We discuss cross-property connections that interrelate effective linear elastic and conductive properties of heterogeneous materials.  More precisely, they relate changes in the properties, as compared to the ones of the bulk material, caused by various inhomogeneities (cracks, pores, inclusions).  They may also be developed for microstructures formed by multiple contacts, such as rough surfaces pressed against each other.  Such connections are especially useful if one property (say, electrical conductivity) is easier to measure than the other (anisotropic elastic constants).

For the properties governed by mathematically similar laws (for example, electrical and thermal conductivities), the connections are straightforward.  However, for the elasticity-conductivity connections – the main focus of the present work – their very existence is non-trivial: not only the governing equations are different, but even the ranks of tensors characterizing the properties are different (fourth-rank tensor of elastic constants vs second-rank conductivity tensor).  We overview various approaches to the problem and then advance the approach rooted in similarity of the microstructural parameters that control the given pair of properties.  This similarity leads to connections that, albeit approximate, have explicit closed form.  They have been experimentally verified on several heterogeneous materials (metal foams, short fiber reinforced composites, metals with fatigue microcracks, sprayed coatings).  On the other hand, for properties controlled by essentially different parameters (such as permeability or fracture of a microcracked material and its elasticity), the correlations may hold only qualitatively, at best.