Asymptotic Expansion Homogenisation and Multiscale Topology Optimisation of Composite Structures

Abstract

Composite materials are among the most prominent materials today, both in terms of applications and development. Nevertheless, their complex structure and heterogeneous nature lead to difficulties, both in the prediction of its properties and on the achievement of the ideal constituent distributions. Homogenisation procedures may provide answers in both cases. With this in mind, the main focus of this chapter is to show the importance of computational procedures for this task, mainly in terms of the different applications of Asymptotic Expansion Homogenisation (AEH) to heterogeneous periodic media and, above all, composite materials. First of all, it is noteworthy that the detailed numerical modelling of the mechanical behaviour of composite material structures tends to involve high computational costs. In this scope, the use of homogenisation methodologies can lead to significant benefits. These techniques allow the simplification of a heterogeneous medium using an equivalent homogenousmedium andmacrostructural behaviour laws obtained from microstructural information. Furthermore, composite materials typically have heterogeneities with characteristic dimensions significantly smaller than the dimensions of the structural component itself. If the distribution of the heterogeneities is roughly periodic, it can usually be approximated by a detailed periodic representative unit-cell. Thus, the Asymptotic Expansion Homogenisation (AEH) method is an excellent methodology to model physical phenomena on media with periodic microstructure, as well as a useful technique to study the mechanical behaviour of structural components built with compositematerials. In terms of computational implementation, the main advantages of this method are (i) the fact that it allows a significant reduction of the number of degrees of freedom and (ii) the capability to find the stress and strain microstructural fields associated with a given macrostructural equilibrium state. In fact, unlike other common homogenisation methods, the AEH leads to explicit mathematical equations to characterise those fields, that is, to perform a localisation. On the other hand, topology optimisation typically deals with material distributions to achieve the best behaviour for a given objective. The common approach to structural topology optimisation uses a variety of compliance minimisation (stiffness maximisation) procedures and functions. When analysing composite materials, these strategies often lead to multiscale procedures, either as a way to relax the initial discrete problem or in an effort to attain both optimal global structure and optimal microstructure. In this sense, the integration of AEH 23