Phenomenological model of elastic materials
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 I 1 , I 2 , I 3 {\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 I 1 {\displaystyle I_{1}} . For compressible rubbers, a dependence on I 3 {\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 [ edit ] Strain energy density function [ edit ] The original model proposed by Yeoh had a cubic form with only I 1 {\displaystyle I_{1}} dependence and is applicable to purely incompressible materials. The strain energy density for this model is written as
W = ∑ i = 1 3 C i ( I 1 − 3 ) i {\displaystyle W=\sum _{i=1}^{3}C_{i}~(I_{1}-3)^{i}} where C i {\displaystyle C_{i}} are material constants. The quantity 2 C 1 {\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 n {\displaystyle n} terms and is written as
W = ∑ i = 1 n C i ( I 1 − 3 ) i . {\displaystyle W=\sum _{i=1}^{n}C_{i}~(I_{1}-3)^{i}~.} When n = 1 {\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
2 ∂ W ∂ I 1 ( 3 ) = μ ( i ≠ j ) {\displaystyle 2{\cfrac {\partial W}{\partial I_{1}}}(3)=\mu ~~(i\neq j)} where μ {\displaystyle \mu } is the shear modulus of the material. Now, at I 1 = 3 ( λ i = λ j = 1 ) {\displaystyle I_{1}=3(\lambda _{i}=\lambda _{j}=1)} ,
∂ W ∂ I 1 = C 1 {\displaystyle {\cfrac {\partial W}{\partial I_{1}}}=C_{1}} Therefore, the consistency condition for the Yeoh model is
2 C 1 = μ {\displaystyle 2C_{1}=\mu \,} Stress-deformation relations [ edit ] The Cauchy stress for the incompressible Yeoh model is given by
σ = − p 1 + 2 ∂ W ∂ I 1 B ; ∂ W ∂ I 1 = ∑ i = 1 n i C i ( I 1 − 3 ) i − 1 . {\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 [ edit ] For uniaxial extension in the n 1 {\displaystyle \mathbf {n} _{1}} -direction, the principal stretches are λ 1 = λ , λ 2 = λ 3 {\displaystyle \lambda _{1}=\lambda ,~\lambda _{2}=\lambda _{3}} . From incompressibility λ 1 λ 2 λ 3 = 1 {\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1} . Hence λ 2 2 = λ 3 2 = 1 / λ {\displaystyle \lambda _{2}^{2}=\lambda _{3}^{2}=1/\lambda } . Therefore,
I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 2 λ . {\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
B = λ 2 n 1 ⊗ n 1 + 1 λ ( n 2 ⊗ n 2 + n 3 ⊗ n 3 ) . {\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
σ 11 = − p + 2 λ 2 ∂ W ∂ I 1 ; σ 22 = − p + 2 λ ∂ W ∂ I 1 = σ 33 . {\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 σ 22 = σ 33 = 0 {\displaystyle \sigma _{22}=\sigma _{33}=0} , we have
p = 2 λ ∂ W ∂ I 1 . {\displaystyle p={\cfrac {2}{\lambda }}~{\cfrac {\partial W}{\partial I_{1}}}~.} Therefore,
σ 11 = 2 ( λ 2 − 1 λ ) ∂ W ∂ I 1 . {\displaystyle \sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.} The engineering strain is λ − 1 {\displaystyle \lambda -1\,} . The engineering stress is
T 11 = σ 11 / λ = 2 ( λ − 1 λ 2 ) ∂ W ∂ I 1 . {\displaystyle T_{11}=\sigma _{11}/\lambda =2~\left(\lambda -{\cfrac {1}{\lambda ^{2}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}~.} Equibiaxial extension [ edit ] For equibiaxial extension in the n 1 {\displaystyle \mathbf {n} _{1}} and n 2 {\displaystyle \mathbf {n} _{2}} directions, the principal stretches are λ 1 = λ 2 = λ {\displaystyle \lambda _{1}=\lambda _{2}=\lambda \,} . From incompressibility λ 1 λ 2 λ 3 = 1 {\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1} . Hence λ 3 = 1 / λ 2 {\displaystyle \lambda _{3}=1/\lambda ^{2}\,} . Therefore,
I 1 = λ 1 2 + λ 2 2 + λ 3 2 = 2 λ 2 + 1 λ 4 . {\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
B = λ 2 n 1 ⊗ n 1 + λ 2 n 2 ⊗ n 2 + 1 λ 4 n 3 ⊗ n 3 . {\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
σ 11 = − p + 2 λ 2 ∂ W ∂ I 1 = σ 22 ; σ 33 = − p + 2 λ 4 ∂ W ∂ I 1 . {\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 σ 33 = 0 {\displaystyle \sigma _{33}=0} , we have
p = 2 λ 4 ∂ W ∂ I 1 . {\displaystyle p={\cfrac {2}{\lambda ^{4}}}~{\cfrac {\partial W}{\partial I_{1}}}~.} Therefore,
σ 11 = 2 ( λ 2 − 1 λ 4 ) ∂ W ∂ I 1 = σ 22 . {\displaystyle \sigma _{11}=2~\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{4}}}\right)~{\cfrac {\partial W}{\partial I_{1}}}=\sigma _{22}~.} The engineering strain is λ − 1 {\displaystyle \lambda -1\,} . The engineering stress is
T 11 = σ 11 λ = 2 ( λ − 1 λ 5 ) ∂ W ∂ I 1 = T 22 . {\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 [ edit ] Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the n 1 {\displaystyle \mathbf {n} _{1}} directions with the n 3 {\displaystyle \mathbf {n} _{3}} direction constrained, the principal stretches are λ 1 = λ , λ 3 = 1 {\displaystyle \lambda _{1}=\lambda ,~\lambda _{3}=1} . From incompressibility λ 1 λ 2 λ 3 = 1 {\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1} . Hence λ 2 = 1 / λ {\displaystyle \lambda _{2}=1/\lambda \,} . Therefore,
I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 1 λ 2 + 1 . {\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
B = λ 2 n 1 ⊗ n 1 + 1 λ 2 n 2 ⊗ n 2 + n 3 ⊗ n 3 . {\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
σ 11 = − p + 2 λ 2 ∂ W ∂ I 1 ; σ 22 = − p + 2 λ 2 ∂ W ∂ I 1 ; σ 33 = − p + 2 ∂ W ∂ I 1 . {\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 σ 22 = 0 {\displaystyle \sigma _{22}=0} , we have
p = 2 λ 2 ∂ W ∂ I 1 . {\displaystyle p={\cfrac {2}{\lambda ^{2}}}~{\cfrac {\partial W}{\partial I_{1}}}~.} Therefore,
σ 11 = 2 ( λ 2 − 1 λ 2 ) ∂ W ∂ I 1 ; σ 22 = 0 ; σ 33 = 2 ( 1 − 1 λ 2 ) ∂ W ∂ I 1 . {\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 λ − 1 {\displaystyle \lambda -1\,} . The engineering stress is
T 11 = σ 11 λ = 2 ( λ − 1 λ 3 ) ∂ W ∂ I 1 . {\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 [ edit ] A version of the Yeoh model that includes I 3 = J 2 {\displaystyle I_{3}=J^{2}} dependence is used for compressible rubbers. The strain energy density function for this model is written as
W = ∑ i = 1 n C i 0 ( I ¯ 1 − 3 ) i + ∑ k = 1 n C k 1 ( J − 1 ) 2 k {\displaystyle W=\sum _{i=1}^{n}C_{i0}~({\bar {I}}_{1}-3)^{i}+\sum _{k=1}^{n}C_{k1}~(J-1)^{2k}} where I ¯ 1 = J − 2 / 3 I 1 {\displaystyle {\bar {I}}_{1}=J^{-2/3}~I_{1}} , and C i 0 , C k 1 {\displaystyle C_{i0},C_{k1}} are material constants. The quantity C 10 {\displaystyle C_{10}} is interpreted as half the initial shear modulus, while C 11 {\displaystyle C_{11}} is interpreted as half the initial bulk modulus.
When n = 1 {\displaystyle n=1} the compressible Yeoh model reduces to the neo-Hookean model for incompressible materials.
History [ edit ] 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 [ edit ] ^ 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 . ^ 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. ^ "Remembering Dr. Graham Johnson Lake (1935–2023)". Rubber Chemistry and Technology . 96 (4): G2–G3. 2023. doi :10.5254/rct-23.498080 . ^ "Biographical Sketch" . ACS Rubber Division. Retrieved 20 February 2024 . ^ "Rubber Division names 3 for awards" . Rubber and Plastics News . Crain. 27 October 2003. Retrieved 16 August 2022 . See also [ edit ]