Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity

Abstract

We show that the averaged response of random isotropic Cauchy elastic material can be described analytically. It leads to a higher gradient model with explicit expressions for the dependence on the second derivatives of the mean field. A subsequent penalty formulation coincides with a linear elastic micro-stretch model with specific choice of constitutive parameters, depending only on the average cut-off length (the internal characteristic length scale L c > 0). Thus the microstretch displacement field can be interpreted as an approximated mean field response for these parameter ranges. The mean field free energy in this micro-stretch formulation is not uniformly pointwise positive, nevertheless, the model is well posed.