Advanced elasticity/Mooney-Rivlin material

A Mooney-Rivlin solid is a generalization of the Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of the Finger tensor $$\mathbf{B}$$:


 * $$W = C_{1} (\overline{I}_1-3) + C_{2} (\overline{I}_2-3)$$,

where $$\overline{I}_1$$ and $$\overline{I}_2$$ are the first and the second invariant of deviatoric component of the Finger tensor:


 * $$I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2$$,


 * $$I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2$$,


 * $$I_3 = \lambda_1^2 \lambda_2^2 \lambda_3^2$$,

where: $$C_{1}$$ and $$C_{2}$$ are constants.

If $$C_1= \frac {1} {2} G$$ (where G is the shear modulus) and $$C_2=0$$, we obtain a Neo-Hookean solid, a special case of a Mooney-Rivlin solid.

The stress tensor $$\mathbf{T}$$ depends upon Finger tensor $$\mathbf{B}$$ by the following equation:


 * $$\mathbf{T} = -p\mathbf{I} +2C_1 \mathbf{B} +2C_2 \mathbf{B}^{-1} $$

The model was proposed by Melvin Mooney and Ronald Rivlin in two independent papers in 1952.

Uniaxial extension
For the case of uniaxial elongation, true stress can be calculated as:


 * $$T_{11} = \left(2C_1 + \frac {2C_2} {\alpha_1} \right) \left( \alpha_1^2 - \alpha_1^{-1} \right)$$

and engineering stress can be calculated as:


 * $$T_{11eng} = \left(2C_1 + \frac {2C_2} {\alpha_1} \right) \left( \alpha_1 - \alpha_1^{-2} \right)$$

The Mooney-Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.

Rubber
Elastic response of rubber-like materials are often modelled based on the Mooney-Rivlin model.

Source

 * C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5