Search

Hashin–Shtrikman based bounds and estimates are obtained for the linear and non-linear effective properties of composites in the form of a thin coating or sandwich layer. It is assumed that the thickness of the layer is of the same order of magnitude as the correlation length between phases, and size effects thereby result. Boundary layers exist within the coating adjacent to the substrate and to the free surface (in the case of a coating). Attention is focused on two-dimensional problems by considering anti-plane shear of an isotropic 2-phase composite on a single-phase substrate, with microstructure prismatic along the direction of anti-plane shear. [ABSTRACT FROM AUTHOR]

Abstract: Effective constitutive relations for waves in composites with random microstructure were proposed by as relations between ensemble averages of stress and momentum, and “effective” strain and velocity which were related to a weighted ensemble average of displacement, and results of an example one-dimensional calculation were presented, explicitly demonstrating the possibility of coupling between mean stress and effective velocity, and mean momentum density and effective strain, even in the long-wavelength (or homogenization) limit. Relations of this type have recently been recognized to be inevitably non-unique, and a quite general prescription for defining unique relations has been advanced (). The present work compares and contrasts the effective relations obtained by either formulation, for the example considered in 2009. The work of 2009 is generalized to the extent that the constituent materials are taken to have some dissipation. It emerges explicitly that an “effective elastic constant” obtained by the method of 2009 can display an apparent energy gain rather than loss. This is not the only term that contributes, however, and it is shown that the effective material remains dissipative, as it should. It is also confirmed, both theoretically and in the computation, that either formulation leads to exactly the same mean stress and momentum density, and to the same dissipation. [Copyright &y& Elsevier]

Abstract: Milton and Willis [Milton, G.W., Willis, J.R., 2007. On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. A 463, 855–880] have recently discussed the general form of constitutive relations that should apply for the description of the dynamics of media with microstructure below the scale of measurement, and concluded that these relations should have the form, somewhat generalized, of effective relations for composites implied by a formulation of Willis [Willis, J.R., 1981. Variational and related methods for the overall properties of composites. In: Yih, C.S. (Ed.), Advances in Applied Mechanics, vol. 21. Academic Press, New York, pp. l–78; Willis, J.R., 1981. Variational principles for dynamic problems for inhomogeneous elastic media, Wave Motion 3, 1–11]. Whereas for a general composite (treated as a random medium), these relations can only be found at some level of approximation, they can be found exactly for a laminate with periodic microstructure, restricted to one-dimensional propagation normal to the lamination, for which Floquet theory is available. Such a medium becomes random if it is randomly translated: the exact position of any one interface is then not known, and ensemble averages may correspondingly be calculated. Example calculations demonstrate, completely explicitly, that “effective modulus” and “effective density” are operators, non-local in space and time; furthermore, if the medium does not display symmetry under reflection, mean stress is coupled not only to mean strain but also to mean velocity, and mean momentum density is likewise coupled not only to mean velocity but also to mean strain. A precisely similar structure applies to the propagation of electromagnetic waves: the effective medium is non-local and bi-anisotropic. The same structure is preserved under a generalized formulation, applicable to cases where only some component of the microstructure is observable. [Copyright &y& Elsevier]

In this paper, composites with a graded distribution of heterogeneities are considered. The heterogeneities vary in statistically non-uniform fashion since in a finite layer (or region) properties such as local volume fraction vary gradually. In order to study this class of composites, a procedure of analysis which leads to the effective constitutive non-local operator of the medium is proposed. For two-phase composites, an approximation of Hashin–Shtrikman type for this operator has been obtained in real space and this has been developed explicitly in the case of laminates. [Copyright &y& Elsevier]