Locally fluctuating elasticity of vulcanized solids

The elastic properties of vulcanized solids, such as rubbery polymeric media, are characterized not only by their small mean shear moduli but also by the random spatial fluctuations of their local elasticity. These fluctuations originate in the randomness locked in, e.g., to a polymer network during the cross-linking process. We examine these fluctuations in the local elasticity of random solids via vulcanization theory, i.e., the replica field theory that describes cross-linking- driven random solidification transitions and their emergent rigidity. To do this we identify the Goldstone excitations of the random solid state as local shear deformations, and interpret their effective free energy in terms of a phenomenological model of inhomogeneous, nonlocal rubber elasticity. Thus, we arrive at predictions for the statistical properties of the fluctuating elasticity of vulcanized solids.