In continuum mechanics, a MooneyâÂÂRivlin solid is a hyperelastic material model where the strain energy density function is a linear combination of two invariants of the left CauchyâÂÂGreen 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 MooneyâÂÂRivlin material is
where and are empirically determined material constants, and and are the first and the second invariant of (the unimodular component of ):
where is the deformation gradient and . For an incompressible material, .
The MooneyâÂÂRivlin 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 MooneyâÂÂRivlin material and we have
If we obtain a neo-Hookean solid, a special case of a MooneyâÂÂRivlin 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.
The Cauchy stress in a compressible hyperelastic material with a stress free reference configuration is given by
For a compressible MooneyâÂÂRivlin material,
Therefore, the Cauchy stress in a compressible MooneyâÂÂRivlin 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
The above equation is often written using the unimodular tensor :
For an incompressible MooneyâÂÂRivlin material with there holds and . Thus
Since the CayleyâÂÂHamilton theorem implies
Hence, the Cauchy stress can be expressed as
where
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
For the case of an incompressible MooneyâÂÂRivlin material under uniaxial elongation, and . Then the true stress (Cauchy stress) differences can be calculated as:
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 MooneyâÂÂRivlin 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 MooneyâÂÂRivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.
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.
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 .
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.
Elastic response of rubber-like materials are often modeled based on the MooneyâÂÂRivlin 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 MooneyâÂÂRivlin model is usually valid for strains less than 100%.