Nonlinear Composite Materials: Effective Constitutive Behavior and Microstructure Evolution
This article is a review of some recent developments in the field of nonlinear composite materials. More precisely, it is an attempt to summarize and relate certain methods that have been developed over the past 10 years for estimating the effective, homo
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P. Ponte Castaneda University of Pennsylvania, Philadelphia, PA, USA
ABSTRACT
This article is a review of some recent developments in the field of nonlinear composite materials. More precisely, it is an attempt to summarize and relate certain methods that have been developed over the past 10 years for estimating the effective, homogenized or overall constitutive behavior of composite materials with nonlinear constituents. In addition to their ability to handle constitutive nonlinearity, these homogenization methods have also been extended in a consistent fashion to account for geometric nonlinearity. In other words, they are able to incorporate the effect of changes in the microstructure, due to the presence of finite deformations, on the effective constitutive behavior of the composites. These homogenization procedures- like some of the earlier, now classical procedures- make use of a "linear comparison composite," for which the effective behavior can be assumed to be readily available from the extensive literature in the field of linear composite materials. In particular, the use of a linear comparison composite provides a way of bringing in higherorder statistics (beyond volume fractions) into the description of the effective behavior of composites with random microstructures, whether they be "particulate" or "granular" in character. However, the choice of the linear comparison composite in these modern homogenization procedures is dictated by appropriate approximations in the context of rigorous variational principles for the effective energy of the composite. This simple fact goes a long way toward explaining the significant improvements achieved by the new methods over the more classical procedures, where the choice of the comparison composite was not derived from a variational principle.
P. Suquet (ed.), Continuum Micromechanics © Springer-Verlag Wien 1997
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P. Ponte Castaneda
1. INTRODUCTION By now, much is known about the characterization of the effective properties of composite materials with linear constitutive behavior. When their microstructure is periodic, the effective properties may be determined exactly in terms of unit cell problems with appropriate boundary conditions [1-2]. For random microstructures, the problem is not deterministic and the effective properties cannot be determined exactly. Instead, the goal becomes to determine the range of possible effective behavior in terms of bounds depending on the known statistics- usually only up to two-point- of the microstructure. Various homogenization methods have been developed for this purpose including the averaging methods of Voigt and Reuss, which were shown to lead to rigorous bounds on the effective moduli of N-phase composites with prescribed volume fractions, and of polycrystals with given orientation distriLution functions, by Hill [3] and Paul [4]. More refined bounds incorporating two- and three-point statistics were made possible by the works of Hashin & Shtrikman [5-7] and Beran [8], respectively (see the reviews of Willis [9-
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