Asymptotic homogenisation in linear elasticity. Part I: Mathematical formulation and finite element modelling

The asymptotic expansion homogenisation method is an excellent methodology to model physical phenomena on media with periodic microstructure and a useful technique to study the mechanical behaviour of structural components built with composite materials. In the first part of this work the authors present a detailed form of the mathematical formulation of the asymptotic expansion homogenisation for linear elasticity problems, as well the explicit mathematical equations that characterise the microstructural stress and strain fields associated with a given macrostructural equilibrium state – the localisation procedure. From this mathematical basis, the authors also present the numerical equations resulting from the finite element modelling of the asymptotic expansion homogenisation method in linear elasticity.