Effective Behavior of Nonlinear Microperiodic Composites with Imperfect Contact Via the Asymptotic Homogenization Method.

Roberto M.s Décio ,Leslie D Pérez-Fernández ,Julián Bravo‐Castillero

doi:10.5540/tcam.2021.022.01.00079

Abstract

The asymptotic homogenization method is applied here to one-dimensional boundary-value problems for nonlinear differential equations with rapidly oscillating piecewise-constant coefficients which model the behavior of nonlinear microperiodic composites, in order to assess the influence of interfacial imperfect contact on the effective behavior. In particular, a nonlinear power-law flux on the gradient of the unknown was considered. Several calculations were performed and are discussed at the end of this work, including a comparison of some results with variational ounds, which is also an important approach of this work.

Highlights

Composite materials can be described like heterogeneous materials formed by the union of two or more homogeneous ones, that will be their phases
Among the mathematical homogenization methods is the asymptotic homogenization method (AHM) [2], wich consider a asymptotic expansion like an approximation of the solution from the problem of interest, in the form of a potential series of a small positive parameter that will be denoted by ε, which characterize the microscale
This paper shows how the AHM can evalueting the effective behavior of biphasics and microperiodc composites, applying the method in boundary values problems with the static heat diffusion equation, onedimensional, considering a relevant case of nonlineaity and comparing the situations of perfect and imperfect adhesions between its phases

Summary

INTRODUCTION

Composite materials can be described like heterogeneous materials formed by the union of two or more homogeneous ones, that will be their phases. Among the mathematical homogenization methods is the asymptotic homogenization method (AHM) [2], wich consider a asymptotic expansion like an approximation of the solution from the problem of interest, in the form of a potential series of a small positive parameter that will be denoted by ε, which characterize the microscale From applying this asymptotic expansion in the original problem, is obtained a recurrent sequence of problems for the coeficients of the ε potences, from what one has locals problems, wich will give the terms of the asymptotic approach. This paper shows how the AHM can evalueting the effective behavior of biphasics and microperiodc composites, applying the method in boundary values problems with the static heat diffusion equation, onedimensional, considering a relevant case of nonlineaity and comparing the situations of perfect and imperfect adhesions between its phases.

PROBLEM’S FORMULATION

THE AHM METHODOLOGY

RESULTS

Some numerical simulations

CONCLUSIONS