Exterior statistics based boundary conditions for establishing statistically equivalent representative volume elements of statistically nonhomogeneous elastic microstructures

The exterior statistics-based boundary conditions or ESBCs have been developed as optimally effective boundary conditions that can be applied to statistically equivalent representative volume elements or SERVEs of heterogeneous microstructures for predicting homogenized response functions in Ghosh and Kubair (2016). However the initial development was for nonuniform, but statistically homogeneous microstructures with no localized features like clustering. In this case, the radial distribution function S2(r) is adequate for the statistically informed Green's functions needed for the development of ESBCs. However, when the microstructure includes statistical inhomogeneities in the form of fiber clusters or matrix-rich regions, the distance-based radial distribution function becomes ineffective. This paper overcomes this shortcoming by introducing the joint, distance (radial) and orientation-based two-point correlation functions S2(r, θ) in the statistically informed Green's functions needed for the ESBCs. The efficacy of the ESBCs is illustrated through validation simulations that compare its results with those generated by affine transformation based displacement and periodicity boundary conditions. Additionally, comparisons are made with the statistical volume elements or SVE methods. It is concluded that the simulations with ESBCs prescribed on the SERVEs have a definite advantage over other methods in defining optimal sized SERVEs without any iteration.