In continuum mechanics, a Mooney–Rivlin solid is a hyperelastic material model where the strain energy density function is a linear combination of two invariants of the left Cauchy–Green deformation tensor. The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948.

The strain energy density function for an incompressible Mooney–Rivlin material is

are the first and the second invariant (the unimodular):

is the deformation gradient.

Derivation

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

are material constants related to the volumetric response. For a compressible and we have

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

is the bulk modulus 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

For a compressible Mooney–Rivlin material,

Therefore, the Cauchy stress in a compressible Mooney–Rivlin material is given by

It can be shown, after some algebra, that the pressure is given by

The stress can then be expressed in the form

:

For an incompressible . Thus

the Cayley–Hamilton theorem implies

Hence, the Cauchy stress can be expressed as

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,

. we can write

Then the expressions for the Cauchy stress differences become

Uniaxial extension

. Then the true stress (Cauchy stress) differences can be calculated as:

Simple tension

. Then we can write

, we can write

and the engineering stress. Hence

If we define

then

. The Mooney–Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.

Equibiaxial tension

. 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

.

Simple shear

The deformation gradient for a simple shear deformation has the form

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

is the shear modulus.

Rubber

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%.

Notes and references

See also

• Finite strain theory

• Continuum mechanics