Hyperelastic material models for rubber

based on [[ A comparison among Neo-Hookean model, Mooney-Rivlin model, and Ogden model for chloroprene rubber ]]

The Neo-Hookean and Mooney-Rivlin models are hyperelastic material models defined by the strain energy density function derived from the invariants of the left Cauchy-Green deformation tensor.

Neo-Hookean Model

The Neo-Hookean model is a hyperelastic material model that predicts the stress-strain behavior of materials, paralleling Hooke’s law. It is one of the simpler models, with the strain energy density function for incompressible Neo-Hookean materials expressed as:

\(W = C_1 (\bar{I}_1 - 3)\) where:

• $C_1$​ is a material constant.
• $\bar{I}_1$ ​ is the first invariant of the left Cauchy-Green deformation tensor.

However, the Neo-Hookean model is known to become inaccurate at large strains.

Mooney-Rivlin Model

The Mooney-Rivlin model, introduced by Melvin Mooney and Ronald Rivlin, improves upon the Neo-Hookean model by including a second invariant of the deformation tensor, allowing better accuracy for larger strains.

The strain energy density function is expressed as:

where:

• $C_1$​ and $C_2$ are empirically determined material constants.
• $\bar{I}_1$ ​​ and $\bar{I}_2$ ​ are the first and second invariant of the deviatoric component of the left Cauchy -Green deformation tensor.

The shear modulus $G$ is related to the material constants by:

The Mooney-Rivlin model is widely used for rubber-like materials and allows the shear modulus to be defined as a function of temperature. Although it has limitations under specific stress states, it is commonly applied for strains up to approximately 200%.

Ogden Model

The Ogden model is highly effective for predicting the nonlinear stress-strain behaviour of materials like rubbers and polymers, particularly under large deformations of up to 700%. It is frequently used for the analysis of components such as O-rings and seals.

Unlike the Neo-Hookean and Mooney-Rivlin models, which are expressed in terms of invariants, the Ogden model uses the principal stretch ratios $\lambda_j$

The strain energy density function is given by:

\(W = \sum_{i=1}^N \frac{\mu_i}{\alpha_i} \left( \lambda_1^{\alpha_i} + \lambda_2^{\alpha_i} + \lambda_3^{\alpha_i} - 3 \right)\) where:

• $\lambda_j$ (for $j=1,2,3$) are the principal stretch ratios.
• $\mu_i$​ and $\alpha_i$​ are empirically determined material constants.
• $N$ is the number of terms used to fit experimental data.

The Ogden model is widely regarded for its accuracy and flexibility in capturing large strain behaviour.