This verification problem is similar to the problem described in “Rolling of thick plates,” Section 1.3.6 of the ABAQUS Example Problems Manual. Here, a two-dimensional, plane strain case of the rolling problem is considered with a much coarser mesh for the steel plate. The plate is modeled using plane strain elements (CPE4R) and 8-node brick elements (C3D8R). In the three-dimensional model all out-of-plane degrees of freedom are prescribed as zero to represent a state of plane strain.

The steel plate has a total thickness of 40 mm and a length of 100 mm. This analysis simulates the rolling of the plate through two roller stands, each of which achieves a reduction in the thickness of the plate of 10 mm. The radius of each roller is 50 mm. The model takes advantage of half-symmetry.

The material is modeled as an elastic, perfectly plastic material with Young's modulus 210 GPa, Poisson's ratio 0.30, yield stress 250 MPa, and density 7500 kg/m³. The two-dimensional case uses the *USER MATERIAL option, along with user subroutine VUMAT. This model can be selected by specifying the material name ABQTEST1 on the *MATERIAL option. The user subroutine has the option to include kinematic hardening. However, this example problem tests the user material only for the case of perfect plasticity and verifies the results by comparison with the results obtained with the standard plasticity model with no hardening (for the three-dimensional case). The rotating cylinder problem of “VUMAT: rotating cylinder,” Section 4.1.27, verifies the hardening case for the user material. The three-dimensional model uses the standard elastic, perfectly plastic material model specified with the *ELASTIC and *PLASTIC options. It also tests the adiabatic heat generation capability using the *DYNAMIC, EXPLICIT, ADIABATIC option, the *SPECIFIC HEAT option, the *EXPANSION option, and the *INELASTIC HEAT FRACTION option. The initial temperature for all nodes in the model is 294°C. The specific heat for this material is 460.46 Joule/kg/°C.

The rolling process is analyzed in two steps. In the first step only the first roller has a prescribed rotational velocity. The second step begins just as the plate is about to reach the second roller. At this time a prescribed velocity boundary condition is added that determines the rotational velocity of the second roller. The coefficient of friction between the rollers and the plate is 0.3. The maximum traction due to friction is assumed to be , or 144.3 MPa.

The contact constraints can be enforced either kinematically or with a penalty method in ABAQUS/Explicit. Kinematic contact gives strict enforcement of the constraints, whereas penalty contact will allow some penetration. However, the two constraint methods will usually give nearly the same results for problems that involve plastic deformation (such as rolling problems), because the contact penetrations with penalty contact will tend to be small. This is related to the fact that the default penalty stiffness is about 10% of the elastic stiffness in the elements along the contact interface. When the material yields, the penalty stiffness will typically be much larger than the effective stiffness of the material, so the penetrations will be rather insignificant. For problems in which the material remains elastic (see “The Hertz contact problem,” Section 1.1.11 of the ABAQUS Benchmarks Manual), the contact penetrations can be significant if the penalty method is used. While kinematic contact is available only with the contact pair capability, penalty contact is available with both the contact pair capability and the general contact capability in ABAQUS/Explicit. For this analysis all three approaches to enforcing the contact constraints are used: kinematic contact with contact pair, penalty contact with contact pair, and general contact. In the first step of the analyses involving contact pairs, when only the first roller has a prescribed rotational velocity, only one contact pair is defined. This contact pair contains the surface of the first roller and the outer surface of the plate. At the start of the second step, when the plate is just about to reach the second roller, a second contact pair is introduced that contains the surface of the second roller and the outer surface of the plate. For the analysis using general contact, the default internally generated all-inclusive contact surface is referenced using the *CONTACT INCLUSIONS option; hence, the contact definitions do not need to be modified from step to step.

The roller speed used for both rollers in this example is 600 rad/s. See “Rolling of thick plates,” Section 1.3.6 of the ABAQUS Example Problems Manual, for a detailed discussion of the choice of rolling speeds.