How to Choose a Hyperelastic Material Model for your FEA?
Hyperelastic materials models are regularly used to represent large deformation behavior of materials. They are commonly used to model mechanical behaviors of unfilled / filled elastomers. In addition to elastomers, hyperelastic material models are also used to approximate the material behavior of biological tissues, polymeric foams etc.
Linearly elastic materials are described through two material constants (like Young’s modulus and Poisson ratio). In contrast, hyperelastic materials are described through a strain energy density function. The strain-energy density can be used to derive a nonlinear constitutive model (i.e. stresses as a function of large strain deformation measures like deformation gradient or Cauchy-Green tensors etc). There are several models proposed in the literature like Neo-Hookean, Mooney-Rivlin, Ogden, Yeoh, Arruda-Boyce (or 8-chain) models. A detailed mathematical description of these hyperelastic strain energy density function can be found on the SimScale documentation. At present, SimScale allows for usage of Neo-Hookean, Mooney-Rivlin, and Signorini models as shown in the Fig. 01 and more are on the way.
Fig 01: Hyperelastic materials models presently offered by SimScale
The Signorini material model can be said to be a more generalized version of the Mooney-Rivlin model. A detailed description of validation of the available models for Uniaxial, Equibiaxial, and Pure shear can be found on the SimScale website.
Effect of Temperature
In general, elastomers demonstrate a strong dependence on strain rate, temperature and loading history. Their mechanical behavior can thus be separated into three regimes: glassy (& brittle), viscoelastic and rubbery. As shown in Fig. 02, below a particular temperature, known as glass transition temperature, the behavior is glassy (& brittle) in nature; while above it, the behavior changes to viscoelastic and further on to rubbery.
Fig 02: Variation of material stiffness w.r.t temperature
Hyperelastic material models are ideally suited to describe the rubbery phase of elastomeric behavior. In this phase, the material can be assumed to be independent of loading history and temperature and displays the pure elastic response.
Tip 01: “What is the glass transition temperature of the elastomer being modeled?“. If the working temperature is much higher than the glass transition temperature of the elastomer, then the hyperelastic material model could be used.
It is important in large deformation mechanics to be certain of what stress measures are being used. Unlike in linear elasticity, while using hyperelastic models, the undeformed (or reference) and deformed (or current) configurations need to be distinguished and addressed. For example: During a uniaxial tensile test, the area of the specimen is measured prior to the test (i.e. in the undeformed configuration) and the force is measured as the test progresses (i.e. in the deformed configuration). Thus, the “stress” plotted in the stress-strain curve obtained experimentally is the First Piola-Kirchhoff (or Nominal) stress.
Fig 03: Parameter fitting for Neo-Hookean, Mooney-Rivlin and Signorini models with Treloar experimental data (uniaxial, equi-biaxial and shear tests)
As can be seen from the strain-energy density function, each hyperelastic material model requires one or more material parameters that need to be identified from experiments. Some of the commonly used experiments are: uniaxial tensile test, equi-biaxial tensile test, and shear (or planar) test. Fig. 03 shows the fitting using all these three experiments. A rule of thumb would be one experiment for each parameter being used by the model. Fitting parameters for a Mooney-Rivlin model with only one experiment, like the uniaxial tensile test, could lead to non-uniqueness of parameters (i.e. fitting procedure can lead to different parameters based on different starting approximations).
Tip 02: “What experimental data is available for parameter fitting?” For a unique set of parameters: just one experiment would be sufficient to obtain parameters for Neo-Hookean; at least two experiments for Mooney-Rivlin; three for Signorini / Arruda-Boyce; three or more for Ogden models.
Magnitude of deformation
The next step would be to identify the magnitude of deformation (or strain) that the component would be subjected to. For example, rubber components are commonly used to suppress vibrations. In such cases where the deformations are moderate (<100%), simple models like Neo-Hookean would suffice. Such a simplification can be invaluable in reducing the computational effort.
Fig 04: Comparison of performance of hyperelastic material models with Treloar experimental data (uniaxial, equi-biaxial and shear tests)
As shown in Fig. 04, Treloar experimental data is compared with different hyperelastic models. As evident from the figure, a useful rule of thumb is that the Ogden model (of order 3) is the most suitable for usage for the entire range (0% up to failure). While Neo-Hookean model performs well up to 100%, Mooney-Rivlin is best up to 150-200%. Yeoh & Arruda-Boyce models are best suitable when fitted for the range of usage.
Tip 03: “What is the magnitude of deformation (in comparison to its size) that is experienced by the component?“ Up to 100% (Neo-Hookean); 150-200% (Mooney-Rivlin); 0-Failure (Full Ogden); particular range of a-b (Yeoh or Arruda-Boyce).
Stability of hyperelastic material models
It is possible that models with multiple parameters are fitted with just one experiment, as discussed earlier. In such a case, depending on the material parameters obtained, the model can be always or conditionally stable or unstable (for other loading conditions).
The Mooney-Rivlin material model (with a polynomial of order one) demonstrates an instability when subjected to biaxial loads. The strain level at which an instability occurs depends on the ratio of C01/C10. If the ratio C01/C10 is decreased, the biaxial strain corresponding to instability increases. For example, for C01 = 1/8 and C10 = 1/24 and application of equi-biaxial loading, an instability sets around an extension of 1.6. As shown in the Fig 05., upon application of equi-biaxial load, the body moves from state A to B. Now, at state B, as the horizontal force is increased with constant vertical force, the body reaches a state C. This is an unphysical response where the displacement decreases as force increases.
Fig 05: Instability of Mooney-Rivlin Model
In contrast, the stability of the Ogden model depends on the chosen parameters. Neo-Hookean model is always stable.
Tip 04: “Do the chosen parameters lead to stable behavior in the range of usage?” If the fitting is done using a particular type(s) test, then it would be useful to try other tests to check the stability of the model parameter(s).
Consistency of parameters used
With regard to Neo-Hookean, Mooney-Rivlin, and Signorini models, for consistency with linear elasticity, the material parameters need to satisfy:
- G = 2 (C10 + C01), where G is the shear modulus
- D1 = K/2, where K is the bulk modulus
Additionally, it is necessary to note that as the Poisson ratio approaches 0.5 (once >0.4), this would necessitate more complicated formulations to account for the incompressible behavior. The stresses measured in experiments (and using models that support incompressibility) are much higher than what one obtains from compressible models. This shall be discussed in a future post along with meshing issues related to modeling hyperelastic/incompressible materials.
Tip 04: “Do you really need an incompressible model?“ Compressible models can significantly underestimate the stresses calculated. Try using reduced integration option available in the mesh tree (of SimScale). Alternatively, we can hope to see elements with mixed or enhanced formulations that are more accurate, in the future.
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