Pressure on infinite geostatic medium

This example simulates a semi-infinite granular medium under initial geostatic stress, subject to pressure suddenly applied to part of its surface.

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ProductsAbaqus/Explicit

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.

Results and discussion

Figure 3 through Figure 5 show contours of pressure at the end of the first step. The pressure has a linear variation, corresponding to the equilibrium geostatic stress field. Immediately after the pressure pulse is applied at the beginning of the second step, elastic waves begin to traverse the medium. These are followed by plastic waves once the stress magnitude exceeds the yield strength. The deviatoric yield stress depends linearly on the magnitude of the pressure and, hence, is augmented in the region of high confinement.

Figure 6 shows the hydrostatic pressure in the axisymmetric case at the end of the second step. The two contour plots are very similar in the region under loading. Likewise, the plastic strain contours in Figure 7 are almost identical. In this case the waves are spherical.

Figure 8 shows the hydrostatic pressure in the plane strain case at the end of the second step, and Figure 9 shows the equivalent plastic strain. Almost identical patterns are again observed. Similar plots of pressure and plastic strain at the end of the second step for the three-dimensional model are shown in Figure 10 and Figure 11. In these plane strain cases the waves are planar.

Figure 12 shows the pressure stress time history for elements 81 and 1361 in the plane strain case. The position of these elements is shown in Figure 13. These elements are at the same position in the small and large models. The results show the effect that the infinite element has in removing wave reflections from the boundaries. The dilatational wave speed for the material is 1160. In the mesh without infinite elements, the wave is expected to return to element 1361 0.033 sec after the pressure is applied, or at a total time of 0.041 sec. Figure 12 shows the wave returning at this time. The mesh with infinite elements shows no wave as significant as this wave returning to element 81.

Figures

Figure 1. Undeformed axisymmetric and plane strain meshes.

Figure 2. Undeformed three-dimensional mesh.

Figure 3. Pressure contours, axisymmetric case, end of Step 1.

Figure 4. Pressure contours, plane strain case, end of Step 1.

Figure 5. Pressure contours, three-dimensional case, end of Step 1.

Figure 6. Hydrostatic pressure contours, axisymmetric case, end of Step 2.

Figure 7. Equivalent plastic strain contours, axisymmetric case, end of Step 2.

Figure 8. Hydrostatic pressure contours, plane strain case, end of Step 2.

Figure 9. Equivalent plastic strain contours, plane strain case, end of Step 2.

Figure 10. Hydrostatic pressure contours, three-dimensional case, end of Step 2.

Figure 11. Equivalent plastic strain contours, three-dimensional case, end of Step 2.

Figure 12. Pressure stress time history for same position in small and large models, plane strain case.

Figure 13. The position of elements 81 and 1361.