Link between the conductivity and elastic moduli of composite materials

We derive relations linking the conductivity σ* and elastic moduli of any two-dimensional, isotropic composite material. Specifically, upper and lower bounds are derived on the effective bulk modulus κ* in terms of σ* and on the effective shear modulus μ* in terms of σ*. In some cases the bounds are attainable by certain microgeometries and thus optimal. Knowledge of the conductivity can yield sharp estimates of the elastic moduli (and vice versa) even for infinite phase contrast.