recent advances in residual stress simulation …

Recent advances in residual stress simulation caused by the welding process

XIII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIII

E. Oñate, D.R.J. Owen, D. Peric and M. Chiumenti (Eds)

RECENT ADVANCES IN RESIDUAL STRESS SIMULATION CAUSED BY THE WELDING PROCESS

H. SALLEM*, E.FEULVARCH*, H.AMIN EL SAYED* B.SOULOUMIAC†, J-B. LEBLOND+, J-M.BERGHEAU*

* Laboratoire de Tribologie et Dynamique des Systèmes (LTDS)Ecole Nationale d’Ingénieur de Saint-Etienne

Université de Lyon58 Rue Jean Parot, 42023 Saint-Étienne, France

E-mail: [email protected], web page: http://ltds.ec-lyon.fr

† ESI-Group, Lyon, France, 70 Rue Robert, 69006 Lyon, France

E-mail: [email protected]

** Institut Jean le Rond d’Alembert (IJLRA)Université Pierre et Marie CURIE

4 Place Jussieu 75252 Paris, France Email: [email protected]

Keywords: Welding, Numerical simulation, Mushy zone, Behavior law, Residual stresses.

Abstract. The purpose of the present paper is to improve the description of residual stress field characteristics generated after welding. The behavior of the material both in the liquidand solid states and during all heating and cooling stages including the solidification in the mushy zone is considered, as well as the surface tension effects in the liquid phase.Simulations are conducted on the finite element software SYSWELD®. A displacement/pressure mixed formulation, based on the linear tetrahedral element of type P1/P1 in the context of elasto-viscoplastic formulation, is used. As regards the structure behaviour, the continuous transition between the liquid and the solid phases during the welding is ensured using a mixture law behavior. Numerical simulations were carried out in the context of Lagrangian approach. In this approach the material over the liquidus temperature is modelled as a Newtonian fluid but the flows in the weld pool are not accounted for.Concerning surface tension modelling, the standard method usually adopted is to apply an external load on the free surface of the weld pool. In the present study, a surface spherical stress state is directly imposed on the surface in membrane elements incorporated in the meshand representing the interface.Since tetrahedral mesh is easily adapted to complex geometry, a discretization of type P1/P1 is used in the case of welding simulation. It shows the relevance of such tetrahedral finite element for the mechanical analysis of elasto-viscoplastic solid metal.A representative simulation of a laser welding case is processed. The material considered in

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H. Sallem, E.Feulvarch, H.Amin El Sayed, B.Souloumiac, J-B Leblond, J-M Bergheau

this study is the Inconel 600 alloy. Computed residual stress distribution reveals the ability of such approaches to predict residual stress states in assessing the integrity of welded components.

1 INTRODUCTION Knowledge of welding residual stress characteristics is essential for structural integrity

assessment containing weld components and may require the use of specific models and numerical methods. However, in order to improve residual stress and distortion predictability, numerical model should include as many physical phenomena occurring during welding as possible.

The numerical simulation of welding rests upon the simulation of interactions between strongly non linear thermal and mechanical phenomena. As known, welding process involves highly localized heat necessary to produce a junction of different materials. Consequently, during heating three distinct zones appears. A pool of molten material is formed (weld pool), it constitutes the liquid phase during heating. The base metal remaining solid constitues the solid phase. These two states can co-exist in the transitional phase, called mushy zone, located between the weld pool and the rest of structure as shown in Figure 1. Note that the metallurgical transformations which could occur in the solid phase are disregarded in this paper.

Figure 1: Schema showing the location of different regions during welding

During welding, in the mushy zone, the material is most sensitive to hot cracking, knownalso as solidification cracking. This phenomena generally occurs during the first solidification[1] and is known to be highly dependent to the solid fraction as well as the amount and distribution of the liquid metal [2]. In fact, as solidification proceeds, generated stresses associated with welding and transmitted by the solid dendritic web, may provoke the deviation of dendrites and then the liquid cannot flow to fill the spaces between solidifying weld metal [3]. Therefore it is important to take into account not only the behavior of the solid at high temperature, but also the behavior of the mushy zone during solidification. Unfortunately only a few studies focus on the mushy zone behavior because of experimental difficulties associated with its study. As far as numerical simulations are concerned, nor the liquid phase, nor the mushy zone behaviors are generally considered for the computation of residual stresses distortions. The approach proposed by Heuze et al. [4] consists of coupling the solid phase and the liquid phase in a single simulation using a Lagrangian approach for the

Base metal

Mushy zone

Weld pool

T solidusT liquidus

Maximum temperature

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solid and an Arbitrary Lagrangian Eulerian formulation for the fluid. The instantaneous conversion from one state to another is determined using the melting temperature without including solid behavior during solidification in the mushy zone. In another study, to model the solidification of alloys in casting, Jaouen [5] has defined a reference temperature called coherence temperature corresponding to liquid fraction 𝑓𝑓𝐿𝐿= 50%, at which the mixture behavior changes. Recently, the continuous transition between the liquid and the solid state was ensured using mixture behavior law [6]. However, when dealing with ALE approach using strong thermo-mechanical coupling, numerical difficulties related to the convergence of the solid element having common borders with several fluid elements, were encountered. In this study, the continuous transition between the liquid and the solid phases during welding is ensured using a mixture law developed in the previous study. The material over the liquidus temperature is modelled as a Newtonian fluid but the flows in the weld pool are not accounted for. Therefore numerical simulations were carried out in the context of a Lagrangian approach. The aim is to adopt the most realistic approach in order to predict residual stress states in welded structures. In addition, due to the huge differences between the mechanical behaviors of the liquid and solid phases, the molten zone needs to be stabilized by including the surface tension effects. In this study, instead of taking the surface tension effects into account through the loads induced on the fluid [7,8], a recent approach proposed by Leblond et al. [9] is adopted. This approach consists in applying directly a surface spherical stress state in membrane elements of zero thickness representing the interface.

The numerical simulation of such problems is known to be very time consuming,especially for industrial cases presenting complex geometries. That’s why linear tetrahedral elements of type P1 seems to be interesting for such applications. However, the treatment of incompressible or nearly incompressible problems, such as elasto(visco)plastic behavior based on von Mises criterion, qualified as quasi-incompressible,can lead to numerical locking phenomena. Numerically, such problems, require specific methods like for example reduced or selective integration technique [10,11]. Yet, these techniques cannot be applied to linear tetrahedral elements of type P1 because these elements have only one integration point.Incompressibility can also be considered by means of mixed formulations which consist inintroducing new variables called Lagrange multipliers such as the pressure [12]. In the case of mixed displacement-pressure formulations, in order to ensure stabilization, the Ladyzenskaya-Babuska-Brezzi (LBB) condition must be fulfilled [13,14]. The latter is a severe condition which ensures the compatibility between the approximation spaces of the displacement and of the pressure [15]. In literature, several tetrahedral finite elements satisfy this condition such asP2/P1 or P1+/P1 elements [16]. But for these elements, additional nodes are included, leading to a significant increase of the size of the problem. In the present study, a discretization of type mixed displacement-pressure P1/P1 is proposed to model an elasto-viscoplastic behavior of a solid metal in the case of welding simulation.

Finally a representative example of a laser welding is processed. The aim is to show the ability of the presented approach to predict the residual stress state of welded components.

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2 MODELLING OF THE MUSHY ZONE: MIXT LAW BEHAVIOR

2.1 Mixture law behavior During welding, the material exists both in the liquid state and the solid state. The liquid

phase is modeled as a Newtonian fluid while a thermo-elasto-viscoplastic behavior is considered for the solid phase. The equation of the Chaboche’s model considered for the solid phase and the values of the parameters used are given in [18]. Note that the parameters given in [18] unable to represent the material for temperature ranging from ambient temperature up to the beginning of the mushy zone.

During welding , in order to ensure a continuous transition between the solid and the liquid phases the recent approach proposed by Amin El Sayed et al. [17] is adopted in the current study. The mixture law combines liquid L-phase and solid S-phase, which are quantified by their volume fractions, 𝑓𝑓𝐿𝐿 and 𝑓𝑓𝑆𝑆. The behavior of the mixture is obtained by combining aparallel scheme for the deviatoric part of the stresses 𝑠𝑠 and a series scheme for the spherical part 𝑝𝑝 (figure 2).

Figure 2: Rheological scheme of the mixture model

The pressure and the deviatoric part of the deformation tensor are supposed to be the same in the solid and the liquid phases. The stress tensor is then obtained by equation 1.

s pI (1)

L sL ss f s f s (2)

s Lp p p (3)

The evolution of the solid phase is determined by the temperature according to the linear mixture rule given by equation 4.

0

1

S L

LS s L

L s

S L

f if T TT Tf if T T TT T

f if T T

(4)

Solid

Liquid

Liquid Solid

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Where 𝑓𝑓𝑠𝑠 = 1 − 𝑓𝑓𝐿𝐿 , 𝑇𝑇𝐿𝐿 and 𝑇𝑇𝑆𝑆 denoting the liquidus and the solidus temperature respectively.

2.2 Application: Inconel 600 alloy’s behavior during welding In the current study, a linear approximation of the solidification path of the Inconel 600

Alloy exposed in [18] is proposed. Consequently, the solidus and the liquidus temperatures obtained are 𝑇𝑇𝑆𝑆 = 1386°𝐶𝐶 and 𝑇𝑇𝐿𝐿 = 1430°𝐶𝐶.Based on high temperature tensile tests, the behavior of the Inconel 600 alloy was characterized experimentally by Bouffier [18] in the solid state till 1340°C. Determined parameters of the elasto-viscoplastic model with mixed isotropic / kinematic hardening werethen extrapolated to higher temperature in the mushy zone.Figure 3 summarizes the behavior of Inconel 600 for the different temperature ranges.

Figure 3 Rheological scheme of the mixture model for the Inconel 600 alloy

3 MIXED DISPLACEMENT-PRESSURE P1/P1 FINITE ELEMENT TO DEAL WITH ELSATO (VISCO) PLASTIC PROBLEM

3.1 Mixed formulation for the problem The Chaboche model is a thermo-elasto-viscoplastic model based on a von Mises criterion.

The formulation of the mechanical problem to deal with this behavior can be written in a mixed format considering two variables; the hydrostatic pressure 𝑝𝑝 and the displacement field 𝑢𝑢. Let us consider a bounded region Ω of R3, and 𝜕𝜕Ω its boundary. Therefore, the weakformulation of the mixed problem to be solved under a small perturbation assumption can be formulated as:

* * * **

* *

, : . 0

, ( . 3 ) 0

d du s dv p u dv u f dv u F ds

pp p u dv

th

(5)

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Where 𝑢 ∗, 𝑝𝑝∗, 𝑠𝑠𝑠, 𝜀𝜀𝑠, 𝜀𝜀𝑡𝑡𝑡, 𝑓𝑓 𝑑𝑑, 𝐹𝐹 𝑑𝑑 and 𝛼𝛼 are respectively the virtual displacement field, thevirtual pressure, the deviatoric part of the Cauchy stress tensor, the strain tensor, the thermal strain, the volume forces, the forces applied on the boundary ∂Ω and the bulk modulus.

3.2 P1 /P1-Finite element discretization To solve these equations, the mixed displacement/pressure finite element method is used.

The approximations of the displacement and pressure are then written (equation 6):4

14

1

( , ) ( ) ( )

( , ) ( ) ( )

i iu

i

ip

i

u x t N x u t

p x t N x p t

i

(6)

𝑢 𝑖𝑖 and 𝑝𝑝𝑖𝑖 denote the vectors of the elementary nodal unknowns at the 4 nodes of thetetrahedral finite element P1/P1 . 𝑁𝑁𝑢𝑢

𝑖𝑖 and 𝑁𝑁𝑝𝑝𝑖𝑖 are the linear shape functions associated to this

node respectively for the displacement and the pressure fields.In order to solve the problem described above, an iterative method such as the Newton-Raphson method is used. The system of linear equations to be solved at each iteration 𝑘𝑘 at time 𝑡𝑡 is of the form:

kk kuuu up

pu pp p

RK K uK K Rp

(7)

Note that the sub-matrix 𝐾𝐾𝑝𝑝𝑝𝑝 appearing on the stiffness matrix is not zero unlike the case of incompressible materials.

4 CONSIDERATION OF THE SURFACE TENSION In order to take into account the surface tension, a surface spherical stress state is directly

imposed in membrane elements incorporated in the mesh and representing the interface. The stress state to impose at any point of the interface, and consequently in membrane elements, can be written:

00

(8)

This stress state is naturally taken into account by including the expression ∫ [𝐵𝐵]𝑇𝑇𝜎𝜎𝑑𝑑𝑑𝑑𝜕𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

in the weak form of the problem where [𝐵𝐵] is the operator linking the

strain to the nodal values of displacement and 𝜎𝜎 = 𝜎𝜎11𝜎𝜎22𝜎𝜎12

.

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5 APPLICATION: NUMERICAL SIMULATION OF THE LASER WELDING OFTHE INCONEL 600

A test is performed using SYSWELD® [19] software to simulate a representative model ofa laser welding. A square plate of 40 mm side and 3 mm thick is considered. The thermal load is represented by an equivalent circular heating source of Goldak type applied on the top center of the plate [20]. As shown in figure 4, this plate admits two planes of symmetry. Thus, only the quarter of the plate is modeled (Figure 4). The quarter of the plate is meshed with the linear tetrahedral elements P1/P1.

Figure 4 Mesh and boundary conditions

The material considered in this application is the Inconel 600. As mentioned in the section 2.2, the parameters of the Lemaître & Chaboche model with mixed isotropic/kinematic hardening [21] are taken from [18] in order to describe the elsto-viscoplastic behavior of the solid phase. The viscosity 𝜇𝜇𝜇of the Newtonian fluid characterizing the liquid phase is taken equal to𝜇6.34𝜇10−3 Pa.s [22]. The compressibility modulus coefficient 𝛼𝛼 𝛼 1

3𝐸𝐸

1−2𝜐𝜐 is computed with 𝜐𝜐 𝛼 0.3 which corresponds to a classical value of the Poisson’s coefficient for solid metals.The BFGS technique (Broyden-Fletcher-Goldfarb-Shanno) is applied to solve the nonlinear problem at each time step.

6 RESULTS AND DISCUSSION

6.1 Thermal results Heating has occurred during the first two seconds. After the first two heating seconds, the

plate is cooled by conduction to reach the ambient temperature. Temperature field is presented on the figure 5. The mushy zone appears delimited by the solidus and the liquidus.

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Figure 5 Temperature distribution (°C) in the plate during welding (1.9s) 6.2 mechanical results

6.2 von Mises stress fields Figure 6 shows von Mises stress field during heating. In the mushy zone, as the liquid

fraction 𝑓𝑓𝐿𝐿 increases, the stress state tends to 0, since only the deviatoric part of the remaining solid contributes (equation 2). In the liquid phase, one can observe that for temperatureshigher than the liquidus, the material is not totally free of stress (≤ 10 MPa). This is due to the remaining solid fraction (fs=1%) necessary to ensure easier convergence.

Figure 6 von Mises stress fields (MPa) in the plate during welding (1.9s)

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6.3 Residual stress fields: Comparison between P1/P1 and Q1 finite elements The residual stress fields generated by welding are obtained after cooling (110 s). Figure 7

shows a comparison between two cases. In the first one (figure7 (a)), the plate is meshed with standard hexahedral elements type Q1 (6480 elements). In the second case (figure 7(b)), the plate is meshed with a linear tetrahedral elements P1/P1 (38880 elements) built by dividing each hexahedron in 6 tetrahedral elements without adding nodes.

Figure 7 Distribution of von Mises stress (MPa) at time t = 110s: (a) hexahedral mesh (Q1), (b) tetrahedral mesh (P1/P1)

Computed residual stress distribution obtained by P1/P1 elements reveals a good similitude with results obtained using hexahedral mesh. In term of computing time, it may be noticed that the element P1/P1, with only one centered point integration, leads to lower computation time (720 s) than the Q1 element (1320 s) which is more difficult to handle complex geometries.

6.4 Effect of surface tensionThe surface tension is considered by incorporating tight membrane elements into the mesh.

To illustrate the application of the proposed approach, a uniform surface tension within the surface of the weld pool is considered. γ is taken equal to 0.002 Nmm-1.Figure 8 compares the vertical displacement fields obtained in the liquid zone at the end of the heating to that obtained without taken into account surface tension. We can see that the maximum vertical displacement of the liquid zone is reduced when considering surface tension. Furthermore, one can note that the upper free surface of the weld pool is smoother (figure 8 (c)) compared to the surface obtained in the figure (figure 8 (c)).

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Figure 8 Displacement fields induced in the liquid zone during welding: (a) the liquid zone size, (b) without considering surface tension, (c) surface tension taking into account through membrane elements

7 CONCLUSIONS The aim of this paper is to improve the simulation of welding process by adopting the most

realistic approach in term of material behavior, meshing and surface tension. The interest of such approach remains in a better description of the residual stress field generated in welded structure. Whereas in classical approaches, nor the liquid phase, nor the mushy zone behaviors are generally considered for the computation of residual stress distortions, in the present approach the continuous transition between the liquid and the solid phases was ensured taking into account material properties at very high temperature.Practical application of the approach was illustrated through a representative case of welding simulation. The convergence was reached properly and leads to satisfactory results compared to reference case. In term of meshing, results show the relevance of the mixed tetrahedral finite element type P1 / P1 to simulate elasto-viscoplastic highly nonlinear problems. The example shows also the effect of surface tension to improve the shape of the free surface of the weld pool.

ACKNOWLEDGMENTS The authors are grateful to MUSICAS project for financial supporting of this research.

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