A symmetric fully optimized second‐order method for nonlinear homogenization

Abstract

AbstractWe consider an alternate formulation of the recently developed ‘fully optimized second‐order’ (FOSO) nonlinear homogenization method, which is based on a stationary variational principle for the macroscopic energy function. In this method, the trial fields are the properties of a suitably designed linear comparison composite (LCC), thus allowing for the estimation of the effective response and field statistics of the nonlinear composite in terms of available estimates for the corresponding quantities in the LCC. The formulation considered in this paper makes use of an alternative choice for the linear comparison composite, leading to homogenization estimates that are more symmetric with respect to Legendre duality than earlier FOSO estimates. The new ‘symmetric’ FOSO method is applied to a class of two‐phase power‐law composites with fibrous microstructures subjected to plane strain loading. The resulting estimates for the effective response and field statistics are found to improve on earlier estimates, and to be in good agreement with full‐field numerical simulations for nonlinear composite cylinder assemblages, as well as with available results for sequentially layered composites.