300px|right|thumb|Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data from PolymerFEM.com
I1,I2,I3
I1
I3
The original model proposed by Yeoh had a cubic form with only
I1
W=
3 | |
\sum | |
i=1 |
Ci~(I
i | |
1-3) |
Ci
2C1
Today a slightly more generalized version of the Yeoh model is used.[3] This model includes
n
W=
n | |
\sum | |
i=1 |
Ci~(I
i | |
1-3) |
~.
When
n=1
For consistency with linear elasticity the Yeoh model has to satisfy the condition
2\cfrac{\partialW}{\partialI1}(3)=\mu~~(i\nej)
\mu
I1=3(λi=λj=1)
\cfrac{\partialW}{\partialI1}=C1
2C1=\mu
The Cauchy stress for the incompressible Yeoh model is given by
\boldsymbol{\sigma}=-p~\boldsymbol{1
For uniaxial extension in the
n1
λ1=λ,~λ2=λ3
λ1~λ2~λ3=1
2=1/λ | |
λ | |
3 |
I1=
2 | |
λ | |
3 |
=λ2+\cfrac{2}{λ}~.
\boldsymbol{B}=
2~n | |
λ | |
1 ⊗ n |
1+\cfrac{1}{λ}~(n2 ⊗ n2+n3 ⊗ n3)~.
\sigma11=-p+2~λ2~\cfrac{\partialW}{\partialI1}~;~~ \sigma22=-p+\cfrac{2}{λ}~\cfrac{\partialW}{\partialI1}=\sigma33~.
\sigma22=\sigma33=0
p=\cfrac{2}{λ}~\cfrac{\partialW}{\partialI1}~.
\sigma11=2~\left(λ2-\cfrac{1}{λ}\right)~\cfrac{\partialW}{\partialI1}~.
λ-1
T11=\sigma11/λ=2~\left(λ-\cfrac{1}{λ2}\right)~\cfrac{\partialW}{\partialI1}~.
For equibiaxial extension in the
n1
n2
λ1=λ2=λ
λ1~λ2~λ3=1
2 | |
λ | |
3=1/λ |
I1=
2 | |
λ | |
3 |
=2~λ2+\cfrac{1}{λ4}~.
\boldsymbol{B}=
2~n | |
λ | |
1 ⊗ n |
1+
2~n | |
λ | |
2 ⊗ n |
2+
4}~n | |
\cfrac{1}{λ | |
3 ⊗ n |
3~.
\sigma11=-p+2~λ2~\cfrac{\partialW}{\partialI1}=\sigma22~;~~ \sigma33=-p+\cfrac{2}{λ4}~\cfrac{\partialW}{\partialI1}~.
\sigma33=0
p=\cfrac{2}{λ4}~\cfrac{\partialW}{\partialI1}~.
\sigma11=2~\left(λ2-\cfrac{1}{λ4}\right)~\cfrac{\partialW}{\partialI1}=\sigma22~.
λ-1
T11=\cfrac{\sigma11
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the
n1
n3
λ1=λ,~λ3=1
λ1~λ2~λ3=1
λ2=1/λ
I1=
2 | |
λ | |
3 |
=λ2+\cfrac{1}{λ2}+1~.
\boldsymbol{B}=
2~n | |
λ | |
1 ⊗ n |
1+
2}~n | |
\cfrac{1}{λ | |
2 ⊗ n |
2+n3 ⊗ n3~.
\sigma11=-p+2~λ2~\cfrac{\partialW}{\partialI1}~;~~ \sigma22=-p+\cfrac{2}{λ2}~\cfrac{\partialW}{\partialI1}~;~~ \sigma33=-p+2~\cfrac{\partialW}{\partialI1}~.
\sigma22=0
p=\cfrac{2}{λ2}~\cfrac{\partialW}{\partialI1}~.
\sigma11=2~\left(λ2-\cfrac{1}{λ2}\right)~\cfrac{\partialW}{\partialI1}~;~~\sigma22=0~;~~\sigma33=2~\left(1-\cfrac{1}{λ2}\right)~\cfrac{\partialW}{\partialI1}~.
λ-1
T11=\cfrac{\sigma11
A version of the Yeoh model that includes
I3=J2
W=
n | |
\sum | |
i=1 |
Ci0
i | |
~(\bar{I} | |
1-3) |
+
n | |
\sum | |
k=1 |
Ck1~(J-1)2k
\bar{I}1=J-2/3~I1
Ci0,Ck1
C10
C11
When
n=1
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]