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 ${\displaystyle I_{1},I_{2},I_{3}}$.[2] The Yeoh model for incompressible rubber is a function only of ${\displaystyle I_{1}}$. For compressible rubbers, an dependence on ${\displaystyle I_{3}}$ 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

The original model proposed by Yeoh had a cubic form with only ${\displaystyle I_{1}}$  dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as

${\displaystyle W=\sum _{i=1}^{3}C_{i}~(I_{1}-3)^{i}}$

where ${\displaystyle C_{i}}$  are material constants. The quantity ${\displaystyle 2C_{1}}$  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 ${\displaystyle n}$  terms and is written as

${\displaystyle W=\sum _{i=1}^{n}C_{i}~(I_{1}-3)^{i}~.}$

When ${\displaystyle n=1}$  the Yeoh model reduces to the neo-Hookean model for incompressible materials.

The Cauchy stress for the incompressible Yeoh model is given by

${\displaystyle {\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {1}}}+2~{\cfrac {\partial W}{\partial I_{1}}}~{\boldsymbol {B}}~;~~{\cfrac {\partial W}{\partial I_{1}}}=\sum _{i=1}^{n}i~C_{i}~(I_{1}-3)^{i-1}~.}$

### Uniaxial extension

For uniaxial extension in the ${\displaystyle \mathbf {n} _{1}}$ -direction, the principal stretches are ${\displaystyle \lambda _{1}=\lambda ,~\lambda _{2}=\lambda _{3}}$ . From incompressibility ${\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1}$ . Hence ${\displaystyle \lambda _{2}^{2}=\lambda _{3}^{2}=1/\lambda }$ . Therefore,

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {2}{\lambda }}~.}$

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

${\displaystyle {\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda }}~(\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3})~.}$

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

${\displaystyle \sigma _{11}=-p+2~\lambda ^{2}~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{22}=-p+{\cfrac {2}{\lambda }}~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{33}~.}$

Since ${\displaystyle \sigma _{22}=\sigma _{33}=0}$ , we have

${\displaystyle p={\cfrac {2}{\lambda }}~{\cfrac {\partial W}{\partial I_{1}}}~.}$

Therefore,

${\displaystyle \sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.}$

The engineering strain is ${\displaystyle \lambda -1\,}$ . The engineering stress is

${\displaystyle T_{11}=\sigma _{11}/\lambda =2~\left(\lambda -{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.}$

### Equibiaxial extension

For equibiaxial extension in the ${\displaystyle \mathbf {n} _{1}}$  and ${\displaystyle \mathbf {n} _{2}}$  directions, the principal stretches are ${\displaystyle \lambda _{1}=\lambda _{2}=\lambda \,}$ . From incompressibility ${\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1}$ . Hence ${\displaystyle \lambda _{3}=1/\lambda ^{2}\,}$ . Therefore,

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=2~\lambda ^{2}+{\cfrac {1}{\lambda ^{4}}}~.}$

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

${\displaystyle {\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda ^{2}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\cfrac {1}{\lambda ^{4}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.}$

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

${\displaystyle \sigma _{11}=-p+2~\lambda ^{2}~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{22}~;~~\sigma _{33}=-p+{\cfrac {2}{\lambda ^{4}}}~{\cfrac {\partial W}{\partial I_{1}}}~.}$

Since ${\displaystyle \sigma _{33}=0}$ , we have

${\displaystyle p={\cfrac {2}{\lambda ^{4}}}~{\cfrac {\partial W}{\partial I_{1}}}~.}$

Therefore,

${\displaystyle \sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{4}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{22}~.}$

The engineering strain is ${\displaystyle \lambda -1\,}$ . The engineering stress is

${\displaystyle T_{11}={\cfrac {\sigma _{11}}{\lambda }}=2~\left(\lambda -{\cfrac {1}{\lambda ^{5}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}=T_{22}~.}$

### Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the ${\displaystyle \mathbf {n} _{1}}$  directions with the ${\displaystyle \mathbf {n} _{3}}$  direction constrained, the principal stretches are ${\displaystyle \lambda _{1}=\lambda ,~\lambda _{3}=1}$ . From incompressibility ${\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1}$ . Hence ${\displaystyle \lambda _{2}=1/\lambda \,}$ . Therefore,

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {1}{\lambda ^{2}}}+1~.}$

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

${\displaystyle {\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda ^{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.}$

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

${\displaystyle \sigma _{11}=-p+2~\lambda ^{2}~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{11}=-p+{\cfrac {2}{\lambda ^{2}}}~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{33}=-p+2~{\cfrac {\partial W}{\partial I_{1}}}~.}$

Since ${\displaystyle \sigma _{22}=0}$ , we have

${\displaystyle p={\cfrac {2}{\lambda ^{2}}}~{\cfrac {\partial W}{\partial I_{1}}}~.}$

Therefore,

${\displaystyle \sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{22}=0~;~~\sigma _{33}=2~\left(1-{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.}$

The engineering strain is ${\displaystyle \lambda -1\,}$ . The engineering stress is

${\displaystyle T_{11}={\cfrac {\sigma _{11}}{\lambda }}=2~\left(\lambda -{\cfrac {1}{\lambda ^{3}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.}$

## Yeoh model for compressible rubbers

A version of the Yeoh model that includes ${\displaystyle I_{3}=J^{2}}$  dependence is used for compressible rubbers. The strain energy density function for this model is written as

${\displaystyle W=\sum _{i=1}^{n}C_{i0}~({\bar {I}}_{1}-3)^{i}+\sum _{k=1}^{n}C_{k1}~(J-1)^{2k}}$

where ${\displaystyle {\bar {I}}_{1}=J^{-2/3}~I_{1}}$ , and ${\displaystyle C_{i0},C_{k1}}$  are material constants. The quantity ${\displaystyle C_{10}}$  is interpreted as half the initial shear modulus, while ${\displaystyle C_{11}}$  is interpreted as half the initial bulk modulus.

When ${\displaystyle n=1}$  the compressible Yeoh model reduces to the neo-Hookean model for compressible materials.

## References

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.