Mooney–Rivlin solid

In continuum mechanics, a Mooney–Rivlin solid[1][2] 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[3][4]

where and are empirically determined material constants, and and are the first and the second invariant of (the unimodular component of [5]):

where is the deformation gradient and . For an incompressible material, .

DerivationEdit

The Mooney–Rivlin model is a special case of the generalized Rivlin model (also called polynomial hyperelastic model[6]) 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 tensorsEdit

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 unimodular tensor   :

 

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 stretchesEdit

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 extensionEdit

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 tensionEdit

 
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 tensionEdit

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 shearEdit

A pure shear deformation can be achieved by applying stretches of the form [7]

 

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

 

Therefore

 

For a pure shear deformation

 

Therefore  .

Simple shearEdit

The deformation gradient for a simple shear deformation has the form[7]

 

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.

RubberEdit

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%.

[8]

Notes and referencesEdit

  1. ^ Mooney, M., 1940, A theory of large elastic deformation, Journal of Applied Physics, 11(9), pp. 582–592.
  2. ^ 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.
  3. ^ 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.
  4. ^ C. W. Macosko, 1994, Rheology: principles, measurement and applications, VCH Publishers, ISBN 1-56081-579-5.
  5. ^ Unimodularity in this context means  .
  6. ^ Bower, Allan (2009). Applied Mechanics of Solids. CRC Press. ISBN 1-4398-0247-5. Retrieved 2018-04-19.
  7. ^ a b Ogden, R. W., 1984, Nonlinear elastic deformations, Dover
  8. ^ Hamza, Muhsin; Alwan, Hassan (2010). "Hyperelastic Constitutive Modeling of Rubber and Rubber-Like Materials under Finite Strain". Eng.& Tech. Journal. 28 (13): 2560–2575.

See alsoEdit