Computational evaluation of strain gradient elasticity constants

Classical effective descriptions of heterogeneous materials fail to capture the influence of the spatial scale of the heterogeneity on the overall response of components. This influence may become important when the scale at which the effective continuum fields vary approaches that of the microstructure of the material and may then give rise to size effects and other deviations from the classical theory. These effects can be successfully captured by continuum theories which include a material length scale, such as strain gradient theories. However, the precise relation between the microstructure on the one hand and the length scale and other properties of the effective modelling are usually unknown. A rigourous link between these two scales of observation is provided by an extension of the classical asymptotic homogenisation theory which was proposed by Smyshlyaev & Cherednichenko (2000) for the scalar problem of antiplane shear. In the present contribution this method is extended to three-dimensional linear elasticity. It requires the solution of a series of boundary value problems on the periodic cell which characterises the microstructure. A finite element solution strategy is developed for this purpose. The resulting fields can be used to determine the effective higher-order elasticity constants required in the Toupin-Mindlin strain gradient theory. The method has been applied to a matrix-inclusion composite, showing that higher-order terms become more important as the stiffness contrast between inclusion and matrix increases.