Mooney-Rivlin hyperelastic model for nonlinear finite element analysis

In previous articles, we have introduced the hyperelastic model in nonlinear finite element analysis, and also introduced the well-known Arruda-Boyce and neo-Hookean models in detail. Today we will be discussing another famous hyperelastic model: Mooney-Rivlin.

Mooney-Rivlin is named after the surname combination of two physicists, M. Mooney and R. S. Rivlin. In 1940, Mooney published a paper entitled “A theory of large elastic deformation” in the Journal of Applied Physics. Eight years later, in 1948, Rivlin published a paper “Large elastic deformations of isotropic materials” in Philosophical Transactions of the Royal Society of London. Those two papers build the essence of the Mooney-Rivlin hyperelastic model, which has dominated the rubber mechanics for about half-century. At the same time, it also lays the foundation for other models based on strain tensor invariants. Another type of hyperelastic model is based on principal stretches, such as the Ogden model, which we will discuss in the future.

Mooney was born in Kansas City, Missouri, the USA in 1893. He received an undergraduate degree from the University of Missouri at the age of 24 and a Ph.D. at 30. He was a fellow of the National Research Council and a physicist at Western Electric and American Rubber Company. Like Rivlin we introduced earlier, Dr. Mooney devoted his life’s work and research to the mechanics of polymer materials. Mooney is more than 20 years older than Rivlin. Dr. Rivlin has been introduced in the last article, so we won’t repeat it here.

Like other hyperelastic models, we use elastic strain energy to characterize mechanical properties. Mooney-Rivlin is based on the order of the level, there are four types: two-parameter, three-parameter, five-parameters, and nine-parameter strain energy.

The form of the strain-energy potential for a two-parameter Mooney-Rivlin model is:

The form of the strain-energy potential for a three-parameter Mooney-Rivlin model is:

The form of the strain-energy potential for a five-parameter Mooney-Rivlin model is:

The form of the strain-energy potential for a nine-parameter Mooney-Rivlin model is:

From these four potentials, we can tell:

• The higher-order potential can model more complex strain-stress constitutive relations but requires more computational efforts, experimental data, and parameter fitting. At the same time, as the nonlinearity becomes stronger, the higher-order may be difficult to converge.
• Mooney-Rivlin model is a special form of Polynomial model. When N = 1, the Polynomial model is reduced to 2-parameter Mooney-Rivlin; when N = 2, the Polynomial model is reduced to 5-parameter Mooney-Rivlin; when N = 3, the Polynomial model is reduced to 9-parameter Mooney-Rivlin.
• The 2-parameter Mooney-Rivlin has the following features: it is reduced to Neo-Hookean when parameter C01=0. (shear module u=2*C10 ). The non-zero C01 term makes the model under uniaxial tension more accurate, but it still cannot accurately simulate the multiaxial stress states. The parameter obtained from one type of deformation experiments, cant not predict another type of deformation accurately. Since the shear modulus of 2-parameter Mooney-Rivlin is constant (u=C10+C01), it cannot model the carbon black rubber. The summation of C10 and C01 must be positive. For most rubber materials, the ratio of C10 / C01 lies between 0.1 and 0.2.
• Three- or more-parameter Mooney-Rivlin models can describe unsteady shear modulus. However, solutions need to be carefully evaluated when the high-order terms are introduced, as it may produce unstable strain energy values and yield non-physical solutions.

Selection and positive definiteness of strain energy potential

Which of these four Mooney-Rivlin models should be used in a practical simulation? It is often determined from the strain-stress curve of material experiments. For single curvature (no inflection point) stress-strain curve, 2 or 3 parameters can be used. Double curvature (including an inflection point) can use 5 parameters. For three curvatures (including 2 inflection points), a 9-parameter model can be selected.

At the same time, to produce effective and correct superelastic material properties, Mooney-Rivlin parameters must meet certain positive definiteness requirements. Failure to meet these conditions may cause a converge difficulty. For Mooney-Rivlin with different numbers of parameters, the requirement of positive definiteness is shown in the figure.

Overall, the Mooney-Rivlin model has been widely recognized and applied. Especially in the small and medium strain applications (0 ~ 100% tensile and 30% compression), Mooney-Rivlin can precisely characterize the mechanical behavior of rubber materials. Four types of potentials also provide users with more choices to model various material behaviors. However, Mooney-Rivlin has some limitations:

• Not suitable for the deformation that exceeds 150%.
• Because higher-order Mooney-Rivlin has many parameters, the parameters are relatively difficult to obtain from the manual or literature, which needs curve fitting of experimental data.
• Not suitable for the analysis of compressible hyperelastic materials, such as foam.
• The error could be large when the strain/stress is beyond the range of input experimental data

An example of Mooney-Rivlin hyperelastic analysis

In this example, we will use a 5-parameter Mooney-Rivlin to analyze the compression state of a rubber cylinder.

Define a Mooney-Rivlin hyperelastic material

Here we define the rubber material and input the parameters as: C10 = -0.55 MPa, C01 = 0.7 MPa, C20 = 1.7 MPa, C11 = 2.5 MPa, C02 = -0.9 MPa, D1 = 0.001 MPa^-1.

Modeling
Create a cylinder with a diameter of 10 mm and a height of 10 mm. Meshing. Fixed bottom constraint. A 5mm downward displacement is applied onto the top surface.

Solve
Since it is a nonlinear analysis, we set 30 substeps for easier convergence. And click the solve button.

View Results
The equivalent stress distribution is shown in the figure. It can be found that the von-Mises stress increases in a nonlinear manner.

The operation video is attached for your reference.