# American Institute of Mathematical Sciences

February  2013, 6(1): 193-214. doi: 10.3934/dcdss.2013.6.193

## Thermodynamics of perfect plasticity

 1 Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8

Received  April 2011 Revised  August 2011 Published  October 2012

Viscoelastic solids in Kelvin-Voigt rheology at small strains exhibiting also stress-driven Prandtl-Reuss perfect plasticity are considered quasistatic (i.e. inertia neglected) and coupled with heat-transfer equation through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Enthalpy transformation is used and existence of a weak solution is proved by an implicit suitably regularized time discretisation.
Citation: Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193
##### References:
 [1] S. Bartels, A. Mielke and T. Roubíček, Quasistatic small-strain plasticity in the limit of vanishing hardening and its numerical approximation,, SIAM J. Numer. Anal., 50 (2012), 951. Google Scholar [2] S. Bartels and T. Roubíček, Thermo-visco-plasticity at small strains,, Zeitschrift angew. Math. Mech., 88 (2008), 735. doi: 10.1002/zamm.200800042. Google Scholar [3] S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion,, Math. Modelling Numer. Anal., 45 (2011), 477. doi: 10.1051/m2an/2010063. Google Scholar [4] L. Boccardo, A. Dall'aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, J. of Funct. Anal., 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar [5] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data,, J. Funct. Anal. \textbf{87} (1989), 87 (1989), 149. doi: 10.1016/0022-1236(89)90005-0. Google Scholar [6] K. Chełmiński, Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening,, Math. Methods Appl. Sci., 24 (2001), 117. Google Scholar [7] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Rational Mech. Anal., 176 (2005), 165. doi: 10.1007/s00205-004-0351-4. Google Scholar [8] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials,, Arch. Ration. Mech. Anal., 180 (2006), 237. doi: 10.1007/s00205-005-0407-0. Google Scholar [9] F. Ebobisse and B. D. Reddy, Some mathematical problems in perfect plasticity,, Computer Meth. Appl. Mech. Engr., 193 (2004), 5071. doi: 10.1016/j.cma.2004.07.002. Google Scholar [10] G. Francfort and A. Mielke, An existence result for a rate-independent material model in the case of nonconvex energies,, J. reine u. angew. Math., 595 (2006), 55. doi: 10.1515/CRELLE.2006.044. Google Scholar [11] J. Frehse and J. Málek, Boundary regularity results for models of elasto-perfect plasticity,, Math. Models Meth. Appl. Sci., 9 (1999), 1307. doi: 10.1142/S0218202599000579. Google Scholar [12] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis I,II,", Kluwer, (1997). Google Scholar [13] P. Krejčí and J. Sprekels, Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity,, Appl. Math., 43 (1998), 173. doi: 10.1023/A:1023224507448. Google Scholar [14] G. A. Maughin, "The Thermomechanics of Plasticity and Fracture,", Cambridge Univ. Press, (1992). doi: 10.1017/CBO9781139172400. Google Scholar [15] A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461. Google Scholar [16] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity,, Math. Modelling Numer. Anal., 43 (2009), 399. doi: 10.1051/m2an/2009009. Google Scholar [17] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. PDE, 31 (2008), 387. doi: 10.1007/s00526-007-0119-4. Google Scholar [18] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, in, (1999), 117. Google Scholar [19] A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151. Google Scholar [20] T. Roubíček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (2005). Google Scholar [21] T. Roubíček, Thermo-visco-elasticity at small strains with $L^1$-data,, Quarterly Appl. Math., 67 (2009), 47. Google Scholar [22] T. Roubíček, Rate independent processes in viscous solids at small strains,, Math. Methods Appl. Sci., 32 (2009), 825. doi: 10.1002/mma.1069. Google Scholar [23] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains,, SIAM J. Math. Anal., 42 (2010), 256. doi: 10.1137/080729992. Google Scholar [24] P. M. Suquet, Existence et régularité des solutions des équations de la plasticité parfaite,, C. R. Acad. Sci. Paris Sér. A, 286 (1978), 1201. Google Scholar [25] R. Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity,, Archive Rat. Mech. Anal., 95 (1986), 137. doi: 10.1007/BF00281085. Google Scholar

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##### References:
 [1] S. Bartels, A. Mielke and T. Roubíček, Quasistatic small-strain plasticity in the limit of vanishing hardening and its numerical approximation,, SIAM J. Numer. Anal., 50 (2012), 951. Google Scholar [2] S. Bartels and T. Roubíček, Thermo-visco-plasticity at small strains,, Zeitschrift angew. Math. Mech., 88 (2008), 735. doi: 10.1002/zamm.200800042. Google Scholar [3] S. Bartels and T. Roubíček, Thermo-visco-elasticity with rate-independent plasticity in isotropic materials undergoing thermal expansion,, Math. Modelling Numer. Anal., 45 (2011), 477. doi: 10.1051/m2an/2010063. Google Scholar [4] L. Boccardo, A. Dall'aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, J. of Funct. Anal., 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar [5] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data,, J. Funct. Anal. \textbf{87} (1989), 87 (1989), 149. doi: 10.1016/0022-1236(89)90005-0. Google Scholar [6] K. Chełmiński, Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening,, Math. Methods Appl. Sci., 24 (2001), 117. Google Scholar [7] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Rational Mech. Anal., 176 (2005), 165. doi: 10.1007/s00205-004-0351-4. Google Scholar [8] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials,, Arch. Ration. Mech. Anal., 180 (2006), 237. doi: 10.1007/s00205-005-0407-0. Google Scholar [9] F. Ebobisse and B. D. Reddy, Some mathematical problems in perfect plasticity,, Computer Meth. Appl. Mech. Engr., 193 (2004), 5071. doi: 10.1016/j.cma.2004.07.002. Google Scholar [10] G. Francfort and A. Mielke, An existence result for a rate-independent material model in the case of nonconvex energies,, J. reine u. angew. Math., 595 (2006), 55. doi: 10.1515/CRELLE.2006.044. Google Scholar [11] J. Frehse and J. Málek, Boundary regularity results for models of elasto-perfect plasticity,, Math. Models Meth. Appl. Sci., 9 (1999), 1307. doi: 10.1142/S0218202599000579. Google Scholar [12] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis I,II,", Kluwer, (1997). Google Scholar [13] P. Krejčí and J. Sprekels, Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity,, Appl. Math., 43 (1998), 173. doi: 10.1023/A:1023224507448. Google Scholar [14] G. A. Maughin, "The Thermomechanics of Plasticity and Fracture,", Cambridge Univ. Press, (1992). doi: 10.1017/CBO9781139172400. Google Scholar [15] A. Mielke, Evolution of rate-independent systems,, in, II (2005), 461. Google Scholar [16] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity,, Math. Modelling Numer. Anal., 43 (2009), 399. doi: 10.1051/m2an/2009009. Google Scholar [17] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. PDE, 31 (2008), 387. doi: 10.1007/s00526-007-0119-4. Google Scholar [18] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis,, in, (1999), 117. Google Scholar [19] A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151. Google Scholar [20] T. Roubíček, "Nonlinear Partial Differential Equations with Applications,", Birkhäuser, (2005). Google Scholar [21] T. Roubíček, Thermo-visco-elasticity at small strains with $L^1$-data,, Quarterly Appl. Math., 67 (2009), 47. Google Scholar [22] T. Roubíček, Rate independent processes in viscous solids at small strains,, Math. Methods Appl. Sci., 32 (2009), 825. doi: 10.1002/mma.1069. Google Scholar [23] T. Roubíček, Thermodynamics of rate independent processes in viscous solids at small strains,, SIAM J. Math. Anal., 42 (2010), 256. doi: 10.1137/080729992. Google Scholar [24] P. M. Suquet, Existence et régularité des solutions des équations de la plasticité parfaite,, C. R. Acad. Sci. Paris Sér. A, 286 (1978), 1201. Google Scholar [25] R. Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity,, Archive Rat. Mech. Anal., 95 (1986), 137. doi: 10.1007/BF00281085. Google Scholar
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