A Mooney-Rivlin solid is a generalization of the w:Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of the w:Finger tensor
B
{\displaystyle \mathbf {B} }
W
=
C
1
(
I
¯
1
−
3
)
+
C
2
(
I
¯
2
−
3
)
{\displaystyle W=C_{1}({\overline {I}}_{1}-3)+C_{2}({\overline {I}}_{2}-3)}
where
I
¯
1
{\displaystyle {\overline {I}}_{1}}
I
¯
2
{\displaystyle {\overline {I}}_{2}}
invariant of w:deviatoric component of the w:Finger tensor :[ 1]
I
1
=
λ
1
2
+
λ
2
2
+
λ
3
2
{\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}
I
2
=
λ
1
2
λ
2
2
+
λ
2
2
λ
3
2
+
λ
3
2
λ
1
2
{\displaystyle I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}}
I
3
=
λ
1
2
λ
2
2
λ
3
2
{\displaystyle I_{3}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}}
where:
C
1
{\displaystyle C_{1}}
C
2
{\displaystyle C_{2}}
If
C
1
=
1
2
G
{\displaystyle C_{1}={\frac {1}{2}}G}
w:shear modulus ) and
C
2
=
0
{\displaystyle C_{2}=0}
w:Neo-Hookean solid , a special case of a Mooney-Rivlin solid .
The stress tensor
T
{\displaystyle \mathbf {T} }
B
{\displaystyle \mathbf {B} }
T
=
−
p
I
+
2
C
1
B
+
2
C
2
B
−
1
{\displaystyle \mathbf {T} =-p\mathbf {I} +2C_{1}\mathbf {B} +2C_{2}\mathbf {B} ^{-1}}
The model was proposed by w:Melvin Mooney and w:Ronald Rivlin in two independent papers in 1952.
Comparison of experimental results (dots) and predictions for w:Hooke's law (1, blue line), w:Neo-Hookean solid (2, red line) and Mooney-Rivlin solid models(3, green line) For the case of uniaxial elongation, true stress can be calculated as:
T
11
=
(
2
C
1
+
2
C
2
α
1
)
(
α
1
2
−
α
1
−
1
)
{\displaystyle T_{11}=\left(2C_{1}+{\frac {2C_{2}}{\alpha _{1}}}\right)\left(\alpha _{1}^{2}-\alpha _{1}^{-1}\right)}
and w:engineering stress can be calculated as:
T
11
e
n
g
=
(
2
C
1
+
2
C
2
α
1
)
(
α
1
−
α
1
−
2
)
{\displaystyle T_{11eng}=\left(2C_{1}+{\frac {2C_{2}}{\alpha _{1}}}\right)\left(\alpha _{1}-\alpha _{1}^{-2}\right)}
The Mooney-Rivlin solid model usually fits experimental data better than w:Neo-Hookean solid does, but requires an additional empirical constant.
Elastic response of rubber-like materials are often modelled based on the Mooney-Rivlin model.
C. W. Macosko Rheology: principles, measurement and applications , VCH Publishers, 1994, ISBN 1-56081-579-5
↑ The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor is usually written
p
B
(
λ
)
=
λ
3
−
a
1
λ
2
+
a
2
λ
−
a
3
{\displaystyle p_{B}(\lambda )=\lambda ^{3}-a_{1}\,\lambda ^{2}+a_{2}\,\lambda -a_{3}}
trace
a
1
{\displaystyle a_{1}}
I
1
{\displaystyle I_{1}}
a
2
{\displaystyle a_{2}}
I
2
{\displaystyle I_{2}}
a
3
{\displaystyle a_{3}}
I
3
{\displaystyle I_{3}}
