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.
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.
Cauchy stress in terms of strain invariants and deformation tensors
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 as
For an incompressible Mooney–Rivlin material with
Note that if then
Then, from the Cayley-Hamilton theorem,
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,
Since . we can write
Then the expressions for the Cauchy stress differences become
For the case of an incompressible Mooney–Rivlin 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 Mooney–Rivlin material under simple tension can be calculated using . Hence
If we define
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.
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
For a pure shear deformation
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
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 Mooney—Rivlin model. The constants are determined by the fitting predicted stress from the above equations to 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
- Mooney, M., 1940, A theory of large elastic deformation, Journal of Applied Physics, 11(9), pp. 582-592.
- Rivlin, R. S., 1948, Large elastic deformations of isotropic materials. IV. Further developments of the general theory, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), pp. 379-397.
- Boulanger, P. and Hayes, M. A., 2001, Finite amplitude waves in Mooney–Rivlin and Hadamard materials, in Topics in Finite Elasticity, ed. M. A Hayes and G. Soccomandi, International Center for Mechanical Sciences.
- C. W. Macosko, 1994, Rheology: principles, measurement and applications, VCH Publishers, ISBN 1-56081-579-5.
- The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor (also called the left Cauchy–Green deformation tensor) is usually written
- Bower, Allan (2009). Applied Mechanics of Solids. CRC Press. ISBN 1-4398-0247-5. Retrieved January 2010.
- Ogden, R. W., 1984, Nonlinear elastic deformations, Dover