- Email: [email protected]
A coupled thermo-viscoplastic formulation at finite strains for the numerical simulation of superplastic forming
Journal of Materials Processing Technology 139 (2003) 514–520
A coupled thermo-viscoplastic formulation at finite strains for the numerical simulation of superplastic forming Laurent Adam, Jean-Philippe Ponthot∗ LTAS—Milieux Continus et Thermomécanique, Université de Liège, 1, Chemin des Chevreuils. B-4000 Liège-1, Belgium
Abstract This paper deals with a thermo-elasto-viscoplastic formulation for the simulation of metal forming involving large deformations. The authors present the governing equations of the model and some applications to superplastic forming. The numerical examples illustrate the potentiality of the finite-element code METAFOR based on this formulation. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Superplastic forming; Large deformation; Visco-plasticity; Diffusion bonding; Finite element
1. Introduction The goal in this paper is to present a complete thermoviscoplastic formulation at finite strains and its implementation in the finite-element code METAFOR [1]. The formulation is derived from the classical J2-plasticity and is extended to take into account thermal and rate-dependent effects. A staggered scheme will be used for the resolution of the thermomechanical problem as well as an extension of the radial return algorithm for the integration of the constitutive law and a penalty approach to manage contact interactions. Finally, some results of superplastic forming (SPF) will be presented and discussed. 2. Non-isothermal large strains formulation 2.1. Constitutive equation In the present model, the authors use an additive decomposition of the rate of deformation [2,3] and assume an hypo-elastic stress–strain relation of the form: ∇
vp
σ ij = H(T)ijkl (Dkl − Dkl − Dth kl )
(1)
where H(T) is the temperature-dependent Hooke stress–strain ∇ tensor; T the temperature; σ an objective rate of the Cauchy stress tensor; D the rate of deformation; Dvp the viscoplastic part of D; De the elastic part of D; and Dth the thermal part of D. ∗ Corresponding author. Tel.: +32-43669310; fax: +32-43669141. E-mail address: [email protected] (J.-P. Ponthot).
In the case of viscoplastic deformation a classical flow rule is used for associative viscoplasticity given by sij (2) Dvp = ΛN, where N ij = √ skl skl is the unit outward normal (N:N = 1) to the yield surface and Λ a positive parameter called the consistency parameter. In a viscoplastic formulation, Λ cannot be determined, as in J2-plasticity, by expressing the so-called consistency condition. Thus one more equation is needed to be able to express that consistency parameter, and thus the viscoplastic part of the rate of deformation. Various models have been proposed to express Λ in terms of the current stress level. For example, classical viscoplastic models of the Perzyna type [4,5] use a consistency parameter of the form: m(T ) 3 σ¯ − σv (T ) Λ= (3) 2 η(T )(vp )1/n(T ) √ where σ¯ is the effective stress, i.e. σ¯ = (3/2)s : s; s the deviator of the stress tensor; σ v (T) the current yield stress; n(T) a hardening exponent; m(T) a rate sensitivity parameter; and η(T) a viscosity parameter. The resulting yield function is the viscoplastic extension of the classical von Mises criterion [6] and is given by vp f¯ = σ¯ − σv (T) − η(vp )1/n (¯˙ )1/m (4) The equations governing the evolution of the thermal part of the tensor of the rate of deformation is a generalisation of the equation used in infinitesimal strain theory, which is given by ˙ Dth (5) ij = βT δij where β is the linear thermal expansion coefficient.
0924-0136/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00529-6
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
515
Fig. 1. Description of the process setup.
2.2. Integration procedure Consider [tn , tn+1 ] a sub-interval of the global time interval of interest, and φ the vector of primary variables defined as φ = (x, v, T )
(6)
where x is the current nodal coordinates vector; v the current nodal velocities vector; and T the current nodal temperature. There is the need to determine φ at time tn+1 and so need to integrate the constitutive law and to update the vector of internal parameters. The governing equations can be formulated as a first order problem of evolution. In the present model, it was decided to use the backward-Euler formula for the approximation of the time derivatives, and the finite-element technique for the spatial discretization. For the time integration a staggered scheme is used, based on a split of the first order operator, which, in comparison with a monolithic scheme, considerably reduces the computational cost of such resolution [7]. As far as the integration of the constitutive law is concerned, there is the need to find the new values of the stress tensor, viscoplastic strains and other internal parameters given the incremental strain history. To compute this part of the formulation the authors rely on the general methodology of an elastic-predictor/plastic-corrector (return mapping algorithm), as synthesised in Simo and Hughes [8]
Fig. 2. Final configurations for different forming temperatures.
Fig. 3. Final distribution of the J2 stress (in MPa) for different forming temperatures. Table 1 Computational features of the forming of a quarter cylinder Forming time CPU (forming at 493 K) Number of time steps Number of iterations
3100 s 1 min 47 s 250 657
Table 2 Material properties for Ti–6Al–4V Young’s modulus (N/mm2 ) Poisson’s ratio Viscoplastic behaviour (N/mm2 ) Friction coefficient (µ)
Fig. 4. SPF of a hemisphere.
E = 15000 η = 0.33 vp σ¯ = 460(˙¯ )0.5 0.2
516
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
Fig. 5. Maximum value of the effective viscoplastic strain rate.
but here, they use an extension of this methodology to the time-dependent case (see [6,9] for details).
3. Numerical illustrations 3.1. Forming of a quarter cylinder This numerical test aims to show especially the importance of thermal spring-back, and so of the unloading phase,
at different forming temperatures. The material parameters are typical of an aluminium alloy which has superplastic behaviour at high temperature. The test conditions are represented in Fig. 1. The right end of the sheet is clamped and the rigid die has a fixed position. The pressure and the temperature are, in a first phase, increased from their room values to their forming values, then they are maintained during the stress relaxation phase, and finally lowered to the room value. In the results exposed below, the thickness to die radius ratio is 1/50π.
Fig. 6. Pressure cycle versus time.
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
517
Table 3 Computational features of the SPF of a hemisphere Forming time CPU (finer mesh) Number of time steps Number of iterations
4650 s 1 min 30 s 251 571
The forming time is 3100 s (300 s for the loading, 2700 s for the stress relaxation and 100 s for the unloading). The mesh is composed of 50 × 4 quadrilateral finite elements with constant pressure. Fig. 2 shows final configurations for the different forming temperatures. One can observe that the cold forming of the sheet induces few irreversible deformations and so the spring-back is almost total. On the other hand, for the forming at high temperature, there is almost no spring-back. The final stress level is a decreasing function of the forming temperature, as can be seen in Fig. 3. These stress distributions are plotted in the initial configuration with a dilatation factor of 15 for the coordinates in the thickness direction. The computational features of this numerical simulation are given in Table 1. 3.2. SPF of a hemisphere
Fig. 7. Effective viscoplastic strain rate distributions at different stages of the process.
This case was proposed by Bellet and Chenot [10] and consists in the forming of a Ti–6Al–4V sheet into a hemispherical rigid die. The dimensions of the sheet and its die are given in Fig. 4. The periphery of the sheet has fixed radial displacements. The material rheology is given in Table 2 the optimum strain rate to induce superplastic deformations being 3 × 10−4 s−1 .
Fig. 8. Thickness distribution in the final configuration versus radial coordinates.
518
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
Fig. 9. Thickness distribution for two different friction coefficients versus radial coordinates.
The next two figures illustrate the automatic control of the maximum effective viscoplastic strain rate (Fig. 5) via the applied pressure (Fig. 6). Fig. 7 shows different configurations of the sheet during the forming process (the last one is the final shape). The iso-values plotted in this figure are those of the effective viscoplastic strain rate. It is observed that during the first stages of the process, the deformation is concentrated on the periphery in order to form the sheet into the die part which has the smallest curvature radius. Next, the maximum effective viscoplastic strain is located around the centre of the sphere inducing the thinning of this area. In this computation, the authors have used two different meshes. The coarser one is made of 30 × 1 quadrilateral axisymmetric elements (still with constant pressure) and the finer one is composed of 100 × 4 elements. Most of the numerical results are those produced using from the finer mesh. Fig. 8 shows the final thickness of the sheet using the two different meshes and are compared to the results of Bellet and Chenot [10]. The coarser mesh was chosen to have almost the same mesh density as those of Bellet and Chenot. One can observe a good agreement between the results. The finer mesh improves the results in the outer part
of the sheet, contrarily to the coarser one, which does not contain enough elements in this region to correctly simulate the forming of the transition radius. Fig. 9 shows the influence of the friction coefficient on the thickness profile. Relatively important differences are observed near the periphery of the sheet. The computational features of this numerical simulation are given in Table 3. The final thickness of the sheet can be modulated using a rigid punch to partially form the sheet instead of applying a pressure during the whole process. The process setup is exposed in Fig. 10. In the following results, the vertical displacement of the punch, Dy, is 170 mm and next a controlled pressure is applied to complete the forming. Since the forming pressure is controlled, the displacement of the punch
Table 4 Computational features of the SPF–DB process Forming time CPU Number of time steps Number of iterations
1500 s 4 min 03 s 299 752
Fig. 10. Forming of a hemisphere using a rigid punch.
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
519
Fig. 11. Thickness profiles in the case of punch + pressure forming.
also needs to be controlled. Fig. 11 shows the final thickness for different friction coefficients of the punch (keeping the friction coefficient between the die and the sheet at 0.2). It is observed that the lowest value of the thickness can be improved using this type of process. It is also evident that Dy and the friction coefficients can be optimised to produce the ideal thickness distribution.
an SPF–diffusion bonding (SPF–DB) process which is used advantageously to produce light sandwich panels. The process consists of three superposed sheets, each 1 mm thick, of titanium alloy bonded together over a part of the total area by DB and formed by the injection of gas within the cells. The material behaviour is the same as in the hemisphere forming simulation (see Table 2).
3.3. SPF–DB forming of multi-cell sandwich sheet This numerical example is also inspired by one proposed in Bellet and Chenot [10]. This example simulates
Fig. 12. Process setup of an SPF–DB of a sandwich panel.
Fig. 13. Different computed configurations of an SPF–DB of a sandwich panel.
520
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
Fig. 12 shows the process setup. The upper side of the panel has fixed vertical displacements, and the left end of the panel is clamped. In this case the DB is simulated by a perfect sticking contact condition. Different computed configurations are represented in Fig. 13. The computational features of this numerical simulation are given in Table 4.
4. Conclusions A complete thermo-elasto-viscoplastic formulation was derived and implemented in a finite-element code. This formulation is able to simulate non-isothermal metal forming involving viscous behaviour. Further developments will have the aim of simulating more accurately the microscopic evolution of the material through a non-isothermal process and its influence over macroscopic constitutive coefficients. Heat transfer computation during contact interactions is also in progress. These two advances will enable the authors to simulate more precisely and completely the DB process.
References [1] J.P. Ponthot, Traitement unifié de la Mécanique des Milieux Continus solides en grandes transformations par la méthode des éléments finis, Ph.D. Thesis, University of Liège, Liège, Belgium, 1995. [2] T.B. Whertheimer, in: Pittman, Wood, Alexander, Zienkiewicz (Eds.), Numerical Methods in Industrial Forming Processes, 1982, pp. 425–434. [3] P. Wriggers, C. Miehe, M. Kleiber, J.C. Simo, in: D.R.J. Owen, E. Hinton, E. Oñate (Eds.), Proceedings of the International Conference on Computational Plasticity (COMPLAS2), Barcelona, Spain, Pineridge Press, 1989, pp. 527–542. [4] P. Perzyna, Fundamental problems in visco-plasticity, in: G. Kuerti (Ed.), Advances in Applied Mechanics, vol. 9, Academic Press, New York, 1966, pp. 243–377. [5] P. Perzyna, Thermodynamic theory of plasticity, in: C.-S. Yih (Ed.), Advances in Applied Mechanics, vol. 11, Academic Press, New York, 1971, pp. 313–355. [6] J.P. Ponthot, Int. J. Plast. 18 (2002) 91–126. [7] F. Armero, J.C. Simo, Int. J. Numer. Meth. Eng. 35 (1992) 737– 766. [8] J.C. Simo, T.J.R Hughes, in: Constitutive Laws for Engineering Materials: Theory and Applications, Elsevier, Amsterdam, 1987. [9] J.P. Ponthot, J. Mater. Process. Technol. 80–81 (1998) 628–634. [10] M. Bellet, J.L. Chenot, in: Numerical Methods in Industrial Forming Processes (NUMIFORM), 1989, pp. 401–406.
A coupled thermo-viscoplastic formulation at finite strains for the numerical simulation of superplastic forming Laurent Adam, Jean-Philippe Ponthot∗ LTAS—Milieux Continus et Thermomécanique, Université de Liège, 1, Chemin des Chevreuils. B-4000 Liège-1, Belgium
Abstract This paper deals with a thermo-elasto-viscoplastic formulation for the simulation of metal forming involving large deformations. The authors present the governing equations of the model and some applications to superplastic forming. The numerical examples illustrate the potentiality of the finite-element code METAFOR based on this formulation. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Superplastic forming; Large deformation; Visco-plasticity; Diffusion bonding; Finite element
1. Introduction The goal in this paper is to present a complete thermoviscoplastic formulation at finite strains and its implementation in the finite-element code METAFOR [1]. The formulation is derived from the classical J2-plasticity and is extended to take into account thermal and rate-dependent effects. A staggered scheme will be used for the resolution of the thermomechanical problem as well as an extension of the radial return algorithm for the integration of the constitutive law and a penalty approach to manage contact interactions. Finally, some results of superplastic forming (SPF) will be presented and discussed. 2. Non-isothermal large strains formulation 2.1. Constitutive equation In the present model, the authors use an additive decomposition of the rate of deformation [2,3] and assume an hypo-elastic stress–strain relation of the form: ∇
vp
σ ij = H(T)ijkl (Dkl − Dkl − Dth kl )
(1)
where H(T) is the temperature-dependent Hooke stress–strain ∇ tensor; T the temperature; σ an objective rate of the Cauchy stress tensor; D the rate of deformation; Dvp the viscoplastic part of D; De the elastic part of D; and Dth the thermal part of D. ∗ Corresponding author. Tel.: +32-43669310; fax: +32-43669141. E-mail address: [email protected] (J.-P. Ponthot).
In the case of viscoplastic deformation a classical flow rule is used for associative viscoplasticity given by sij (2) Dvp = ΛN, where N ij = √ skl skl is the unit outward normal (N:N = 1) to the yield surface and Λ a positive parameter called the consistency parameter. In a viscoplastic formulation, Λ cannot be determined, as in J2-plasticity, by expressing the so-called consistency condition. Thus one more equation is needed to be able to express that consistency parameter, and thus the viscoplastic part of the rate of deformation. Various models have been proposed to express Λ in terms of the current stress level. For example, classical viscoplastic models of the Perzyna type [4,5] use a consistency parameter of the form: m(T ) 3 σ¯ − σv (T ) Λ= (3) 2 η(T )(vp )1/n(T ) √ where σ¯ is the effective stress, i.e. σ¯ = (3/2)s : s; s the deviator of the stress tensor; σ v (T) the current yield stress; n(T) a hardening exponent; m(T) a rate sensitivity parameter; and η(T) a viscosity parameter. The resulting yield function is the viscoplastic extension of the classical von Mises criterion [6] and is given by vp f¯ = σ¯ − σv (T) − η(vp )1/n (¯˙ )1/m (4) The equations governing the evolution of the thermal part of the tensor of the rate of deformation is a generalisation of the equation used in infinitesimal strain theory, which is given by ˙ Dth (5) ij = βT δij where β is the linear thermal expansion coefficient.
0924-0136/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00529-6
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
515
Fig. 1. Description of the process setup.
2.2. Integration procedure Consider [tn , tn+1 ] a sub-interval of the global time interval of interest, and φ the vector of primary variables defined as φ = (x, v, T )
(6)
where x is the current nodal coordinates vector; v the current nodal velocities vector; and T the current nodal temperature. There is the need to determine φ at time tn+1 and so need to integrate the constitutive law and to update the vector of internal parameters. The governing equations can be formulated as a first order problem of evolution. In the present model, it was decided to use the backward-Euler formula for the approximation of the time derivatives, and the finite-element technique for the spatial discretization. For the time integration a staggered scheme is used, based on a split of the first order operator, which, in comparison with a monolithic scheme, considerably reduces the computational cost of such resolution [7]. As far as the integration of the constitutive law is concerned, there is the need to find the new values of the stress tensor, viscoplastic strains and other internal parameters given the incremental strain history. To compute this part of the formulation the authors rely on the general methodology of an elastic-predictor/plastic-corrector (return mapping algorithm), as synthesised in Simo and Hughes [8]
Fig. 2. Final configurations for different forming temperatures.
Fig. 3. Final distribution of the J2 stress (in MPa) for different forming temperatures. Table 1 Computational features of the forming of a quarter cylinder Forming time CPU (forming at 493 K) Number of time steps Number of iterations
3100 s 1 min 47 s 250 657
Table 2 Material properties for Ti–6Al–4V Young’s modulus (N/mm2 ) Poisson’s ratio Viscoplastic behaviour (N/mm2 ) Friction coefficient (µ)
Fig. 4. SPF of a hemisphere.
E = 15000 η = 0.33 vp σ¯ = 460(˙¯ )0.5 0.2
516
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
Fig. 5. Maximum value of the effective viscoplastic strain rate.
but here, they use an extension of this methodology to the time-dependent case (see [6,9] for details).
3. Numerical illustrations 3.1. Forming of a quarter cylinder This numerical test aims to show especially the importance of thermal spring-back, and so of the unloading phase,
at different forming temperatures. The material parameters are typical of an aluminium alloy which has superplastic behaviour at high temperature. The test conditions are represented in Fig. 1. The right end of the sheet is clamped and the rigid die has a fixed position. The pressure and the temperature are, in a first phase, increased from their room values to their forming values, then they are maintained during the stress relaxation phase, and finally lowered to the room value. In the results exposed below, the thickness to die radius ratio is 1/50π.
Fig. 6. Pressure cycle versus time.
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
517
Table 3 Computational features of the SPF of a hemisphere Forming time CPU (finer mesh) Number of time steps Number of iterations
4650 s 1 min 30 s 251 571
The forming time is 3100 s (300 s for the loading, 2700 s for the stress relaxation and 100 s for the unloading). The mesh is composed of 50 × 4 quadrilateral finite elements with constant pressure. Fig. 2 shows final configurations for the different forming temperatures. One can observe that the cold forming of the sheet induces few irreversible deformations and so the spring-back is almost total. On the other hand, for the forming at high temperature, there is almost no spring-back. The final stress level is a decreasing function of the forming temperature, as can be seen in Fig. 3. These stress distributions are plotted in the initial configuration with a dilatation factor of 15 for the coordinates in the thickness direction. The computational features of this numerical simulation are given in Table 1. 3.2. SPF of a hemisphere
Fig. 7. Effective viscoplastic strain rate distributions at different stages of the process.
This case was proposed by Bellet and Chenot [10] and consists in the forming of a Ti–6Al–4V sheet into a hemispherical rigid die. The dimensions of the sheet and its die are given in Fig. 4. The periphery of the sheet has fixed radial displacements. The material rheology is given in Table 2 the optimum strain rate to induce superplastic deformations being 3 × 10−4 s−1 .
Fig. 8. Thickness distribution in the final configuration versus radial coordinates.
518
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
Fig. 9. Thickness distribution for two different friction coefficients versus radial coordinates.
The next two figures illustrate the automatic control of the maximum effective viscoplastic strain rate (Fig. 5) via the applied pressure (Fig. 6). Fig. 7 shows different configurations of the sheet during the forming process (the last one is the final shape). The iso-values plotted in this figure are those of the effective viscoplastic strain rate. It is observed that during the first stages of the process, the deformation is concentrated on the periphery in order to form the sheet into the die part which has the smallest curvature radius. Next, the maximum effective viscoplastic strain is located around the centre of the sphere inducing the thinning of this area. In this computation, the authors have used two different meshes. The coarser one is made of 30 × 1 quadrilateral axisymmetric elements (still with constant pressure) and the finer one is composed of 100 × 4 elements. Most of the numerical results are those produced using from the finer mesh. Fig. 8 shows the final thickness of the sheet using the two different meshes and are compared to the results of Bellet and Chenot [10]. The coarser mesh was chosen to have almost the same mesh density as those of Bellet and Chenot. One can observe a good agreement between the results. The finer mesh improves the results in the outer part
of the sheet, contrarily to the coarser one, which does not contain enough elements in this region to correctly simulate the forming of the transition radius. Fig. 9 shows the influence of the friction coefficient on the thickness profile. Relatively important differences are observed near the periphery of the sheet. The computational features of this numerical simulation are given in Table 3. The final thickness of the sheet can be modulated using a rigid punch to partially form the sheet instead of applying a pressure during the whole process. The process setup is exposed in Fig. 10. In the following results, the vertical displacement of the punch, Dy, is 170 mm and next a controlled pressure is applied to complete the forming. Since the forming pressure is controlled, the displacement of the punch
Table 4 Computational features of the SPF–DB process Forming time CPU Number of time steps Number of iterations
1500 s 4 min 03 s 299 752
Fig. 10. Forming of a hemisphere using a rigid punch.
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
519
Fig. 11. Thickness profiles in the case of punch + pressure forming.
also needs to be controlled. Fig. 11 shows the final thickness for different friction coefficients of the punch (keeping the friction coefficient between the die and the sheet at 0.2). It is observed that the lowest value of the thickness can be improved using this type of process. It is also evident that Dy and the friction coefficients can be optimised to produce the ideal thickness distribution.
an SPF–diffusion bonding (SPF–DB) process which is used advantageously to produce light sandwich panels. The process consists of three superposed sheets, each 1 mm thick, of titanium alloy bonded together over a part of the total area by DB and formed by the injection of gas within the cells. The material behaviour is the same as in the hemisphere forming simulation (see Table 2).
3.3. SPF–DB forming of multi-cell sandwich sheet This numerical example is also inspired by one proposed in Bellet and Chenot [10]. This example simulates
Fig. 12. Process setup of an SPF–DB of a sandwich panel.
Fig. 13. Different computed configurations of an SPF–DB of a sandwich panel.
520
L. Adam, J.-P. Ponthot / Journal of Materials Processing Technology 139 (2003) 514–520
Fig. 12 shows the process setup. The upper side of the panel has fixed vertical displacements, and the left end of the panel is clamped. In this case the DB is simulated by a perfect sticking contact condition. Different computed configurations are represented in Fig. 13. The computational features of this numerical simulation are given in Table 4.
4. Conclusions A complete thermo-elasto-viscoplastic formulation was derived and implemented in a finite-element code. This formulation is able to simulate non-isothermal metal forming involving viscous behaviour. Further developments will have the aim of simulating more accurately the microscopic evolution of the material through a non-isothermal process and its influence over macroscopic constitutive coefficients. Heat transfer computation during contact interactions is also in progress. These two advances will enable the authors to simulate more precisely and completely the DB process.
References [1] J.P. Ponthot, Traitement unifié de la Mécanique des Milieux Continus solides en grandes transformations par la méthode des éléments finis, Ph.D. Thesis, University of Liège, Liège, Belgium, 1995. [2] T.B. Whertheimer, in: Pittman, Wood, Alexander, Zienkiewicz (Eds.), Numerical Methods in Industrial Forming Processes, 1982, pp. 425–434. [3] P. Wriggers, C. Miehe, M. Kleiber, J.C. Simo, in: D.R.J. Owen, E. Hinton, E. Oñate (Eds.), Proceedings of the International Conference on Computational Plasticity (COMPLAS2), Barcelona, Spain, Pineridge Press, 1989, pp. 527–542. [4] P. Perzyna, Fundamental problems in visco-plasticity, in: G. Kuerti (Ed.), Advances in Applied Mechanics, vol. 9, Academic Press, New York, 1966, pp. 243–377. [5] P. Perzyna, Thermodynamic theory of plasticity, in: C.-S. Yih (Ed.), Advances in Applied Mechanics, vol. 11, Academic Press, New York, 1971, pp. 313–355. [6] J.P. Ponthot, Int. J. Plast. 18 (2002) 91–126. [7] F. Armero, J.C. Simo, Int. J. Numer. Meth. Eng. 35 (1992) 737– 766. [8] J.C. Simo, T.J.R Hughes, in: Constitutive Laws for Engineering Materials: Theory and Applications, Elsevier, Amsterdam, 1987. [9] J.P. Ponthot, J. Mater. Process. Technol. 80–81 (1998) 628–634. [10] M. Bellet, J.L. Chenot, in: Numerical Methods in Industrial Forming Processes (NUMIFORM), 1989, pp. 401–406.