In this series of tests the free rolling angular velocity, , of a circular disk in contact with a flat rigid surface is calculated for different disk geometries, contact pressures, friction coefficients, material models, and element types. The ground velocity is specified as either a straight-line translational velocity of = 2.0 or as a cornering angular velocity of = 0.02. By specifying a large cornering radius, = 100.0, straight line rolling with velocity = 2.0 is recovered. The results obtained with ABAQUS are compared to numerical results published by Faria (1989).

The model consists of a ring with outer radius = 2.0 and variable inner radius . Three different geometries ( = 0.2, 1.0, 1.7) are considered. The model is fully fixed on the inside, and plane strain boundary conditions are imposed along the axial direction.

Two material models are considered: a linear elastic material with = 800.0 and = 0.3 and an incompressible hyperelastic material with = 80.0 and = 20.0. The friction coefficients considered are = 0.02 and = 0.2. The first analysis step is a *STATIC analysis where the rigid surface is displaced a distance = 0.05 or = 0.1 to establish a contact pressure. The friction coefficient during this step is held constant at zero. This step is followed by a *STEADY STATE TRANSPORT analysis where the ground traveling velocity and spinning angular velocity are applied and the friction coefficient is ramped to its final value.

The problem is discretized with different types of three-dimensional elements. The models that are discretized with first-order elements use 34 element divisions along the circumference and 5 element divisions in the radial direction. The second-order and cylindrical element models use 18 elements along the circumference and 3 elements in the radial direction. All the models are discretized with one element in the axial direction. A first-order finite element mesh for the case = 1.0 is shown in Figure 1.5.1–1.

In this series of tests the effects of inertia on a free spinning and/or cornering structure are verified for different element types and angular velocities. An incompressible hyperelastic material with = 80.0, = 20.0, and = 0.036 is used. The model consists of a ring with outer radius = 2.0 and inner radius = 1.0. Each input file contains two models with identical geometry: one model is loaded using *DLOAD with a centrifugal load (load type CENT) and serves as the reference solution; the loading on the other model is caused by steady-state rolling inertia effects. The difference between the two loading conditions is that deformation of the structure is accounted for in steady-state rolling but not for load type CENT.

The problem is discretized with different types of three-dimensional elements. The models that are discretized with first-order elements use 24 element divisions along the circumference and 2 element divisions in the radial direction. The second-order and cylindrical element models use 12 elements along the circumference and 1 element in the radial direction. All the models are discretized with 1 element in the axial direction.

The models are fully fixed on the inside. Plane strain boundary conditions are imposed along the axial direction in the first step. Since the material is incompressible, the loading does not give rise to any deformation. As a consequence, the stress state caused by free spinning can be compared directly to the stress state obtained using *DLOAD with load type CENT. In the second step the plane strain boundary condition is released, and the disk is allowed to deform. Since the deformation of the structure is accounted for in steady-state rolling, the stress states in the two models are no longer comparable in Step 2.

In this series of tests the effect of material convection with a viscoelastic material model is verified for different element types. The model consists of a ring with outer radius = 2.0 and inner radius = 1.0. The model is fully fixed on the inside, and plane strain boundary conditions are imposed along the axial direction.

An incompressible hyperelastic material is used, with long-term moduli = 80.0, = 20.0, shear relaxation coefficient of = 0.2, and relaxation time = 0.1.

The first analysis step is a *STATIC, LONG TERM analysis where the rigid surface is displaced a distance = 0.2 to establish a contact pressure. This step is followed by a *STEADY STATE TRANSPORT analysis step where viscoelastic material effects are considered. No frictional stresses are transmitted so that the disk spins without translating along the foundation.

The problem is discretized with different types of three-dimensional elements. The models that are discretized with first-order elements use 30 element divisions along the circumference and 5 element divisions in the radial direction. The second-order and cylindrical element models use 20 elements along the circumference and 3 elements in the radial direction. All the models are discretized with one element in the axial direction.

Some other tests were performed to verify the effects of material convection when the material response is slightly compressible and allows for relaxation of the pressure stress; when viscoelastic effects take place in plane stress elements; when a hyperfoam material with relaxation is used; when the model contains viscoelastic rebars embedded in an elastic or viscoelastic material; and when an incompressible hyperelastic material with two terms defining the Prony series is used.