Thermo-visco-elasticity for models with growth conditions in Orlicz spaces

Filip Klawe

DOI: http://dx.doi.org/10.12775/TMNA.2016.006

Abstract


We study a quasi-static evolution of the thermo-visco-elastic model. We act with external forces on a non-homogeneous material body, which is a subject of our research. Such action may cause deformation of this body and may change its temperature. Mechanical part of the model contains two kinds of deformation: elastic and visco-elastic. The mechanical deformation is coupled with temperature and both of them may influence each other. Since the constitutive function on evolution of the visco-elastic deformation depends on temperature, the visco-elastic properties of material also depend on temperature. We consider the thermodynamically complete model related to a hardening rule with growth condition in generalized Orlicz spaces. We provide the proof of existence of solutions for such class of models.

Keywords


Visco-elasticity; thermal effects; Galerkin approximation; monotonicity method; renormalizations; generalized Orlicz space

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References


R.A. Adams, Sobolev Spaces, Academic Press, 1975.

H.-D. Alber and K. Chelmiński, Quasistatic problems in viscoplasticity theory II: Models with nonlinear hardening, Math. Models Methods Appl. Sci. 17 (2007), 189–213.

H.W. Alt, Lineare Funktionalanalysis, Springer–Verlag, 2006.

G. Anzellotti and M. Giaquinta, Existence of the displacements field for an elastoplastic body subject to Hencky’s law and von mises yield condition, Manuscripta Math. 32 (1980), 101–136.

J.M. Ball and F. Murat, Remarks on Chacon’s biting lemma, Proc. Amer. Math. Soc. 107 (3), Nov. 1989.

L. Bartczak, Mathematical analysis of a thermo-visco-plastic model with Bodnera–Partom constitutive equations, J. Math. Anal. Appli. 385 (2012), 961–974.

D. Blanchard Truncations and monotonicity methods for parabolic equations, Nonlinear Anal. 21 (1993), 725–743.

D. Blanchard and F. Murat, Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A Math. 127 (1997), 1137–1152.

L. Boccardo and T. Gallouët, Non-linear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149–169.

M. Brokate, P. Krejčı́ and D. Rachinskiı̆, Some analytical properties of the multidimensional continuous Mróz model of plasticity, Control Cybernet. 27(2) (1998), 199–215.

M. Bulı́ček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal. 44 (2012), 2756–2801.

K. Chelmiński, On large solutions for the quasistatic problem in non-linear viscoelasticity with the constitutive equations of Bodner–Partom, Math. Methods Appl. Sci. 19 (1996), 933–942.

K. Chelmiński and P. Gwiazda, On the model of Bodner–Partom with nonhomogeneous boundary data, Math. Nachr. 214 (2000), 5–23.

K. Chelmiński and P. Gwiazda, Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory, Math. Methods Appl. Sci. 30 (2007), 1357–1374.

K. Chelmiński and R. Racke, Mathematical analysis of a model from thermoplasticity with kinematic hardening, J. Appl. Anal. 12, (2006), 37–57.

E. Emmrich and A. Wróblewska-Kamińska, Convergence of a full discretization of quasi-linear parabolic equations in isotropic and anisotropic Orlicz spaces, SIAM J. Numer. Anal. 51 (2013), 1163–1184.

L.C. Evans, Partial Differential Equations, Amer. Math. Soc., 1998.

M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Publications of the Scuola Normale Superiore, vol. 11, Scuola Normale Superiore, 2012.

A. E. Green and P. M. Naghdi, A general theory of an elastic-plastic continuum, Arch. Rational Mech. Anal. 18 (1965), 251–281.

P. Gwiazda, F.Z. Klawe and A. Świerczewska-Gwiazda, Thermo-visco-elasticity for Norton–Hoff-type models, Nonlinear Anal. Real World Appl. 26 (2015), 199–228.

P. Gwiazda, F.Z. Klawe and A. Świerczewska-Gwiazda, Thermo-visco-elasticity for the Mróz model in the framework of thermodynamically complete systems, Discrete Contin. Dyn. Syst. Ser. S 7 (2014), 981–991.

P. Gwiazda, P. Minakowski and A. Świerczewska-Gwiazda, On the anisotropic Orlicz spaces applied in the problems of continuum mechanics, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 1291–1306.

P. Gwiazda and A. Świerczewska, Large eddy simulation turbulence model with Young measures, Appl. Math. Lett. 18 (2005), 923–929.

P. Gwiazda and A. Świerczewska-Gwiazda, On non-Newtonian fluids with the property of rapid thickening, Math. Models Methods Appl. Sci. 18 (2008), 1073–1092.

P. Gwiazda and A. Świerczewska-Gwiazda, On steady non-Newtonian flows with growth conditions in generalized Orlicz spaces, Topol. Methods Nonlinear Anal. 32 (2008), 103–114.

P. Gwiazda, A. Świerczewska-Gwiazda and A. Wróblewska, Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Math. Methods Appl. Sci. 33 (2010), 125–248.

P. Gwiazda, Generalized Stokes system in Orlicz spaces, Discrete Contin. Dyn. Syst. Ser. A [27] 32 (6) (2012), 2125–2146.

P. Gwiazda, P. Wittbold, A. Wróblewska and A. Zimmermann, Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differential Equations 253 (2012), 635–666.

P. Gwiazda, P. Wittbold, A. Wróblewska and A. Zimmermann, Renormalized solutions of nonlinear parabolic problems in generalized Musielak–Orlicz spaces, Nonlinear Anal. 129 (2015), 1–36.

D. Hömberg, A mathematical model for induction hardening including mechanical effects, Nonlinear Anal. 5 (2004), 55–90.

F.Z. Klawe, Mathematical analysis of thermo-visco-elastic models, PhD thesis, University of Warsaw, 2015.

M.A. Krasnosel’skiĭ and Y.B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, Netherlands, 1961.

L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Pergamon Press, 1970 (7th ed.).

J. Málek, J. Nečas, M. Rokyta and M. Růžicka, Weak and measure-valued solutions to evolutionary PDEs, Chapman & Hall, London, 1996.

P.A. Meyer, Probability and potentials, Blaisdell, 1966.

S. Müller, Variational models for microstructure and phase transitions, Lecture Notes in Math.: Calculus of Variations and Geometric Evolution Problems, vol. 1713, Springer–Verlag, Berlin, Heidelberg, 1999.

J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin, Heidelberg, 1983.

M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Pure Appl. Math., Dekker, New York, 1991.

W. Strauss, Partial Differential Equations, Wiley, 2007.

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

A. Świerczewska-Gwiazda, Anisotropic parabolic problems with slowly or rapidly growing terms, Colloq. Math. 134 (2014), 113–130.

A. Świerczewska-Gwiazda, Nonlinear parabolic problems in Musielak–Orlicz spaces, Nonlinear Anal. 98 (2014), 48–65.

R. Temam and G. Strang, Functions of bounded deformation, Arch. Rational Mech. Anal. 75 (1980/1981), 7–21.

T. Valent, Boundary Value Problems of Finite Elasticity: Local Theorems on Existence, Uniqueness, and Analytic Dependence on Data, Springer Publishing Company, Incorporated, 1st ed., 2011.

A. Wróblewska, Steady flow of non-Newtonian fluids – monotonicity methods in generalized Orlicz spaces, Nonlinear Anal. 72 (2010), 4136–4147.

A. Wróblewska-Kamińska, An application of Orlicz spaces in partial differential equations, PhD thesis, University of Warsaw, 2012.

E. Zeidler, Nonlinear Functional Analysis II/B – Nonlinear Monotone Operators, Springer–Verlag, Berlin, Heidelberg, New York, 1990.


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