Product: ABAQUS/Standard
This example is intended to serve two functions: to verify the coding of a standard rate-independent plasticity theory for metals and to assess the accuracy of the integration of the plasticity equations, especially in the case of nonproportional stressing. Integration of elastic-plastic material models is a potential source of error in numerical structural analysis. See, for example, the discussions by Krieg and Krieg (1977) and Schreyer et al. (1979). Usually the error is most severe when kinematic hardening is used in plane stress with nonproportional stressing (perhaps because of the complexity of the motion of the stress point and yield surface in stress space in this theory). This example contains two such problems. The exact solutions are available for both problems (Foster Wheeler report, 1972). Experience with a number of other computer programs has suggested that the second example, in particular, is a severe test of the numerical implementation of the plasticity theory. Both problems involve states of uniform plane stress and, hence, are done here by using a single plane stress element.
The material models for the unixially and biaxially loaded cases are described below.
Figure 3.2.1–1 shows the material model for this case. The elastic modulus is 68.94 GPa (10.0 × 106 lb/in2), the yield stress is 68.94 MPa (10.0 × 103 lb/in2), and the work hardening slope is 34.50 GPa (5.0 × 106 lb/in2). This is specified by giving a yield stress of 34.57 GPa (5.01 × 106 lb/in2) at a plastic strain of 0.5. The *SECTION FILE option is used to output the total force and the total moment on the loaded face of the model.
Figure 3.2.1–1 shows the material model for this case. The elastic modulus is 207 GPa (30.0 × 106 lb/in2), the yield stress is 207 MPa (30.0 × 103 lb/in2), and the work hardening slope is 10.41 GPa (1.51 × 106 lb/in2). This is specified by giving a yield stress of 10.62 GPa (1.53 × 106 lb/in2) at a plastic strain of 0.95.
The geometries and loading distributions for the unixial and biaxial cases are described below.
Figure 3.2.1–1 shows the geometry for this case. Two types of meshes are provided: a single-element mesh using higher-order plane stress and shell elements (CPS8R, S8R5, S9R5, and STRI65) and a mesh using linear shell and continuum shell elements (S4R and SC8R). Two edges have simple support. The load history is shown in Figure 3.2.1–2 and is prescribed with the *AMPLITUDE option (“Amplitude curves,” Section 27.1.2 of the ABAQUS Analysis User's Manual). The load distribution is a uniform, direct stress on the element edge. Since the strain should be uniform, the edge nodes are constrained using the *EQUATION option (“Linear constraint equations,” Section 28.2.1 of the ABAQUS Analysis User's Manual) to move together in the direction normal to the edge. Then the total load on the edge is simply given on one of the edge nodes.
The case is set up with the same geometric model (Figure 3.2.1–1). However, the loading is more complex (see Figure 3.2.1–2).
First, the plate is loaded into the plastic range in uniaxial tension in the x-direction, unloaded slightly, and reloaded. Biaxial loading then follows, with 
and 
prescribed, as shown in Figure 3.2.1–2, so that the quantity 
remains constant at 276 MPa (40000 lb/in2). This loading is defined by the *AMPLITUDE option by reading in a file of values previously calculated in the small program AMP (see elasticplasticplate_amplitude.f).
Exact solutions for these two problems have been developed by Chern in a Foster Wheeler report (1972), where they are documented as Problems 8 and 9. These solutions provide a basis for the comparison of the ABAQUS results.
The plastic strains are the basic solution in these cases (since stress is prescribed). The results for this case are summarized in Table 3.2.1–1. The ABAQUS results agree with the exact solution. Table 3.2.1–1 also records the number of iterations required to achieve equilibrium.
The results in this case are best represented by the versus 
plot shown in Figure 3.2.1–3. The agreement with the exact solution is again very close.
Uniaxial loading case using the CPS8R element.
Biaxial loading case using the CPS8R element.
Program used to generate the amplitude data records.
Uniaxial loading case using the S8R5 element.
Biaxial loading case using the S8R5 element.
Uniaxial loading case using the S9R5 element.
Biaxial loading case using the S9R5 element.
Uniaxial loading case using the STRI65 element.
Biaxial loading case using the STRI65 element.
Uniaxial loading case using the S4R element.
Biaxial loading case using the S4R element.
Uniaxial loading case using the SC8R element.
Biaxial loading case using the SC8R element.
Foster Wheeler Corporation, “Intermediate Heat Exchanger for Fast Flux Test Facility: Evaluation of the Inelastic Computer Programs,” report prepared for Westinghouse ARD, Foster Wheeler Corporation, Livingston, NJ, 1972.
Krieg, R. D., and D. B. Krieg, “Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model,” ASME Journal of Pressure Vessel Technology, vol. 99, no.4, pp. 510–515, 1977.
Schreyer, H. L., R. F. Kulak, and J. M. Kramer, “Accurate Numerical Solutions for Elastic-Plastic Models,” ASME Journal of Pressure Vessel Technology, vol. 101, no.3, pp. 226–234, 1979.
Table 3.2.1–1 Some results for uniaxial load.
| Load increment | Number of iterations | |  (10–3) | ||
|---|---|---|---|---|---|
| (MPa) | (lb/in2) | (ABAQUS) | (exact) | ||
| 1 | 1 | 68.947 | 10000 | 0 | 0 |
| 2 | 1 | 103.422 | 15000 | 0.500 | 0.500 |
| 3 | 1 | 137.895 | 20000 | 1.000 | 1.000 |
| 4 | 1 | 172.369 | 25000 | 1.500 | 1.500 |
| 5 | 3 | 86.529 | 12550 | 1.500 | 1.500 |
| 6 | 2 | 0.69 | 100 | 1.010 | 1.010 |
| 7 | 3 | 103.77 | 15050 | 1.010 | 1.010 |
| 8 | 2 | 206.83 | 30000 | 2.000 | not shown |
| 9 | 3 | 103.77 | 15050 | 2.000 | not shown |
| 10 | 2 | 0.69 | 100 | 1.010 | 1.010 |
