October  2014, 7(5): 981-991. doi: 10.3934/dcdss.2014.7.981

Thermo-visco-elasticity for the Mróz model in the framework of thermodynamically complete systems

1. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa

2. 

Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, Warszawa 02-097, Poland

Received  March 2013 Revised  October 2013 Published  May 2014

We derive a thermomechanical model for the evolution of the visco-elastic body subject to the action of external forces. The presented framework captures elastic and visco-elastic deformation as well as thermal effects occurring in the material. Consequently we couple the momentum balance with the heat equation. The system is supplemented by the constitutive relation for the Cauchy stress tensor and visco-elastic strain tensor. As an example exhibiting the proposed approach one can mention the Mróz model. We establish existence of solutions to the quasi-static version of the derived system.
Citation: Piotr Gwiazda, Filip Z. Klawe, Agnieszka Świerczewska-Gwiazda. Thermo-visco-elasticity for the Mróz model in the framework of thermodynamically complete systems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 981-991. doi: 10.3934/dcdss.2014.7.981
References:
[1]

H.-D. Alber, Materials with Memory, Lecture Notes in Math, 1682, Springer, Berlin, Heidelberg, New York, 1998.

[2]

H.-D. Alber and K. Chełmiński, Quasistatic problems in viscoplasticity theory. II: Models with nonlinear hardening, Mathematical Models and Methods in Applied Sciences, 17 (2007), 189-213. doi: 10.1142/S0218202507001887.

[3]

L. Bartczak, Mathematical analysis of a thermo-visco-plastic model with Bodnera-Partom constitutive equations, Journal of Mathematical Analysis and Applications, 385 (2012), 961-974. doi: 10.1016/j.jmaa.2011.07.023.

[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]

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.

[6]

M. Bulíček, Navier's Slip and Evolutionary Navier-Stokes-Fourier-like Systems with Pressure, Shear-Rate and Temperature Dependent Viscosity, Ph.D thesis, Charles University in Prague, 2006.

[7]

M. Bulíček, E. Feireisl and J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Analysis: Real World Applications, 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018.

[8]

M. Bulíček and P. Pustějovská, On existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index, Journal of Mathematical Analysis and Applications, 402 (2013), 157-166. doi: 10.1016/j.jmaa.2012.12.066.

[9]

K. Chełmiński, On large solutions for the quasistatic problem in non-linear viscoelasticity with the constitutive equations of Bodner-Partom, Mathematical Methods in the Applied Sciences, 19 (1996), 933-942. doi: 10.1002/(SICI)1099-1476(199608)19:12<933::AID-MMA802>3.0.CO;2-M.

[10]

K. Chełmiński and P. Gwiazda, On the model of Bodner-Partom with nonhomogeneous boundary data, Mathematische Nachrichten, 214 (2000), 5-23. doi: 10.1002/1522-2616(200006)214:1<5::AID-MANA5>3.0.CO;2-O.

[11]

K. Chełmiński and P. Gwiazda, Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory, Mathematical Methods in the Applied Sciences, 30 (2007), 1357-1374. doi: 10.1002/mma.844.

[12]

K. Chełmiński and R. Racke, Mathematical analysis of a model from thermoplasticity with kinematic hardening, J. Appl. Anal., 12 (2006), 37-57. doi: 10.1515/JAA.2006.37.

[13]

G. Duvaut and J.L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.

[14]

C. Johnson, Existence theorems for plasticity problem, J. Math. Pures Appl., 55 (1976), 431-444.

[15]

C. Johnson, On plasticity with hardening, J. Math. Anal. Appl., 62 (1978), 325-336. doi: 10.1016/0022-247X(78)90129-4.

[16]

A. E. Green and P. M. Naghdi, A general theory of an elastic-plastic continuum, Archive for Rational Mechanics and Analysis, 18 (1965), 251-281. doi: 10.1007/BF00251666.

[17]

P. Gwiazda and A. Świerczewska, Large eddy simulation turbulence model with Young measures, Applied Mathematics Letters, 18 (2005), 923-929. doi: 10.1016/j.aml.2004.07.035.

[18]

S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Chapman & Hall/CRC, Boca Raton, 2000.

[19]

L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Course of Theoretical Physics, 7, Pergamon Press, London-Paris-Frankfurt, 1959.

[20]

Z. Mróz, On the description of anisotropic workhardening, J. Mech. Phys. Solids, 15 (1967), 163-175.

[21]

J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier Scientific Publisher Company, 1980.

[22]

P. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, J. Mécanique, 20 (1981), 3-39.

[23]

P. Suquet, Plasticité et Homogénéisation, Ph.D thesis, Univerité de Paris VI, 1982.

[24]

A. Świerczewska, A dynamical approach to large eddy simulation of turbulent flows: Existence of weak solutions, Math. Methods Appl. Sci., 29 (2006), 99-121.

[25]

K. R. Rajagopal, Implicit constitutive relations, in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford, UK.

[26]

R. Temam, Mathematical Problems in Plasticity, (in French) Dunod, Paris, 1983; (in English) Gauthier-Villars, Paris-New York, 1984.

[27]

R. Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Archive for Rational Mechanics and Analysis, 95 (1986), 137-183. doi: 10.1007/BF00281085.

[28]

T. Valent, Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3736-5.

show all references

References:
[1]

H.-D. Alber, Materials with Memory, Lecture Notes in Math, 1682, Springer, Berlin, Heidelberg, New York, 1998.

[2]

H.-D. Alber and K. Chełmiński, Quasistatic problems in viscoplasticity theory. II: Models with nonlinear hardening, Mathematical Models and Methods in Applied Sciences, 17 (2007), 189-213. doi: 10.1142/S0218202507001887.

[3]

L. Bartczak, Mathematical analysis of a thermo-visco-plastic model with Bodnera-Partom constitutive equations, Journal of Mathematical Analysis and Applications, 385 (2012), 961-974. doi: 10.1016/j.jmaa.2011.07.023.

[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]

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0.

[6]

M. Bulíček, Navier's Slip and Evolutionary Navier-Stokes-Fourier-like Systems with Pressure, Shear-Rate and Temperature Dependent Viscosity, Ph.D thesis, Charles University in Prague, 2006.

[7]

M. Bulíček, E. Feireisl and J. Málek, A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Analysis: Real World Applications, 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018.

[8]

M. Bulíček and P. Pustějovská, On existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index, Journal of Mathematical Analysis and Applications, 402 (2013), 157-166. doi: 10.1016/j.jmaa.2012.12.066.

[9]

K. Chełmiński, On large solutions for the quasistatic problem in non-linear viscoelasticity with the constitutive equations of Bodner-Partom, Mathematical Methods in the Applied Sciences, 19 (1996), 933-942. doi: 10.1002/(SICI)1099-1476(199608)19:12<933::AID-MMA802>3.0.CO;2-M.

[10]

K. Chełmiński and P. Gwiazda, On the model of Bodner-Partom with nonhomogeneous boundary data, Mathematische Nachrichten, 214 (2000), 5-23. doi: 10.1002/1522-2616(200006)214:1<5::AID-MANA5>3.0.CO;2-O.

[11]

K. Chełmiński and P. Gwiazda, Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory, Mathematical Methods in the Applied Sciences, 30 (2007), 1357-1374. doi: 10.1002/mma.844.

[12]

K. Chełmiński and R. Racke, Mathematical analysis of a model from thermoplasticity with kinematic hardening, J. Appl. Anal., 12 (2006), 37-57. doi: 10.1515/JAA.2006.37.

[13]

G. Duvaut and J.L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.

[14]

C. Johnson, Existence theorems for plasticity problem, J. Math. Pures Appl., 55 (1976), 431-444.

[15]

C. Johnson, On plasticity with hardening, J. Math. Anal. Appl., 62 (1978), 325-336. doi: 10.1016/0022-247X(78)90129-4.

[16]

A. E. Green and P. M. Naghdi, A general theory of an elastic-plastic continuum, Archive for Rational Mechanics and Analysis, 18 (1965), 251-281. doi: 10.1007/BF00251666.

[17]

P. Gwiazda and A. Świerczewska, Large eddy simulation turbulence model with Young measures, Applied Mathematics Letters, 18 (2005), 923-929. doi: 10.1016/j.aml.2004.07.035.

[18]

S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Chapman & Hall/CRC, Boca Raton, 2000.

[19]

L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Course of Theoretical Physics, 7, Pergamon Press, London-Paris-Frankfurt, 1959.

[20]

Z. Mróz, On the description of anisotropic workhardening, J. Mech. Phys. Solids, 15 (1967), 163-175.

[21]

J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier Scientific Publisher Company, 1980.

[22]

P. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, J. Mécanique, 20 (1981), 3-39.

[23]

P. Suquet, Plasticité et Homogénéisation, Ph.D thesis, Univerité de Paris VI, 1982.

[24]

A. Świerczewska, A dynamical approach to large eddy simulation of turbulent flows: Existence of weak solutions, Math. Methods Appl. Sci., 29 (2006), 99-121.

[25]

K. R. Rajagopal, Implicit constitutive relations, in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford, UK.

[26]

R. Temam, Mathematical Problems in Plasticity, (in French) Dunod, Paris, 1983; (in English) Gauthier-Villars, Paris-New York, 1984.

[27]

R. Temam, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Archive for Rational Mechanics and Analysis, 95 (1986), 137-183. doi: 10.1007/BF00281085.

[28]

T. Valent, Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-3736-5.

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