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 ${\displaystyle \mathbf {B} }$:

${\displaystyle W=C_{1}({\overline {I}}_{1}-3)+C_{2}({\overline {I}}_{2}-3)}$,

where ${\displaystyle {\overline {I}}_{1}}$ and ${\displaystyle {\overline {I}}_{2}}$ are the first and the second invariant of w:deviatoric component of the w:Finger tensor:[1]

$\displaystyle I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2$ ,
${\displaystyle I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}}$,
${\displaystyle I_{3}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}}$,

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

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

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

${\displaystyle \mathbf {T} =-p\mathbf {I} +2C_{1}\mathbf {B} +2C_{2}\mathbf {B} ^{-1}}$

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:

${\displaystyle T_{11}=\left(2C_{1}+{\frac {2C_{2}}{\alpha _{1}}}\right)\left(\alpha _{1}^{2}-\alpha _{1}^{-1}\right)}$

and w:engineering stress can be calculated as:

${\displaystyle T_{11eng}=\left(2C_{1}+{\frac {2C_{2}}{\alpha _{1}}}\right)\left(\alpha _{1}-\alpha _{1}^{-2}\right)}$

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
${\displaystyle p_{B}(\lambda )=\lambda ^{3}-a_{1}\,\lambda ^{2}+a_{2}\,\lambda -a_{3}}$
In this article, the trace ${\displaystyle a_{1}}$ is written ${\displaystyle I_{1}}$, the next coefficient ${\displaystyle a_{2}}$ is written ${\displaystyle I_{2}}$, and the determinant ${\displaystyle a_{3}}$ would be written ${\displaystyle I_{3}}$.