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The polynomial hyperelastic material model [ 1] rubber elasticity . In this model, the strain energy density function is of the form of a polynomial in the two invariants
I
1
,
I
2
{\displaystyle I_{1},I_{2}}
The strain energy density function for the polynomial model is [ 1]
W
=
∑
i
,
j
=
0
n
C
i
j
(
I
1
−
3
)
i
(
I
2
−
3
)
j
{\displaystyle W=\sum _{i,j=0}^{n}C_{ij}(I_{1}-3)^{i}(I_{2}-3)^{j}}
where
C
i
j
{\displaystyle C_{ij}}
C
00
=
0
{\displaystyle C_{00}=0}
For compressible materials, a dependence of volume is added
W
=
∑
i
,
j
=
0
n
C
i
j
(
I
¯
1
−
3
)
i
(
I
¯
2
−
3
)
j
+
∑
k
=
1
m
1
D
k
(
J
−
1
)
2
k
{\displaystyle W=\sum _{i,j=0}^{n}C_{ij}({\bar {I}}_{1}-3)^{i}({\bar {I}}_{2}-3)^{j}+\sum _{k=1}^{m}{\frac {1}{D_{k}}}(J-1)^{2k}}
where
I
¯
1
=
J
−
2
/
3
I
1
;
I
1
=
λ
1
2
+
λ
2
2
+
λ
3
2
;
J
=
det
(
F
)
I
¯
2
=
J
−
4
/
3
I
2
;
I
2
=
λ
1
2
λ
2
2
+
λ
2
2
λ
3
2
+
λ
3
2
λ
1
2
{\displaystyle {\begin{aligned}{\bar {I}}_{1}&=J^{-2/3}~I_{1}~;~~I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}~;~~J=\det({\boldsymbol {F}})\\{\bar {I}}_{2}&=J^{-4/3}~I_{2}~;~~I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\end{aligned}}}
In the limit where
C
01
=
C
11
=
0
{\displaystyle C_{01}=C_{11}=0}
Neo-Hookean solid model. For a compressible Mooney-Rivlin material
n
=
1
,
C
01
=
C
2
,
C
11
=
0
,
C
10
=
C
1
,
m
=
1
{\displaystyle n=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},m=1}
W
=
C
01
(
I
¯
2
−
3
)
+
C
10
(
I
¯
1
−
3
)
+
1
D
1
(
J
−
1
)
2
{\displaystyle W=C_{01}~({\bar {I}}_{2}-3)+C_{10}~({\bar {I}}_{1}-3)+{\frac {1}{D_{1}}}~(J-1)^{2}}
References
^ a b Rivlin, R. S. and Saunders, D. W., 1951, Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phi. Trans. Royal Soc. London Series A, 243(865), pp. 251-288.
See also
