Composites with imperfect interface

New variational principles and bounds are introduced, describing the effective conductivity tensor for anisotropic two-phase heat conducting composites with interfacial surface resistance between phases. The new upper bound is given in terms of the two-point correlation function, component volume fractions and moment of inertia tensor for the surface of each heterogeneity. The new lower bound is given in terms of the interfacial surface area, component volume fractions and a scale-free matrix of parameters. This matrix corresponds to the effective conductivity associated with the same geometry but with non-conducting inclusions. The bounds are applied to theoretically predict the occurrence of size effect phenomena. We identify a parameter R(cr) that measures the relative importance of interfacial resistance and contrast between phase resistivities. The scale at which size effects occur is determined by this parameter. For isotropic conducting spheres in a less conducting isotropic matrix we show that for monodisperse suspensions of spheres of radius R(cr) the effective conductivity equals that of the matrix. For polydisperse suspensions of spheres it is shown that, when the mean radius lies below R(cr), the effective conductivity lies below that of the matrix.