Green’s function and stress fields in stochastic heterogeneous continua

Abstract

Many engineering materials used today are heterogenous in composition e.g. Composites – Polymer Matrix Composites, Metal Matrix Composites. Even, conven-tional engineering materials – metals, plastics, alloys etc. – may develop heterogenei-ties, like inclusions and residual stresses, during the manufacturing process. Moreover, these materials may also have intrinsic heterogeneities at a nanoscale in the form of grain boundaries in metals, crystallinity in amorphous polymers etc. While, the ho-mogenized constitutive models for these materials may be satisfactory at a macroscale, recent studies of phenomena like fatigue failure, void nucleation, size-dependent brittle-ductile transition in polymeric nanofibers reveal a major play of mi-cro/nanoscale physics in these phenomena. At this scale, heterogeneities in a material may no longer be ignored. Thus, this demands a study into the effects of various material heterogeneities.

In this work, spatial heterogeneities in two material properties – elastic modulus and yield stress – have been investigated separately. The heterogeneity in the elastic modulus is studied in the context of Green’s function. The Stochastic Finite Element method is adopted to get the mean statistics of the Green’s function defined on a stochastic heterogeneous 2D infinite space. A study of the elastic-plastic transition in a domain having stochastic heterogenous yield stress was done using Mont-Carlo meth-ods. The statistics for various stress and strain fields during the transition were ob-tained. Further, the effects of size of the domain and the strain-hardening rate on the stress fields during the heterogeneous elastic-plastic transition were investigated. Finally, a case is made for the role of the heterogenous elastic-plastic transition in damage nucleation and growth.