2006 Volume 62 Issue 3 Pages 667-678
Multi-scale modeling based on the mathematical homogenization theory explains to us that the micro-scale problem in multi-scale modeling provide a constitutive (stress-strain) relationship for the macro-scale problem of a overall structure. We focus on this explanation to evaluate the macroscopic loading surfaces, and carry out three-dimensional micro-scale analysis of a granular material. At first, we give an overview of a multi-scale model to show that the micro-scale problem provides a constitutive relationship for the macro-scale problem. Next, we apply three-dimensional micro-scale analysis of idealized granular materials to evaluate and examine loading surfaces and the flow rule which is relevant to the loading surface. By the numerical experiments, we obtained the following main conclusions: Loading surfaces of granular materials are curved surfaces containing inflection points and exhibit kinematic and rotational hardening in p-q plane; plastic strain increments exhibit incremental nonlinearity near the inflection point; in π plane, it is difficult to uniquely define loading surfaces, because the incremental nonlinearity of plastic strain increments is conspicuous.
