Uniform deformation of a granular medium theory and experiment

Abstract

The solution of most continuum mechanics problems requires the selection or the creation of a mathematical model of the medium. If we are talking about an elastic body or a linearly viscous fluid, the model construction problem does not exist, since the classical models have been created and the corresponding Lam6 and Navier-Stokes equations have been formulated. For the more complex media, for example the elastoplastic and granular media and the nonlinear fluids, there are no generally accepted equations. Therefore the model selection question must be resolved separately for the problems of each class. Here much depends on the specific conditions of loading and the objectives which are posed in formulating and solving the specific problems [1].Practically all the continuum working models are of a phenomenological nature. This means that they are constructed on the basis of some fundamental experiments. These experiments must determine both the constructive equations themselves and the values of the parameters appearing in these equations.The question of selection of the basis experiments is far from simple. In principle, any experiments, for example the penetration of dies, can be used to construct the mathematical models. However, for the interpretation of this experiment it is necessary to select in advance the mathematical model, solve the boundary-value problem, compare the results with the experimental data, correct the model, and so on. This is a very complex procedure. However, there is a special class of loadings for which preliminary data on the mathematical model are not required for the interpretation of the experiment. This is the class of quasi-static loadings, in which the distributions of the stresses and the strains in space are uniform. Here the deformation process reduces to the realization of a sequence of affine transformations. Therefore, in the stable processes the kinematics of the deformation will be the same for both the elastoplastic media and for the viscous or any other media. Thus, the experiments that realize the uniform states are ideal as the basis experiments for the construction and analysis of the mathematical models of continuous media.It is not technically possible to achieve precisely uniform loading. Therefore the most that we can do is to realize deformation processes that are as close to uniform as is possible. Such processes and the corresponding techniques are well known for the metals [2, 3]. These include the twisting and stretching of thin-wall tubular specimens. However, this classical technique is not applicable for the granular media, liquids with complex rheology, and other such materials. Here it is necessary to seek new fundamental experiments.A general classification of the uniform deformation processes is given in [4, 5]. In this classification the well-known processes appear as particular cases: uniform tension, torsion, and so on. But, in spite of the near triviality of the problem formulation, new classes of loadings have been found. Several devices and in...