# American Institute of Mathematical Sciences

December  2017, 10(6): 1413-1466. doi: 10.3934/dcdss.2017075

## Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity

 DIMI -Universitá degli studi di Brescia, V. Branze, 38, Brescia, I-25133, Italy

Dedicated to Tomáš Roubíček on the occasion of his sixtieth birthday

Received  September 2016 Revised  December 2016 Published  June 2017

Fund Project: The author has been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

In this paper we address a model coupling viscoplasticity with damage in thermoviscoelasticity. The associated PDE system consists of the momentum balance with viscosity and inertia for the displacement variable, at small strains, of the plastic and damage flow rules, and of the heat equation. It has a strongly nonlinear character and in particular features quadratic terms on the right-hand side of the heat equation and of the damage flow rule, which have to be handled carefully. We propose two weak solution concepts for the related initial-boundary value problem, namely 'entropic' and 'weak energy' solutions. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Finally, in the case of a prescribed temperature profile, and under a strongly simplifying condition, we provide a continuous dependence result, yielding uniqueness of weak energy solutions.

Citation: Riccarda Rossi. Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1413-1466. doi: 10.3934/dcdss.2017075
##### References:
 [1] R. Alessi, J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), 575-615.  doi: 10.1007/s00205-014-0763-8. [2] R. Alessi, J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity: variational formulation and main properties, Mech. Mater., 80 (2015), 351-367.  doi: 10.1016/j.mechmat.2013.12.005. [3] S. Bartels and T. Roubíček, Thermoviscoplasticity at small strains, ZAMM Z. Angew. Math. Mech., 88 (2008), 735-754.  doi: 10.1002/zamm.200800042. [4] S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, ESAIM Math. Model. Numer. Anal., 45 (2011), 477-504.  doi: 10.1051/m2an/2010063. [5] E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1187-1208.  doi: 10.1016/j.anihpc.2007.05.009. [6] E. Bonetti, E. Rocca, R. Rossi and M. Thomas, A rate-independent gradient system in damage coupled with plasticity via structured strains, ESAIM: Proceedings and Surveys, 54 (2016), 54-69.  doi: 10.1051/proc/201654054. [7] M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11 (2004), 111-130. [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. [9] V. Crismale, Globally stable quasistatic evolution for a coupled elastoplastic-damage model, ESAIM Control Optim. Calc. Var., 22 (2016), 883-912.  doi: 10.1051/cocv/2015037. [10] V. Crismale, Globally stable quasistatic evolution for strain gradient plasticity coupled with damage, Ann. Mat. Pura Appl., 196 (2017), 641-685.  doi: 10.1007/s10231-016-0590-7. [11] V. Crismale and G. Lazzaroni, Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model Calc. Var. Partial Differential Equations, 55 (2016), Art. 17, 54pp. doi: 10.1007/s00526-015-0947-6. [12] G. Dal Maso, A. DeSimone and M. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Rational Mech. Anal., 180 (2006), 237-291.  doi: 10.1007/s00205-005-0407-0. [13] E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490.  doi: 10.1016/j.camwa.2006.02.042. [14] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.  doi: 10.1002/mma.1089. [15] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin Heidelberg, 2002. [16] G. Geymonat and P. Suquet, Functional spaces for Norton-Hoff materials, Math. Methods Appl. Sci., 8 (1986), 206-222.  doi: 10.1002/mma.1670080113. [17] C. Heinemann and C. Kraus, Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., 21 (2011), 321-359. [18] C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24 (2013), 179-211.  doi: 10.1017/S095679251200037X. [19] C. Heinemann and E. Rocca, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, Math. Methods Appl. Sci., 38 (2015), 4587-4612.  doi: 10.1002/mma.3393. [20] R. Herzog, C. Meyer and A. Stötzner, Existence of solutions of a (sonsmooth) thermoviscoplastic model and sssociated optimal control problems, Nonlinear Anal. Real World Appl., 35 (2017), 75-101.  doi: 10.1016/j.nonrwa.2016.10.008. [21] A. D. Ioffe, On lower semicontinuity of integral functionals. Ⅰ, SIAM J. Control Optimization, 15 (1977), 521-538.  doi: 10.1137/0315035. [22] R. Klein, Laser Welding of Plastics, John Wiley & Sons Inc., New York, 2012.  doi: 10.1002/9783527636969. [23] D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616.  doi: 10.1142/S021820251250056X. [24] P. Krejčí and J. Sprekels, On a system of nonlinear PDEs with temperature-dependent hysteresis in one-dimensional thermoplasticity, J. Math. Anal. Appl., 209 (1997), 25-46.  doi: 10.1006/jmaa.1997.5304. [25] P. Krejčí, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal., 34 (2002), 409-434.  doi: 10.1137/S0036141001387604. [26] P. Krejčí, J. Sprekels and U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl., 13 (2003), 695-712. [27] G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, WIAS Preprint 2025. [28] M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), 217-229.  doi: 10.1016/0022-1236(79)90113-7. [29] A. Mielke and R. Rossi, Existence and uniqueness results for a class of rate-independent hysteresis problems, Math. Models Methods Appl. Sci., 17 (2007), 81-123.  doi: 10.1142/S021820250700184X. [30] E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341.  doi: 10.1142/S021820251450002X. [31] E. Rocca and R. Rossi, "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM J. Math. Anal., 47 (2015), 2519-2586.  doi: 10.1137/140960803. [32] R. Rossi, From visco to perfect plasticity in thermoviscoelastic materials, Preprint arXiv: 1609.07232. [33] T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.  doi: 10.1137/080729992. [34] T. Roubíček, Nonlinear Partial Differential Equations with Applications, vol. 153 of International Series of Numerical Mathematics, 2nd edition, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1. [35] T. Roubíček, Thermodynamics of perfect plasticity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 193-214.  doi: 10.3934/dcdss.2013.6.193. [36] T. Roubíček, O. Souček and R. Vodička, A model of rupturing lithospheric faults with reoccurring earthquakes, SIAM J. Appl. Math., 73 (2013), 1460-1488.  doi: 10.1137/120870396. [37] T. Roubíček and G. Tomassetti, Thermomechanics of damageable materials under diffusion: Modelling and analysis, Z. Angew. Math. Phys., 66 (2015), 3535-3572.  doi: 10.1007/s00033-015-0566-2. [38] T. Roubíček and J. Valdman, Perfect plasticity with damage and healing at small strains, its modeling, analysis, and computer implementation, SIAM J. Appl. Math., 76 (2016), 314-340.  doi: 10.1137/15M1019647. [39] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [40] F. Solombrino, Quasistatic evolution problems for nonhomogeneous elastic plastic materials, J. Convex Anal., 16 (2009), 89-119.

show all references

Dedicated to Tomáš Roubíček on the occasion of his sixtieth birthday

##### References:
 [1] R. Alessi, J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity and nucleation of cohesive cracks, Arch. Ration. Mech. Anal., 214 (2014), 575-615.  doi: 10.1007/s00205-014-0763-8. [2] R. Alessi, J. Marigo and S. Vidoli, Gradient damage models coupled with plasticity: variational formulation and main properties, Mech. Mater., 80 (2015), 351-367.  doi: 10.1016/j.mechmat.2013.12.005. [3] S. Bartels and T. Roubíček, Thermoviscoplasticity at small strains, ZAMM Z. Angew. Math. Mech., 88 (2008), 735-754.  doi: 10.1002/zamm.200800042. [4] S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion, ESAIM Math. Model. Numer. Anal., 45 (2011), 477-504.  doi: 10.1051/m2an/2010063. [5] E. Bonetti and G. Bonfanti, Well-posedness results for a model of damage in thermoviscoelastic materials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 1187-1208.  doi: 10.1016/j.anihpc.2007.05.009. [6] E. Bonetti, E. Rocca, R. Rossi and M. Thomas, A rate-independent gradient system in damage coupled with plasticity via structured strains, ESAIM: Proceedings and Surveys, 54 (2016), 54-69.  doi: 10.1051/proc/201654054. [7] M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11 (2004), 111-130. [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. [9] V. Crismale, Globally stable quasistatic evolution for a coupled elastoplastic-damage model, ESAIM Control Optim. Calc. Var., 22 (2016), 883-912.  doi: 10.1051/cocv/2015037. [10] V. Crismale, Globally stable quasistatic evolution for strain gradient plasticity coupled with damage, Ann. Mat. Pura Appl., 196 (2017), 641-685.  doi: 10.1007/s10231-016-0590-7. [11] V. Crismale and G. Lazzaroni, Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model Calc. Var. Partial Differential Equations, 55 (2016), Art. 17, 54pp. doi: 10.1007/s00526-015-0947-6. [12] G. Dal Maso, A. DeSimone and M. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Rational Mech. Anal., 180 (2006), 237-291.  doi: 10.1007/s00205-005-0407-0. [13] E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490.  doi: 10.1016/j.camwa.2006.02.042. [14] E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.  doi: 10.1002/mma.1089. [15] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin Heidelberg, 2002. [16] G. Geymonat and P. Suquet, Functional spaces for Norton-Hoff materials, Math. Methods Appl. Sci., 8 (1986), 206-222.  doi: 10.1002/mma.1670080113. [17] C. Heinemann and C. Kraus, Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., 21 (2011), 321-359. [18] C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24 (2013), 179-211.  doi: 10.1017/S095679251200037X. [19] C. Heinemann and E. Rocca, Damage processes in thermoviscoelastic materials with damage-dependent thermal expansion coefficients, Math. Methods Appl. Sci., 38 (2015), 4587-4612.  doi: 10.1002/mma.3393. [20] R. Herzog, C. Meyer and A. Stötzner, Existence of solutions of a (sonsmooth) thermoviscoplastic model and sssociated optimal control problems, Nonlinear Anal. Real World Appl., 35 (2017), 75-101.  doi: 10.1016/j.nonrwa.2016.10.008. [21] A. D. Ioffe, On lower semicontinuity of integral functionals. Ⅰ, SIAM J. Control Optimization, 15 (1977), 521-538.  doi: 10.1137/0315035. [22] R. Klein, Laser Welding of Plastics, John Wiley & Sons Inc., New York, 2012.  doi: 10.1002/9783527636969. [23] D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616.  doi: 10.1142/S021820251250056X. [24] P. Krejčí and J. Sprekels, On a system of nonlinear PDEs with temperature-dependent hysteresis in one-dimensional thermoplasticity, J. Math. Anal. Appl., 209 (1997), 25-46.  doi: 10.1006/jmaa.1997.5304. [25] P. Krejčí, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal., 34 (2002), 409-434.  doi: 10.1137/S0036141001387604. [26] P. Krejčí, J. Sprekels and U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl., 13 (2003), 695-712. [27] G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, WIAS Preprint 2025. [28] M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal., 33 (1979), 217-229.  doi: 10.1016/0022-1236(79)90113-7. [29] A. Mielke and R. Rossi, Existence and uniqueness results for a class of rate-independent hysteresis problems, Math. Models Methods Appl. Sci., 17 (2007), 81-123.  doi: 10.1142/S021820250700184X. [30] E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341.  doi: 10.1142/S021820251450002X. [31] E. Rocca and R. Rossi, "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage, SIAM J. Math. Anal., 47 (2015), 2519-2586.  doi: 10.1137/140960803. [32] R. Rossi, From visco to perfect plasticity in thermoviscoelastic materials, Preprint arXiv: 1609.07232. [33] T. Roubíček, Thermodynamics of rate-independent processes in viscous solids at small strains, SIAM J. Math. Anal., 42 (2010), 256-297.  doi: 10.1137/080729992. [34] T. Roubíček, Nonlinear Partial Differential Equations with Applications, vol. 153 of International Series of Numerical Mathematics, 2nd edition, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0513-1. [35] T. Roubíček, Thermodynamics of perfect plasticity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 193-214.  doi: 10.3934/dcdss.2013.6.193. [36] T. Roubíček, O. Souček and R. Vodička, A model of rupturing lithospheric faults with reoccurring earthquakes, SIAM J. Appl. Math., 73 (2013), 1460-1488.  doi: 10.1137/120870396. [37] T. Roubíček and G. Tomassetti, Thermomechanics of damageable materials under diffusion: Modelling and analysis, Z. Angew. Math. Phys., 66 (2015), 3535-3572.  doi: 10.1007/s00033-015-0566-2. [38] T. Roubíček and J. Valdman, Perfect plasticity with damage and healing at small strains, its modeling, analysis, and computer implementation, SIAM J. Appl. Math., 76 (2016), 314-340.  doi: 10.1137/15M1019647. [39] J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360. [40] F. Solombrino, Quasistatic evolution problems for nonhomogeneous elastic plastic materials, J. Convex Anal., 16 (2009), 89-119.
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