Stiffness tensor random fields through upscaling of planar random materials

Unique effective material properties are not possible for random heterogeneous materials at intermediate length scales, which is to say at some mesoscale above the microscale yet prior to the attainment of the representative volume element (RVE). Focusing on elastic moduli in particular, a micromechanical analysis based on the Hill-Mandel condition leads to the conclusion that two fields, stiffness and compliance, are required to bound the response of the material. In particular, we analyze means and correlation coefficients of a random planar material with a two-phase microstructure of random checkerboard type. We employ micromechanics, which can be viewed as an upscaling, smoothing procedure using the concept of a mesoscale "window", and random field theory to compute the correlation structure of 4th-rank tensor fields of stiffness and compliance for a given mesoscale. Results are presented for various correlation distances, volume fractions, and contrasts in stiffness between phases. The main contribution of this research is to provide the data for developing analytical correlation functions, which can then be used at any mesoscale to generate micromechanically based inputs into analytical and computational mechanics models.