Second-order homogenisation of functionally graded materials

The homogenisation theory for periodic composites is generalised to the case of quasi-periodic composites. In quasi-periodic composites, the unit cell does not repeat throughout the medium but gradually changes along one or more directions of periodicity (grading directions). Quasi-periodic composites are thus to functionally graded materials (FGMs) what periodic composites are to statistically uniform composite materials. Contrarily to most of the homogenisation methods applied to FGMs, the proposed second-order homogenisation theory takes explicitly into account the grading at the micro-level. The derived equivalent material happens to be a particular second gradient material in which few components of the strain gradient (second gradient of the displacement) should be taken into account in addition to the classical strains (first gradient of displacement). The second gradient theory therefore appears as the natural framework to appropriately handle functionally graded materials at the macro-level. It is worth mentioning that the presented second-order homogenisation procedure is somehow analogous to the one developed for periodic composite materials submitted to rapidly varying macroscopic strain fields as in regions of high gradients. In fact, both are a generalisation of the first-order homogenisation theory for periodic media and lead to a second gradient equivalent material. However, besides their different domains of application, they exhibit further substantial differences, which are highlighted in the paper.