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 $𝐁$:

$W=C_1({\bar{I}}_1-3)+C_2({\bar{I}}_2-3)$,

where ${\bar{I}}_1$ and ${\bar{I}}_2$ are the first and the second invariant of w:deviatoric component of the w: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: $C_1$ 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 $𝐓$ depends upon Finger tensor $𝐁$ by the following equation:

$𝐓=-p𝐈+2C_1𝐁+2C_2{𝐁}^{-1}$

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:

$T_{11}=(2C_1+\frac{2C_2}{{\alpha }_1})({\alpha }_1^2-{\alpha }_1^{-1})$

and w:engineering stress can be calculated as:

$T_{11eng}=(2C_1+\frac{2C_2}{{\alpha }_1})({\alpha }_1-{\alpha }_1^{-2})$

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, Template:ISBN