G-convergence and Homogenization of Viscoelastic Flows
Robert Pertsch Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko, Klaus Hackl · 2020
This chapter considers the definition of G-convergence and some basic facts about it. It introduces a corrector operator approach which is related to the method of oscillating test functions and also to the ”condition N”. The chapter utilizes the corrector approach to derive effective equations of viscoelastic mixtures with time-varying interface. It shows that effective constitutive equations contain long-memory terms, and that the effective material parameters satisfy certain auxiliary problems which resemble cell problems of periodic homogenization. The chapter aims to derive homogenized models for flows of incompressible viscoelastic mixtures with time-varying interface. Both phases are modeled as Kelvin-Voight materials: the elastic stresses satisfy Hook&s;s law written in the spatial formulation, and the viscous stresses obey Newton&s;s law. The chapter considers the fluid-structure interaction problem, where one phase is a solid-like viscoelastic material, and the other is a viscous Newtonian fluid.