# 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 & Continuous Dynamical Systems - S, 2017, 10 (6) : 1413-1466. doi: 10.3934/dcdss.2017075
##### References:
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Non Linéaire, 25 (2008), 1187-1208.  doi: 10.1016/j.anihpc.2007.05.009.  Google Scholar [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.  Google Scholar [7] M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11 (2004), 111-130.   Google Scholar [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin Heidelberg, 2002.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] A. D. Ioffe, On lower semicontinuity of integral functionals. Ⅰ, SIAM J. Control Optimization, 15 (1977), 521-538.  doi: 10.1137/0315035.  Google Scholar [22] R. Klein, Laser Welding of Plastics, John Wiley & Sons Inc., New York, 2012.  doi: 10.1002/9783527636969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [27] G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, WIAS Preprint 2025. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] R. Rossi, From visco to perfect plasticity in thermoviscoelastic materials, Preprint arXiv: 1609.07232. Google Scholar [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.  Google Scholar [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.  Google Scholar [35] T. Roubíček, Thermodynamics of perfect plasticity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 193-214.  doi: 10.3934/dcdss.2013.6.193.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] F. Solombrino, Quasistatic evolution problems for nonhomogeneous elastic plastic materials, J. Convex Anal., 16 (2009), 89-119.   Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [3] S. Bartels and T. Roubíček, Thermoviscoplasticity at small strains, ZAMM Z. Angew. Math. Mech., 88 (2008), 735-754.  doi: 10.1002/zamm.200800042.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. Brokate, P. Krejčí and H. Schnabel, On uniqueness in evolution quasivariational inequalities, J. Convex Anal., 11 (2004), 111-130.   Google Scholar [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin Heidelberg, 2002.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] A. D. Ioffe, On lower semicontinuity of integral functionals. Ⅰ, SIAM J. Control Optimization, 15 (1977), 521-538.  doi: 10.1137/0315035.  Google Scholar [22] R. Klein, Laser Welding of Plastics, John Wiley & Sons Inc., New York, 2012.  doi: 10.1002/9783527636969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [27] G. Lazzaroni, R. Rossi, M. Thomas and R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, WIAS Preprint 2025. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] R. Rossi, From visco to perfect plasticity in thermoviscoelastic materials, Preprint arXiv: 1609.07232. Google Scholar [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.  Google Scholar [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.  Google Scholar [35] T. Roubíček, Thermodynamics of perfect plasticity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 193-214.  doi: 10.3934/dcdss.2013.6.193.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [40] F. Solombrino, Quasistatic evolution problems for nonhomogeneous elastic plastic materials, J. Convex Anal., 16 (2009), 89-119.   Google Scholar
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