Asymptotic Homogenization Technique for Stress and Strength for Some Non-Linear Elastic Problems with Near Periodic Structure J. Orlik Fraunhofer Institut Techno- und Wirtschaftsmathematik, Gottlieb-Daimler-Str. 49, 67663 Kaiserslautern, Germany orlik@itwm.fhg. de 1. SUMMARY An asymptotic homogenization procedure is employed to obtain effective constitutive law, a homogenized macro-stresses and -deformation gradient and a first approximation to the micro-stresses and deformation gradient in non-linear elastic composite materials, from their microstructure, properties of the components and applied macro-loads. Using the approximate micro-deformation gradient, a non-local macro- strength condition for the composite materials is determined by elastic and strength micro-properties of the periodicity cell and expressed in terms of the homogenized macro-deformation gradient. The procedure is illustrated by formal asymptotic expansion for incompressible Neo-Hooke material. 2. INTRODUCTION The homogenization technique is an asymptotic method, which allows to find local approximate stresses and deformation gradient in a composite material with near periodic structure. This method is useful for composites structured by different size-scales, when direct application of the finite element method (FEM) becomes very expensive. The main idea of the asymptotic homogenization is clustering of the problem with the microstructure on two one-scale problems, namely an auxiliary periodic problem on the unit cell with the same structure as the periodicity cell and a boundary value problem (BVP) in the macro-domain with averaged over the cell mechanical characteristics. The problems can be further solved by the FEM. The first approximation to the local deformation gradient is then represented by a combination of the solutions to the auxiliary cell- problems and the macro-problem. 133