A neo-Hookean solid [1] [2] is a hyperelastic material model, similar to Hooke's law , that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations . The model was proposed by Ronald Rivlin in 1948 using invariants, though Mooney had already described a version in stretch form in 1940, and Wall had noted the equivalence in shear with the Hooke model in 1942.
In contrast to linear elastic materials, the stress-strain curve of a neo-Hookean material is not linear . Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the dissipative release of energy as heat while straining the material, and perfect elasticity is assumed at all stages of deformation.
The neo-Hookean model is based on the statistical thermodynamics of cross-linked polymer chains and is usable for plastics and rubber -like substances. Cross-linked polymers will act in a neo-Hookean manner because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain point the polymer chains will be stretched to the maximum point that the covalent cross links will allow, and this will cause a dramatic increase in the elastic modulus of the material. The neo-Hookean material model does not predict that increase in modulus at large strains and is typically accurate only for strains less than 20%.[3] The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model.
The strain energy density function for an incompressible neo-Hookean material in a three-dimensional description is
W
=
C
1
(
I
1
−
3
)
{\displaystyle W=C_{1}(I_{1}-3)}
where
C
1
{\displaystyle C_{1}}
is a material constant, and
I
1
{\displaystyle I_{1}}
is the first invariant (trace ), of the right Cauchy-Green deformation tensor , i.e.,
I
1
=
λ
1
2
+
λ
2
2
+
λ
3
2
{\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}
where
λ
i
{\displaystyle \lambda _{i}}
are the principal stretches .[2]
For a compressible neo-Hookean material the strain energy density function is given by
W
=
C
1
(
I
1
−
3
−
2
ln
J
)
+
D
1
(
J
−
1
)
2
;
J
=
det
(
F
)
=
λ
1
λ
2
λ
3
{\displaystyle W=C_{1}~(I_{1}-3-2\ln J)+D_{1}~(J-1)^{2}~;~~J=\det({\boldsymbol {F}})=\lambda _{1}\lambda _{2}\lambda _{3}}
where
D
1
{\displaystyle D_{1}}
is a material constant and
F
{\displaystyle {\boldsymbol {F}}}
is the deformation gradient . It can be shown that in 2D, the strain energy density function is
W
=
C
1
(
I
1
−
2
−
2
ln
J
)
+
D
1
(
J
−
1
)
2
{\displaystyle W=C_{1}~(I_{1}-2-2\ln J)+D_{1}~(J-1)^{2}}
Several alternative formulations exist for compressible neo-Hookean materials, for example
W
=
C
1
(
I
¯
1
−
3
)
+
(
C
1
6
+
D
1
4
)
(
J
2
+
1
J
2
−
2
)
{\displaystyle W=C_{1}~({\bar {I}}_{1}-3)+\left({\frac {C_{1}}{6}}+{\frac {D_{1}}{4}}\right)\!\left(J^{2}+{\frac {1}{J^{2}}}-2\right)}
where
I
¯
1
=
J
−
2
/
3
I
1
{\displaystyle {\bar {I}}_{1}=J^{-2/3}I_{1}}
is the first invariant of the isochoric part
C
¯
=
(
det
C
)
−
1
/
3
C
=
J
−
2
/
3
C
{\displaystyle {\bar {\boldsymbol {C}}}=(\det {\boldsymbol {C}})^{-1/3}{\boldsymbol {C}}=J^{-2/3}{\boldsymbol {C}}}
of the right Cauchy–Green deformation tensor .
For consistency with linear elasticity,
C
1
=
μ
2
;
D
1
=
λ
L
2
{\displaystyle C_{1}={\frac {\mu }{2}}~;~~D_{1}={\frac {{\lambda }_{L}}{2}}}
where
λ
L
{\displaystyle {\lambda }_{L}}
is the first Lamé parameter and
μ
{\displaystyle \mu }
is the shear modulus or the second Lamé parameter .[4] Alternative definitions of
C
1
{\displaystyle C_{1}}
and
D
1
{\displaystyle D_{1}}
are sometimes used, notably in commercial finite element analysis software such as Abaqus .[5]
Cauchy stress in terms of deformation tensors
edit
Compressible neo-Hookean material
edit
For a compressible Ogden neo-Hookean material the Cauchy stress is given by
σ
=
J
−
1
P
F
T
=
J
−
1
∂
W
∂
F
F
T
=
J
−
1
(
2
C
1
(
F
−
F
−
T
)
+
2
D
1
(
J
−
1
)
J
F
−
T
)
F
T
{\displaystyle {\boldsymbol {\sigma }}=J^{-1}{\boldsymbol {P}}{\boldsymbol {F}}^{T}=J^{-1}{\frac {\partial W}{\partial {\boldsymbol {F}}}}{\boldsymbol {F}}^{T}=J^{-1}\left(2C_{1}({\boldsymbol {F}}-{\boldsymbol {F}}^{-T})+2D_{1}(J-1)J{\boldsymbol {F}}^{-T}\right){\boldsymbol {F}}^{T}}
where
P
{\displaystyle {\boldsymbol {P}}}
is the first Piola–Kirchhoff stress. By simplifying the right hand side we arrive at
σ
=
2
C
1
J
−
1
(
F
F
T
−
I
)
+
2
D
1
(
J
−
1
)
I
=
2
C
1
J
−
1
(
B
−
I
)
+
2
D
1
(
J
−
1
)
I
{\displaystyle {\boldsymbol {\sigma }}=2C_{1}J^{-1}\left({\boldsymbol {F}}{\boldsymbol {F}}^{T}-{\boldsymbol {I}}\right)+2D_{1}(J-1){\boldsymbol {I}}=2C_{1}J^{-1}\left({\boldsymbol {B}}-{\boldsymbol {I}}\right)+2D_{1}(J-1){\boldsymbol {I}}}
which for infinitesimal strains is equal to
≈
4
C
1
ε
+
2
D
1
tr
(
ε
)
I
{\displaystyle \approx 4C_{1}{\boldsymbol {\varepsilon }}+2D_{1}\operatorname {tr} ({\boldsymbol {\varepsilon }}){\boldsymbol {I}}}
Comparison with Hooke's law shows that
C
1
=
μ
2
{\displaystyle C_{1}={\tfrac {\mu }{2}}}
and
D
1
=
λ
L
2
{\displaystyle D_{1}={\tfrac {\lambda _{L}}{2}}}
.
For a compressible Rivlin neo-Hookean material the Cauchy stress is given by
J
σ
=
−
p
I
+
2
C
1
dev
(
B
¯
)
=
−
p
I
+
2
C
1
J
2
/
3
dev
(
B
)
{\displaystyle J~{\boldsymbol {\sigma }}=-p~{\boldsymbol {I}}+2C_{1}\operatorname {dev} ({\bar {\boldsymbol {B}}})=-p~{\boldsymbol {I}}+{\frac {2C_{1}}{J^{2/3}}}\operatorname {dev} ({\boldsymbol {B}})}
where
B
{\displaystyle {\boldsymbol {B}}}
is the left Cauchy–Green deformation tensor, and
p
:=
−
2
D
1
J
(
J
−
1
)
;
dev
(
B
¯
)
=
B
¯
−
1
3
I
¯
1
I
;
B
¯
=
J
−
2
/
3
B
.
{\displaystyle p:=-2D_{1}~J(J-1)~;~\operatorname {dev} ({\bar {\boldsymbol {B}}})={\bar {\boldsymbol {B}}}-{\tfrac {1}{3}}{\bar {I}}_{1}{\boldsymbol {I}}~;~~{\bar {\boldsymbol {B}}}=J^{-2/3}{\boldsymbol {B}}~.}
For infinitesimal strains (
ε
{\displaystyle {\boldsymbol {\varepsilon }}}
)
J
≈
1
+
tr
(
ε
)
;
B
≈
I
+
2
ε
{\displaystyle J\approx 1+\operatorname {tr} ({\boldsymbol {\varepsilon }})~;~~{\boldsymbol {B}}\approx {\boldsymbol {I}}+2{\boldsymbol {\varepsilon }}}
and the Cauchy stress can be expressed as
σ
≈
4
C
1
(
ε
−
1
3
tr
(
ε
)
I
)
+
2
D
1
tr
(
ε
)
I
{\displaystyle {\boldsymbol {\sigma }}\approx 4C_{1}\left({\boldsymbol {\varepsilon }}-{\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }}){\boldsymbol {I}}\right)+2D_{1}\operatorname {tr} ({\boldsymbol {\varepsilon }}){\boldsymbol {I}}}
Comparison with Hooke's law shows that
μ
=
2
C
1
{\displaystyle \mu =2C_{1}}
and
κ
=
2
D
1
{\displaystyle \kappa =2D_{1}}
.
Proof:
The Cauchy stress in a compressible hyperelastic material is given by
σ
=
2
J
[
1
J
2
/
3
(
∂
W
∂
I
¯
1
+
I
¯
1
∂
W
∂
I
¯
2
)
B
−
1
J
4
/
3
∂
W
∂
I
¯
2
B
⋅
B
]
+
[
∂
W
∂
J
−
2
3
J
(
I
¯
1
∂
W
∂
I
¯
1
+
2
I
¯
2
∂
W
∂
I
¯
2
)
]
I
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}\left({\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}+{\bar {I}}_{1}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}\right){\boldsymbol {B}}-{\cfrac {1}{J^{4/3}}}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+\left[{\cfrac {\partial {W}}{\partial J}}-{\cfrac {2}{3J}}\left({\bar {I}}_{1}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}\right)\right]~{\boldsymbol {I}}}
For a compressible Rivlin neo-Hookean material,
∂
W
∂
I
¯
1
=
C
1
;
∂
W
∂
I
¯
2
=
0
;
∂
W
∂
J
=
2
D
1
(
J
−
1
)
{\displaystyle {\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}=C_{1}~;~~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}=0~;~~{\cfrac {\partial {W}}{\partial J}}=2D_{1}(J-1)}
while, for a compressible Ogden neo-Hookean material,
∂
W
∂
I
¯
1
=
C
1
;
∂
W
∂
I
¯
2
=
0
;
∂
W
∂
J
=
2
D
1
(
J
−
1
)
−
2
C
1
J
{\displaystyle {\cfrac {\partial {W}}{\partial {\bar {I}}_{1}}}=C_{1}~;~~{\cfrac {\partial {W}}{\partial {\bar {I}}_{2}}}=0~;~~{\cfrac {\partial {W}}{\partial J}}=2D_{1}(J-1)-{\cfrac {2C_{1}}{J}}}
Therefore, the Cauchy stress in a compressible Rivlin neo-Hookean material is given by
σ
=
2
J
[
1
J
2
/
3
C
1
B
]
+
[
2
D
1
(
J
−
1
)
−
2
3
J
C
1
I
¯
1
]
I
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}~C_{1}~{\boldsymbol {B}}\right]+\left[2D_{1}(J-1)-{\cfrac {2}{3J}}~C_{1}{\bar {I}}_{1}\right]{\boldsymbol {I}}}
while that for the corresponding Ogden material is
σ
=
2
J
[
1
J
2
/
3
C
1
B
]
+
[
2
D
1
(
J
−
1
)
−
2
C
1
J
−
2
3
J
C
1
I
¯
1
]
I
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}\left[{\cfrac {1}{J^{2/3}}}~C_{1}~{\boldsymbol {B}}\right]+\left[2D_{1}(J-1)-{\cfrac {2C_{1}}{J}}-{\cfrac {2}{3J}}~C_{1}{\bar {I}}_{1}\right]{\boldsymbol {I}}}
If the isochoric part of the left Cauchy-Green deformation tensor is defined as
B
¯
=
J
−
2
/
3
B
{\displaystyle {\bar {\boldsymbol {B}}}=J^{-2/3}{\boldsymbol {B}}}
, then we can write the Rivlin neo-Heooken stress as
σ
=
2
C
1
J
[
B
¯
−
1
3
I
¯
1
I
]
+
2
D
1
(
J
−
1
)
I
=
2
C
1
J
dev
(
B
¯
)
+
2
D
1
(
J
−
1
)
I
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2C_{1}}{J}}\left[{\bar {\boldsymbol {B}}}-{\tfrac {1}{3}}{\bar {I}}_{1}{\boldsymbol {I}}\right]+2D_{1}(J-1){\boldsymbol {I}}={\cfrac {2C_{1}}{J}}\operatorname {dev} ({\bar {\boldsymbol {B}}})+2D_{1}(J-1){\boldsymbol {I}}}
and the Ogden neo-Hookean stress as
σ
=
2
C
1
J
[
B
¯
−
1
3
I
¯
1
I
−
I
]
+
2
D
1
(
J
−
1
)
I
=
2
C
1
J
[
dev
(
B
¯
)
−
I
]
+
2
D
1
(
J
−
1
)
I
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2C_{1}}{J}}\left[{\bar {\boldsymbol {B}}}-{\tfrac {1}{3}}{\bar {I}}_{1}{\boldsymbol {I}}-{\boldsymbol {I}}\right]+2D_{1}(J-1){\boldsymbol {I}}={\cfrac {2C_{1}}{J}}\left[\operatorname {dev} ({\bar {\boldsymbol {B}}})-{\boldsymbol {I}}\right]+2D_{1}(J-1){\boldsymbol {I}}}
The quantities
p
:=
−
2
D
1
J
(
J
−
1
)
;
p
∗
=
−
2
D
1
J
(
J
−
1
)
+
2
C
1
{\displaystyle p:=-2D_{1}~J(J-1)~;~~p^{*}=-2D_{1}~J(J-1)+2C_{1}}
have the form of pressures and are usually treated as such. The Rivlin neo-Hookean stress can then be expressed in the form
τ
=
J
σ
=
−
p
I
+
2
C
1
dev
(
B
¯
)
{\displaystyle {\boldsymbol {\tau }}=J~{\boldsymbol {\sigma }}=-p{\boldsymbol {I}}+2C_{1}\operatorname {dev} ({\bar {\boldsymbol {B}}})}
while the Ogden neo-Hookean stress has the form
τ
=
−
p
∗
I
+
2
C
1
dev
(
B
¯
)
{\displaystyle {\boldsymbol {\tau }}=-p^{*}{\boldsymbol {I}}+2C_{1}\operatorname {dev} ({\bar {\boldsymbol {B}}})}
Incompressible neo-Hookean material
edit
For an incompressible neo-Hookean material with
J
=
1
{\displaystyle J=1}
σ
=
−
p
I
+
2
C
1
B
{\displaystyle {\boldsymbol {\sigma }}=-p~{\boldsymbol {I}}+2C_{1}{\boldsymbol {B}}}
where
p
{\displaystyle p}
is an undetermined pressure.
Cauchy stress in terms of principal stretches
edit
Compressible neo-Hookean material
edit
For a compressible neo-Hookean hyperelastic material , the principal components of the Cauchy stress are given by
σ
i
=
2
C
1
J
−
5
/
3
[
λ
i
2
−
I
1
3
]
+
2
D
1
(
J
−
1
)
;
i
=
1
,
2
,
3
{\displaystyle \sigma _{i}=2C_{1}J^{-5/3}\left[\lambda _{i}^{2}-{\cfrac {I_{1}}{3}}\right]+2D_{1}(J-1)~;~~i=1,2,3}
Therefore, the differences between the principal stresses are
σ
11
−
σ
33
=
2
C
1
J
5
/
3
(
λ
1
2
−
λ
3
2
)
;
σ
22
−
σ
33
=
2
C
1
J
5
/
3
(
λ
2
2
−
λ
3
2
)
{\displaystyle \sigma _{11}-\sigma _{33}={\cfrac {2C_{1}}{J^{5/3}}}(\lambda _{1}^{2}-\lambda _{3}^{2})~;~~\sigma _{22}-\sigma _{33}={\cfrac {2C_{1}}{J^{5/3}}}(\lambda _{2}^{2}-\lambda _{3}^{2})}
Proof:
For a compressible hyperelastic material , the principal components of the Cauchy stress are given by
σ
i
=
λ
i
λ
1
λ
2
λ
3
∂
W
∂
λ
i
;
i
=
1
,
2
,
3
{\displaystyle \sigma _{i}={\cfrac {\lambda _{i}}{\lambda _{1}\lambda _{2}\lambda _{3}}}~{\frac {\partial W}{\partial \lambda _{i}}}~;~~i=1,2,3}
The strain energy density function for a compressible neo Hookean material is
W
=
C
1
(
I
¯
1
−
3
)
+
D
1
(
J
−
1
)
2
=
C
1
[
J
−
2
/
3
(
λ
1
2
+
λ
2
2
+
λ
3
2
)
−
3
]
+
D
1
(
J
−
1
)
2
{\displaystyle W=C_{1}({\bar {I}}_{1}-3)+D_{1}(J-1)^{2}=C_{1}\left[J^{-2/3}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})-3\right]+D_{1}(J-1)^{2}}
Therefore,
λ
i
∂
W
∂
λ
i
=
C
1
[
−
2
3
J
−
5
/
3
λ
i
∂
J
∂
λ
i
(
λ
1
2
+
λ
2
2
+
λ
3
2
)
+
2
J
−
2
/
3
λ
i
2
]
+
2
D
1
(
J
−
1
)
λ
i
∂
J
∂
λ
i
{\displaystyle \lambda _{i}{\frac {\partial W}{\partial \lambda _{i}}}=C_{1}\left[-{\frac {2}{3}}J^{-5/3}\lambda _{i}{\frac {\partial J}{\partial \lambda _{i}}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+2J^{-2/3}\lambda _{i}^{2}\right]+2D_{1}(J-1)\lambda _{i}{\frac {\partial J}{\partial \lambda _{i}}}}
Since
J
=
λ
1
λ
2
λ
3
{\displaystyle J=\lambda _{1}\lambda _{2}\lambda _{3}}
we have
λ
i
∂
J
∂
λ
i
=
λ
1
λ
2
λ
3
=
J
{\displaystyle \lambda _{i}{\frac {\partial J}{\partial \lambda _{i}}}=\lambda _{1}\lambda _{2}\lambda _{3}=J}
Hence,
λ
i
∂
W
∂
λ
i
=
C
1
[
−
2
3
J
−
2
/
3
(
λ
1
2
+
λ
2
2
+
λ
3
2
)
+
2
J
−
2
/
3
λ
i
2
]
+
2
D
1
J
(
J
−
1
)
=
2
C
1
J
−
2
/
3
[
−
1
3
(
λ
1
2
+
λ
2
2
+
λ
3
2
)
+
λ
i
2
]
+
2
D
1
J
(
J
−
1
)
{\displaystyle {\begin{aligned}\lambda _{i}{\frac {\partial W}{\partial \lambda _{i}}}&=C_{1}\left[-{\frac {2}{3}}J^{-2/3}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+2J^{-2/3}\lambda _{i}^{2}\right]+2D_{1}J(J-1)\\&=2C_{1}J^{-2/3}\left[-{\frac {1}{3}}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2})+\lambda _{i}^{2}\right]+2D_{1}J(J-1)\end{aligned}}}
The principal Cauchy stresses are therefore given by
σ
i
=
2
C
1
J
−
5
/
3
[
λ
i
2
−
I
1
3
]
+
2
D
1
(
J
−
1
)
{\displaystyle \sigma _{i}=2C_{1}J^{-5/3}\left[\lambda _{i}^{2}-{\cfrac {I_{1}}{3}}\right]+2D_{1}(J-1)}
Incompressible neo-Hookean material
edit
In terms of the principal stretches , the Cauchy stress differences for an incompressible hyperelastic material are given by
σ
11
−
σ
33
=
λ
1
∂
W
∂
λ
1
−
λ
3
∂
W
∂
λ
3
;
σ
22
−
σ
33
=
λ
2
∂
W
∂
λ
2
−
λ
3
∂
W
∂
λ
3
{\displaystyle \sigma _{11}-\sigma _{33}=\lambda _{1}~{\cfrac {\partial {W}}{\partial \lambda _{1}}}-\lambda _{3}~{\cfrac {\partial {W}}{\partial \lambda _{3}}}~;~~\sigma _{22}-\sigma _{33}=\lambda _{2}~{\cfrac {\partial {W}}{\partial \lambda _{2}}}-\lambda _{3}~{\cfrac {\partial {W}}{\partial \lambda _{3}}}}
For an incompressible neo-Hookean material,
W
=
C
1
(
λ
1
2
+
λ
2
2
+
λ
3
2
−
3
)
;
λ
1
λ
2
λ
3
=
1
{\displaystyle W=C_{1}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3)~;~~\lambda _{1}\lambda _{2}\lambda _{3}=1}
Therefore,
∂
W
∂
λ
1
=
2
C
1
λ
1
;
∂
W
∂
λ
2
=
2
C
1
λ
2
;
∂
W
∂
λ
3
=
2
C
1
λ
3
{\displaystyle {\cfrac {\partial {W}}{\partial \lambda _{1}}}=2C_{1}\lambda _{1}~;~~{\cfrac {\partial {W}}{\partial \lambda _{2}}}=2C_{1}\lambda _{2}~;~~{\cfrac {\partial {W}}{\partial \lambda _{3}}}=2C_{1}\lambda _{3}}
which gives
σ
11
−
σ
33
=
2
(
λ
1
2
−
λ
3
2
)
C
1
;
σ
22
−
σ
33
=
2
(
λ
2
2
−
λ
3
2
)
C
1
{\displaystyle \sigma _{11}-\sigma _{33}=2(\lambda _{1}^{2}-\lambda _{3}^{2})C_{1}~;~~\sigma _{22}-\sigma _{33}=2(\lambda _{2}^{2}-\lambda _{3}^{2})C_{1}}
Uniaxial extension
edit
Compressible neo-Hookean material
edit
The true stress as a function of uniaxial stretch predicted by a compressible neo-Hookean material for various values of
C
1
,
D
1
{\displaystyle C_{1},D_{1}}
. The material properties are representative of natural rubber .
For a compressible material undergoing uniaxial extension, the principal stretches are
λ
1
=
λ
;
λ
2
=
λ
3
=
J
λ
;
I
1
=
λ
2
+
2
J
λ
{\displaystyle \lambda _{1}=\lambda ~;~~\lambda _{2}=\lambda _{3}={\sqrt {\tfrac {J}{\lambda }}}~;~~I_{1}=\lambda ^{2}+{\tfrac {2J}{\lambda }}}
Hence, the true (Cauchy) stresses for a compressible neo-Hookean material are given by
σ
11
=
4
C
1
3
J
5
/
3
(
λ
2
−
J
λ
)
+
2
D
1
(
J
−
1
)
σ
22
=
σ
33
=
2
C
1
3
J
5
/
3
(
J
λ
−
λ
2
)
+
2
D
1
(
J
−
1
)
{\displaystyle {\begin{aligned}\sigma _{11}&={\cfrac {4C_{1}}{3J^{5/3}}}\left(\lambda ^{2}-{\tfrac {J}{\lambda }}\right)+2D_{1}(J-1)\\\sigma _{22}&=\sigma _{33}={\cfrac {2C_{1}}{3J^{5/3}}}\left({\tfrac {J}{\lambda }}-\lambda ^{2}\right)+2D_{1}(J-1)\end{aligned}}}
The stress differences are given by
σ
11
−
σ
33
=
2
C
1
J
5
/
3
(
λ
2
−
J
λ
)
;
σ
22
−
σ
33
=
0
{\displaystyle \sigma _{11}-\sigma _{33}={\cfrac {2C_{1}}{J^{5/3}}}\left(\lambda ^{2}-{\tfrac {J}{\lambda }}\right)~;~~\sigma _{22}-\sigma _{33}=0}
If the material is unconstrained we have
σ
22
=
σ
33
=
0
{\displaystyle \sigma _{22}=\sigma _{33}=0}
. Then
σ
11
=
2
C
1
J
5
/
3
(
λ
2
−
J
λ
)
{\displaystyle \sigma _{11}={\cfrac {2C_{1}}{J^{5/3}}}\left(\lambda ^{2}-{\tfrac {J}{\lambda }}\right)}
Equating the two expressions for
σ
11
{\displaystyle \sigma _{11}}
gives a relation for
J
{\displaystyle J}
as a function of
λ
{\displaystyle \lambda }
, i.e.,
4
C
1
3
J
5
/
3
(
λ
2
−
J
λ
)
+
2
D
1
(
J
−
1
)
=
2
C
1
J
5
/
3
(
λ
2
−
J
λ
)
{\displaystyle {\cfrac {4C_{1}}{3J^{5/3}}}\left(\lambda ^{2}-{\tfrac {J}{\lambda }}\right)+2D_{1}(J-1)={\cfrac {2C_{1}}{J^{5/3}}}\left(\lambda ^{2}-{\tfrac {J}{\lambda }}\right)}
or
D
1
J
8
/
3
−
D
1
J
5
/
3
+
C
1
3
λ
J
−
C
1
λ
2
3
=
0
{\displaystyle D_{1}J^{8/3}-D_{1}J^{5/3}+{\tfrac {C_{1}}{3\lambda }}J-{\tfrac {C_{1}\lambda ^{2}}{3}}=0}
The above equation can be solved numerically using a Newton–Raphson iterative root-finding procedure.
Incompressible neo-Hookean material
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Comparison of experimental results (dots) and predictions for Hooke's law (1), neo-Hookean solid(2) and Mooney-Rivlin solid models(3)
Under uniaxial extension,
λ
1
=
λ
{\displaystyle \lambda _{1}=\lambda \,}
and
λ
2
=
λ
3
=
1
/
λ
{\displaystyle \lambda _{2}=\lambda _{3}=1/{\sqrt {\lambda }}}
. Therefore,
σ
11
−
σ
33
=
2
C
1
(
λ
2
−
1
λ
)
;
σ
22
−
σ
33
=
0
{\displaystyle \sigma _{11}-\sigma _{33}=2C_{1}\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)~;~~\sigma _{22}-\sigma _{33}=0}
Assuming no traction on the sides,
σ
22
=
σ
33
=
0
{\displaystyle \sigma _{22}=\sigma _{33}=0}
, so we can write
σ
11
=
2
C
1
(
λ
2
−
1
λ
)
=
2
C
1
(
3
ε
11
+
3
ε
11
2
+
ε
11
3
1
+
ε
11
)
{\displaystyle \sigma _{11}=2C_{1}\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)=2C_{1}\left({\frac {3\varepsilon _{11}+3\varepsilon _{11}^{2}+\varepsilon _{11}^{3}}{1+\varepsilon _{11}}}\right)}
where
ε
11
=
λ
−
1
{\displaystyle \varepsilon _{11}=\lambda -1}
is the engineering strain . This equation is often written in alternative notation as
T
11
=
2
C
1
(
α
2
−
1
α
)
{\displaystyle T_{11}=2C_{1}\left(\alpha ^{2}-{\cfrac {1}{\alpha }}\right)}
The equation above is for the true stress (ratio of the elongation force to deformed cross-section). For the engineering stress the equation is:
σ
11
e
n
g
=
2
C
1
(
λ
−
1
λ
2
)
{\displaystyle \sigma _{11}^{\mathrm {eng} }=2C_{1}\left(\lambda -{\cfrac {1}{\lambda ^{2}}}\right)}
For small deformations
ε
≪
1
{\displaystyle \varepsilon \ll 1}
we will have:
σ
11
=
6
C
1
ε
=
3
μ
ε
{\displaystyle \sigma _{11}=6C_{1}\varepsilon =3\mu \varepsilon }
Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is
3
μ
{\displaystyle 3\mu }
, which is in concordance with linear elasticity (
E
=
2
μ
(
1
+
ν
)
{\displaystyle E=2\mu (1+\nu )}
with
ν
=
0.5
{\displaystyle \nu =0.5}
for incompressibility).
Equibiaxial extension
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Pure dilation
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Simple shear
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References
edit
^ Treloar, L. R. G. (1943). "The elasticity of a network of long-chain molecules—II" . Transactions of the Faraday Society . 39 : 241–246.
^ a b c Ogden, R. W. (26 April 2013). Non-Linear Elastic Deformations . Courier Corporation. ISBN 978-0-486-31871-4 .
^ Gent, A. N., ed., 2001, Engineering with rubber , Carl Hanser Verlag, Munich.
^ Pence, T. J., & Gou, K. (2015). On compressible versions of the incompressible neo-Hookean material. Mathematics and Mechanics of Solids , 20(2), 157–182. [1]
^ Abaqus (Version 6.8) Theory Manual
See also
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