Yeoh hyperelastic model

The Yeoh hyperelastic material model^[1] is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber. The model is based on Ronald Rivlin's observation that the elastic properties of rubber may be described using a strain energy density function which is a power series in the strain invariants ${\displaystyle I_{1},I_{2},I_{3}}$ of the Cauchy-Green deformation tensors.^[2] The Yeoh model for incompressible rubber is a function only of ${\displaystyle I_{1}}$. For compressible rubbers, a 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.

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 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.

For consistency with linear elasticity the Yeoh model has to satisfy the condition

${\displaystyle 2{\cfrac {\partial W}{\partial I_{1}}}(3)=\mu ~~(i\neq j)}$

where ${\displaystyle \mu }$ is the shear modulus of the material. Now, at ${\displaystyle I_{1}=3(\lambda _{i}=\lambda _{j}=1)}$,

${\displaystyle {\cfrac {\partial W}{\partial I_{1}}}=C_{1}}$

Therefore, the consistency condition for the Yeoh model is

${\displaystyle 2C_{1}=\mu \,}$

The Cauchy stress for the incompressible Yeoh model is given by

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

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

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

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 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

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

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

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

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 incompressible materials.

The model is named after Oon Hock Yeoh. Yeoh completed his doctoral studies under Graham Lake at the University of London.^[4] Yeoh held research positions at Freudenberg-NOK, MRPRA (England), Rubber Research Institute of Malaysia (Malaysia), University of Akron, GenCorp Research, and Lord Corporation.^[5] Yeoh won the 2004 Melvin Mooney Distinguished Technology Award from the ACS Rubber Division.^[6]

• Hyperelastic material
• Strain energy density function
• Mooney-Rivlin solid
• Finite strain theory
• Stress measures