American Institute of Mathematical Sciences

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

Thermodynamics of perfect plasticity

Tomáš Roubíček 1,

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.
Keywords: thermodynamics, thermal expansion, Kelvin-Voigt rheology, adiabatic effects, small strains, bounded-deformation-valued solution., Prandtl-Reuss plasticity.
Mathematics Subject Classification: Primary: 35K85; Secondary: 49S05, 74C05, 80A1.
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

show all references

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

[1]

Miroslav Bulíček, Josef Málek, K. R. Rajagopal. On Kelvin-Voigt model and its generalizations. Evolution Equations & Control Theory, 2012, 1 (1) : 17-42. doi: 10.3934/eect.2012.1.17
[2]

Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110
[3]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[4]

Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029
[5]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[6]

Xiaobin Yao, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247
[7]

Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020007
[8]

Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269
[9]

Manil T. Mohan. Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020067
[10]

Weixia Zhao. The expansion of gas from a wedge with small angle into a vacuum. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2319-2330. doi: 10.3934/cpaa.2013.12.2319
[11]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
[12]

Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure & Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267
[13]

Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012
[14]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the one-dimensional version of the dynamical Marguerre-Vlasov system with thermal effects. Conference Publications, 2009, 2009 (Special) : 536-547. doi: 10.3934/proc.2009.2009.536
[15]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141
[16]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations & Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411
[17]

Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317
[18]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations & Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016
[19]

Asim Aziz, Wasim Jamshed, Yasir Ali, Moniba Shams. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020142
[20]

Marc Bonnet. Inverse acoustic scattering using high-order small-inclusion expansion of misfit function. Inverse Problems & Imaging, 2018, 12 (4) : 921-953. doi: 10.3934/ipi.2018039

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences