Instability of Two-Term Mooney-Rivlin Model
In engineering practice, rubber material models are widely used in large deformation nonlinear response simulations. One such model is the well-known Mooney-Rivlin model, described with other rubber material models in the ADINA Theory and Modeling Guide .
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 Newton-Raphson 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.
Neo-Hookean material model: C1 = 1/6
Mooney-Rivlin material model: C1 = 1/8, C2 = 1/24
Ogden material model: μ1 = 0.497, α1 = 1.3;
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.
Figure 1 (continued)
However, when looking deeper and using the ADINA option to plot the stability curves, see Figure 2, we see that the neo-Hookean and Ogden material descriptions are stable, but the Mooney-Rivlin description is unstable (stability indicator
< 0.0) in biaxial tension for strains greater than about 0.6.
(c) Material 3 described by Ogden model
Consider the following two analyses in which the instability in the Mooney-Rivlin 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 Mooney-Rivlin 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 Mooney-Rivlin Model
Results using Ogden Model
A cylinder of material 2, modeled with 27/4 3-D elements and the Mooney-Rivlin model, is stretched by an axial force, then expanded by a deformation-dependent 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.
The neo-Hookean material model (with only one constant) is always stable.
For the Mooney-Rivlin material model, the strain level at which instability occurs depends on the ratio of C2/ C1. If the ratio C2/ C1 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 , it is important to look at the stability curves plotted in ADINA prior to performing a general nonlinear simulation using these models.