A high-order three-scale reduced asymptotic approach for thermo-mechanical problems of nonlinear heterogeneous materials with multiple spatial scales

An effective high-order three-scale reduced asymptotic (HTRA) approach is developed to simulate the coupled thermo-mechanical problems of nonlinear heterogeneous materials with multiple spatial scales. In this paper, the heterogeneous structures are constructed by periodical layout of unit cells on microscopic and mesoscopic scale. At first, the linear and nonlinear higher-order local cell functions defined at microscale and mesoscale are derived by multiscale asymptotic expansion approach. Further, two types of homogenized parameters are introduced, and related nonlinear homogenized solutions are obtained on the macroscopic domain. Then, the high-order three-scale temperature and displacement solutions are established by assembling the various multiscale local solutions and homogenization solutions. The key characteristic of the presented methods is an efficient reduced form for evaluating higher-order local cell problems, and hence greatly reducing the calculating amount in comparison to direct numerical simulations. Also, the new high-order nonlinear homogenization solutions that do not require higher-order continuity of the macroscale solutions. Finally, the effectiveness and accuracy of the multiscale algorithms are verified by some typical numerical examples.