# Advanced elasticity/Yeoh material

The Yeoh w:hyperelastic material model 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 $I_{1},I_{2},I_{3}$. The Yeoh model for incompressible rubber is a function only of $I_{1}$. For compressible rubbers, an dependence on $I_{3}$ 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 $I_{1}$ dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as
    $W=\sum _{i=1}^{3}C_{i}~(I_{1}-3)^{i}$
where $C_{i}$ are material constants. The quantity $2C_{1}$ can be interpreted as the initial w:shear modulus.
Today a slightly more generalized version of the Yeoh model is used. This model includes $n$ terms and is written as
    $W=\sum _{i=1}^{n}C_{i}~(I_{1}-3)^{i}~.$
When $n=1$ the Yeoh model reduces to the neo-Hookean model for incompressible materials.
The Cauchy stress for the incompressible Yeoh model is given by
    ${\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {1}}}+2~{\cfrac {\partial W}{\partial I_{1}}}~{\boldsymbol {B}}~;~~{\cfrac {\partial W}{\partial I_{1}}}=\sum _{i=1}^{n}i~C_{i}~(I_{1}-3)^{i-1}~.$

### Uniaxial extension

For uniaxial extension in the $\mathbf {n} _{1}$-direction, the principal stretches are $\lambda _{1}=\lambda ,~\lambda _{2}=\lambda _{3}$. From incompressibility $\lambda _{1}~\lambda _{2}~\lambda _{3}=1$. Hence $\lambda _{2}^{2}=\lambda _{3}^{2}=1/\lambda$. Therefore,
    $I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {2}{\lambda }}~.$
The left Cauchy-Green deformation tensor can then be expressed as
    ${\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda }}~(\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3})~.$
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
    $\sigma _{11}=-p+2~\lambda ^{2}~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{22}=-p+{\cfrac {2}{\lambda }}~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{33}~.$
Since $\sigma _{22}=\sigma _{33}=0$, we have
    $p={\cfrac {2}{\lambda }}~{\cfrac {\partial W}{\partial I_{1}}}~.$
Therefore,
    $\sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.$
The engineering strain is $\lambda -1\,$. The engineering stress is
    $T_{11}=\sigma _{11}/\lambda =2~\left(\lambda -{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.$

### Equibiaxial extension

For equibiaxial extension in the $\mathbf {n} _{1}$ and $\mathbf {n} _{2}$ directions, the principal stretches are $\lambda _{1}=\lambda _{2}=\lambda \,$. From incompressibility $\lambda _{1}~\lambda _{2}~\lambda _{3}=1$. Hence $\lambda _{3}=1/\lambda ^{2}\,$.
Therefore,
    $I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=2~\lambda ^{2}+{\cfrac {1}{\lambda ^{4}}}~.$
The left Cauchy-Green deformation tensor can then be expressed as
    ${\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda ^{2}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\cfrac {1}{\lambda ^{4}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.$
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
    $\sigma _{11}=-p+2~\lambda ^{2}~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{22}~;~~\sigma _{33}=-p+{\cfrac {2}{\lambda ^{4}}}~{\cfrac {\partial W}{\partial I_{1}}}~.$
Since $\sigma _{33}=0$, we have
    $p={\cfrac {2}{\lambda ^{4}}}~{\cfrac {\partial W}{\partial I_{1}}}~.$
Therefore,
    $\sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{4}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{22}~.$
The engineering strain is $\lambda -1\,$. The engineering stress is
    $T_{11}={\cfrac {\sigma _{11}}{\lambda }}=2~\left(\lambda -{\cfrac {1}{\lambda ^{5}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}=T_{22}~.$

### Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the $\mathbf {n} _{1}$ directions with the $\mathbf {n} _{3}$ direction constrained, the principal stretches are $\lambda _{1}=\lambda ,~\lambda _{3}=1$. From incompressibility $\lambda _{1}~\lambda _{2}~\lambda _{3}=1$. Hence $\lambda _{2}=1/\lambda \,$.
Therefore,
    $I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {1}{\lambda ^{2}}}+1~.$
The left Cauchy-Green deformation tensor can then be expressed as
    ${\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda ^{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.$
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
    $\sigma _{11}=-p+2~\lambda ^{2}~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{11}=-p+{\cfrac {2}{\lambda ^{2}}}~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{33}=-p+2~{\cfrac {\partial W}{\partial I_{1}}}~.$
Since $\sigma _{22}=0$, we have
    $p={\cfrac {2}{\lambda ^{2}}}~{\cfrac {\partial W}{\partial I_{1}}}~.$
Therefore,
    $\sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~;~~\sigma _{22}=0~;~~\sigma _{33}=2~\left(1-{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.$
The engineering strain is $\lambda -1\,$. The engineering stress is
    $T_{11}={\cfrac {\sigma _{11}}{\lambda }}=2~\left(\lambda -{\cfrac {1}{\lambda ^{3}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.$

## Yeoh model for compressible rubbers

A version of the Yeoh model that includes $I_{3}=J^{2}$ dependence is used for compressible rubbers. The strain energy density function for this model is written as
    $W=\sum _{i=1}^{n}C_{i0}~({\bar {I}}_{1}-3)^{i}+\sum _{k=1}^{n}C_{k1}~(J-1)^{2k}$
where ${\bar {I}}_{1}=J^{-2/3}~I_{1}$, and $C_{i0},C_{k1}$ are material constants. The quantity $C_{10}$ is interpreted as half the initial shear modulus, while $C_{11}$ is interpreted as half the initial bulk modulus.
When $n=1$ the compressible Yeoh model reduces to the neo-Hookean model for compressible materials.