Locally Periodic Unfolding Method and Two-Scale Convergence on Surfaces of Locally Periodic Microstructures

Mariya Ptashnyk

doi:10.1137/140978405

Abstract

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of nonperiodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed nonperiodically. Using the methods of locally periodic two-scale convergence on oscillating surfaces and the locally periodic boundary unfolding operator, we are able to analyze differential equations defined on boundaries of nonperiodic microstructures and consider nonhomogeneous Neumann conditions on the boundaries of perforations, distributed nonperiodically.

Highlights

Many natural and man-made composite materials comprise non-periodic microscopic structures, e.g. fibrous microstructures in heart muscles [23, 48], exoskeletons [27], industrial filters [52], or spacedependent perforations in concrete [50]
An important special case of non-periodic microstructures is that of the so-called locally periodic microstructures, where spatial changes are observed on a scale smaller than the size of the domain under consideration, but larger than the characteristic size of the microstructure
For many locally periodic microstructures spatial changes cannot be represented by periodic functions depending on slow and fast variables, e.g. plywood-like structures of gradually rotated planes of parallel aligned fibers [13]

Summary

Introduction

Many natural and man-made composite materials comprise non-periodic microscopic structures, e.g. fibrous microstructures in heart muscles [23, 48], exoskeletons [27], industrial filters [52], or spacedependent perforations in concrete [50]. For many locally periodic microstructures spatial changes cannot be represented by periodic functions depending on slow and fast variables, e.g. plywood-like structures of gradually rotated planes of parallel aligned fibers [13] In these situations the standard two-scale convergence and periodic unfolding method cannot be applied. For a multiscale analysis of problems posed in domains with non-periodic perforations, in this paper we extend the periodic unfolding method and two-scale convergence on oscillating surfaces to locally periodic situations (see Definition 3.2–3.5). To illustrate the difference between the formulation of non-periodic microstructure by using periodic functions and the locally periodic formulation of the problem, we consider a plywood-like structure, given as the superposition of gradually rotated planes of aligned parallel fibers.

Locally periodic microstructures and locally periodic perforated domains

The l-p unfolding operator

Micro-macro decomposition

The l-p unfolding operator in perforated domains

10. Discussions