at any time t

(isotropy)

From Equation (B.6) and applying, the standard Coleman–Noll procedure leads to a constitutiveequation expressed in terms of material variables as follows:



S D 2 @O @C C

D 2 ln.J /J

@J@C � 6 ˛

.1�ln.J //J

.� � �0/@J@C C �

@tr. NC/@C

D hln.J / � 3 ˛ .1�ln.J //

J.� � �0/

iC�1 C 2 � J�

23

I � 1

3tr.C/C�1

� (B.11)

or its terms in spatial variables as follows:

� D 1J

FSFT

D �J

hln.J / � 3 ˛ .1�ln.J //

J.� � �0/

i1C 2 � J�

53 dev.b/

(B.12)

B.2. Constitutive thermo-hyperelastoplastic model

Consistent with the assumption of isotropy and extending the hyperelastic model to plasticity, wecharacterize the stress response by a stored energy with the form

O D OU .J e/C OW . Nbe/C OM.�; J e/C OT .�/C OK. Ne p; �/ (B.13)

The elastic part of the free energy is uncoupled into volumetric/deviatoric response describedby the functions OU.J e/ and OW . Nbe/, respectively. The function OM.�; J e/ describes the thermo-mechanical coupling due to thermal expansion and provides the potential for the associated elasticstructural entropy, while the function OT .�/ is the potential for the purely thermal entropy. Thefunction OK. Ne p; �/ is a nonlinear function of the equivalent plastic strain Ne p and temperature � thatdescribes the isotropic strain hardening via the relation ˇ D �@ Ne p OK. Ne

p; �/. To make matters asconcrete as possible, we consider the following explicit forms [20, 81].

OU.J e/ D1

2 ln2.J e/

OW . Nbe/ D1

2� t r. Nbe/ � 3

�D1

2� t r. NCe/ � 3

�OT .�/ D c

�.� � �0/ � � ln

��

�0

�

OM.�; J e/ D �3 ˛ ln.J e/

J e.� � �0/

(B.14)

where � > 0, > 0, c > 0 and ˛ can be interpreted as the shear modulus, the bulk modulus, theheat capacity, and the thermal expansion coefficient, respectively.

Some remarks can be made about the structure of the free-energy function (B.13):

(1) The structure of the free energy is usually restricted to temperature-independent materialproperties.

(2) The thermoelastic free energy is decoupled from the plastic contribution @ Ne p OK. Nep; �/ asso-

ciated with the hardening variable Ne p (this assumption is motivated by the experimentalobservation that the lattice structure remains unaffected by the plastic deformation) [20].

(3) The functions OU .J e/ and OW . Nbe/ generalize the linear isotropic elastic model.(4) The function OK. Ne p; �/ represents the visible (macroscopic) plastic deformations that are the

results of microscopic dislocation (crystallographic defects in the crystal structure) motionand multiplication. Generally, the material exhibits high strength if there are either high lev-els of dislocations or no dislocations. In addition, the function OK. Ne p; �/ represents the yieldstress decreasing as the grain size is increased [82]. Also, OK. Ne p; �/ represents the decreasein dislocation density because of the heating of the material above its critical temperature(thermal softening).

There are four main strengthening mechanisms for metals, each one is a method to preventdislocation motion and propagation, or make it energetically unfavorable for the dislocation tomove (work hardening, solid solution strengthening, precipitation hardening, and grain boundarystrengthening).



In addition, there are other factors that affect the shape and the magnitude of the hardening poten-tial among them [83]: (i) material composition; (ii) previous heat treatment; (iii) the type of crystalstructure; and (iv) prior history of plastic deformation. Different hardening potentials that representthe work hardening phenomenon have been proposed in the literature, which reflect some of thestrain hardening patterns observed in the experiments. Among them are the following:

B.2.1. Voce and Simo hardening potential. Voce [68] presented and Simo [20] applied the followingpotential describing isotropic hardening:

OK. Ne p; �/ D 12h.�/. Ne p/2 � Œ�0.�/ � �1.�/� OH. Ne

p/

OH. Ne p/ D

´Ne p � 1�exp�ı Ne

p

ıfor ı ¤ 0

0 for ı D 0

(B.15)

where ı is the saturation exponent and the functions h.�/, �0.�/ and �1.�/ describe linearthermal softening.

�0.�/ D �0.�0/ .1 � w0.� � �0//

�1.�/ D �1.�0/ .1 � wh.� � �0//

h.�/ D h.�0/ .1 � wh.� � �0//

(B.16)

where �0.�0/ is the initial yield stress, �1.�0/ is the final saturation hardening stress, h.�0/ is thelinear hardening modulus, all obtained at the reference temperature �0, while w0 and wh are theflow stress softening and hardening softening parameter, respectively.

The previous potential allows us to study materials exhibiting a combination of linear andsaturation-type hardening.

B.3. Yield condition

Accurate flow stress models are considered extremely necessary to represent work materialconstitutive behavior under high-strain rate deformation conditions. We consider the classicalMises–Hubber yield conditions, expressed in terms of the Kirchhoff stress tensor, for the case ofrate-independent plasticity:

ˆ. ; Ne p; �/ D kdev. /k �

r2

3

��y � OK

0. Ne p; �/�D kdev. /k �

r2

3

�y C ˇ

�6 0 (B.17)

and for rate-dependent plasticity

f . ; Ne p; �/ D kdev. /k �

r2

3

�y C ˇ

� �1C g. PNe

p/�D 0

or

f . ; Ne p; �/ D ˆ. ; Ne p; �/ �

r2

3

�y C ˇ

�g. PNe

p/ D 0

if ˆ. ; Ne p; �/ > 0

(B.18)

where �y denotes the flow stress, �y0 denotes the flow stress at � D �0, ˇ D �K 0. Ne p; �/ theisotropic nonlinear hardening modulus, ˇ0 the isotropic hardening at � D �0, g. PNe

p/ the strain

rate hardening modulus, and Ne p the hardening parameter. The expressions .�y C ˇ/ y and g. PNep/

depend on the hardening law used. Numerous empirical and semi-empirical flow stress models havebeen proposed. Some examples of strain rate-dependent models have been developed by Johnsonand Cook [70] and Bäker [71]. Simo [20] proposed the strain-dependent model that will be used inthis work.

B.3.1. Simo flow model. In the Simo flow model a particular expression is proposed to define the

hardening and thermal softening condition .�y C ˇ/:



Table A.1. Simo yield function. Material properties.

Yield stress �y 450 MPaFlow stress softening w0 0.002 K�1

Reference temperature �0 293.15 KLinear hardening H 129.24 MPaHardening softening wh 0.002 K�1

Saturation hardening Kinf 715 MPaHardening exponent ı 16.93

.�y C ˇ/ D O�y C OH. Nep/C . OKinf � O�y/ .1 � exp.�ı Ne p//

whereO�y D �y .1 � w0.� � �0//

OH D H .1 � wh.� � �0//OKinf D Kinf .1 � wh.� � �0//

(B.19)

This model describes the strain hardening and thermal softening for most steels in temperaturerange from 300 to 400 K [20]. Common values of material constants of the Simo yield function areshown in Table A.1.

B.4. Associative flow rule

The functional form of the corresponding associate flow rule is uniquely determined analyzing theevolution equations and the plastic dissipation. For the Mises–Hubber yield function (B.21) and thefree-energy function (B.20), Simo [20] and Ibrahimbegovic [78] show that the flow rule takes theform based on the principle of maximum plastic dissipation.

O D OT .�/C OM.�; J e/C OU.J e/C OW . Nbe/C OK. Ne p; �/ (B.20)

f . ; Ne p; �/ D kdev. /k �

r2

3

�y C ˇ

� �1C g. PNe

p/�D 0 (B.21)

A detailed procedure about how to get the flow rule is shown in the following lines. First, definingthe plastic mechanical dissipation and then the evolution equations.

B.4.1. Mechanical dissipation. Because of the restriction to isotropy implied by the thermo-elasticdomain, the functional form of the internal energy function e can be written as

e D Oe. Nbe; Ne p; �e/ with �e D � � �p (B.22)

where � is the entropy of the system, Ne p is the equivalent plastic strain, and Nbe is the elastic leftCauchy–Green tensor. The free energy can be expressed in terms of the internal energy via theLegendre transformation (A.13) as

O . Nbe; Ne p; �/ D Oe. Nbe; Ne p; �/ � �p � (B.23)

Applying the second law of thermodynamics, constitutive equations consistent with the assumedfree-energy function are derived. This gives the expression of the energy dissipation as

D D � dC � P� � Pe D � dC � P� � PO � P�e � � �e P� (B.24)

differentiating the free-energy function O of the Equation (B.20) with respect to time



PO D@ O

@ NbePNbe

C@ O

@ Ne pPNep C

@ O

@�P� (B.25)

and taking the derivative of be with respect to time.

Pbe D PFF�1F.Cp/�1FT C F.Cp/�1FTF�T PFT C F. PCp/�1FT (B.26)

Using the definition of the spatial velocity gradient l D PFF�1 and the Lie derivative of the elasticleft Cauchy–Green tensor Lvbe (B.28), the time derivative of be is written as

Pbe D lbe C belT C Lvbe (B.27)

Remark 1The Lie derivative for the tensor be is defined as

Lvbe D F²@

@tŒF�1beF�T �

³FT D F

@

@tŒ.Cp/�1�FT D F. PCp/�1FT (B.28)

The Lie derivative of be tensor is exactly the push forward of the time derivative of the pull-backof the spatial tensor be . More information about push-forward and pull-back operations is given inreferences [73, 74]. Inserting Equation (B.27) into Equation (B.25), the derivative of the free-energyfunction O becomes

PO D@ O

@be.2lbe C Lvbe/C

@ O

@ Ne pPNep C

@ O

@�P� (B.29)

By inserting the relation d D symŒl� into (B.29) and using the Legendre transformation (B.23),the dissipation inequality becomes

D D P�p � C �@ O

@�� C �p

!� C

� 2

@ O

@bebe!

d �@ O

@beLvbe �

@ O

@ Ne pPNep > 0 (B.30)

By demanding that (B.30) hold for all admissible processes, the Kirchhoff stress tensor is obtainedby the general expression:

� D 2 @O

@be be D 2Fe @W@Ce FeT

D J eh�3 ˛ .1�ln.J e//

.J e/2.� � �0/ �

ln.J e/J e

i1C � dev. Nbe/

(B.31)

The hydrostatic and deviatoric parts of the Kirchhoff stress tensor are

p WD��3 ˛

.1 � ln.J e//

J e.� � �0/ � ln.J e/

1 (B.32)

s WD � dev. Nbe/ (B.33)

and the entropy constitutive equation

� D �p �@ O

@�D �p � @� OT .�/ � @� OM.�; J e/ � @� OK. Ne

p; �/ (B.34)

The dissipation inequality becomes

Dmech WD D D P�p � � @ O

@beLvbe �

@ O

@ Ne pPNep > 0 (B.35)



B.4.2. Evolution equations and maximum plastic dissipation. Now, we need to define the evolutionequations for the internal variables in the model in order to complete the constitutive theory ofplasticity at finite strains.

Based on the thermo-mechanical principle of maximum dissipation, the problem is to find thevalues of the stress, the isotropic nonlinear hardening, and the temperature . ; ˇ; �/ such that thedissipation function (B.35) attains a maximum subject to the constraint ˆ. ; Ne p; �/ 6 0 (rate-independent plasticity), prescribed the intermediate configuration (be is fixed), and prescribed therates .Lvbe; PNep; P�/. The problem can be reformulated as a constrained minimization of the negativevalue of the dissipation

. ; ˇ; �/ D arg

�min

ˆ.�; Ne p ;�/60.�D/

(B.36)

D arg

"min

ˆ.�; Ne p ;�/60

�P�p � C

@ O

@beLvbe C

@ O

@ Ne pPNep

!#(B.37)

But the problem can be expressed as an unconstrained minimization problem by introducing aLagrangian functional

@�Lp. ; ˇ; � I�/ D �D. ; ˇ; �/C �ˆ. ; Ne p; �/ (B.38)

D �P�p� C1

2 � Lvbebe�1 � PNe

pC �ˆ. ; Ne p; �/ (B.39)

The solution to the problem is given by

@�Lp. ; ˇ; � I�/ D

1

2Lvbebe�1 C � @�ˆ. ; Ne p; �/ D 0 (B.40)

@ˇLp. ; ˇ; � I�/ D � PNep C � @ˇˆ. ; Nep; �/ D 0 (B.41)

@�Lp. ; ˇ; � I�/ D �P�p C � @�ˆ. ; Nep; �/ D 0 (B.42)

where the consistency parameter � is the Lagrange multiplier satisfying the Kuhn–Tucker conditions

� > 0 ˆ. ; Ne p; �/ 6 0 �ˆ. ; Ne p; �/ D 0 (B.43)

It is important to remark that the Kuhn–Tucker conditions are equivalent to the loading–unloadingconditions. In summary, the evolution equations of the internal variables are

Lvbe D �2 � @�ˆ. ; Ne p; �/be (B.44)

PNepD �� @ˇˆ. ; Ne

p; �/ (B.45)

P�p D � @�ˆ. ; Nep; �/ (B.46)

From expressions (B.26) and (B.28), the Lie derivative of the elastic left Cauchy–Green tensorcan be expressed in material description as

. PCp/�1 D �2 � @�f .dev. /; Ne p; �/ .Cp/�1 (B.47)

D �2 �dev. /

kdev. /k.Cp/�1 D �2 �

sksk

.Cp/�1 (B.48)

PNepD �

r2

3(B.49)

P� p D �

r2

3.@�� C @�ˇ/ (B.50)

Using the specific constitutive equations and decomposing be into its spherical and deviatoricparts, the exact flow rule (B.44) becomes



Lvbe D �2 � J�23 n2ksk�� 2 � J�

231

3tr. Nbe/n (B.51)

The first term in (B.51) can be neglected in most metals, because this term is of the order of theflow stress over the shear modulus, which for metal plasticity, is of the order of 10�3 [20]. UsingNF D J�

13F at the modified flow rule

Lvbe D �2 � J�231

3tr. Nbe/n (B.52)

NF.Cp/�1 NFT D �2 �1

3tr. Nbe/ n (B.53)

APPENDIX C: TIME INTEGRATION OF THE CONSTITUTIVE LAW

The problem of integrating numerically the initial-value ODE equations represented by (B.47) inconjunction with the condition (B.43) is the focus of this appendix.

C.1. Implicit backward Euler integration scheme

Let .Cpn /�1; Ne p; �n denote the initial state at time tn, and assume that the deformation gradient andtemperature field FnC1; �nC1 at time tnC1 are prescribed. Let us focus on the time step tn ! tnC1,where �t D tnC1 � tn. Using an implicit unconditionally stable scheme on (B.52) and the scalarequations of (B.44) gives

NFnC1.CpnC1/

�1 � .Cpn /�1�NFTnC1 D �2 �nC1�t

1

3tr. NbenC1/nnC1

NepnC1 � Ne

pn D �nC1�t

r2

3

�pnC1 � �

pn D �nC1�t

r2

3

@��y C @�ˇ

�(C.1)

The right-hand side of Equation (C.1) in terms of spatial variables becomes

NbenC1 � NFn;nC1 NbenNFTn;nC1 D �2 �nC1�t

1

3tr. NbenC1/nnC1 (C.2)

along with the following counterpart of the loading–unloading conditions:

��t > 0 fnC1 nC1; Ne

pnC1; �nC1

�6 0 �nC1�t fnC1

nC1; Ne

pnC1; �nC1

�D 0 (C.3)

where the yield condition is defined by the Mises criterion

fnC1 nC1; Ne

pnC1; �nC1

�D kdev. nC1/k �

r2

3

�y;nC1 C ˇnC1

�(C.4)

A closed-form solution of these equations is obtained by defining the thermo-elastic state bythe relationships

Nbe;trialnC1 D

NFn;nC1 Cpn�1 NFTn;nC1 D NFn;nC1 ben NF

Tn;nC1

strialnC1 D � dev

�Nbe;trialnC1

�f trialnC1 D

��strialnC1

�� r2

3


pn /� (C.5)

We observe that the trial state is determined solely in terms of the initial conditions ben; Nepn ; �n and

the given incremental deformation gradient NFn;nC1. We remark that this state may not correspondto any actual state, unless the incremental process is elastic. An analysis of Equation (C.5) revealstwo alternative situations:

First, we consider the case for which



f trialnC1 < 0 (C.6)

It follows that the trial state is admissible in the sense that

NbenC1 D Nbe;trialnC1 D NFn;nC1 Cpn

�1 NFTn;nC1 D NFn;nC1 ben NFTn;nC1

snC1 D strialnC1

strialnC1 D � dev�Nbe;trialnC1

�NepnC1 D Ne

pn

(C.7)

and satisfy

(1) The stress–strain relationship(2) The flow rule and the hardening law with ��nC1 D �nC1�t D 0(3) The Kuhn–Tucker conditions, because

fnC1. nC1; NepnC1; �nC1/ D f

trialnC1 6 0 ��nC1 D 0 (C.8)

satisfy (C.3).

Next, we consider the case for which f trialnC1 > 0. Clearly, the trial state cannot be asolution to the incremental problem because Nbe;trialnC1 ; Ne

pn ; �n violates the constraint condition

fnC1. nC1; NepnC1; �nC1/ 6 0. As a result, we require that ��nC1 > 0 so that Ne pnC1 ¤ Ne

pn to obtain

snC1 ¤ strialnC1 .To summarize our results, the conclusion that an incremental process for given incremental defor-

mation gradient is elastic or plastic is drawn solely on the basis of the trial state according tothe criterion

f trialnC1

²6 0 ) elastic step ��nC1 D 0> 0 ) plastic step ��nC1 > 0

Here, we focus on the algorithmic problem for an incremental plastic process characterized bythe conditions

f trialnC1 > 0, fnC1. nC1; Ne

pnC1; �nC1/ D 0 (C.9)

and��nC1 > 0 (C.10)

The objective is to determine the solution . NbenC1; NepnC1; �nC1; snC1; ��nC1/ to the problem (C.2),

(C.3), and (C.4). To accomplish this, we express the isochoric Kirchhoff stress tensor snC1 in termsof strialnC1 and ��nC1 as follows

snC1 D � devNbenC1

�D � dev

NFn;nC1 ben NF

Tn;nC1

�� 2��nC1 �

1

3trNbenC1

�nnC1

D strialnC1 � 2��nC1 �1

3trNbenC1

�nnC1

(C.11)

The update of the isochoric Kirchhoff stress tensor and the tensor benC1 need the determinationof the trace of benC1. By taking the trace of Equation (C.2) and using (C.5), we conclude that

tr. NbenC1/ D tr. Nbe;trialnC1 / (C.12)

Then, replacing (C.12) in (C.2), we obtain

NbenC1 D Nbe;trialnC1 � 2 �nC1�t

1

3tr. Nbe;trialnC1 /nnC1 (C.13)

and using the hyperelastic relationships yields



snC1 D strialnC1 � 2��nC1 �1

3tr. Nbe;trialnC1 /nnC1 (C.14)

From (C.11) and the definition snC1 D ksnC1knnC1, the normal nnC1 is determined in terms ofthe trial stress strial

nC1

ksnC1knnC1 D kstrialnC1 knnC1 � 2��nC1 �1

3tr. Nbe;trialnC1 /nnC1

kstrialnC1 kntrialnC1 D

�ksnC1k C 2��nC1 �

1

3tr. Nbe;trialnC1 /

nnC1

kstrialnC1 knnC1 D kstrialnC1 kn

trialnC1

nnC1 D ntrialnC1

(C.15)

By taking the dot product of (C.11) with nnC1 and using (C.4), we obtain the following scalarnonlinear equations that determine the consistency parameter ��nC1:

g.��nC1/ D ksnC1k � 2��nC1 �1

3tr. Nbe;trialnC1 / �

r2

3


pnC1/

�D f trialnC1 � 2��nC1 �

1

3tr. Nbe;trialnC1 /C

r2

3

�y;n C ˇn. Ne

pn /�

�

r2

3


pnC1/

�D 0

(C.16)

Equation (C.16) is effectively solved by a local Newton iterative procedure because g.��nC1/ isa convex function for the isotropic hardening functions used in this work, and then convergence ofthe Newton–Raphson is guaranteed.

Once ��nC1 is determined from (C.16) the intermediate configuration, the hardening variableand plastic entropy are updated from (C.1).

C.2. Algorithmic constitutive tensorIn the following lines, we provide an expression for the algorithmic tangent moduli, which is a keyaspect in the linearization of the weak form of the momentum equation. The algorithmic constitutivetensor is developed for the implicit integration scheme and for the IMPL-EX scheme.

C.2.1. Algorithmic constitutive tensor: implicit integration scheme. The expression for the tangentmoduli for the implicit stress update algorithm will be presented in the following lines

@SnC1@CnC1

D ı1Ctrialdev C ı2NnC1 ˝ Dev.N2nC1/ C ı3NnC1 ˝ NnC1 (C.17)

where the coefficients ı1; ı2; and ı3 are defined by the expressions

ı1 D

1 �

2��nC1

kStrialnC1 k

!(C.18)

ı2 D 2

0@ N��nC1 � N� kStrialnC1 k�

2 N�C 23

dŒ�yCˇ

d�nC1

�1A (C.19)

ı3 D

0@2 N�2��nC1kStrialnC1 k

CN� 23��nC1kStrialnC1 k � 2 N�

2�2 N�C 2

3

dŒ�yCˇ

d�nC1

� �2

3��nC1kStrialnC1 k

1A (C.20)

and where trial Ctrialdev

is given by

CtrialnC1 D@StrialnC1

@CnC1D N�nC1

�1

3.CnC1/�1 ˝ .CnC1/�1 C InC1

�

�1

3�J�

23 .Cpn /

�1 ˝ .CnC1/�1 C .CnC1/�1 ˝ .Cpn /�1

(C.21)



where InC1 the operator has the following component form

InC1;ijkl D �1

2.CnC1;ik/

�1.CnC1;jl/�1 C .CnC1;il/

�1.CnC1;jk/�1 (C.22)

It is important to remark that, the consistent deviatoric tangent modulus is nonsymmetrical.The last point to complete the derivation of the consistent tangent modulus is to calculate the

derivatives of the isotropic hardening function used in this work with respect to the plastic multiplier.The following equations present the derivatives of the Voce and Simo model (B.19).

@.�y C ˇ/nC1

@ NepnC1

D H .1 � w0.� � �0//

CKinf .1 � wh.� � �0// �K0 .1 � w0.� � �0//

�ı exp.�ı Ne pnC1/

(C.23)

Because the stress update formula is cast in terms of spatial quantities, it is convenient to trans-form the material algorithmic tangent moduli (C.17) into the spatial configuration via a pull-forwardoperation as follows

Ncc dev;ijkl D@snC1@1nC1

D FnC1;iAFnC1;kCFnC1;lDFnC1;jBCdev;ABCD (C.24)

@snC1@1nC1

D ı1@strialnC1

@1nC1C ı2nnC1 ˝ dev.n2nC1/C ı3nnC1 ˝ nnC1 (C.25)

C.2.2. Algorithmic constitutive tensor: IMPL-EX integration scheme. The derivation of the algorith-mic tangent moduli for the IMPL-EX stress update algorithm follows a similar procedure to thatused for the implicit scheme.

The nonsymmetrical expression for the consistent deviatoric elastoplastic module for the IMPL-EX stress update scheme is given by

NC D@SnC1@CnC1

D Nı1Ctrialdev C

Nı2NnC1 ˝ Dev.N2nC1/C Nı3NnC1 ˝ NnC1 (C.26)

where the coefficients Nı1; Nı2; and Nı3 are defined by the expressions

Nı1 D

1 �

2 N�� N�nC1

kStrialnC1 k

!(C.27)

Nı2 D 2 N�� N�nC1 (C.28)

Nı3 D

2 N�2

kStrialnC1 kC2

3kStrialnC1 k

!� N�nC1 (C.29)

where Ctrialdev

is given by (C.21) and

� N�nC1 D ��nC1�tnC1

�tn(C.30)

As was said earlier, a comparison of the coefficients of Equations (C.27) and (C.18) shows thatthe algorithmic tangent modulus is simpler in IMPL-EX scheme that in implicit scheme. Also,Equation (C.27) shows that the tangent moduli of the IMPL-EX scheme is independent of theisotropic hardening function used; by the previous reason, the task of implementing a new hard-ening function inside the IMPL-EX scheme is simpler than in the implicit scheme. Because thestress update formula is cast in terms of spatial quantities, it is convenient to transform the mate-rial algorithmic tangent moduli (C.26) into the spatial configuration via a pull-forward operationas follows



Ncc dev;ijkl D@NsnC1@1nC1

D FnC1;iAFnC1;kCFnC1;lDFnC1;jB NCdev;ABCD (C.31)

@NsnC1@1nC1

D Nı1@strialnC1

@1nC1C Nı2nnC1 ˝ dev.n2nC1/C Nı3nnC1 ˝ nnC1 (C.32)

C.3. Linearization of the algorithmic dissipation

In the same way, the solution of the mechanical problem using an implicit integration schemerequires the algorithmic elastoplastic tangent moduli, and the solution of the thermal problemrequires the linearization of the algorithmic dissipation.

The mechanical dissipation (B.35) that comes from the free energy (B.20) depends only onthe initial flow stress �y . This feature, however, is not consistent with the experimental observa-tion on metals that suggests that part of the work hardening possess a dissipative character. Inorder to accommodate the experimental observations introduced earlier into the phenomenologi-cal thermoplastic constitutive model, an additional dissipation hypothesis concerning the amount ofmechanical dissipation must be introduced. In practice, this is accomplished by assuming that themechanical dissipation is a fraction of the total plastic power.

Dmech D �r2

3.�y C ˇ/ P� (C.33)

where � 2 Œ0; 1� is a constant dissipation factor chosen in the range of Œ0:85; 0:95�.

C.3.1. Linearization of the algorithmic dissipation: implicit integration scheme. An implicitbackward Euler time discretization of the plastic dissipation is shown in the next equation

DnC1mech

D �

r2

3.�y C ˇ/nC1

��nC1

�t(C.34)

The derivative of the dissipation with respect to the temperature is given by the followingexpression

@DnC1mech

@�D a

264��nC1 �

q23.�y C ˇ/nC12 N�C 2

3b�

375 (C.35)

where the coefficients a and b are given by the expressions

a D�

�t

r2

3

@.�y C ˇ/nC1

@�(C.36)

b D@.�y C ˇ/nC1

@��nC1(C.37)

The terms a and b depend on the yield functions .�y Cˇ/nC1. The term b has been calculated inthe previous section. Therefore, it is only necessary to calculate the derivative of the yield functionswith respect to the temperature field, as is shown in the following lines.

First, the derivative with respect to temperature of the Simo and Voce yield function is

@.�y C ˇ/nC1

@�D ��y CK0

1 � exp.�ı Ne pnC1/w0

��H CKinf

1 � exp.�ı Ne pnC1/w0

� (C.38)

C.3.2. Linearization of the algorithmic dissipation: IMPL-EX integration scheme. Starting from theextrapolated value of the plastic multiplier, the plastic dissipation at tnC1 could be written as



NDnC1mech

D �

r2

3.�y C ˇ/nC1

� N�nC1

�t(C.39)

As the extrapolated value of the plastic multiplier is held constant during the time increment, thelinearization of the IMPL-EX dissipation is given by

@ NDnC1mech

@�D �

r2

3

@.�y C ˇ/nC1

@�

� N�nC1

�t(C.40)

A comparison of Equations (C.41) and (C.35) shows how simple it is to linearize the plasticdissipation in case of using IMPL-EX.

The derivative of the yield function with respect to the temperature field for the model used inthis work has been presented in Equation (C.38).

Using the coefficients introduced in Equations (C.36) and (C.37), the linearization in case ofIMPL-EX is simplified as

@ NDnC1mech

@�D a Œ� N�nC1� (C.41)

APPENDIX D: TIME INTEGRATION OF THE IBVP

The implicit scheme is unconditionally stable; it means that there is no restriction on the time stepused in the numerical simulation. In implicit formulations, mechanical problem can be solved ina static or dynamic way. Furthermore, implicit formulations can be used with standard and mixed(displacement/pressure) finite elements. However, implicit schemes need the solution of a linearsystem of equations, a certain number of times, within each time step. Usually, the solution ofthe linear system represents most of the computing time. Furthermore, in the implementation ofa new constitutive equation, the implicit time integration has the requirement of an algorithmicconstitutive tensor. Moreover, in some cases, an implicit scheme does not converge, because of thehigh nonlinearities involved in the problem.

The explicit formulation solves the mechanical problem in a dynamical way. The solution of eachtime step in an explicit scheme is simple and computationally efficient, provided the use of a lumpedmass matrix in the simulation. Explicit schemes do not need the solution of a linear system; this topicis an advantage if the numerical solution is carried out using parallel computing. Implementation ofa new constitutive equation is an easy task; it allows to implement simple or complex constitutiveequations without a big effort. Explicit schemes are conditionally stable, it means that the time stepused in the simulations should be less or equal than a given critical time step, the critical time stepcorresponds to the time that take to a wave to travel through the small finite element of the mesh. Incase of an elastic material, the critical time step depends on the mesh size, elastic modulus, Poissonratio, density of the material, and a constant that depends on the finite element used.

�tc D �xq3�.1��/ .1C /

D �xq2G.1��/ .1�2 /

(D.1)

The restriction imposed on the time step by the explicit schemes allows concluding that for numer-ical simulation that involves long period of computing time or low speeds, implicit schemes aremore favorable in comparison with explicit schemes.

There is no a reference comparison between explicit and implicit time integration schemes inthe literature. There are no clear rules to determine in which condition one scheme is better thanthe other.

In the literature, implicit schemes have been used in [3, 43, 84, 85] and explicit schemes in [1, 2,54]. Also, there are some mixed schemes in which the hydrostatic part of the balance of momentumis integrated implicitly and the deviatoric part is integrated explicitly. Some examples of mixed timeintegration schemes are given in the definition of the The Characteristic-based Split [76] and TheFinite Calculus [62].



An implicit coupled algorithm is presented next.

D.1. Implicit coupled algorithm (monolithic scheme)

For simplicity, a partition of the time domain I WD Œ0; T � into N time steps, of the same length �tis considered. Let us focus on the time step tn ! tnC1, where �t D tnC1 � tn. The applicationof an implicit backward Euler time integration scheme to the problem (displacements, pressures,and temperatures), (16), (14), (15) yields the algorithm described in Box 7 defined by the initialconditions described in (A.18).

COUPLED SYSTEM OF EQUATIONS

1. Momentum

Fu;dyn. RunC1/ D

Fu;int .� nC1.unC1; pnC1; �nC1/I�nC1.unC1; �nC1// � Fu;ext .unC1/

2. Incompressibility

�Mp CMstab

�pnC1 D Fp;vol.JnC1.unC1; �nC1//

where Mp and Mstab are the mass-type matrices of the linearized pressure and stabilizationforces, respectively.

3. Energy

F�;dyn. R�/ D

F�;int .q.�nC1/IDint .unC1; �nC1/I�nC1.unC1; �nC1// � F�;ext

———————————————————–



unC1 D un C vnC1�t

pnC1 D pn C�pnC1�nC1 D �n C P�nC1�t

The definition of the variables that appear in this boxis explained in Sections 4 and 6.

Box 7: Implicit coupled solution scheme.

The set of equations presented in Box 7 show a simultaneous solution scheme of the coupledsystems of equations where the temperature varies during the mechanical step and the configura-tion varies during the thermal step. At first glance, the simultaneous solution is the obvious one, buta depth analysis shows that is a computationally intensive procedure [20]. The monolithic schemeis unconditionally stable because of its fully implicit character. The different time scales associ-ated with the thermal and mechanical fields suggested that an effective numerical integration of thecoupled problem should take advantage of these different time scales. One of the effective integra-tion schemes is the so-called staggered algorithms, whereby the problem is partitioned into several



smaller sub-problems that are solved sequentially (splitting each time step in several pseudo-timesteps). Most of the time, this technique is especially attractive from a computational point of view,because the large and no symmetric system that results from a simultaneous solution scheme isreplaced by a much smaller, subsystem.

ACKNOWLEDGEMENTS

The research leading to these results has received funding from the European Research Council under theEuropean Union’s Seventh Framework Programme FP/2007-2013)/ERC grant agreement number 320815,Advanced Grant Project COMP-DES-MAT.

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