zbMATH — the first resource for mathematics

Nonlinear composites. (English) Zbl 0889.73049
Adv. Appl. Mech. 34, 171-302 (1998).
From the introduction: The article is concerned with some closely related methods that have been developed recently to estimate the effective behavior of nonlinear composite with random microstructures. After a short introduction, section II deals with some preliminaries, including the definition of effective potentials, by means of the classical minimum energy principles, and their application to the determination of bounds of the Voigt-Reuss type for nonlinear composites. Section III gives a brief presentation of the variational principles which provide an extension of the Hashin-Shtrikman variational principles for nonlinear composites and utilising a linear homogeneous reference material, and allow the computation of improved bounds for nonlinear composites with random microstructures incorporating statistical information of order two. Section IV presents new variational principles for isotropic and power-law composites, respectively, which allow the restatement of the effective energy function of a nonlinear composite in terms of that of an appropriately chosen linear heterogeneous reference material, where the distribution of moduli in the reference material is determined by the variational principle itself. Application of suitable approximations in the context of these variational principles then permits the determination of bounds for nonlinear composites directly from corresponding bounds for linear comparison composites with the same microstructures as the nonlinear composites.
Section V starts with an asymptotic expansion which is exact to the second order in the contrast of the properties of the phases and shows that the bounds and estimates obtained by the variational procedures of section IV (and therefore those of section III) are only exact to the first order in the contrast. A selection of results for linear composites is presented in section VI. Finally, section VII provides some applications to sample nonlinear composites, including porous materials, rigidly reinforced composites, and two-phase power-law and ideally plastic composites.
For the entire collection see [Zbl 0881.00023].

74E30 Composite and mixture properties
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74A40 Random materials and composite materials