Objective evaluation of linearization procedures in nonlinear homogenization: A methodology and some implications on the accuracy of micromechanical schemes

Amna Rekik,François Auslender,Michel Bornert,André Zaoui

doi:10.1016/j.ijsolstr.2006.10.001

Abstract

A systematic methodology for an accurate evaluation of various existing linearization procedures sustaining mean fields theories for nonlinear composites is proposed and applied to recent homogenization methods. It relies on the analysis of a periodic composite for which an exact resolution of both the original nonlinear homogenization problem and the linear homogenization problems associated with the chosen linear comparison composite (LCC) with an identical microstructure is possible. The effects of the sole linearization scheme can then be evaluated without ambiguity. This methodology is applied to three different two-phase materials in which the constitutive behavior of at least one constituent is nonlinear elastic (or viscoplastic): a reinforced composite, a material in which both phases are nonlinear and a porous material. Comparisons performed on these three materials between the considered homogenization schemes and the reference solution bear out the relevance and the performances of the modified second-order procedure introduced by Ponte Castañeda in terms of prediction of the effective responses. However, under the assumption that the field statistics (first and second moments) are given by the local fields in the LCC, all the recent nonlinear homogenization procedures still fail to provide an accurate enough estimate of the strain statistics, especially for composites with high contrast.

Highlights

Most of the nonlinear homogenization procedures for heterogeneous materials implicitly or explicitly rely on two separates stages
The methodology and the numerical tool which are proposed in this paper enable a systematic and accurate evaluation of the linearization procedures without suffering from the limitations mentioned in introduction
Note that a similar approach has independently been used by Lahellec and Suquet (2004) to evaluate the specific model proposed by these authors, as well as by Moulinec and Suquet (2004) for a comparison between the classical and the modified secant linearization procedures

Summary

Introduction

Most of the nonlinear homogenization procedures for heterogeneous materials implicitly or explicitly rely on two separates stages. A less rigorous but more efficient linearization scheme might be preferred and has to be selected among the numerous classical or more recently proposed formulations, such as Hill’s incremental scheme (Hill, 1965), the classical (Hutchinson, 1976; Berveiller and Zaoui, 1979) or modified (Ponte Castaneda, 1991; Suquet, 1995) secant approaches, the affine formulation (Masson and Zaoui, 1999; Masson et al, 2000) and its variants (Ponte Castaneda, 2002; Chaboche and Kanoute, 2003), the second-order procedures (Ponte Castaneda, 1996, 2002) or the Lahellec and Suquet procedure (Lahellec and Suquet, 2004) These models are based on more or less sophisticated derivations and generate various predictions, which deserve an objective and systematic comparison to each other and with exact solutions, both at the local and global levels to appreciate their respective merits and limitations. It is the purpose of the present study to provide a methodology for such a systematic and objective comparison, from which, in addition, guidelines for improved formulations might be deduced

Objectives

Methods

Results

Conclusion