Yeoh hyperelastic model

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

The Yeoh hyperelastic material model ^[1] is a phenomenological model for the deformation of nearly incompressible, nonlinear ^{[ disambiguation needed ]} 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 .

Yeoh model for incompressible rubbers

Strain energy density function

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

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

Stress-deformation relations

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

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

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

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

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 incompressible materials.

History

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]

References

1. Yeoh, O. H. (November 1993). "Some forms of the strain energy function for rubber". Rubber Chemistry and Technology . 66 (5): 754–771. doi:10.5254/1.3538343.
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.
4. "Remembering Dr. Graham Johnson Lake (1935–2023)". Rubber Chemistry and Technology. 96 (4): G2–G3. 2023. doi:10.5254/rct-23.498080.
5. "Biographical Sketch". ACS Rubber Division. Retrieved 20 February 2024.
6. "Rubber Division names 3 for awards". Rubber and Plastics News. Crain. 27 October 2003. Retrieved 16 August 2022.

See also

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