Problem description
To model a semi-infinite half-space, one option is to generate a mesh that extends far away from the region of interest so that there are no reflections from the farthest boundaries of the model back into the region of interest. However, this is computationally expensive, because the solution must be computed in a large part of the model in which the user has no interest. Infinite elements allow the region of interest (the interior) to be modeled with a suitable mesh, while the far field is simulated with a set of infinite elements that are added to the perimeter of the interior mesh.
In each case considered (axisymmetric, plane strain, and three-dimensional), two meshes are used: (1) a small mesh defining the interior, surrounded by infinite elements, and (2) a larger model of ordinary finite elements, extended to a sufficient distance so that no waves are reflected back into the interior during the time of analysis. The purpose of having these two meshes is to verify the infinite elements. The larger model has exactly the same discretization in the interior region as the smaller mesh. If the infinite elements are performing properly, the solution should be nearly identical in the interior portion of both meshes.
The initial geostatic stress field is defined using initial conditions. One of the features of the infinite elements is that they will apply the proper tractions on the boundary to maintain an initial equilibrium stress field. The first step in this problem is of a duration of 5 milliseconds. Only the gravitational (self-weight) load corresponding to the geostatic field is applied. There should be no accelerations and no changes in the stresses during this step. The step is carried out to verify that the infinite elements do in fact maintain an equilibrium state of stress.
In the second step of the analysis a pressure is applied instantaneously over a portion of the top surface and held constant through the 8 milliseconds of response.
The granular material is simulated with the extended Drucker-Prager model. The frictional angle is 40°, while the material is nondilatational (the dilation angle is 0°). The yield surface in the deviatoric plane is assumed to be noncircular, with the parameter K, which defines the dependency on the third stress invariant, being 0.9. Perfect plasticity is assumed, with a yield stress in uniaxial compression of 5 × 103. Young's modulus is 1 × 109, and Poisson's ratio is 0.3.
Axisymmetric, plane strain, and three-dimensional models with the corresponding infinite elements are studied. The meshes are shown in Figure 1 and Figure 2. In the plane strain and three-dimensional cases the model assumes symmetry about a center plane. The three-dimensional model has one layer of elements, with the displacement in the x-direction constrained to give plane strain response.