Stability of Rubber Models

In engineering practice, various rubber models are used extensively. We present here the use of a new model and the stability indicators available in ADINA.

As an example, the rubber component shown above is analyzed using different rubber material models. The analysis does not converge when a rubber material model with negative stability indicators is used. The stability characteristics of rubber material models can be visualized using the new stability plot feature of ADINA 8.5⁽¹⁾.

Figures 1 and 2 show the user-input data for uniaxial and biaxial tension of the rubber material considered here. These figures plot engineering stress vs. stretch, which are typically used for the presentation of material data for rubber-like materials. The figures also show the fitted response of two rubber material models: the 9-term Mooney-Rivlin material model and the Sussman-Bathe material model. Both material models fit the user-input data very well.

Figure 3 shows the same data presented in terms of uniaxial tension / compression, and using true stress - logarithmic strain. Note that the response covers tension and compression.

Figure 1  Engineering stress vs. stretch, uniaxial tension

Figure 2  Engineering stress vs. stretch, biaxial tension

Figure 3  True stress vs. log strain. These curves are obtained, as is standard, from the uniaxial tension and biaxial tension curves shown in Figures 1 and 2

These material models are used in the analysis of the rubber component shown above. The rubber component is pulled by equal applied displacements, as shown in Figure 4.

Figure 4  Rubber component in initial (original) and final state

Figure 5 shows the force-deflection curves obtained with the material models. When the 9-term Mooney-Rivlin material model is used, no solution can be obtained for applied displacements larger than about 1.4. But when the Sussman-Bathe material model is used, solutions can be obtained for much larger applied displacements.

Figure 5  Force-deflection curves

The new stability plot feature gives insight into the convergence behavior of the rubber component. The stability plots for the material models considered here are shown in Figures 6 and 7.

Figure 6  Stability curves from material data, Sussman-Bathe material model
(Material 1 refers to the fact that only one material is used)

Figure 7  Stability curves from material data, Mooney-Rivlin material model

The idea behind the stability plots is as follows. Consider a uniform sheet of rubber subjected to uniaxial tension. For each strain level, the incremental stiffness matrix (corresponding to perturbation of forces due to perturbation of displacements) is determined. The eigenvalues of the incremental stiffness matrix are calculated and the smallest eigenvalue is taken as the stability indicator. If the stability indicator is greater than zero, then the material is stable (with respect to perturbations in the applied forces), otherwise the material is unstable. The same procedure is employed for pure shear and biaxial tension.

The stability plots show that the Sussman-Bathe model is stable for all three modes of deformation, but the 9-term Mooney-Rivlin model becomes unstable in biaxial tension for true strains > 0.4. Since the rubber component is in biaxial tension, it is not surprising that the analysis of this component using the 9-term Mooney-Rivlin model fails at rather small loads/deformations (for rubber).

It is a desirable characteristic of a material model that, if the underlying experimental stress-strain data corresponds to a stable material, then the material model should also be stable. Clearly, in this instance, the 9-term Mooney-Rivlin material model does not have this characteristic.

Of course, different material constants could be chosen for the Mooney-Rivlin model, in order to make the stability indicators positive, but then this model would not fit the input data as well.

Keywords:
Rubber material, ADINA, stability, Mooney-Rivlin, Sussman-Bathe

Reference

• T. Sussman & K.J. Bathe, "A model of incompressible isotropic hyperelastic material behavior using spline interpolations of tension-compression test data", Commun. Num. Meth. Engng (2008), in press