Keywords: inverse analysis, multiscale finite element method, homogenization, composite materials, nonlinear elasticity.

Within the scope of this contribution, a combination of the Levenberg-Marquardt method with the multiscale finite element method is used in order to determine the material parameters of composite materials. The first of the mentioned methods represents an improved version of the steepest descent method, where the Taylor approximation is used in order to improve the convergence in the vicinity of the solution. The method is used to minimize a merit function which depends on the experimental and numerical results. The latter are obtained by using the multiscale finite element approach where for the determination of the effective material parameters and the constitutive law, an analysis of the behaviour of a representative volume element (RVE) is introduced [1]. The bridge between the scales is realized by introducing the Hill-Mandel macrohomogeneity condition requiring an energy balance between the scales [2]. The special significance of this condition lies in the fact that it allows the derivation of the boundary conditions for the RVE.

The concept presented as well as the software which has been developed for it have a general character and the chosen numerical examples should be understood just as possible illustrations. These examples deal with the investigation of the parameters of two-phase composites consisting of nonlinear elastic materials. The microstructure of these materials is depicted by a unit square RVE with an elliptic void or an inclusion embedded in the matrix material. The examples are concerned with the investigation of the material parameters related to the matrix and the inclusion separately as well as for both of them. The behavior of each phase is described by two parameters, the shear and the bulk modulus. In the examples, the following tendencies are observed. For a small number of material parameters (here up to four are considered) the unique solution is achieved independently from the initial trial value. The problem of determining the global minimum does not arise. A different number of experimental data in general leads to different iteration paths and possibly to a decrease in the number of iteration steps.

1

S. Ilic, K. Hackl, "Application of the Multiscale FEM to the Modeling of Nonlinear Multiphase Materials", Journal of Theoretical and Applied Mechanics, 47, 537-551, 2009.

2

R. Hill, "On Constitutive Macro-Variables for Heterogeneous Solids at Finite strain", Proceedings of the Royal Society A, 326, 131-147, 1972. doi:10.1098/rspa.1972.0001

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