Composites with imperfect interface

New variational principles are introduced, describing the eective conductivity tensor for anisotropic two-phase heat conducting composites with interfacial surface resistance between phases.These principles reduce to those of Hashin and Shtrikman in the limit of vanishing surface resistance.The new principles allow for an unconstrained class of trial elds.New bounds are derived on the eective heat conductivity tensor for an anisotropic two-phase composite with interfacial surface resistance between phases.The new upper bound is given in terms of the two-point correlation function, component v olume 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 v olume fractions and a scale free matrix of parameters.This matrix corresponds to the eective conductivity associated with the same geometry but with nonconducting inclusions.The bounds are applied to theoretically predict the occurence of size eect phenomena in the effective heat transport behaviour of two-phase suspensions with interfacial resistance between phases.For monodisperse suspension of spheres a critical radius is found for which the eective heat conductivity equals that of the matrix.For polydisperse suspensions of spheres it is shown that, when the mean radius lies below the critical value, the eective conductivity lies below that of the matrix.