Practical Issues in the Development and Implementation of Hyperelastic Models
Hubert Lobo and Twylene Bethard, Datapoint Testing Services


Hyperelastic models are used extensively in the finite element analysis of rubber and elastomers. These models need to be able to describe elastomeric behavior at large deformations and under different modes of deformation. In order to accomplish this daunting task, material models have been presented that can mathematically describe this behavior [1]. There are several in common use today, notably, the Mooney-Rivlin, Ogden and Arruda Boyce. Each of these has advantages that we will discuss in this article. Further, we will examine the applicability of a particular material model for a given modeling situation.

General Model Selection Guidelines

The purpose of a material model is to provide the finite element analysis with a description of material behavior that matches the conditions the product sees in real life. This can be a complex matter because the real life scenario may have the product responding simultaneously to a multiplicity of conditions such as rate, temperature and the environment. A classic example may be the rubber boot of an automotive CV joint that is simultaneously seeing large deformation, temperature, cyclic loading and oil or grease. To completely describe the material behavior would require a hyperelastic model on an oil soaked boot rubber over a range of temperatures with some consideration given to rate dependency. It becomes highly impractical to attempt to model all these situations. Accordingly, one often adopts a strategy that seeks to use the simplest acceptable model that achieves a reasonable approximation of the actual scenario. This strategy may be weighted to include a more detailed modeling of the greatest potential sources of failure. Careful thought given to material modeling at the start of the FEA project results in considerable savings in time, money and effort, and will dramatically improve the likelihood of success of the project. Some of these issues are considered below.

Mode of Deformation

The deformation of rubber and elastomeric materials is very complex. Products often tend to be rather thick so that tension, compression and shear modes of deformation all come into effect. Since the deformation of rubber materials is generally large, and the behavior in each of the above modes is different, we have a somewhat complex problem describing this behavior to the simulation. It is for this reason that the hyperelastic model is the model of choice for rubber. It must be remembered however, that the hyperelastic model, for all its robustness, is not an easy model to come by or to use. Accordingly, it makes sense to examine whether the product under consideration indeed requires such a complex model. In the case of applications where the product is in a relatively simple state of deformation, it may be feasible to achieve a suitable model using just tensile or compressive data. It is important to ensure that there is no artifact present in the measurement. Measurements in more than one mode of deformation should be made equivalent strain rates to ensure that there is no potential of artifact due to rate dependency. It is generally advisable to base measurements on national or international standards, making exceptions to adapt the techniques to the need at hand, in this case, the development of the appropriate hyperelastic model.

In the case of tensile tests, non-contact extensometry is needed to ensure that there is no artifact from excessive stretch in the grip region. Video or laser methods are commonly used. Dogbones prevent artifacts from the point of attachment affect the measurements in the gauge region. They become critical when measurements to failure are required, because use of non-contact extensometers obviates the other reason.

The planar tension test utilizes a large aspect ratio of width to length. By ensuring that the material contracts only in the thickness direction in response to the applied tensile deformation, it is possible to achieve a state of pure shear at an angle of 45 degrees to the direction of extension. Methods of clamping of the specimen become crucial in the elimination of artifacts, because the gauge region is now so close to the clamps that stress states in the clamped material can easily extend into the gauge region rendering a lower than expected stress-strain response. Again, non-contact extensometry is essential to obtain accurate results.

The biaxial tension test is yet another means to provide stress-strain data for a hyperelastic model. Here, the specimen in sheet form is stretched either radially or in the rectangular (x-y) plane achieving a stress state that can prove particularly important in modeling situations where there are large multi-axial strains. There are several important aspects of the test. First, the issue of clamping is fundamental. The x-y techniques require exceedingly complex devices to ensure that clamps separate uniformly from each other as the specimen is deformed. Second, strains must be measured precisely, using non-contact means and far away from the grip sections to prevent artifacts. Cross head separation will not yield displacement data of sufficient accuracy. Such factors contribute significantly to the cost of biaxial data making it useful in situations where the data is essential but not capable of replacing the conventional techniques. Two computer controlled devices are described in [2 & 3]. These devices are excellent for determination of properties in the biaxial stress state to very large stretches. They work well in the prediction of situations such as film manufacture, and stretching where extremely large draw ratios are observed. Bubble inflation presents a simpler experimental approach to achieve the same end. Here, however, the analysis of the resulting data is complex in addition to the test being quite sensitive to parameters such as temperature and uniformity of the film thickness. The technique of Miller [4] has shows an alternate means to radially pull a specially cut circular specimen to achieve an elegant biaxial stress state in the center of the specimen. In their paper, the authors show good correlation between experiment and simulation for the prediction of the biaxial stress state to 70 to 80% strain. They further suggest that the data may be used instead of compressive measurements.

The uniaxial compressive measurements are achieved by means of a standard compression test, eg. ASTM D575 [5] with the following important exceptions. The test is performed at strain rates that match those used in the tensile and planar tension tests. Moreover, contrary to the ASTM requirement that requires a no-slip condition between the specimen and platens, it is important now to achieve a lubricated state so as to minimize shear states in the specimen that can occur if excessive friction exists between the specimen and the platen (Figure 1). Carefully performed lubricated squeezing experiments monitored by video show that it is possible to achieve fairly large strains with very little bowing in the specimen. The video experiments also show that the compressive test is deceptively simple. Reversal of the platens at the end of a test appears to show that the specimen can undergo buckling behavior when strained beyond 50%. Accordingly, the issue of equivalency to biaxial tension needs careful evaluation, particularly at large strains.

Figure1. Effect of lubrication on compression data for two materials

Of the four techniques described above, at least two and preferably three modes are needed to develop a good hyperelastic model. In the selection of tests, careful consideration should be given to the modes of deformation that will be seen in the product. For situations where the rubber is being stretched, uniaxially as in the case of retaining straps and simple drive belts, uniaxial tension data may prove adequate. For applications where the component is being stretched in two directions, e.g., a balloon catheter, the data should focus more on the uniaxial tension, biaxial tension and planar tension modes. The magnitude of deformation may also dictate the type of test that is needed. In contrast, in applications where compressive forces are significant such as in gasket and sealing applications, the uniaxial tension, planar tension and compressive modes are important. In confined applications, rubber can no longer be assumed to be incompressible. The determination of volumetric stress-strain data is important and can seriously impact the simulation. Simplistic measurements of volumetric compression use a well dimensioned specimen placed within a piston-cylinder apparatus. Because the hydrostatic stress is so much greater than the deviatoric stress, the resulting is believed to give data with little or no artifact [6]. Pressure volume temperature measurements using a high pressure dilatometer permit the measurement of volumetric data without this artifact. True bulk modulus measurements are possible using this technique [7].

Magnitude of Deformation

Sometimes, rubber components see only small deformation such as components subject to vibration. In such cases, simpler neo-hookean models may prove more than adequate to describe this behavior [8]. Such simplifications can prove invaluable in reducing computation time and reducing the cost of material model creation. Further, they permit us to account for other effects that might be more important to the simulation, such as environmental or visco-elastic effects, which, if combined with hyperelastic models, might severely tax the abilities of the computer and the analysis. Large deformations will require the use of hyperelastic models. If the magnitude of strain in the product is known, it is possible to optimize the hyperelastic model for the region of interest. Such approximations may be obtained by running simple linear elastic or neo-hookean simulations. If the objective of the study includes the modeling of failure, it becomes necessary to adopt approaches that include fracture mechanics.

Choosing a Material Model

The hyperelastic material models present a number of options to aid in a best fit of the material data. Mooney-Rivlin model is the by far, the most common model in use today. It presents many advantages in terms of being able to handle the different kinds of behavior seen in rubbers. The ability to increase the number of modes permits the handling of large strain behaviors with some level of dexterity. The objective of any model development effort however, is to fit the data at as low a number of modes as possible. As stated above, an important guideline in model fitting is advance knowledge of the strain range of interest. The Ogden model is another model that has great versatility in fitting the complex behavior of rubber. The choice between these models is often governed by the goodness of fit to the actual data. However, as pointed out by Boyce [1], these empirical models may not be as robust as those based on statistical mechanics. The more recent Arruda-Boyce model is a promising model that has a statistical mechanics basis. This model presupposes a certain kind of rubber behavior so that it has the capability to work well for materials that obey its laws. A big advantage of this approach is that the model can then predict behavior in multiple modes from data taken in only one mode, e.g., tension. We find in our work, however, that there are some materials that do not obey its laws so that the need for more detailed characterizations cannot be eliminated. Additionally, the Arruda-Boyce model does not handle the initial deflection region well, as seen in Figure 2. In comparison, the Mooney Rivlin model does a better job (Figure 3), because it has the potential of a greater number of coefficients is able to handle such behavior. Further, we observed that the Arruda-Boyce model has difficulty with biaxial (compressive) data [8].

Figure 2. Arruda-Boyce fit of precycled data

Mullin’s Effect

The Mullin’s effect describes the change in stress-strain behavior that occurs between an initial deformation of a rubber as compared to subsequent loadings. The effect is attributed to microscopic breakage of links in the rubber that weaken it during its initial deformation so that it is weaker in subsequent loadings. For the purposes of simulation, it is particularly important to question whether the Mullin’s effect is a relevant for the situation under consideration because it can have a considerable impact on the material testing as well as on the material model that is used to represent the data. Accordingly, components that are subject to repeated load-unload cycles to the same strain should utilize material data that have been developed on samples after they have been subject to the Mullin’s effect. Here, test specimens are subjected to repeated load-unload cycles to the strain of interest, until there is no change in the shape of the curves. The resulting ultimate stress-strain behavior is used for the model development. It is important to note that these models are not suitable to describe first time deformations. Here, the curve of the first deformation is needed. Analogously, products that are subject to varying deformations are not be properly described by either one individual curve. It is important to note that conventional hyperelastic models cannot describe such complex behaviors as seen in Figures 4 and 5. In such cases, a simpler piecewise approach may prove adequate. Again, being aware of the needs of the simulation will permit the analyst to request the appropriate information at little or no additional cost. Failure to anticipate this need results in expensive retesting. In the worst case, the wrong data is used resulting in erroneous simulations.

Figure 4. Mooney-Rivlin fit of uncycled material

Figure 5. Arruda Boyce fit of uncycled material

Effect of Temperature

Generally speaking, materials become stiffer as we lower the temperature. Consequently, they will deform less and can even suffer brittle failure if a ductile-brittle transition is crossed. For example, many rubber materials can be shattered if dipped in a liquid nitrogen environment. In contrast, thermoplastic elastomers will weaken considerably in the vicinity of their melting point. Knowing the practical application temperature range of the material will help us determine whether such issues are of importance to the project. At the very least, one will seek to determine properties in the temperature range where there is the greatest likelihood of failure. There are many ways to determine the applicable temperature range of a material. A convenient technique is dynamic mechanical analysis, which monitors the change in modulus with temperature. This test easily identifies glass transitions, usually harbingers of brittle transitions, as well as modulus changes at the melting point.

Effect of the Environment

The effect of weathering or the presence of oil, gasoline, body fluids or other chemicals can significantly affect the behavior of a material. The consequence of the environment is often unpredictable and may improve or adversely affect the performance of the product. Environmental effects are often neglected in the development of the material model. While this kind of data is not typically available, off the shelf, it is useful to note that since most hyperelastic models are derived from testing, advance consideration to environment will result in the development of the right kind of data, often at little or no additional cost. For example, it is commonplace to routinely subject to the environment, specimens of materials that are being evaluated as part of a selection specification. It is well worth the additional effort then, to utilize these specimens to determine necessary properties as against those of the virgin specimens.


[1] M.C. Boyce and E.M. Arruda, J. Rubber Chem. Tech v.73, pp 504-523 (2000)
[2] C.Gerlach, C.P.Buckley and D.P.Jones, Trans I Chem E 76 Part A, 38 (1998).
[4] J. Day, K. Miller, Testing and Analysis, (2000)
[5] ASTM Standards, v9.1 , pg 119 (1999)
[6] ABAQUS Standard User’s Manual 10.5.1
[7] Y.A. Fakhreddine and P. Zoller, SPE ANTEC Proceedings, pg 1642 (1991)
[8] D.J. Siebert and N. Schoche, J. Rubber Chem. Tech, v.73 pp 366-384 (2000)