Two-scale homogenization for evolutionary variational inequalities via the energetic formulation

This paper is devoted to the two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using so-called joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals

On evolutionary Gamma convergence for gradient systems

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional Eε and the dissipation potential Rε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions uε. We investigate the question under which conditions the limits u of (subsequences of) the solutions uε are solutions of the gradient system generated by the Γ-limits E0 and R0. Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where Rε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results

Stochastic homogenization of Λ\Lambda-convex gradient flows

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $\Lambda$-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $p$-Laplace operator with $p\in (1,\infty)$. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($\Lambda$-)convex functionals.Comment: arXiv admin note: text overlap with arXiv:1805.0954

Homogenization of some degenerate pseudoparabolic variational inequalities

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem

Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence

In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savar'e 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches

Stochastic unfolding and homogenization of evolutionary gradient systems

The mathematical theory of homogenization deals with the rigorous derivation of effective models from partial differential equations with rapidly-oscillating coefficients. In this thesis we deal with modeling and homogenization of random heterogeneous media. Namely, we obtain stochastic homogenization results for certain evolutionary gradient systems. In particular, we derive continuum effective models from discrete networks consisting of elasto-plastic springs with random coefficients in the setting of evolutionary rate-independent systems. Also, we treat a discrete counterpart of gradient plasticity. The second type of problems that we consider are gradient flows. Specifically, we study continuum gradient flows driven by λ-convex energy functionals. In stochastic homogenization the derived deterministic effective equations are typically hardly-accessible for standard numerical methods. For this reason, we study approximation schemes for the effective equations that we obtain, which are well-suited for numerical analysis. For the sake of a simple treatment of these problems, we introduce a general procedure for stochastic homogenization – the stochastic unfolding method. This method presents a stochastic counterpart of the well-established periodic unfolding procedure which is well-suited for homogenization of media with periodic microstructure. The stochastic unfolding method is convenient for the treatment of equations driven by integral functionals with random integrands. The advantage of this strategy in regard to other methods in homogenization is its simplicity and the elementary analysis that mostly relies on basic functional analysis concepts, which makes it an easily accessible method for a wide audience. In particular, we develop this strategy in the setting that is suited for problems involving discrete-to-continuum transition as well as for equations defined on a continuum physical space. We believe that the stochastic unfolding method may also be useful for problems outside of the scope of this work

Variational approaches and methods for dissipative material models with multiple scales

In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit.   As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory

Stochastic homogenization of Lambda-convex gradient flows

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals

Stochastic homogenization of Λ-convex gradient flows

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the p-Laplace operator with p∈(1,∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals