Computational plasticity for composite structures based on mathematical homogenization: Theory and practice

This paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains. Starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases. The overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro-constituent, and history data is updated at two points (reinforcement and matrix) in the microstructure, one for each phase. Macroscopic history data is stored in the database and then subjected in the post-processing stage onto the unit cell in the critical locations. For numerical examples considered, the CPU time obtained by means of the two-point averaging scheme with variational micro-history recovery with 30 seconds on SPARC 10/51 as opposed to 7 hours using classical mathematical homogenization theory. At the same time the maximum error in the microstress field in the critical unit cell was only 3.5% in comparison with the classical mathematical homogenization theory.