A high-order three-scale reduced homogenization for nonlinear heterogeneous materials with multiple configurations

Abstract

A novel high-order three-scale reduced homogenization (HTRH) approach is introduced for analyzing the nonlinear heterogeneous materials with multiple periodic microstructure. The first-order, second-order and high-order local cell solutions at microscale and mescoscale gotten by solving the distinct multiscale cell functions are derived at first. Then, two kinds of homogenized parameters are calculated, and the nonlinear homogenization equations defined on global structure are evaluated, successively. Further, the displacement and stress fields are established as high-order multiscale approximate solutions by assembling the various unit cell solutions and homogenization solutions. The significant characteristics of the presented approach are an efficient reduced model form for solving high-order nonlinear local cell problems, and hence reducing the computational cost in comparison to direct computational homogenization. Besides, the new asymptotic high-order nonlinear homogenization does not need higher order continuities of the coarse-scale (or macroscale) solutions. Finally, by some representative examples, the efficiency and accuracy of the presented algorithms are verified. The numerical results clearly illustrate that the HTRH approach reported in this work is effective and accurate to predict the macroscopic nonlinear properties, and can capture the microscale and mesoscale behavior of the composites accurately.