A Mooney-Rivlin solid is a generalization of the w:Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of the w: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 w:deviatoric component of the w:Finger tensor:[1]

$I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2$,
$I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2$,
$I_3 = \lambda_1^2 \lambda_2^2 \lambda_3^2$,

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

If $C_1= \frac {1} {2} G$ (where G is the w:shear modulus) and $C_2=0$, we obtain a w: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 w:Melvin Mooney and w:Ronald Rivlin in two independent papers in 1952.

## Uniaxial extension

Comparison of experimental results (dots) and predictions for w:Hooke's law(1, blue line), w:Neo-Hookean solid(2, red line) and Mooney-Rivlin solid models(3, green line)

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 w: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 w: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

## Notes and References

1. The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor is usually written
$p_B (\lambda) = \lambda^3 - a_1 \, \lambda^2 + a_2 \, \lambda - a_3$
In this article, the trace $a_1$ is written $I_1$, the next coefficient $a_2$ is written $I_2$, and the determinant $a_3$ would be written $I_3$.