Mean stresses in microstructures due to interface stresses: A generalization of a capillary equation for solids

We derive a general expression for the mean stress in a solid induced by the stresses of areal and linear defects such as grain boundaries and triple junction lines. The mean stress is a function of measurable stereological quantities, weighted areas of interfaces and lengths of the linear defects, not their curvatures. Generally the stress is non-hydrostatic. For anisotropic and textured microstructures, the mean stress depends on stereological moments of the interface orientation distribution function, but otherwise it is independent of the geometry of the microstructure. In two limiting cases, isolated spherical particles (in a matrix or not) and layer structures, the solution for the stress reduces to results which have been derived previously. These solutions can be used to determine average interface stresses from experimental lattice constant data. Although the specific interface area is always the important quantity in this relation, for the spheres this area/volume reduces to the particle curvatures commonly used in the literature for many microstructures. (C) 1997 Acta Metallurgica Inc.