Effect of boundary condition and statistical volume element size on inhomogeneity and anisotropy of apparent properties

We use square Statistical Volume Elements (SVEs) to homogenize elastic and fracture properties of a carbon-nanofiber reinforced polymer. The effect of SVE size and boundary condition (BC) on homogenized properties is investigated. Three different criteria are proposed for the convergence of properties and reaching the Representative Volume Element (RVE) limit: (1) Variation-based criterion examines the trend at which the variation of a property decreases versus SVE size; (2) Mean-based criterion examines the rate at which the SVE-mean of property converges to its terminal value; (3) BC-based criterion examines whether the homogenized properties can be independent of the choice of BC. These criteria show that the geometric property of fiber volume fraction/mass density converges to its homogeneous limit first followed by the elasticity tensor. Fracture strengths have the most complex response as they remain highly inhomogeneous, random, anisotropic, and size-dependent for the SVE sizes considered. This is further discussed in the context of stochastic multiscale material modeling and well-known size effect models for fracture strength. Furthermore, the square shape of the SVE is shown to induce nonphysical bias angles wherein elastic and fracture properties often take their maximum or minimum values at integer multiples of 45 degrees. The traction BC is shown to be superior to displacement BC by having a weaker form of this anisotropy and higher strength values. Moreover, for the anisotropic domain anisotropy indices converge from below and above for traction and displacement BCs, respectively; thus resembling Reuss and Voigt hierarchy of bounds for elastic properties.