Thermo-mechanical analysis of nonlinear heterogeneous materials by second-order reduced asymptotic expansion approach

An effective second-order reduced asymptotic expansion (SRAE) approach is proposed to analyze the thermo-mechanical coupling problems of nonlinear periodic heterogeneous materials. The first-order and second-order nonlinear unit cell solutions at a microscale obtained by calculating the different multiscale cell functions are given at first. Then, the homogenization coefficients are evaluated, and the nonlinear homogenized equations defined on whole structure are solved, successively. Also, the temperature and displacement fields are constructed as second-order multiscale approximate solutions by assembling the distinct local cell solutions and homogenized solutions. The main features of the proposed approach are: (i) an effective model reduction scheme for computing high-order nonlinear unit cell problems and (ii) an asymptotic high-order homogenization solution that does not need high-order continuity of the macroscale solutions. Further, the corresponding finite element-difference algorithms based on the SRAE approach are given in detail. Finally, by some typical numerical examples, the availability and correctness of the proposed algorithms are confirmed. The computational results clearly demonstrate that the SRAE approach reported in this work are efficient and valid to predict the macroscopic thermo-mechanical properties, and can catch the microscale information of the heterogenous materials accurately.