Asymptotic behavior of a randomly voided material with elastic properties gradient
Résumé
The asymptotic expansion analysis was developed in the framework of homogenization technique, which is applicable for three-dimensional composites made up of inclusions randomly embedded within a matrix. The so-called asymptotic expansion homogenization (AEH) method was developed by Francfort [1] for the case of linear thermoelasticity in periodic structures. The AEH method has been employed to calculate the homogenized thermomechanical properties of composite materials (elastic moduli and coefficient of thermal expansion) [2, 3]. This technique of homogenization enables to replace heterogeneous materials by a homogeneous equivalent medium including second-order displacement gradients [4]. The displacement vector and the stress tensor are considered as functions of macroscopic (x) and microscopic (y) variables. They may be expanded in a series of powers of small (material) parameter , which is the ratio between macroscopic and microscopic scales. í More precisely, the present work is devoted to linear stochastic homogenization and Γ-convergence [5] problems for variational functional. This Γ-convergence allows us to study the corresponding variational problem and to prove the convergence of the minimums and of the minimizers. By combining variational convergence with ergodic theory, we study the macroscopic behavior of linear elastic heterogeneous materials. The inclusions are randomly distributed within a matrix, their size is of order η. The variational limit functional energy obtained when η tends to 0 is deterministic and non-local [6, 7]. By including the characteristic displacement vectors, or correctors, the problem can be solved in order to evidence some links with second gradient theory. The extended form of our variational formulation can be expressed as follows.In this equation, the functionalmodels the overall elastic energy, the functions U n represent the displacement fields within the structure subjected to a given load, denotes the first gradient of the (symmetric) macroscopic strain tensor and Ω denotes the volume of the structure. In addition, the dual quantities of the first and second gradients of the displacement field, represent the simple and double force stress tensors, respectively. Computational results in stochastic cases are presented. Comparison of the model with classical bounds