Wave speed-porosity scaling for cellular solids

We present general arguments for the dependence of wave speed upon the porosity of cellular solids. In particular, we explore this relationship in the context of the topology of the network architecture. We show that the wave speed in cellular solids having rigid network topology equals 0.58 times the speed in the porosity filled continuum made of the same material. This is a very universal result which is independent of the material properties and is valid asymptotically in the long wavelength and thin cell wall limits. This scaling is shown to be linear with volume fraction for mechanism topologies in a high porosity medium. We further show that a scaling law with a single exponent violates the Hashin-Shtrikhman bounds, hence untenable. The advantage of stiffness for kinematically rigid networks over mechanism topology networks disappears for a volume fraction of about 0.56. An extended version of this work will be published elsewhere.