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 userinput 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 rubberlike materials.
The figures also show the fitted response of two rubber material models: the 9term
MooneyRivlin material model and the SussmanBathe material model. Both material models
fit the userinput 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 forcedeflection curves obtained with the material models. When
the 9term MooneyRivlin material model is used, no solution can be obtained for
applied displacements larger than about 1.4. But when the SussmanBathe material model is used, solutions can be
obtained for much larger applied displacements.
Figure 5 Forcedeflection 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, SussmanBathe material model
(Material 1 refers to the fact that only one material is used)
Figure 7 Stability curves from material data, MooneyRivlin 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 SussmanBathe model is stable for all three modes
of deformation, but the 9term MooneyRivlin 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 9term MooneyRivlin model
fails at rather small loads/deformations (for rubber).
It is a desirable characteristic of a material model that, if the underlying experimental
stressstrain data corresponds to a stable material, then the material model should
also be stable. Clearly, in this instance, the 9term MooneyRivlin material model
does not have this characteristic.
Of course, different material constants could be chosen for the MooneyRivlin 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, MooneyRivlin, SussmanBathe
Reference

T. Sussman & K.J. Bathe, "A model of incompressible isotropic hyperelastic material
behavior using spline interpolations of tensioncompression test data",
Commun. Num. Meth. Engng (2008), in press
^{(1)} ADINA version 8.5.2 and higher
