A Finite Element Method of the Boundary Value Problems of Elasto-Plastic Deformations for Granular Materials

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  • Ueno Masami
    Laboratory of Agricultural Machinery, Faculty of Agriculture, Kyushu University
  • Hashiguchi Koichi
    Laboratory of Agricultural Machinery, Faculty of Agriculture, Kyushu University

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  • 粒状体の弾塑性変形境界値問題に対する有限要素法
  • リュウジョウタイ ノ ダンソセイ ヘンケイ キョウカイチ モンダイ ニ タイス

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Abstract

In this paper, an incremental finite element method was formulated and a program for plane strain problems was prepared in order to analyze numerically boundary value problems of the elasto-plastic deformations for granular materials like soils. Elasto-plastic constitutive equations proposed by one of the authors were adopted to this finite element formulation and programming. A constitutive tensor D_<ijkl> plays very important roll on the finite element method. In elastic stress states, the elastic constitutive tensor E_<ijkl> was used for D_<ijkl>. On the other hand, the elasto-plastic constitutive tensor D^^-_<ijkl> was used for D_<ijkl> in plastic stress states. By composing these constitutive tensors and the shape function for triangular element, a stiffness equation is obtained as follows. K_<ij>Δu_j =ΔF_i where K_<ij> is a stiffness tensor, Δu_i is a nodal displacement increment and ΔF_i is a nodal load increment. Nodal displacement increments and nodal load increments are obtained as the solutions of this equation under the given boundary conditions, and then, strain increments and stress increments are calculated. Four examples for the condition of triaxial c o mpression were analized. These examples were composed of two kinds of soils, i. e., Weald clay and Toyoura sand, and of two different stress paths where the initial conditions at the beginning of compression are an isotropic plastic state or an elastic state. The strain increment was 0.0025 in one stage of the iterative calculations, and iterative numbers were 60-200 times. These calculated results were compared with experimental results of the triaxial compression tests for Weald clay (by Parry 1959) and Toyoura sand (by Miura 1975). The comparisons show that the calculated results can roughly trace the experimental curves. According to these comparisons, the finite element method formulated in this paper seems to be applicable to arbitrary boundary value problems of elasto-plastic deformations for granular materials. Henceforth, by use of this finite element method, we shall analyze the practical problems, the load-sinkage problem of farm tractors, for example.

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