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# Mooney-Rivlin solid

In continuum mechanics, 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 Finger tensor $mathbf\left\{B\right\}$:

$W = C_\left\{10\right\} \left(overline\left\{I\right\}_1-3\right) + C_\left\{01\right\} \left(overline\left\{I\right\}_2-3\right)+ frac\left\{1\right\}\left\{d\right\}\left(J_\left\{el\right\}-1\right)^2$,

where $overline\left\{I\right\}_1$ and $overline\left\{I\right\}_2$ 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: $overline\left\{I_p\right\} = J^\left\{-2/3\right\}I_p$.

and $C_\left\{10\right\}$, $C_\left\{01\right\}$, and $d$ are constants.

If $C_1= frac \left\{1\right\} \left\{2\right\} G$ (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 $mathbf\left\{T\right\}$ depends upon Finger tensor $mathbf\left\{B\right\}$ by the following equation:

$mathbf\left\{T\right\} = -pmathbf\left\{I\right\} +2C_1 mathbf\left\{B\right\} +2C_2 mathbf\left\{B\right\}^\left\{-1\right\}$

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:

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

and engineering stress can be calculated as:

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

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

## Brain tissues

Elastic response of soft tissues like that in the brain is 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

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