Product: ABAQUS/Standard
This example is intended to verify the ORNL plasticity theory (Oak Ridge, 1981) model in ABAQUS under conditions of plane stress with biaxial stressing. An exact solution is developed to verify the ABAQUS results. The problem involves a state of uniform plane stress, so the geometric model is a single element, constrained to respond uniformly. This capability can be invoked by using the *ORNL option (“ORNL – Oak Ridge National Laboratory constitutive model,” Section 11.2.12 of the ABAQUS Analysis User's Manual).
The case is set up with the same geometric and virgin material model as in Case 2 in “Uniformly loaded, elastic-plastic plate,” Section 3.2.1. The plate is first loaded elastically to the virgin yield surface in the 
-direction and then loaded into the plastic range in uniaxial tension in the -direction to a stress, 
, of 276 MPa (40000 lb/in2). Biaxial loading then follows, with 
and 
prescribed, as shown in Figure 3.2.2–1, so that 
. This loading is defined by the *AMPLITUDE option (“Amplitude curves,” Section 19.1.2 of the ABAQUS Analysis User's Manual). ABAQUS reads in two files (ORNL2.AMP and ORNL3.AMP) of values, which are calculated in the small program AMP (see ornlbiaxialload_ampdata.f).
An “exact” solution is developed by first defining the total strain rates, 
and 
, as functions of the stress rates 
and 
. The resulting rate equations are then integrated numerically with high accuracy to give a reference solution.
Ziegler's kinematic hardening gives, under isothermal conditions,
is the yield stress and 
is the slope of the stress versus plastic strain curve under uniaxial loading conditions.Under plane stress conditions (
) and with 
,
and 
Hence,
, 
The stress rate-strain rate relation is, therefore,
, 
, and 
Inverting this relationship gives the total strain rates as
The center of the yield surface translates according to Ziegler's kinematic hardening rule, so that
Hence,
Given the values of the variables , , 
, 
, 
and 
, at the beginning of the increment, together with the prescribed stress increments and , the total strain rate equation and the translation rate equation for the center of the yield surface provide the values of , , 
, and 
.
A small program for calculating the required variables is given in ornlbiaxialload_exact.f. The main program provides prescribed stress increments 
and 
equal to those used in the finite element analysis. Each of these increments is then split into 1000 subincrements, and the total strain rate equation and the translation rate equation for the center of the yield surface are integrated over each subincrement to provide virtually exact values of 
, 
, 
, and 
, corresponding to the prescribed values of and used in the analysis. In each of the subincrements, a test is made to determine if 
When this test is satisfied, the yield surface is expanded from the virgin properties to the 10th cycle properties so that is increased from its virgin value of 207 MPa (30000 lb/in2) to its 10th cycle value of 234 MPa (34000 lb/in2). This value of is used in each subincrement following the initial satisfaction of the test 
, in accordance with the ORNL plasticity algorithm.
The loading path in stress space is shown in Figure 3.2.2–1. When the stress contacts point 
, the yield surface starts to translate so that at point 
the yield surface occupies the position shown by the dashed curve. At point the stress path changes direction, and elastic loading along path 
occurs. At point the stress point pierces the yield surface, and since 
, the ORNL algorithm prescribes an expansion of the yield surface from the virgin properties to the 10th cycle properties. The expanded yield surface is indicated in Figure 3.2.2–1 by the dashed and dotted curve. Continuing loading along path 
produces an elastic response since point lies inside the 10th cycle yield surface. At point 
the stress point contacts the expanded yield surface, and active plastic yielding occurs along path 
. A comparison between the “exact” results and the finite element results in Table 3.2.2–1 and Table 3.2.2–2 shows very close agreement.
Biaxial loading test.
Program used for generating the data records for the *AMPLITUDE option.
Program used for generating the “exact” solution.
Nuclear Standard NE F9–5T, “Guidelines and Procedures for Design of Class 1 Elevated Temperature Nuclear System Components,” USDOE Technical Information Center, Oak Ridge, Tennessee, March 1981.
Table 3.2.2–1 Comparison of “exact” and numerical results for biaxial plate using ORNL plasticity theory—stresses and strains in the -direction.
| Numerical solution | "Exact" solution | Increment | ||
|---|---|---|---|---|
| , MPa (103 lb/in2) | (%) | , MPa (103 lb/in2) | (%) | type |
| 206.84 (30.00) | 0.1000 | 206.84 (30.00) | 0.1000 | Elastic |
| 224.08 (32.50) | 0.2656 | 224.08 (32.50) | 0.2655 | Plastic |
| 241.32 (35.00) | 0.4312 | 241.32 (35.00) | 0.4311 | Plastic |
| 258.55 (37.50) | 0.5968 | 258.55 (37.50) | 0.5968 | Plastic |
| 275.79 (40.00) | 0.7624 | 275.79 (40.00) | 0.7624 | Plastic |
| 258.55 (37.50) | 0.7516 | 258.55 (37.50) | 0.7516 | Elastic |
| 68.95 (10.00)* | 0.6324 | 68.95 (10.00) | 0.6324 | Elastic |
| 51.71 (7.50) | 0.6216 | 51.71 (7.50) | 0.6214 | Elastic |
| 34.48 (5.00) | 0.4702 | 34.47 (5.00) | 0.4744 | Plastic |
| 17.24 (2.50) | 0.2999 | 17.24 (2.50) | 0.3076 | Plastic |
| 3.65 (0.53) | 0.1191 | 0.00 (0.00) | 0.1292 | Plastic |
| –17.24 (–2.50) | –0.0709 | –17.24 (–2.50) | –0.0591 | Plastic |
| –34.48 (–5.00) | –0.2687 | –34.47 (–5.00) | –0.2558 | Plastic |
| –51.71 (–7.50) | –0.4731 | –51.71 (–7.50) | –0.4598 | Plastic |
*The yield surface is pierced at  68.95 MPa (10000 lb/in2),  206.84 MPa (30000 lb/in2. The next increment is elastic due to the expansion of the yield surface. | ||||
Table 3.2.2–2 Comparison of “exact” and numerical results for biaxial plate using ORNL plasticity theory—stresses and strains in the 
-direction.
| Numerical solution | "Exact" solution | Increment | ||
|---|---|---|---|---|
| , MPa (103 lb/in2) | (%) | , MPa (103 lb/in2) | (%) | type |
| 0.00 (0.00) | –0.0300 | 0.00 (0.00) | –0.0300 | Elastic |
| 0.04 (0.01) | –0.1111 | 0.00 (0.00) | –0.1108 | Plastic |
| 0.04 (0.01) | –0.1923 | 0.00 (0.00) | –0.1922 | Plastic |
| 0.04 (0.01) | –0.2734 | 0.00 (0.00) | –0.2734 | Plastic |
| 0.04 (0.01) | –0.3545 | 0.00 (0.00) | –0.3545 | Plastic |
| 17.24 (2.50) | –0.3437 | 17.24 (2.50) | –0.3437 | Elastic |
| 206.84 (30.00)* | –0.2245 | 206.84 (30.00) | –0.2245 | Elastic |
| 224.08 (32.50) | –0.2137 | 224.08 (32.50) | –0.2135 | Elastic |
| 241.32 (35.00) | 0.0313 | 241.32 (35.00) | 0.0319 | Plastic |
| 258.55 (37.50) | 0.2914 | 258.55 (37.50) | 0.2930 | Plastic |
| 275.79 (40.00) | 0.5537 | 275.79 (40.00) | 0.5564 | Plastic |
| 293.03 (42.50) | 0.8172 | 293.03 (42.50) | 0.8211 | Plastic |
| 310.26 (45.00) | 1.0811 | 310.26 (45.00) | 1.0860 | Plastic |
| 327.51 (47.50) | 1.3450 | 327.50 (47.50) | 1.3508 | Plastic |
| *The yield surface is pierced at 68.95 MPa (10000 lb/in2), 206.84 MPa (30000 lb/in2. The next increment is elastic due to the expansion of the yield surface. | ||||
