# 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 and 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), 951-976. [2] S. Bartels and T. Roubíček, Thermo-visco-plasticity at small strains, Zeitschrift angew. Math. Mech., 88 (2008), 735-754. doi: 10.1002/zamm.200800042. [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), 477-504. doi: 10.1051/m2an/2010063. [4] L. Boccardo, A. Dall'aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. of Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040. [5] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. [6] K. Chełmiński, Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening, Math. Methods Appl. Sci., 24 (2001), 117-136. [7] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4. [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), 237-291. doi: 10.1007/s00205-005-0407-0. [9] F. Ebobisse and B. D. Reddy, Some mathematical problems in perfect plasticity, Computer Meth. Appl. Mech. Engr., 193 (2004), 5071-5094. doi: 10.1016/j.cma.2004.07.002. [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), 55-91. doi: 10.1515/CRELLE.2006.044. [11] J. Frehse and J. Málek, Boundary regularity results for models of elasto-perfect plasticity, Math. Models Meth. Appl. Sci., 9 (1999), 1307-1321. doi: 10.1142/S0218202599000579. [12] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis I,II," Kluwer, Dordrecht, Part I: 1997, Part II: 2000. [13] P. Krejčí and J. Sprekels, Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity, Appl. Math., 43 (1998), 173-205. doi: 10.1023/A:1023224507448. [14] G. A. Maughin, "The Thermomechanics of Plasticity and Fracture," Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9781139172400. [15] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [16] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, Math. Modelling Numer. Anal., 43 (2009), 399-428. doi: 10.1051/m2an/2009009. [17] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. PDE, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4. [18] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in "Models of Continuum Mechanics in Analysis and Engineering" (Eds.: H.D. Alber, et al.), Shaker Ver., Aachen, (1999), 117-129. [19] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189. [20] T. Roubíček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005 (2nd edition 2012). [21] T. Roubíček, Thermo-visco-elasticity at small strains with $L^1$-data, Quarterly Appl. Math., 67 (2009), 47-71. [22] T. Roubíček, Rate independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862. doi: 10.1002/mma.1069. [23] 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. [24] P. M. Suquet, Existence et régularité des solutions des équations de la plasticité parfaite, C. R. Acad. Sci. Paris Sér. A, 286 (1978), 1201-1204. [25] R. Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Archive Rat. Mech. Anal., 95 (1986), 137-183. doi: 10.1007/BF00281085.

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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), 951-976. [2] S. Bartels and T. Roubíček, Thermo-visco-plasticity at small strains, Zeitschrift angew. Math. Mech., 88 (2008), 735-754. doi: 10.1002/zamm.200800042. [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), 477-504. doi: 10.1051/m2an/2010063. [4] L. Boccardo, A. Dall'aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data, J. of Funct. Anal., 147 (1997), 237-258. doi: 10.1006/jfan.1996.3040. [5] L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. [6] K. Chełmiński, Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening, Math. Methods Appl. Sci., 24 (2001), 117-136. [7] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Rational Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4. [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), 237-291. doi: 10.1007/s00205-005-0407-0. [9] F. Ebobisse and B. D. Reddy, Some mathematical problems in perfect plasticity, Computer Meth. Appl. Mech. Engr., 193 (2004), 5071-5094. doi: 10.1016/j.cma.2004.07.002. [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), 55-91. doi: 10.1515/CRELLE.2006.044. [11] J. Frehse and J. Málek, Boundary regularity results for models of elasto-perfect plasticity, Math. Models Meth. Appl. Sci., 9 (1999), 1307-1321. doi: 10.1142/S0218202599000579. [12] S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis I,II," Kluwer, Dordrecht, Part I: 1997, Part II: 2000. [13] P. Krejčí and J. Sprekels, Temperature-dependent hysteresis in one-dimensional thermovisco-elastoplasticity, Appl. Math., 43 (1998), 173-205. doi: 10.1023/A:1023224507448. [14] G. A. Maughin, "The Thermomechanics of Plasticity and Fracture," Cambridge Univ. Press, Cambridge, 1992. doi: 10.1017/CBO9781139172400. [15] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," (Edited by C. M. Dafermos and E. Feireisl), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559. [16] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, Math. Modelling Numer. Anal., 43 (2009), 399-428. doi: 10.1051/m2an/2009009. [17] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. PDE, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4. [18] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in "Models of Continuum Mechanics in Analysis and Engineering" (Eds.: H.D. Alber, et al.), Shaker Ver., Aachen, (1999), 117-129. [19] A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189. [20] T. Roubíček, "Nonlinear Partial Differential Equations with Applications," Birkhäuser, Basel, 2005 (2nd edition 2012). [21] T. Roubíček, Thermo-visco-elasticity at small strains with $L^1$-data, Quarterly Appl. Math., 67 (2009), 47-71. [22] T. Roubíček, Rate independent processes in viscous solids at small strains, Math. Methods Appl. Sci., 32 (2009), 825-862. doi: 10.1002/mma.1069. [23] 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. [24] P. M. Suquet, Existence et régularité des solutions des équations de la plasticité parfaite, C. R. Acad. Sci. Paris Sér. A, 286 (1978), 1201-1204. [25] R. Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Archive Rat. Mech. Anal., 95 (1986), 137-183. doi: 10.1007/BF00281085.
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