A second-order multiscale reduced homogenization for nonlinear statistically heterogeneous materials

This work introduces an effective second-order multiscale reduced homogenization (SMRH) approach to analyze the nonlinear statistically heterogeneous materials. In these kinds of composites, the microscale information of particles, including their shapes, sizes, orientations, spatial distributions, volume fractions and so on, changes with position of the structures. At first, the micro-configurations of the heterogeneous structure with random distributions are briefly described. Then, the SMRH formulations for nonlinear problems are constructed, along with detailed statistical multiscale methods for statistically heterogeneous materials. The key characteristics of the new statistical multiscale methods include: (i) innovative reduced models designed to solve inelastic problems in random composites with significantly lower computational cost, (ii) high-order homogenized solutions that sidesteps the need for higher-order continuity in the macro solutions, and (iii) statistical high-order multiscale algorithms developed for investigating nonlinear statistically heterogeneous materials. Finally, several representative numerical examples are presented to validate the effectiveness of nonlinear random materials under different probability distribution models. The computational results clearly demonstrates that the statistical second-order multiscale reduced homogenization is valid for analyzing the nonlinear problems of statistically heterogeneous materials and proves beneficial for the development of random composites with multiscale arrangements.