Advanced elasticity/Yeoh material

The Yeoh w:hyperelastic material model[1] is a phenomenological model for the deformation of nearly w:incompressible, w:nonlinear w:elastic materials such as w:rubber. The model is based on Ronald Rivlin's observation that the elastic properties of rubber may be described using a w:strain energy density function which is a power series in the strain invariants .[2] The Yeoh model for incompressible rubber is a function only of . For compressible rubbers, an dependence on is added on. Since a polynomial form of the strain energy density function is used but all the three invariants of the left Cauchy-Green deformation tensor are not, the Yeoh model is also called the reduced polynomial model.

Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data from PolymerFEM.com

Yeoh model for incompressible rubbers

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The original model proposed by Yeoh had a cubic form with only   dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as

 

where   are material constants. The quantity   can be interpreted as the initial w:shear modulus.

Today a slightly more generalized version of the Yeoh model is used.[3] This model includes   terms and is written as

 

When   the Yeoh model reduces to the neo-Hookean model for incompressible materials.

The Cauchy stress for the incompressible Yeoh model is given by

 

Uniaxial extension

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For uniaxial extension in the  -direction, the principal stretches are  . From incompressibility  . Hence  . Therefore,

 

The left Cauchy-Green deformation tensor can then be expressed as

 

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

 

Since  , we have

 

Therefore,

 

The engineering strain is  . The engineering stress is

 

Equibiaxial extension

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For equibiaxial extension in the   and   directions, the principal stretches are  . From incompressibility  . Hence  . Therefore,

 

The left Cauchy-Green deformation tensor can then be expressed as

 

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

 

Since  , we have

 

Therefore,

 

The engineering strain is  . The engineering stress is

 

Planar extension

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Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the   directions with the   direction constrained, the principal stretches are  . From incompressibility  . Hence  . Therefore,

 

The left Cauchy-Green deformation tensor can then be expressed as

 

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

 

Since  , we have

 

Therefore,

 

The engineering strain is  . The engineering stress is

 

Yeoh model for compressible rubbers

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A version of the Yeoh model that includes   dependence is used for compressible rubbers. The strain energy density function for this model is written as

 

where  , and   are material constants. The quantity   is interpreted as half the initial shear modulus, while   is interpreted as half the initial bulk modulus.

When   the compressible Yeoh model reduces to the neo-Hookean model for compressible materials.

References

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  1. Yeoh, O. H., 1993, "Some forms of the strain energy function for rubber," Rubber Chemistry and technology, Volume 66, Issue 5, November 1993, Pages 754-771.
  2. Rivlin, R. S., 1948, "Some applications of elasticity theory to rubber engineering", in Collected Papers of R. S. Rivlin vol. 1 and 2, Springer, 1997.
  3. Selvadurai, A. P. S., 2006, "Deflections of a rubber membrane", Journal of the Mechanics and Physics of Solids, vol. 54, no. 6, pp. 1093-1119.