Derivation
The Mooney–Rivlin model is a special case of the generalized Rivlin model (also called polynomial hyperelastic model) which has the form

with
where
are material constants related to the distortional response and
are material constants related to the volumetric response. For a compressible Mooney–Rivlin material
and we have

If
we obtain a neo-Hookean solid, a special case of a Mooney–Rivlin solid.
For consistency with linear elasticity in the limit of small strains, it is necessary that

where
is the bulk modulus and
is the shear modulus.
Cauchy stress in terms of strain invariants and deformation tensors
The Cauchy stress in a compressible hyperelastic material with a stress free reference configuration is given by
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}\left({\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}+{\bar {I}}_{1}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}\right){\boldsymbol {B}}-{\cfrac {1}{J^{4/3}}}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+\left[{\cfrac {\partial {W}}{\partial J}}-{\cfrac {2}{3J}}\left({\bar {I}}_{1}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}\right)\right]~{\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02006f31ec08f5ff11d32479fcc80ae15ffb0ea)
For a compressible Mooney–Rivlin material,

Therefore, the Cauchy stress in a compressible Mooney–Rivlin material is given by
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}\left(C_{1}+{\bar {I}}_{1}~C_{2}\right){\boldsymbol {B}}-{\cfrac {1}{J^{4/3}}}~C_{2}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+\left[2D_{1}(J-1)-{\cfrac {2}{3J}}\left(C_{1}{\bar {I}}_{1}+2C_{2}{\bar {I}}_{2}~\right)\right]{\boldsymbol {I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de809e7ffcb8f8f629278882f7f4c6108b144979)
It can be shown, after some algebra, that the pressure is given by

The stress can then be expressed in the form
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {1}{J}}\left[-p~{\boldsymbol {I}}+{\cfrac {2}{J^{2/3}}}\left(C_{1}+{\bar {I}}_{1}~C_{2}\right){\boldsymbol {B}}-{\cfrac {2}{J^{4/3}}}~C_{2}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}-{\cfrac {2}{3}}\left(C_{1}\,{\bar {I}}_{1}+2C_{2}\,{\bar {I}}_{2}\right){\boldsymbol {I}}\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5323a3351d65641f91631a027d9536c483fe2794)
The above equation is often written using the unimodular tensor
:
![{\displaystyle {\boldsymbol {\sigma }}={\cfrac {1}{J}}\left[-p~{\boldsymbol {I}}+2\left(C_{1}+{\bar {I}}_{1}~C_{2}\right){\bar {\boldsymbol {B}}}-2~C_{2}~{\bar {\boldsymbol {B}}}\cdot {\bar {\boldsymbol {B}}}-{\cfrac {2}{3}}\left(C_{1}\,{\bar {I}}_{1}+2C_{2}\,{\bar {I}}_{2}\right){\boldsymbol {I}}\right]\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1c604deb0a6e20f4b6e7f857a5b50d36994a8b)
For an incompressible Mooney–Rivlin material with
there holds
and
. Thus

Since
the Cayley–Hamilton theorem implies

Hence, the Cauchy stress can be expressed as

where
Cauchy stress in terms of principal stretches
In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by

For an incompressible Mooney-Rivlin material,

Therefore,

Since
. we can write

Then the expressions for the Cauchy stress differences become

Uniaxial extension
For the case of an incompressible Mooney–Rivlin material under uniaxial elongation,
and
. Then the true stress (Cauchy stress) differences can be calculated as:

Simple tension
Comparison of experimental results (dots) and predictions for
Hooke's law(1, blue line),
neo-Hookean solid(2, red line) and Mooney–Rivlin solid models(3, green line)
In the case of simple tension,
. Then we can write

In alternative notation, where the Cauchy stress is written as
and the stretch as
, we can write

and the engineering stress (force per unit reference area) for an incompressible Mooney–Rivlin material under simple tension can be calculated using
. Hence

If we define

then

The slope of the
versus
line gives the value of
while the intercept with the
axis gives the value of
. The Mooney–Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.
Equibiaxial tension
In the case of equibiaxial tension, the principal stretches are
. If, in addition, the material is incompressible then
. The Cauchy stress differences may therefore be expressed as

The equations for equibiaxial tension are equivalent to those governing uniaxial compression.
Pure shear
A pure shear deformation can be achieved by applying stretches of the form

The Cauchy stress differences for pure shear may therefore be expressed as

Therefore

For a pure shear deformation

Therefore
.
Simple shear
The deformation gradient for a simple shear deformation has the form

where
are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by

In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as

Therefore,

The Cauchy stress is given by

For consistency with linear elasticity, clearly
where
is the shear modulus.
Rubber
Elastic response of rubber-like materials are often modeled based on the Mooney–Rivlin model. The constants
are determined by fitting the predicted stress from the above equations to the experimental data. The recommended tests are uniaxial tension, equibiaxial compression, equibiaxial tension, uniaxial compression, and for shear, planar tension and planar compression. The two parameter Mooney–Rivlin model is usually valid for strains less than 100%.