Instability of TwoTerm MooneyRivlin Model
In engineering practice, rubber material models are widely used in large deformation nonlinear response simulations. One such model is the wellknown MooneyRivlin model, described with other rubber material models in the ADINA Theory and Modeling Guide [1].
In this News we want to focus on the fact that some rubber material
models are unstable, in the sense that an unstable material response for
certain strain levels may be obtained. If then, in the simulation of a structure, strain levels corresponding to instability are reached, the NewtonRaphson nonlinear iterative solution may have difficulty to converge, and the calculated response may be physically unrealistic, even if no difficulties in the iterative solution have been encountered.
(a) Material 1 described by neoHookean model
Figure 1 Uniaxial and biaxial tension plots


For illustration, consider three incompressible materials with Young’s modulus E = 1.0 at small strains. The three materials are represented, respectively, using the neoHookean, twoterm MooneyRivlin, and Ogden models. The large strain response is obtained using the following constants in the material characterizations:
NeoHookean material model: C_{1} = 1/6
MooneyRivlin material model:
C_{1} = 1/8, C_{2} = 1/24
Ogden material model: μ_{1} = 0.497, α_{1} = 1.3; μ_{2} = 0.001, α_{2} = 5.0; μ_{3} = 0.0079, α_{3} = 2.0.
These material constants are taken from equation (7.2.31) of Ref. [2], except that the μ_{1}, μ_{2} and μ_{3} are scaled so that the smallstrain Young's modulus E = 1.0.
Using these constants, the uniaxial and biaxial tension plots for the three materials described by the models are as shown in Figure 1. Note that these curves have positive slopes over the complete strain
ranges, which seems to imply that no instabilities are present and hence will be encountered in a complex nonlinear 2D or 3D simulation with these material constants.
(b) Material 2 described by MooneyRivlin model 

(c) Material 3 described by Ogden model 
Figure 1 (continued)
However, when looking deeper and using the ADINA option to plot the stability curves, see Figure 2, we see that the neoHookean and Ogden material descriptions are stable, but the MooneyRivlin description is unstable (stability indicator
< 0.0) in biaxial tension for strains greater than about 0.6.
(a) Material 1 described by neoHookean model 

(b) Material 2 described by MooneyRivlin model 
(c) Material 3 described by Ogden model
Figure 2 Stability plots
Consider the following two analyses in which the instability in the MooneyRivlin model is triggered, and in which the subsequent response is physically unrealistic.
Plane stress sheet in biaxial tension
A rubber sheet of material 2 is modeled with one plane stress element and the MooneyRivlin model, and two independent forces are applied. First, equal forces are applied to stretch the sheet (point A in the graph). Then the horizontal force is increased, while the vertical force remains constant. As the movie and figure below show, the horizontal displacement decreases as the horizontal force increases. This is surely a physically unrealistic material response.
Results using MooneyRivlin Model
The next movie shows the same numerical experiment, except that the material 3 described by the Ogden model (with constants given above) is used. The horizontal displacement increases as the horizontal force increases, which is physically realistic.
Results using Ogden Model
Cylinder under axial force and internal pressure
A cylinder of material 2, modeled with 27/4 3D elements and the MooneyRivlin model, is stretched by an axial force, then expanded by a deformationdependent internal pressure. Initially, as the internal pressure is increased, the response is continuous and realistic. But when the internal pressure is increased to above a critical value, the response suddenly jumps in an unrealistic way. A movie is shown at the top of this page.
We can give some general observations:
The neoHookean material model (with only one constant) is always stable.
For the MooneyRivlin material model, the strain level at which instability occurs depends on the ratio of C_{2}/ C_{1}. If the ratio C_{2}/ C_{1} is decreased, the biaxial strain corresponding to instability increases, but the material is always unstable after this strain level is reached.
The Ogden material model is stable for the material constants used above, but for different choices of material constants, the
material model may become unstable.
Hence, for any of the rubber material models available, including the material model presented in Reference [3], it is important to look at the stability curves plotted in ADINA prior to performing a general nonlinear simulation using these models.
Keywords:
Rubber material model, MooneyRivlin, Ogden, neoHookean, hyperelastic, material instability
References

ADINA Theory and Modeling Guide, Report ARD 087.

R. W. Ogden, Nonlinear elastic deformations, Ellis Horwood Limited, 1984.

T. Sussman and K.J. Bathe, “A model of incompressible hyperelastic material behavior using spline interpolations of tensioncompression test data”, Communications in Numerical Methods in Engineering, 25,
5363, 2009.
