RD-E: 4200 Rubber Ring: Crush and Slide

A rubber ring resting on a flat rigid surface is pushed down by a circular roller to produce self-contact on the inside surface of the ring. Then the roller is simultaneously rolled and translated so that crushed ring rolls along the flat surface.

The model consists of a deformed rubber ring resting on a flat, rigid surface. Another circular rigid roller rests at the top of the ring and is in contact with the ring at just a point. Contact is defined between the rigid surfaces and the outside surface of the ring and self-contact is defined in the inside surface of the ring. The loading is applied in two steps - in the first step, the circular roller is pushed down enough to produce self-contact of the inside surface of the ring. In the second step, the roller is simultaneously translated and rotated such that the crushed ring rolls along the flat rigid surface producing a constantly changing region of contact.

This example is considered a static problem and the nonlinear implicit solver is used.

Options and Keywords Used

Nonlinear implicit large displacement analysis
Self-contact
Hyper-elastic material
Hyper-elastic rubber material (/MAT/LAW42 (OGDEN))
Boundary conditions (/BCS)
Releasing of dof (/BCSR)
Imposed displacement (/IMPDISP)
Incompressible solid element (/PROP/TYPE14 (SOLID))
Contact definition (/INTER/TYPE7)
Implicit analysis (Implicit Controls)

The hyper-elasticity and contact causes major nonlinearities. Therefore, a static nonlinear analysis is performed using the arc-length displacement strategy. The time step is determined by a displacement norm control.

Nonlinear Implicit Parameters
Implicit type: Static nonlinear
Nonlinear solver: BFGS Quasi-Newton method
Termination criteria: Relative residual in energy
Tolerance: 0.001
Update of stiffness matrix: 5 iterations maximum
Time step control method: Arc-length
Initial time step: 0.001
Minimum time step: 1e-6
Maximum time step: 0.001
Line search method: AUTO
Special residual force computation with contact interfaces present: 5
Desired convergence iteration number: 6
Maximum convergence iteration number: 15
Decreasing time step factor: 0.8
Maximum increasing time step scale factor: 1.1
Arc-length: Automatic computation
Spring-back option: No

A solver method is required to resolve Ax=b in each iteration of a nonlinear cycle. It is defined in the option /IMPL/SOLVER.

Linear Implicit Options
Linear solver: Direct
Precondition methods: Factored approximate Inverse
Maximum iterations number: System dimension (NDOF)
Stop criteria: Relative residual of preconditioned matrix
Tolerance for stop criteria: Machine precision

A restart analysis is performed for the second load step.

The input implicit options set in both the Engine files are:

/IMPL/PRINT/NONLIN/-1 --- Printout frequency for nonlinear iteration
/IMPL/NONLIN/2 ---- Static nonlinear computation
 5 1 0.001
/IMPL/SOLVER/3 ----- Solver method (solve Ax=b)
5 0 3 0.0
/IMPL/DTINI ----- Initial time step determines initial loading increment
0.001
/IMPL/DT/STOP ------- Min Max values for time step
 1e-6 0.001
/IMPL/DT/2 ------ Time step control method 2 - Arc-length + Line-search will be used with this method to accelerate and control convergence.
 6 0 15 0.8 1.1
/IMPL/AUTOSPC/ALL ---- Constraining automatically zero stiffness dof
/IMPL/LSEARCH/3 ---- Line search method for nonlinear analysis
/IMPL/RREF/INTER/5 ------ Special Reference residual computation with contact

Refer to the Radioss manual for more details about implicit options.

Input Files

The input files used in this example include:

Rubber_ring: <install_directory>/hwsolvers/demos/radioss/example/42_Rubber_ring/rubber_ring*

Model Description

The purpose of this example is to demonstrate a nonlinear large displacement implicit analysis involving hyper-elastic material and contacts using Radioss.

The deformed ring has an inner diameter of six units and an outer diameter of 8 units and the thickness of the ring is 0.67 units. The diameter of the circular rigid roller is 2 units. The thickness of the rigid flat surface and the circular roller is 0.01units. A question that might come to mind is: "Why a half-symmetric model is not be used in this example?" Now, even though the loading in the first step is symmetric, the loading in the second step is not, so the full model had to be used for the example to simulate the loading of both the steps.

The hyper-elastic rubber ring has been modeled using the Ogden, Mooney-Rivlin material (/MAT/LAW42) with the following properties:

Material Properties
$μ_{1}$: 0.7
$μ_{2}$: -0.5
$α_{1}$: 2.0
$α_{2}$: 2.0
Poisson's ratio: 0.495

The rigid barriers (roller and flat surface) have been modeled using elastic material, but a 1D rigid link has been connected to all the nodes of each of the barriers, making them essentially rigid. The material used for the rigid barriers has the following properties:

Material Properties
Density: 7.9e^-9
Young's modulus: 600 $[MPa]$
Poisson's ratio: 0.3

Geometric Linear (NLSTAT) or Geometric Nonlinear (NLGEOM) Analysis

In geometric linear analysis all deformations and rotations are small - displacements of 5% of the model dimension are considered small.

For this rubber ring example, the final deformations and strains after crushing the rubber ring are much larger than the above mentioned limit. So, the geometrically linear static NLSTAT analysis could not be considered for this example.

Model Method

Mesh and properties
The ring mesh is a regular solid mesh modeled with four elements (dimension of 0.25 units) along the width and two elements (dimension of 0.335 units) through the thickness. The flat rigid surface and the circular roller are both modeled as a regular shell mesh of 0.01mm thickness with the flat surface around 0.75 units in dimension and the roller being around 0.467 units in dimension.

Figure 3. Geometry of the Rubber Ring and Roller Model
The ring has been modeled using first order fully-integrated solid elements.
```
/PROP/SOLID/5
WHEEL 
          14     10      1    222
```
The flat surface and roller have been modeled using the first order reduced integration shell elements with three integration points through the thickness. Full integration elements were not considered as you are not interested in any detailed post-processing of the barrier.
```
/PROP/SHELL/6
BARRIER
 1    2
 3    0.01
```
Load and boundary conditions
The boundary conditions applied to the rubber ring in step 1 are shown in Figure 4.

Figure 4. Boundary Conditions Applied to Ring in First Step

The boundary conditions applied to the flat rigid surface and circular roller in step 1, are shown in Figure 5.

The flat surface is constrained in all DOF's, while the roller is pushed down by 6.22 units in Y-axis so much that self-contact is established within the inner surface of the ring.

Figure 5. Boundary Conditions Applied to Flat Surface and Roller in 1st Step
In the second step, the top roller is to be simultaneously translated and rotated such that the wheel in the crushed configuration rolls along the flat rigid surface in -X direction. So, the X translation and ZZ rotations of the circular roller have to be released from the primary node of /RBODY. Additionally, the center nodes of the ring that were constrained in X DOF (Figure 4) need to be released for the ring to roll along the flat surface. So, the Engine file for the second step has the following cards representing release of the above-mentioned degrees of freedom.
```
/BCSR/TRA/X/
5 6 8 9 15 16 17 18
87 88 89 93 94 95 241 242
243 244 245 246 5717 5699 5681 5663
5662 5537 5517 5497 5477 5476 2269
/BCSR/ROT/Z
2269
```
Contact definition
Several contacts have been defined: i) contact between the circular roller and rubber ring, (ii) contact between the flat rigid surface and rubber ring, and (iii) self-contact within the inner surface of the rubber ring.
A small physical gap (0.05 units) has been introduced between the circular roller and the rubber ring and also between rubber ring and the flat rigid surface. The minimum gap specified for the contact is slightly higher than the physical gap for contact to take effect. Static Coulomb friction of 0.5 is defined for all the interfaces. The definition of one such interface is:
```
/INTER/TYPE7/14
TOP_Rubber
        25        30         4                                     0
       

                                     0.5               0.055
       000                             0 
         0                                       2
```
Also, since the contact involved is between a rigid part and a very soft hyper-elastic material, it is advisable that the E*h (Young's modulus * thickness) of the rigid part be approximately the same order as the bulk modulus of the rubber material.

Results

Animations

The deformed shape of the rubber ring after the circular roller is pushed down enough is shown in Figure 6.

Figure 7 shows the slide of the crushed rubber ring along the flat rigid surface after the roller has been simultaneously translated and rotated.

The stresses in the rubber ring after it has been crushed and sliding along the flat rigid surface are shown in Figure 8.