# 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 for incompressible rubbers[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

- ↑ 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. - ↑ 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. - ↑ 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.