The polynomial hyperelastic material model is a phenomenological model of 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
of the left Cauchy-Green deformation tensor.
The strain energy density function for the polynomial model is
W
=
∑
i
,
j
=
0
n
C
i
j
(
I
1
−
3
)
i
(
I
2
−
3
)
j
where
C
i
j
are material constants and
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
D
k
(
J
−
1
)
2
k
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
In the limit where
C
01
=
C
11
=
0
, the polynomial model reduces to the 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
and we have
W
=
C
01
(
I
¯
2
−
3
)
+
C
10
(
I
¯
1
−
3
)
+
D
1
(
J
−
1
)
2
