TFA and HS based homogenization techniques for nonlinear composites

The paper presents two reduced order homogenization techniques for studying the response of nonlinear composite materials. The first approach is based on Transformation Field Analysis, which considers the presence of eigenstrains to account for inelastic strains, while the second approach is derived from Hashin–Shtrikman variational principle, which introduces eigenstresses, namely polarization stresses, on a homogeneous elastic reference material to account for the heterogeneity and the inelastic response. In particular, the case of elasto-plastic periodic composites is investigated. A Unit Cell is identified and divided in subsets. In each subset, the eigenstrains or eigenstresses are assumed uniform. A very effective technique is derived for the Hashin–Shtrikman approach updating the elastic reference material properties during the inelastic strain evolution. Numerical procedures are implemented to derive the nonlinear response of composites with plastic constituents. Several numerical applications are carried out to assess the effectiveness of the two presented reduced order models. A deep investigation on the differences and similarities of the two approaches is presented, proving their equivalence under particular circumstances. Simple and complex loading histories are considered, comparing the results with a finite element solution, considered as the reference solution.