Homogenization of periodic materials with viscoelastic phases using the generalized FVDAM theory

Abstract

In the last decades, new generations of advanced materials have been designed and manufactured for specific applications. The micromechanics plays an important role in the development of heterogeneous materials, enabling efficient analyses of composite materials with complex geometries, circumventing the traditional trial-and-error approach, producing substantial cost savings. The unit cell problem to the analysis of periodic heterogeneous media can be solved by the well-established 0th order version of the finite-volume theory, named finite-volume direct averaging micromechanics (FVDAM) theory. This standard version of the FVDAM theory employs an incomplete second-order displacement field within individual subvolumes of a discretized analysis domain together with a surface-averaging framework, which does not enforce displacement or traction continuity in a point-wise manner. This, in turn, produces interfacial interpenetrations and non-traction stress discontinuities, thereby demanding very refined meshes in order to produce good interfacial conformability and pointwise stress continuity between adjacent subvolumes. To overcome these shortcomings, a generalized FVDAM theory has been proposed to enable analysis of periodic heterogeneous materials in the finite-deformation domain. The generalization is based on a higher-order displacement field representation and on the definition of elasticity-based surface-averaged kinematic and static variables related through a local stiffness matrix. Herein, we specialize the generalized FVDAM theory to the infinitesimal analysis of periodic materials with viscoelastic phases, where a total or secant formulation is employed, with the viscoelastic strains evaluated incrementally using an algorithm based on the concept of state variables. The generalized or 2nd order version considerably improves interfacial conformability and pointwise traction and non-traction stress continuity between adjacent subvolumes in comparison with the 0th order version, but with a higher computational cost. Furthermore, for the same mesh discretization, these two versions provide comparable macroscopic response. Considering these features, the 0th order version is recommended to evaluate the effective elastic properties and the homogenized creep and relaxation functions, while the 2nd order version is more efficient in the evaluation of the microscopic displacement and stress fields.