Stochastic homogenization of rate-dependent models of monotone type in plasticity

In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Γ-convergence theory.

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

[2]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

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

[4]  On Two-Scale Convergence , 2004 .

[5]  M. Heida.  Stochastic homogenization of rate-independent systems , 2016 .

[6]  Stochastic homogenization of plasticity equations , 2016, 1604.02291.

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

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

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

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

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

[12]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[13]  J. Mecke,et al.  Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen , 1967 .

[14]  Ben Schweizer,et al.  Homogenization of plasticity equations with two-scale convergence methods , 2015 .

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

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

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

[18]  Viorel Barbu,et al.  Differential equations in Banach spaces , 1976 .

[19]  S. Fitzpatrick.  Representing monotone operators by convex functions , 1988 .

[20]  J. K. Hunter,et al.  Measure Theory , 2007 .

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

[22]  Sergiy Nesenenko,et al.  Homogenization of rate-dependent inelastic models of monotone type , 2013, Asymptot. Anal..

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

[24]  R. Rockafellar.  Integrals which are convex functionals. II , 1968 .

[25]  P. Neff,et al.  Well-posedness for dislocation-based gradient viscoplasticity, II: general nonassociative monotone plastic flows , 2013 .

[26]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[27]  M. Heida.  Stochastic homogenization of rate-independent systems and applications , 2017 .

[28]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[29]  Augusto Visintin,et al.  Scale-transformations and homogenization of maximal monotone relations with applications , 2013, Asymptot. Anal..

[30]  J. Aubin.  Set-valued analysis , 1990 .

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

[32]  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.

[33]  Ben Schweizer,et al.  Homogenization of the Prager model in one-dimensional plasticity , 2009 .

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

[35]  G. Francfort,et al.  On periodic homogenization in perfect elasto-plasticity , 2014 .

[36]  V. Zhikov,et al.  Homogenization of random singular structures and random measures , 2006 .