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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 $\mathbf{B}$:

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

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

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

where: $\overline{I_p} = J^{-2/3}I_p$.

and C10, C01, and d are constants.

(Explanatory gloss for students of applied mathematics, physics, or other disciplines: 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 a1 is written I1, the next coefficient a2 is written I2, and the determinant a3 would be written I3.)

If $C_1= \frac {1} {2} G$ (where G is the shear modulus) and C2 = 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 elongation

Neo-Hookean solid model is an extension of Hooke's law for the case of large deformations. The model of neo-Hookean solid is usable for plastics and rubber-like substances.

The response of a neo-Hookean material, or hyperelastic material, to an applied stress differs from that of a linear elastic material. While a linear elastic material has a linear relationship between applied stress and strain, a neo-Hookean material does not. A hyperelastic material will initially be linear, but at a certain point, the stress-strain curve will plateau due to the release of energy as heat while straining the material. Then, at another point, the elastic modulus of the material will increase again.

This hyperelasticity, or rubber elasticity, is often observed in polymers. Cross-linked polymers will act in this way because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain point the polymer chains will be stretched to the maximum point that the covalent cross links will allow, and this will cause a dramatic increase in the elastic modulus of the material. One can also use thermodynamics to explain the elasticity of polymers.

## Neo-Hookean Solid Model

The model of neo-Hookean solid assumes that the extra stresses due to deformation are proportional to Finger tensor:

$\mathbf {T} = -p \mathbf {I} + G \mathbf {B}$,

where $\mathbf {T}$ - stress tensor, p - pressure, $\mathbf {I}$ - is the unity tensor, G is a constant equal to shear modulus, $\mathbf {B}$ is the Finger tensor.

The strain energy for this model is:

$W = \frac{1}{2} G I_B$,

where W is potential energy and $I_B=\mathrm{tr}(\mathbf{B})$ is the trace (or first invariant) of Finger tensor $\mathbf {B}$.

Usually the model is used for incompressible media.

The model was proposed by Ronald Rivlin in 1948.

## Uni-axial 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.

## 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