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An approximate solution to the title problem for a scalar model elastic medium is obtained via a variational formulation. The layer of random composite is a mixture of two materials both of which have the same elastic modulus but different densities while the adjoining homogeneous half-spaces may have any moduli and densities. No information on the composite is assumed, other than statistical uniformity with the one- and two-point probabilities assigned. With the additional assumption of an exponential two-point correlation function, the composite supports exactly two mean plane waves and the solution is correspondingly explicit. Particular attention is paid to the total fluxes of energy. The system is excited by a plane harmonic wave incident from the lower half-space. All of the disturbance so generated is random and can be split into a mean disturbance (which is independent of the sample of the random composite) plus a disturbance whose mean is zero. Both contribute to the mean flux of energy, which conforms to the requirement of conservation of energy, both exactly and in the variational approximation. The entire solution is studied without any approximation other than that in the variational formulation. [ABSTRACT FROM AUTHOR]

Strain-gradient plasticity theories are reviewed in which some measure of the plastic strain rate is treated as an independent kinematic variable. Dislocation arguments are invoked in order to provide a physical basis for the hardening at interfaces. A phenomenological, flow theory version of gradient plasticity is constructed in which stress measures, work-conjugate to plastic strain and its gradient, satisfy a yield condition. Plastic work is also done at internal interfaces and a yield surface is postulated for the work-conjugate stress quantities at the interface. Thereby, the theory has the potential to account for grain size effects in polycrystals. Both the bulk and interfacial stresses are taken to be dissipative in nature and due attention is paid to ensure that positive plastic work is done. It is shown that the mathematical structure of the elasto-plastic strain-gradient theory has similarities to conventional rigid-plasticity theory. Uniqueness and extremum principles are constructed for the solution of boundary value problems. [Copyright &y& Elsevier]

Abstract: In this paper the mechanical behaviour of finite random heterogeneous bodies is considered. The analysis of non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. Microstructures can be periodic but generically they are random. In the first case, an exact calculation can be performed but in the second case recourse has to be made either to simulation or to some scheme of approximation. One such scheme is based on a stochastic variational principle. The novelty of the present work is that a stochastic variational principle is projected directly onto a finite-element basis so that all subsequent analysis is performed within a finite-element framework. The proposed formulation provides expressions for the local stress and strain fields in any realization of the medium, from which expressions for statistically-averaged quantities can be derived. Then an approximation of Hashin–Shtrikman type is developed, which generates a FE-based numerical procedure able to take account of interactions between random inclusions and boundary layer effects in finite composite structures. Finally, two examples are presented, namely a cylinder with square cross-section subjected to mixed boundary conditions of different types on different faces and a rectangular body containing a centre crack. The results show that in the vicinity of the boundary or close to the crack tip, the strain and the stress in the matrix and in the inclusions differ considerably from those obtained by the formal application of conventional homogenization. [Copyright &y& Elsevier]