A set of enhanced formulations for existing nonlinear homogenization schemes and their evaluation

• We propose six modifications to some of the linearization procedures used in nonlinear homogenization.
• Main idea consists in proposing alternatives for reference strains used to define the linear comparison composites.
• The proposed new linear comparison composites are tested for rigidly-reinforced and porous power-law composites.
• The modified secant scheme provides good results for strongly nonlinear rigidly-reinforced composites at intermediate volume fractions.
• Two variants of the enhanced tangent second-order formulation lead to accurate estimates of the exact effective response.

On the basis of non-biased comparative evaluations of various linearization procedures used in nonlinear homogenization, performed both at the global and local scales for power-law composites (Rekik et al., 2005, 2007, 2012), we propose in this paper six ad hoc enhancements of some of the linearization procedures considered in Rekik et al. (2007). Both “stress–strain” approaches (the secant and affine formulations) or “variational formulations” (the tangent second-order method (Ponte Castañeda, 1996)) are considered. The main idea consists in proposing alternatives for the usual reference strains used by the secant, affine and tangent second-order procedures. The new linear comparison composites generated by the linearization step around the chosen alternative descriptors of the strain field statistics explicitly account for either intraphase strain fluctuations or both inter- and intraphase strain fluctuations. As a first illustration, the relevance and limitations of the enhanced linearization procedures are tested for rigidly-reinforced and porous power-law composites. For isochoric loadings, it is shown that two variants of the enhanced tangent second-order formulation lead to accurate estimates of the exact effective response which are in good agreement with the efficient second-order scheme of Ponte Castañeda (2002a). Further, the modified secant formulation provides good results for strongly nonlinear rigidly-reinforced composites away from low particulate volume fraction and the percolation threshold; however some new inherent limitations of secant formulations are also established. At last, a very discriminant situation is tested: it consists of a porous medium submitted to a pure hydrostatic loading at low pore concentrations. It is shown that one variant of the proposed enhanced second-order formulations leads to accurate estimates alike the efficient and more sophisticated formulations proposed in Bilger et al. (2002); Danas et al. (2008).