Abstract
Abstract 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 Castaneda, 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 Castaneda (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).
Highlights
Nonlinear homogenization techniques are powerful methods allowing the derivation of bounds or estimates for the effective properties of heterogeneous nonlinear composites from both their local constitutive laws and the statistical description of their microstructure
The evolutions with respect to the work-hardening exponent m of the normalized effective flow stresses derived from the proposed SEC-Kcov, SOE-1-Fcov, SOE-1-Fmom2 and SOE-1-(E-F)cov formulations are reported in Fig. 1-a together with the ones associated with the secant, affine and second-order procedures and with the nonlinear (NL) exact solutions
It is observed that the predictions of the AFF-ANI-Fcov are, as expected, much softer than those provided by the classical affine AFF-ANI formulation and the secant formulations (SEC, variational procedure (VAR)), especially for low values of the work-hardening parameter m
Summary
Nonlinear homogenization techniques are powerful methods allowing the derivation of bounds or estimates for the effective properties of heterogeneous nonlinear composites from both their local constitutive laws and the statistical description of their microstructure. In [38, 39, 40], various nonlinear homogenization schemes such as the classical secant scheme (referred to as SEC), the variational procedure (VAR), the original affine formulation (AFF-ANI) and its isotropic simplification (AFF-ISOT), the original (SOE-1) and improved (SOE-2) second-order procedures as well as the Lahellec and Suquet (LS) formulation were compared with regard to their predictions in terms of overall responses and local field statistics for the special case of power-law two-phase composites with different contrast between the phases ranging from the rigidly-reinforced composites to porous materials. Reminder of general principles of nonlinear homogenization schemes and their evaluation by a non-biased methodology
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