## Limitations of Hyperelastic Material Models

**Introduction:**

Polymeric rubber components are widely used in automotive, aerospace and biomedical systems in the form of vibration isolators, suspension components, seals, o-rings, gaskets etc. Finite element analysis (FEA) is a common tool used in the design and development of these components and hyperelastic material models are used to describe these polymer materials in the FEA methodology. The quality of the CAE carried out is directly related to the input material property and simulation technology. Nonlinear materials like polymers present a challenge to successfully obtain the required input data and generate the material models for FEA. In this brief article we review the limitations of the hyperleastic material models used in the analysis of polymeric materials.

** Theory:**

A material model describing the polymer as isotropic and hyperelastic is generally used and a strain energy density function (*W*) is used to describe the material behavior. The strain energy density functions are mainly derived using statistical mechanics, and continuum mechanics involving invariant and stretch based approaches.

**Statistical Mechanics Approach**

The statistical mechanics approach is based on the assumption that the elastomeric material is made up of randomly oriented molecular chains. The total end to end length of a chain (*r*) is given by

Where *µ* and *l** _{m}* are material constants obtained from the curve-fitting procedure and

*J*is the elastic volume ratio.

^{el}**Invariant Based Continuum Mechanics Approach**

The Invariant based continuum mechanics approach is based on the assumption that for a isotropic, hyperelastic material the strain energy density function can be defined in terms of the Invariants. The three different strain invariants can be defined as

*I*_{1} = l_{1}^{2}+l_{2}^{2}+l_{3}^{2}

*I*_{2} = l_{1}^{2}l_{2}^{2}+l_{2}^{2}l_{3}^{2}+l_{1}^{2}l_{3}^{2}

*I*_{3} = l_{1}^{2}l_{2}^{2}l_{3}^{2}

With the assumption of material incompressibility, *I _{3}=1*, the strain energy function is dependent on

*I*and

_{1}*I*only. The Mooney-Rivlin form can be derived from Equation 3 above as

_{2}*W(I _{1},I_{2})*

*= C*+

_{10}(I_{1}-3)*C*

_{01}*(I*–

_{2}*3)*…………………………………………………………

**(4)**

With *C _{01 }= 0* the above equation reduces to the Neo-Hookean form.

**Stretch Based Continuum Mechanics Approach**

The Stretch based continuum mechanics approach is based on the assumption that the strain energy potential can be expressed as a function of the principal stretches rather than the invariants. The Stretch based Ogden form of the strain energy function is defined as

where *µ _{i} *and α

*are material parameters and for an incompressible material*

_{i}*D*

_{i}=0.Neo-Hookean and Mooney-Rivlin models described above are hyperelastic material models where, the strain energy density function is calculated from the invariants of the left Cauchy-Green deformation tensor, while in the Ogden material model the strain energy density function is calculated from the principal deformation stretch ratios.

The Neo-Hookean model, one of the earliest material model is based on the statistical thermodynamics approach of cross-linked polymer chains and as can be studied is a first order material model. The first order nature of the material model makes it a lower order predictor of high strain values. It is thus generally accepted that Neo-Hookean material model is not able to accurately predict the deformation characteristics at large strains.

The material constants of Mooney-Rivlin material model are directly related to the shear modulus ‘G’ of a polymer and can be expressed as follows:

*G *= 2(*C _{10} *+

*C*) …………………………….…

_{01}**(6)**

Mooney-Rivlin model defined in equation (4) is a 2nd order material model, that makes it a better deformation predictor that the Neo-Hookean material model. The limitations of the Mooney-Rivlin material model makes it usable upto strain levels of about 100-150%.

Ogden model with N=1,2, and 3 constants is the most widely used model for the analysis of suspension components, engine mounts and even in some tire applications. Being of a different formulation that the Neo-Hookean and Mooney-Rivlin models, the Ogden model is also a higher level material models and makes it suitable for strains of upto 400 %. With the third order constants the use of Ogden model make it highly usable for curve-fitting with the full range of the tensile curve with the typical ‘S’ upturn.

**Discussion and Conclusions:**

The choice of the material model depends heavily on the material and the stretch ratios (strains) to which it will be subjected during its service life. As a rule-of-thumb for small strains of approximately 100 % or *l*=2.0, simple models such as Mooney-Rivlin are sufficient but for higher strains a higher order material model as the Ogden model may be required to successfully simulate the ”upturn” or strengthening that can occur in some materials at higher strains.

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