Thermodynamic analysis of brittle fracture specimens near the threshold developed by Rice (Thermodynamics of quasi-static growth of Griffith cracks, J Mech Phys Solid 26:61–78, 1978) is extended to specimens undergoing microstructural changes. The proposed extension gives rise to a generalization of the threshold concept that mirrors the way the resistance curve generalizes the fracture toughness. In the absence of experimental data, the resistance curve near the threshold is constructed using a basic lattice model.

It is demonstrated that for an isolated Mode I planar crack embedded in an infinite body, the stress intensity factor along the crack front is a function independent of the elastic constants.

We present generalizations of Hill's classical results concerned with the macroscopic strain and stress measures. Generalizations involve polynomial boundary conditions and polynomial moments of the microscopic fields. It is shown that for higher-order polynomials certain boundary conditions and moments should be excluded from considerations in order to guarantee unique relationships between boundary data and macroscopic measures. Particularly simple relationships are obtained for spherical specimens, for which higher-order macroscopic measures are defined in terms of spherical harmonics. Also it is demonstrated that higher-order macroscopic measures and constitutive equations can be useful in multi-scale analysis of problems formulated in terms of integral equations.

Interactions in linear elastic solids containing inhomogeneities are examined using integral equations. Direct and reflected interactions are identified. Direct interactions occur simply because elastic fields emitted by inhomogeneities affect each other. Reflected interactions occur because elastic fields emitted by inhomogeneities are reflected by the specimen boundary back to the individual inhomogeneities. It is shown that the reflected interactions are of critical importance to analysis of representative volume elements. Further, the reflected interactions are expressed in simple terms, so that one can obtain explicit approximate expressions for the effective stiffness tensor for linear elastic solids containing ellipsoidal and non-ellipsoidal inhomogeneities. For ellipsoidal inhomogeneities, the new approximation is closely related to that of Mori and Tanaka. In general, the new approximation can be used to recover Ponte Castañeda–Willis׳ and Kanaun–Levin׳s approximations. Connections with Maxwell׳s approximation are established.