1.1.7 Pressurized rubber disc

Products: ABAQUS/Standard  ABAQUS/Explicit  

In this example a rubber disc, pinned around its outside edge, is subjected to pressure so that it bulges into a spherical shape. The example is an illustration of a rubber elasticity problem involving finite strains on a membrane-like structure. The published results of Oden (1972) and Hughes and Carnoy (1981) are used to verify the ABAQUS quasi-static solution.

The example shows that ABAQUS can solve this type of problem. The ABAQUS/Standard results also demonstrate that, because of the treatment of the pinned-edge condition, the load stiffness matrix associated with the pressure loading is not symmetric at the outer edge of the pressurized face of the disc. It is found that, after a small amount of straining, these nonsymmetric terms must be included in the stiffness matrix for the solution to be numerically efficient.

Both a thick and a thin disc are tested. The solutions obtained using ABAQUS/Explicit show dynamic effects when compared to the quasi-static solution found by ABAQUS/Standard. The thin disc model in ABAQUS/Explicit demonstrates the ability of ABAQUS/Explicit to handle volume expansion of membrane-like structures; the application of fluid cavity elements in ABAQUS/Explicit is also demonstrated.

Problem description

Loading and solution method

The loading consists of a uniform pressure applied to the bottom surface of the disc. The modified Riks method is used in ABAQUS/Standard since the loading is proportional and because the solution may exhibit instability. A pressure magnitude of 1.38 MPa (200 lb/in2) is specified: this magnitude is somewhat arbitrary since the Riks method is chosen. From other studies we expect that an initial pressure of about 0.014 MPa (2 lb/in2) should take the disc a reasonable way into the nonlinear regime. Hence, an initial increment of 0.01 and a period of 1 are specified on the *STATIC option to achieve this level of pressure in the initial increment. (Since the Riks algorithm is used, the actual pressure magnitude at the end of the first increment will differ somewhat from the initial value of 0.014 MPa, depending on the extent of nonlinearity in that increment. See the descriptions of the Riks option in Unstable collapse and postbuckling analysis, Section 6.2.4 of the ABAQUS Analysis User's Manual, and Modified Riks algorithm, Section 2.3.2 of the ABAQUS Theory Manual, for more details.)

Since the surface to which the pressure is applied rotates and stretches, there is a stiffness contribution associated with the pressure (a “load stiffness matrix”). Because of the treatment of the pinned outer edge, the perimeter of the surface to which the pressure is applied is not fully constrained and, hence, gives rise to a nonsymmetric contribution in the local stiffness matrix (see Hibbitt, 1979). During that part of the solution where strains and rotations are not very large, it makes little difference to the number of iterations needed to solve the equilibrium equations if this nonsymmetric contribution is ignored. However, to continue the analysis beyond a pressure of about 0.07 MPa (10 lb/in2)—when the displacement at the center of the disc is about half the radius—it is essential that these terms are included. This requires that UNSYMM=YES be used on the *STEP option in the ABAQUS/Standard analysis. In practical cases, if this parameter is omitted in the initial run, it can be introduced on a restarted run if necessary. An example using S4R elements with enhanced hourglass control is also included.

The effect of uniform tensile prestress in ABAQUS/Standard is also investigated. The prestress is applied as equal radial and circumferential stresses through *INITIAL CONDITIONS, TYPE=STRESS. Prestress values of 0.35, 0.7, and 1.4 MPa (50, 100, and 200 lb/in2) are investigated.

In the explicit dynamic analysis the pressure is ramped up over the duration of the step. The maximum applied pressure for the thick disc case is 0.317 MPa (46 psi) and is applied by using the *DLOAD option or by prescribing the pressure directly to a fluid cavity reference node. In the fluid-driven case the fluid cavity is modeled either with hydrostatic fluid elements (see Modeling fluid-filled cavities, Section 11.5.1 of the ABAQUS Analysis User's Manual) or with the surface-based fluid cavity capability (see Defining fluid cavities, Section 11.6.2 of the ABAQUS Analysis User's Manual). The fluid cavity elements or surfaces are defined underneath the disc so that the initial volume of the fluid cavity is zero. For both load cases the 0.317 MPa pressure value was chosen based on the final value obtained in the quasi-static simulation via ABAQUS/Standard utilizing the Riks method for incrementation control. The maximum pressure for the thin disc is 0.036 MPa (4.5 psi) and is prescribed at a fluid cavity reference node as in the thick disc case. The rate of loading was observed to affect the simulation for all cases in ABAQUS/Explicit.

A thick disc example for the two-dimensional axisymmetric continuum case in ABAQUS/Explicit illustrates the use of the *EXTREME VALUE option to control the duration of the analysis and to force output when an extreme value criterion is reached. Using the *EXTREME NODE VALUE option, the end of the analysis is specified to occur when the center of the plate has bulged out to twice its initial radius. Thickness strain is monitored in the bottom row of elements with the *EXTREME ELEMENT VALUE option, and an output state is written when the strain falls below the specified value. Additional examples using S4R and M3D4R elements with enhanced hourglass control are included.

Results and discussion

Input files

References

Figures

Figure 1.1.7–1 Mesh for pressurized rubber disk.

Figure 1.1.7–2 Displaced shapes of pressurized rubber disk, ABAQUS/Standard analysis.

Figure 1.1.7–3 Displaced shapes for the axisymmetric continuum mesh, thick disc model, ABAQUS/Explicit analysis.

Figure 1.1.7–4 Central thickness versus central displacement, ABAQUS/Standard analysis.

Figure 1.1.7–5 Thickness strain versus central displacement for the axisymmetric continuum mesh, thick disc model, ABAQUS/Explicit analysis.

Figure 1.1.7–6 Comparison of pressure-deflection results, ABAQUS/Standard analysis.

Figure 1.1.7–7 Pressure versus deflection results for load ramp duration of 0.01 sec, thick disc model with *DLOAD loading, ABAQUS/Explicit analysis.

Figure 1.1.7–8 Pressure versus deflection results for load ramp duration of 0.01 sec, thick disc model with fluid pressure loading, ABAQUS/Explicit analysis.

Figure 1.1.7–9 Pressure versus deflection results for load ramp duration of 0.01 sec, thin disc model with fluid pressure loading, ABAQUS/Explicit analysis.

Figure 1.1.7–10 Pressure versus deflection results for load ramp duration of 0.10 sec, thick disc model with *DLOAD loading, ABAQUS/Explicit analysis.