Two-Scale Analysis for Dynamic Coupled Thermoelasticity Problems of Periodical Composite Materials

The dynamic propagation of mechanical and thermal disturbances in composite materials and their structure has been one of the subjects of recent investigations. The coupled mechnical-thermal behavior of periodicity composites under dynamic loads in classical thermoelasticity theory can be expressed as a PDE system with periodic oscillating coefficients with period ε. For this kind of multi-scale problems arisen in material science and engineering computation, engineers are often interested in evaluating the physics and mechanics properties of different scales, such as macroscopic scale, mesoscopic scale and microscopic scale. On macroscopic sacle,Gilles [1] and William [2] had given the homogenized procedure for dynamic coupled thermoelasticity problems of periodical composites. On mesosopic scale, to solve the problems by analytical methods is difficult because the dynamic displacement field and temperature field must be solved simultaneously. At the same time, since the material coefficients sharply vary with e-periodicity, then the solution, especially their derivatives vary sharply too. While FEMs or FDMs method is applied to solving this kind of problems, in order to capture the small-scale behaviors of the solution and its derivatives, the mesh size must be very small. So it leads to very large scale computation, and the big computation cost. Feng and Cui [3] studied the static coupled thermoelastic problem of periodical composites and gave an efficient Two-Scale algorithm to obtain the numerical solution, that can capture the meso-scale property of composites effectively. In this paper, using the Two-Scale Method proposed by Cao and Cui [4], we obtain the two-order and two-scale asymptotic expansion formula for dynamic coupled thermoelasticity problems of periodical composites in three dimension case. We also define the effective physical and mechanical parameters of the composites. When the fore three terms in two-scale asymptotic expansion are taken as approximate solution, it can be proved that the approximate order is O(e). Based on the asymptotic expansion formula, we propose a Two-Scale finite element algorithm to numerically solve the dynamic coupled thermoelasticity problems previously. At last, the numerical experiments are given and the validity of our algorithm is verified.