Product: ABAQUS/Standard
The verification problems contained in this section test the steady-state transport analysis capability in ABAQUS. The verification concentrates on frictional effects, inertia effects, and material convection. Frictional effects are verified by comparing results obtained with ABAQUS to results published by Faria (1989). Inertia effects are verified by comparing special cases of steady-state transport analyses with results obtained from an ABAQUS analysis where centrifugal loads are applied using a distributed load with load type CENT. Material convection is verified by comparison with a transient Lagrangian analysis.
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 E = 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.
Table 1.5.1–1 and Table 1.5.1–2 compare the free rolling angular velocity, , obtained from the ABAQUS simulation with the reference solution. The results presented in Table 1.5.1–2 are obtained using C3D8RH elements.
Additional verification tests are performed to verify contact between a spinning deformable body and a spinning rigid body. In all these tests the deformable body uses the properties and discretization described earlier. The rotating rigid body is in contact either with the inside surface of the deformable body (such as in the case where a tire is mounted on a rigid rim) or with the outside surface of the deformable body (such as in the case where a tire is in contact with a rotating drum). No reference solutions are available for the case where the rigid body is in contact with the inside surface of the deformable body. By specifying a large radius for the rigid body in the case where a rigid spinning drum is in contact with the outside surface of the deformable body, straight line rolling is recovered. We selected a rigid body radius of 
= 1000.0 and an angular velocity of = 0.002, which corresponds to straight line rolling with a velocity 
= 2.0.
Another verification test is performed to verify contact between a rolling gear-like thick cylinder with an outer radius of 8.5 and a flat rigid surface. The model is generated by revolving a single three-dimensional 15° sector about the symmetry axis. The gear-like cylinder travels at a ground velocity of 2.7778 with an angular velocity varying from 0.2 to 0.5. The results are compared to those obtained from a transient Lagrangian analysis.
C3D8H elements, hyperelastic material, = 1.0, = 0.1, = 0.02, straight line rolling with = 2.0 (requires two-dimensional input file pstca4shhfa.inp).
C3D8RH elements, hyperelastic material, = 1.0, = 0.05, = 0.2, straight line rolling with = 2.0 (requires two-dimensional input file pstca4syhfa.inp).
C3D20 elements, elastic material, = 0.2, = 0.10, = 0.02, straight line rolling with = 2.0 (requires two-dimensional input file pstca8sfefa.inp).
C3D20R elements, elastic material, = 1.7, = 0.05, = 0.02, straight line rolling with = 2.0 (requires two-dimensional input file pstca8srefa.inp).
C3D8I elements, elastic material, = 10.2, = 0.05, = 0.02, cornering with = 0.02 and = 100.0 (requires two-dimensional input file pstca4siefa.inp).
C3D6H elements, hyperelastic material, = 1.0, = 0.10, = 0.02, cornering with = 0.02 and = 100.0 (requires two-dimensional input file pstca3shhfa.inp).
Contact between a rigid drum and the outside surface of a deformable body, C3D8H elements; similar to pstc38shhfs.inp (requires two-dimensional input file pstca4shhfa.inp).
Contact between a rigid drum and the outside surface of a deformable body, C3D8RH elements; similar to pstc38syhfs.inp (requires two-dimensional input file pstca4syhfa.inp).
Contact between a rigid “rim” and the inside surface of a deformable body and contact between a flat rigid foundation and the outside surface of a deformable body, C3D6H elements; similar to pstc36shhfc.inp (requires two-dimensional input file pstca3shhfr.inp).
CCL12H elements, hyperelastic material, = 1.0, = 0.1, = 0.02, straight line rolling with = 2.0 (requires two-dimensional input file pstca4shhfa.inp).
Contact between a flat rigid surface and a gear-like deformable cylinder (requires three-dimensional input file sstransp_per_hyper_preload.inp).
Faria, L. O., Tire Modeling by Finite Elements, Ph.D. dissertation, The University of Texas at Austin, 1989.
Table 1.5.1–1 Comparison of ABAQUS results with reference solutions for the free rolling angular velocity.
| Input file | Reference solution | ABAQUS | % Difference |
|---|---|---|---|
| pstc38shhfs.inp | 0.95009 | 0.94635 | 0.39 |
| pstc38syhfs.inp | 0.98006 | 0.98213 | 0.21 |
| pstc3ksfefs.inp | 1.02970 | 1.02726 | 0.24 |
| pstc3ksrefs.inp | 1.00297 | 1.00283 | 0.01 |
| pstc38siefc.inp | 1.02180 | 1.00674 | 1.47 |
| pstc36shhfc.inp | 0.95195 | 0.94568 | 0.66 |
| Material type | | | | Reference solution | ABAQUS | % Difference |
|---|---|---|---|---|---|---|
| Hyperelastic | 0.05 | 0.2 | 0.02 | 0.99349 | 0.99219 | 0.13 |
| 1.0 | 0.02 | 0.97977 | 0.98053 | 0.08 | ||
| 1.7 | 0.02 | 0.87183 | 0.84974 | 2.53 | ||
| 1.0 | 0.20 | 0.98066 | 0.98212 | 0.15 | ||
| 0.10 | 0.2 | 0.02 | 0.98558 | 0.98422 | 0.14 | |
| 1.0 | 0.02 | 0.95009 | 0.95059 | 0.05 | ||
| 1.7 | 0.02 | 0.73057 | 0.65790 | 9.95 | ||
| 1.0 | 0.20 | 0.95195 | 0.95100 | 0.10 | ||
| Linear elastic | 0.05 | 0.2 | 0.02 | 1.02180 | 1.02332 | 0.15 |
| 1.0 | 0.02 | 1.02415 | 1.02574 | 0.16 | ||
| 1.7 | 0.02 | 1.00297 | 1.00263 | 0.03 | ||
| 0.10 | 0.2 | 0.02 | 1.02970 | 1.03410 | 0.43 | |
| 1.0 | 0.02 | 1.02810 | 1.02872 | 0.06 | ||
| 1.7 | 0.02 | 0.99156 | 0.99542 | 0.39 |
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 a distributed 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 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.
For the straight line rolling tests, the results match the reference solution. For the cornering tests without free spinning, the results match the reference solution. For the cornering tests with free spinning, the results do not match the reference solution because the steady-state inertia loading includes Coriolis effects due to a spinning wheel in a rotating reference frame; these effects are not accounted for in the reference solution.
C3D6H elements; requires two-dimensional input file pstca3shhia.inp.
C3D8H elements; requires two-dimensional input file pstca4shhia.inp.
C3D15H elements; requires two-dimensional input file pstca6shhia.inp.
C3D20H elements; requires two-dimensional input file pstca8shhia.inp.
M3D4R elements; requires two-dimensional input file pstma2srhia.inp.
M3D8R elements; requires two-dimensional input file pstma3srhia.inp.
S4R elements; requires two-dimensional input file pstsa2shhia.inp.
Postprocesssing analysis.
S8R elements; requires two-dimensional input file pstsa3sheia.inp.
CCL9H elements; requires two-dimensional input file pstca3shhia.inp.
CCL12H elements; requires two-dimensional input file pstca4shhia.inp.
CCL18H elements; requires two-dimensional input file pstca6shhia.inp.
CCL24RH elements; requires two-dimensional input file pstca8shhia.inp.
C3D6H elements; requires two-dimensional input file pstca3shhia.inp. Cornering only; no free spinning.
C3D8RH elements; requires two-dimensional input file pstca4syhia.inp. Cornering only; no free spinning.
C3D8RH elements; requires two-dimensional input file pstca4shhia.inp. Cornering and free spinning.
C3D20H elements; requires two-dimensional input file pstca8shhia.inp. Cornering and free spinning.
C3D20RH elements; requires two-dimensional input file pstca8yhhia.inp. Cornering and free spinning.
M3D4R elements; requires two-dimensional input file pstma2srhia.inp. Cornering and free spinning.
M3D8R elements; requires two-dimensional input file pstma3srhia.inp. Cornering only; no free spinning.
S4R elements; requires two-dimensional input file pstsa2shhia.inp. Cornering only; no free spinning.
S8R elements; requires two-dimensional input file pstsa3sheia.inp. Cornering and free spinning.
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 step where the long-term response is requested and 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.
The reaction force normal to the foundation and the torque around the axle are compared to results obtained from a transient Lagrangian analysis using the quasi-static analysis procedure that is run until steady-state conditions are achieved. A model with fine meshing (C3D8RH elements) along the entire circumference is used to obtain this reference solution.
Table 1.5.1–3 compares the solutions obtained using different element types with the reference solution.
Additional tests were performed to verify the effects of material convection with an elastic-plastic or a viscoplastic material model. Both the Mises metal plasticity model with kinematic hardening and the two-layer viscoelastic-elastoplastic model, which is best suited for modeling the response of materials with significant time-dependent behavior as well as plasticity at elevated temperature, have been used. The model consists of a disc with outer radius of 4.0 mm, inner radius of 1.0 mm, and thickness of 3.0 mm. The disc is generated either by revolving the cross-section of an axisymmetric mesh about the symmetry axis or by revolving a single three-dimensional repetitive sector of the model about the symmetry axis. The bottom surface of the disc is fixed. The top surface is subjected either to a nonuniform distributed load or a nonuniform contact pressure and frictional stress due to a pad being applied to the top surface of the disc. For each of these tests the disc is assumed to rotate at an angular velocity of 87.2 rad/sec or 5 rad/sec, respectively.
Each model has been analyzed using both a quasi steady-state transport solution technique through the use of the steady-state transport pass-by-pass analysis technique and a directly sought steady-state solution technique. For each of the tests the circumferential stress and circumferential plastic strain are compared to results obtained from a transient Lagrangian analysis.
C3D6H elements; requires two-dimensional input file pstca3shhma.inp.
C3D8RH elements; requires two-dimensional input file pstca4syhma.inp.
C3D8H elements; requires two-dimensional input file pstca4shhma.inp.
C3D8IH elements; requires two-dimensional input file pstca4sjhma.inp.
C3D15H elements; requires two-dimensional input file pstca6shhma.inp.
C3D20RH elements; requires two-dimensional input file pstca8srhma.inp.
C3D20H elements; requires two-dimensional input file pstca8sfhma.inp.
CCL9H elements; requires two-dimensional input file pstca3shhma.inp.
CCL12H elements; requires two-dimensional input file pstca4shhma.inp.
CCL18H elements; requires two-dimensional input file pstca6shhma.inp.
CCL24H elements; requires two-dimensional input file pstca8sfhma.inp.
M3D4R elements; requires two-dimensional input file pstma2srhma.inp.
M3D8R elements; requires two-dimensional input file pstma3srhma.inp.
M3D4R elements with viscoelastic rebar; requires two-dimensional input file pstma2rbema.inp.
C3D20RH elements with viscoelastic rebar; requires two-dimensional input file pstca8rbema.inp.
Viscoelastic continuum and viscoelastic rebar; requires two-dimensional input file pstrebara.inp.
Pressure stress relaxation; requires two-dimensional input file pstpressa.inp.
Hyperfoam material; requires two-dimensional input file psthfoama.inp.
Two-term Prony series; requires two-dimensional input file pst2pronya.inp.
Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
Table 1.5.1–3 Reaction forces and torques for analyses using different elements.
| Element Type | Force | Torque |
|---|---|---|
| C3D6H | 373.12 | –9.55 |
| C3D8RH | 370.46 | –9.49 |
| C3D8H | 373.98 | –9.55 |
| C3D8IH | 376.92 | –9.62 |
| C3D15H | 372.74 | –9.66 |
| C3D20RH | 372.53 | –9.91 |
| C3D20H | 373.43 | –9.97 |
| CCL9H | 357.3 | –10.02 |
| CCL12H | 375.5 | –10.33 |
| CCL18H | 371.3 | –10.04 |
| CCL24H | 373.3 | –9.97 |
| Reference solution | 374.47 | –9.77 |
