Asymptotic homogenisation in linear elasticity. Part II: Finite element procedures and multiscale applications

The asymptotic expansion homogenisation (AEH) method can be used to solve problems involving physical phenomena on continuous media with periodic microstructures. In particular, the AEH is a useful technique to study of the behaviour of structural components built with composite materials. The main advantages of this approach lie on the fact that (i) it allows a significant reduction of the problem size and (ii) it has the capability to characterise stress and deformation microfields. In fact, specific equations can be developed to define these fields, in a process designated by localisation and not found on typical homogenisation methods. In the AEH methodology, overall material properties can be derived from the mechanical behaviour of selected periodic microscale representative volumes (also known as representative unit-cells, RUC). Nevertheless, unit-cell based modelling requires the control of some parameters, such as reinforcement volume fraction, geometry and distribution within the matrix material. The need for variety and flexibility leads to the development of automatic geometry generation algorithms. Additionally, the unstructured finite element meshes required by these RUC are usually non-periodic and involve the control of specific periodic boundary conditions. This work presents some numerical procedures developed in order to support finite element AEH implementations, rendering them more efficient and less user-dependent. The authors also present a numerical study of the influence of the reinforcement volume fraction on the overall material properties for a metal matrix composite (MMC) reinforced with spherical ceramic particles. A general multiscale application is shown, with both the homogenisation and localisation procedures.