Homogenization of rate-dependent inelastic models of monotone type

Using the periodic unfolding method we derive the homogenized equations for the quasi-static initial boundary value problem with internal variables, which model the deformation behavior of viscoplastic materials with a periodic microstructure. The free energy associated with the problem is allowed to be positive semi-definite.

[1]  Alexander Pankov,et al.  G-Convergence and Homogenization of Nonlinear Partial Differential Operators , 1997 .

[2]  Gian-Carlo Rota.  Opérateurs maximaux monotones: H. Brézis, North-Holland, 1983, 183 pp. , 1985 .

[3]  Sergiy Nesenenko.  Homogenization in Viscoplasticity , 2007, SIAM J. Math. Anal..

[4]  H. Alber.  Materials with Memory , 1998 .

[5]  Shouchuan Hu,et al.  Handbook of multivalued analysis , 1997 .

[6]  Quoc Son Nguyen,et al.  Sur les matériaux standard généralisés , 1975 .

[7]  H. Alber.  Evolving Microstructure and Homogenization , 2000 .

[8]  Augusto Visintin,et al.  Homogenization of nonlinear visco-elastic composites , 2008 .

[9]  Quasistatic problems in viscoplasticity theory , 2002 .

[10]  A. Visintin.  Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity , 2008, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  D. Pascali,et al.  Nonlinear mappings of monotone type , 1979 .

[12]  Doina Cioranescu,et al.  Periodic unfolding and homogenization , 2002 .

[13]  Periodic homogenization of Prandtl-Reuss plasticity equations in arbitrary dimension , 2010 .

[14]  H. Attouch.  Variational convergence for functions and operators , 1984 .

[15]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

[16]  V. Barbu.  Nonlinear Semigroups and di erential equations in Banach spaces , 1976 .

[17]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[18]  T. Roubíček.  Nonlinear partial differential equations with applications , 2005 .

[19]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[20]  H. Alber,et al.  Quasistatic Problems in Viscoplasticity Theory I: Models with Linear Hardening , 2007 .

[21]  H. Alber.  Justification of homogenized models for viscoplastic bodies with microstructure , 2003 .

[22]  Doina Cioranescu,et al.  The Periodic Unfolding Method in Homogenization , 2008, SIAM J. Math. Anal..

[23]  A. Mielke,et al.  On rate-independent hysteresis models , 2004 .

[24]  S. Nesenenko,et al.  JUSTIFICATION OF HOMOGENIZATION IN VISCOPLASTICITY: FROM CONVERGENCE ON TWO SCALES TO AN ASYMPTOTIC SOLUTION IN L2(Ω) , 2009 .

[25]  H. Alber.  Materials with Memory: Initial-Boundary Value Problems for Constitutive Equations with Internal Variables , 1998 .

[26]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[27]  Nicolas Meunier,et al.  Periodic homogenization of monotone multivalued operators , 2007 .

[28]  D. Varberg.  Convex Functions , 1973 .

[29]  Alexander Mielke,et al.  Two-Scale Homogenization for Evolutionary Variational Inequalities via the Energetic Formulation , 2007, SIAM J. Math. Anal..

[30]  Shouchuan Hu,et al.  Handbook of Multivalued Analysis: Volume I: Theory , 1997 .

[31]  E. Giusti.  Direct methods in the calculus of variations , 2003 .

[32]  T. Valent,et al.  Boundary Value Problems of Finite Elasticity , 1988 .