An incremental-secant mean-field homogenization method with second statistical moments for elasto-visco-plastic composite materials

Abstract

This paper presents an extension of the recently developed incremental-secant mean-field homogenization (MFH) procedure in the context of elasto-plasticity to elasto-visco-plastic composite materials while accounting for second statistical moments. In the incremental-secant formulation, a virtual elastic unloading is performed at the composite level in order to evaluate the residual stress and strain states in the different phases, from which a secant MFH formulation is applied. When applying the secant MFH process, the linear-comparison-composite (LCC) is built from the piece-wise heterogeneous residual strain-stress state using naturally isotropic secant tensors defined using either first or second statistical moment values. As a result non-proportional and non-radial loading conditions can be considered because of the incremental-secant formulation, and accurate predictions can be obtained as no isotropization step is required. The limitation of the incremental-secant formulation previously developed was the requirement in case of hard inclusions to cancel the residual stress in the matrix phase, resulting from the composite material unloading, to avoid over-stiff predictions. It is shown in this paper that in the case of hard inclusions by defining a proper second statistical moment estimate of the von Mises stress, the residual stress can be kept in the different composite phases. Moreover it is shown that the method can be extended to visco-plastic behaviors without modifying the homogenization process as the incremental-secant formulation only requires the definition of the secant operator of the different phase material models. Finally, it is shown that although it is also possible to define a proper second statistical moment estimate of the von Mises stress in the case of soft inclusions, this does not improve the accuracy as compared to the increment-secant method with first order statistical moment estimates.