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 $mathbf\{B\}$:

:$W\; =\; C\_\{10\}\; (overline\{I\}\_1-3)\; +\; C\_\{01\}\; (overline\{I\}\_2-3)+\; frac\{1\}\{d\}(J\_\{el\}-1)^2$,

where $overline\{I\}\_1$ and $overline\{I\}\_2$ are the first and the second invariant of deviatoric component of the Finger tensor: [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 $I\_3$.]

:$I\_1\; =\; lambda\_1^2\; +\; lambda\_2\; ^2+\; lambda\_3\; ^2$,

:$I\_2\; =\; lambda\_1^2\; lambda\_2^2\; +\; lambda\_2^2\; lambda\_3^2\; +\; lambda\_3^2\; lambda\_1^2$,

:$I\_3\; =\; lambda\_1^2\; lambda\_2^2\; lambda\_3^2$,

where: $overline\{I\_p\}\; =\; J^\{-2/3\}I\_p$.

and $C\_\{10\}$, $C\_\{01\}$, and $d$ 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:

:$mathbf\{T\}\; =\; -pmathbf\{I\}\; +2C\_1\; mathbf\{B\}\; +2C\_2\; mathbf\{B\}^\{-1\}$

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:

:$T\_\{11\}\; =\; left(2C\_1\; -\; frac\; \{2C\_2\}\; \{alpha\_1\}\; ight)\; left(\; alpha\_1^2\; -\; alpha\_1^\{-1\}\; ight)$

and engineering stress can be calculated as:

:$T\_\{11eng\}\; =\; left(2C\_1\; -\; frac\; \{2C\_2\}\; \{alpha\_1\}\; ight)\; left(\; alpha\_1\; -\; alpha\_1^\{-2\}\; ight)$

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.

ource

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

Notes and References