At first, we explain the background in which the global-local numerical method for granular media [1,2], which was based on mathematical homogenization theory [3,4], was developed. We also explain the feature of the numerical method. Especially, the governing equations of the method show that the role of the constitutive equation in a global structure is performed by micro-scale analysis. Next, we describe the purpose of this study that we discuss the mechanical properties of granular materials by focusing on the distinguished idea of the global-local method, that the micro-scale analysis provides a macroscopic constitutive relationship. We give a particular focus to the relationship of elastic properties (one of the basic items of the constitutive model) and induced anisotropy.

Next, we present an overview of the global-local analysis method for granular media based on mathematical homogenization theory [3,4], showing how the micro-scale problem provides the constitutive relationship. Mainly, the governing equations in two-scales and the concept of the method are presented.

We show numerical experiments to examine the relationships between the elastic properties and the induced anisotropy. A series of the numerical tests composed of the shear loading test and stress probe test are carried out. At first, the analysis condition such as the model, loading paths, parameters are expressed. Next, we show the results of the numerical experiments. Stress-strain curves and micro-structural anisotropy in shear loading tests are presented. Incremental elastic response by the stress probe tests are also presented.

We formulate the induced anisotropic elastic matrix by introducing the fabric tensor directly. We show that the induced anisotropy of the incremental elastic response can be reproduced for the most part by the equation.

Finally, we summarize and arrange conclusions. The main conclusions of this study are as follows:

1. The incremental elastic response of idealized dry granular materials in initial isotropic stress state shows the near isotropy.
2. The incremental elastic response in shear loadings indicates the induced anisotropy along with the degree of the shear stress.
3. The anisotropic response of the incremental elastic strain is closely related to the micro-structural geometrical anisotropy and can be represented by the simple anisotropic elastic matrix that introduces the fabric tensor.

1

K. Kaneko, K. Terada, T. Kyoya and Y. Kishino, "Global-Local Analysis of Granular Media in Quasi-Static Equilibrium", Int. J. Solids Struct., 40, 4043-4069, 2003. doi:10.1016/S0020-7683(03)00209-9

2

K. Kaneko, S. Tsutsumi, T. Ishii, K. Kumagai, Y. Kishino and K. Hashiguchi, "Macroscopic Constitutive Relation of Granular Materials Predicted by Global-Local Analysis", Powders and Grains 2005, 159-163, 2005.

3

J. L. Lions, "Some Methods in the Mathematical Analysis of Systems and Their Control", Science Press, Beijing, China, 1981.

4

G. Allaire, "Homogenization and Two-Scale Convergence", SIAM, J. Math. Anal., 23, 1482-1518, 1992. doi:10.1137/0523084

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