Advanced elasticity/Mooney-Rivlin material

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

↑ 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 $I_{3}$ .

← Neo-Hookean material

Advanced elasticity

Ogden material →

[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 $I_{3}$ .

[1]