In a model undergoing inelastic deformation, both normal and shear components of stress undergo change under combined stress fields, corresponding to the increment of either normal or shear component of strain. It is well known that finite element plasticity solutions often become highly inaccurate, especially in the fully plastic range. Approximations occur not only due to the numerical assumptions but also due to the basic incremental character of the plasticity law. As a starting point, a small-scale test problem is chosen for incorporating the material plasticity model into the finite element scheme. Results of these calculations are compared with those from Abaqus, a commercial software package.
Figure 1: The Test Problem for Elasto-plastic Computations by a Linear Tetrahedral Mesh.
Figure 2: Validation of the Computations with ABAQUS/Explicit by a Linear Tetrahedral Mesh.
The physical problem under study is a solid cube with one side fixed and the opposite side subject to dynamic load, which is contained in a vertical plane, with two components of unit magnitude applied at A in the x and z directions as shown in Fig. 1. The pulse in time varies as a Ricker waveform. The material is modelled by elasto-perfectly-plastic Drucker-Prager law of zero friction angle and the problem domain is meshed by linear tetrahedra. The displacements at the location B are calculated for Case a and Case b by our plasticity code, Plasto-quake, and compared to Abaqus in Fig. 2. The explicit method was employed by both Plasto-quake and Abaqus, and element-by-element calculations was performed by Plasto-quake. The dynamic loads of Ricker waveform in these two cases have different central frequencies, 0.4 Hz for Case a and 2.0 Hz for Case b. Moreover, the yielding threshold value is set to be ten times smaller in Case a than in Case b. For each case, the three curves show how the displacement components in x, y and z directions vary with time. In Case a, larger plastic deformations are developed due to the smaller yielding threshold value. Comparing the results between Plasto-quake and Abaqus in each case, it is seen that they agree very closely both in amplitude and in frequency. Additional tests have been conducted using similar sample problems to ensure that the selected element type, algorithm, integration step, and the implementation of the constitutive model are appropriate for the accuracy and efficiency of our large-scale elasto-plastic simulations on parallel computers.
To further substantiate the validity of the computational model, Plasto-quake, for elastoplastic analysis, another major type of elements, 10-node quadratic tetrahedra, are used in a similar test problem, shown in Fig. 3. This type of elements is available only in the standard library of elements of Abaqus, which uses an implicit time integration method. Since the explicit method is desired for efficient parallel computing, it is retained in Plasto-quake. For numerical quadrature in Plasto-quake, four Gauss points are adopted in every element. For this test problem, a uniform pressure loading is applied on a central area of the top surface, as shown in Fig. 3. This load varies in time as a Ricker waveform. The calculated dynamic responses, including both acceleration and displacement components, at locations B, C and D are compared in Fig. 4, 5 and 6, respectively. Notice the rapid oscillation of acceleration but not of displacement, with both techniques. At C and D the displacements along x and y coincide due to symmetry. It is obvious that a very good agreement exists between Plasto-quake and ABAQUS/Standard, though they employ different time-integration methods and may have some unknown minor differences in the implementations.
Figure 3: The Test Problem for Elasto-plastic Computations by a Quadratic Tetrahedral Mesh.
Figure 4: Comparison of Dynamic Response at B: (a) by ABAQUS/Standard with
implicit method; (b) by Plasto-quake with explicit method.
Figure 5: Comparison of Dynamic Response at C: (a) by ABAQUS/Standard with
implicit method; (b) by Plasto-quake with explicit method.
Figure 6: Comparison of Dynamic Response at D: (a) by ABAQUS/Standard with
implicit method; (b) by Plasto-quake with explicit method.
