The Mooney–Rivlin model is a special case of the generalized Rivlin 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 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 The above equation is often written using the unimodulartensor : 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
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 differences can be calculated as:
Simple tension
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 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 fitsexperimental 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%.
