Critical Properties of an Elastic Fractal

Some solvable fractal models of the percolating backbone are used to investigate the critical behavior of a random, $d$-dimensional, isotropic, elastic medium. The critical exponent $T$ for the elastic moduli is found to be appreciably greater than the conductivity exponent $t$, and the ratio of bulk to shear modulus is found to have the universal value $\frac{4}{d}$. A comparison with the effective-medium and the Clausius-Mossotti-type approximations leads to the conjecture that the result $\frac{4}{d}$ is in fact exact.