An important consider in nonlinear finite element analysis is
material frame indifference or objectivity of the material response .
The idea is that the position of the observer frame should not affect the
constitutive relations of a material. You can find more details and a history of
the idea in Truesdell and Noll (1992) - sections 17, 18, 19, and 19A.
[ 1]
We have already talked about the objectivity of kinematic quantities and stress rates.
Let us now discuss the same ideas with a particular constitutive model in mind.
Hyperelastic materials
edit
A detailed description of thermoelastic materials can be found in Continuum mechanics/Thermoelasticity . In this discussion we will avoid the complications induced by
including the temperature.
In the material configuration, a hyperelastic material satisfies two requirements:
a stored energy function (
W
{\displaystyle W}
the stored energy function depends locally only on the deformation gradient. Given these requirements, if
N
=
P
T
{\displaystyle {\boldsymbol {N}}={\boldsymbol {P}}^{T}}
P
{\displaystyle {\boldsymbol {P}}}
N
T
(
X
,
t
)
=
P
(
X
,
t
)
=
ρ
0
∂
W
∂
F
[
X
,
F
(
X
,
t
)
]
{\displaystyle {\boldsymbol {N}}^{T}(\mathbf {X} ,t)={\boldsymbol {P}}(\mathbf {X} ,t)=\rho _{0}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\left[\mathbf {X} ,{\boldsymbol {F}}(\mathbf {X} ,t)\right]}
The stored energy function
W
(
X
,
F
)
{\displaystyle W(\mathbf {X} ,{\boldsymbol {F}})}
W
(
X
,
Q
⋅
F
)
=
W
(
X
,
F
)
{\displaystyle W(\mathbf {X} ,{\boldsymbol {Q}}\cdot {\boldsymbol {F}})=W(\mathbf {X} ,{\boldsymbol {F}})}
where
Q
{\displaystyle {\boldsymbol {Q}}}
S
⋅
Q
T
=
1
{\displaystyle {\boldsymbol {S}}\cdot {\boldsymbol {Q}}^{T}={\boldsymbol {\mathit {1}}}}
This objectivity condition can be achieved only if (in the material configuration)
W
(
X
,
F
)
=
W
^
(
X
,
C
)
;
C
=
F
T
⋅
F
{\displaystyle W(\mathbf {X} ,{\boldsymbol {F}})={\hat {W}}(\mathbf {X} ,{\boldsymbol {C}})~;~~{\boldsymbol {C}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}}
since
C
r
=
F
T
⋅
Q
T
⋅
Q
⋅
F
=
C
{\displaystyle {\boldsymbol {C}}_{r}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {Q}}^{T}\cdot {\boldsymbol {Q}}\cdot {\boldsymbol {F}}={\boldsymbol {C}}}
We can show that
Constitutive relations for hyperelastic materials
N
=
2
ρ
0
∂
W
^
∂
C
⋅
F
T
P
=
2
ρ
0
F
⋅
∂
W
^
∂
C
S
=
2
ρ
0
∂
W
^
∂
C
σ
=
2
ρ
F
⋅
∂
W
^
∂
C
⋅
F
T
τ
=
2
ρ
0
F
⋅
∂
W
^
∂
C
⋅
F
T
{\displaystyle {\begin{aligned}{\boldsymbol {N}}&=2~\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}\\{\boldsymbol {P}}&=2~\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\\{\boldsymbol {S}}&=2~\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\\{\boldsymbol {\sigma }}&=2~\rho ~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}\\{\boldsymbol {\tau }}&=2~\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}\end{aligned}}}
Proof:
The stress strain relation for a hyperelastic material is
P
=
ρ
0
∂
W
∂
F
[
X
,
F
(
X
,
t
)
]
{\displaystyle {\boldsymbol {P}}=\rho _{0}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\left[\mathbf {X} ,{\boldsymbol {F}}(\mathbf {X} ,t)\right]}
The chain rule then implies that
P
:
T
=
ρ
0
∂
W
^
∂
C
:
(
∂
C
∂
F
:
T
)
{\displaystyle {\boldsymbol {P}}:{\boldsymbol {T}}=\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:\left({\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}\right)}
for any second order tensor
T
{\displaystyle {\boldsymbol {T}}}
Now, using the product rule of differentiation,
∂
C
∂
F
:
T
=
∂
∂
F
(
F
T
⋅
F
)
:
T
=
(
∂
F
T
∂
F
:
T
)
⋅
F
+
F
T
⋅
(
∂
F
∂
F
:
T
)
{\displaystyle {\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}={\frac {\partial }{\partial {\boldsymbol {F}}}}\left({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}\right):{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}^{T}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot \left({\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}\right)}
or,
∂
C
∂
F
:
T
=
(
I
T
:
T
)
⋅
F
+
F
T
⋅
(
I
:
T
)
=
T
T
⋅
F
+
F
T
⋅
T
{\displaystyle {\frac {\partial {\boldsymbol {C}}}{\partial {\boldsymbol {F}}}}:{\boldsymbol {T}}=\left({\boldsymbol {\mathsf {I}}}^{T}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot \left({\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}\right)={\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}}}
where
I
{\displaystyle {\boldsymbol {\mathsf {I}}}}
P
:
T
=
ρ
0
∂
W
^
∂
C
:
(
T
T
⋅
F
+
F
T
⋅
T
)
{\displaystyle {\boldsymbol {P}}:{\boldsymbol {T}}=\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:\left({\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}}+{\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}}\right)}
Using the identity
A
:
(
B
T
⋅
C
)
=
(
C
⋅
A
T
)
:
B
=
(
B
⋅
A
)
:
C
{\displaystyle {\boldsymbol {A}}:({\boldsymbol {B}}^{T}\cdot {\boldsymbol {C}})=({\boldsymbol {C}}\cdot {\boldsymbol {A}}^{T}):{\boldsymbol {B}}=({\boldsymbol {B}}\cdot {\boldsymbol {A}}):{\boldsymbol {C}}}
we have
∂
W
^
∂
C
:
(
T
T
⋅
F
)
=
[
F
⋅
(
∂
W
^
∂
C
)
T
]
:
T
and
∂
W
^
∂
C
:
(
F
T
⋅
T
)
=
[
F
⋅
∂
W
^
∂
C
]
:
T
{\displaystyle {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:({\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}})=\left[{\boldsymbol {F}}\cdot \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}\right]:{\boldsymbol {T}}\quad {\text{and}}\quad {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:({\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}})=\left[{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right]:{\boldsymbol {T}}}
Therefore, invoking the arbitrariness of
T
{\displaystyle {\boldsymbol {T}}}
P
=
ρ
0
F
⋅
(
∂
W
^
∂
C
)
T
+
ρ
0
F
⋅
∂
W
^
∂
C
{\displaystyle {\boldsymbol {P}}=\rho _{0}~{\boldsymbol {F}}\cdot \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}+\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}}
Since
C
=
C
T
{\displaystyle {\boldsymbol {C}}={\boldsymbol {C}}^{T}}
(
∂
W
^
∂
C
)
T
=
∂
W
^
∂
C
{\displaystyle \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}={\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}}
which implies that
P
=
2
ρ
0
F
⋅
∂
W
^
∂
C
or
N
=
2
ρ
0
∂
W
^
∂
C
⋅
F
T
{\displaystyle {\boldsymbol {P}}=2~\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\quad {\text{or}}\quad {\boldsymbol {N}}=2~\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}}
Recall the relations between the 2nd Piola-Kirchhoff stress tensor and the
first Piola-Kirchhoff stress tensor (and the nominal stress tensor)
S
=
F
−
1
⋅
P
{\displaystyle {\boldsymbol {S}}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {P}}}
Therefore, we have
S
=
2
ρ
0
∂
W
^
∂
C
{\displaystyle {\boldsymbol {S}}=2~\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}}
Also from the relation between the Cauchy stress and the 2nd Piola-Kirchhoff stress
tensor
σ
=
J
−
1
F
⋅
S
⋅
F
T
=
ρ
ρ
0
F
⋅
S
⋅
F
T
{\displaystyle {\boldsymbol {\sigma }}=J^{-1}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}={\cfrac {\rho }{\rho _{0}}}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}}
we have
σ
=
2
ρ
F
⋅
∂
W
^
∂
C
⋅
F
T
or
τ
=
2
ρ
0
F
⋅
∂
W
^
∂
C
⋅
F
T
{\displaystyle {\boldsymbol {\sigma }}=2~\rho ~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}\quad {\text{or}}\quad {\boldsymbol {\tau }}=2~\rho _{0}{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}}
We may also express these relations in terms of the Lagrangian Green strain
E
=
1
2
(
F
T
⋅
F
−
1
)
=
1
2
(
C
−
1
)
{\displaystyle {\boldsymbol {E}}={\frac {1}{2}}({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}-{\boldsymbol {\mathit {1}}})={\frac {1}{2}}({\boldsymbol {C}}-{\boldsymbol {\mathit {1}}})}
Then we have
∂
W
∂
C
=
∂
W
∂
E
:
∂
E
∂
C
=
1
2
∂
W
∂
E
{\displaystyle {\frac {\partial W}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial {\boldsymbol {E}}}}:{\frac {\partial {\boldsymbol {E}}}{\partial {\boldsymbol {C}}}}={\frac {1}{2}}~{\frac {\partial W}{\partial {\boldsymbol {E}}}}}
Hence, we can write
N
=
ρ
0
∂
W
^
∂
E
⋅
F
T
P
=
ρ
0
F
⋅
∂
W
^
∂
E
S
=
ρ
0
∂
W
^
∂
E
σ
=
ρ
F
⋅
∂
W
^
∂
E
⋅
F
T
τ
=
ρ
0
F
⋅
∂
W
^
∂
E
⋅
F
T
{\displaystyle {\begin{aligned}{\boldsymbol {N}}&=\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {E}}}}\cdot {\boldsymbol {F}}^{T}\\{\boldsymbol {P}}&=\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {E}}}}\\{\boldsymbol {S}}&=\rho _{0}~{\frac {\partial {\hat {W}}}{\partial {\boldsymbol {E}}}}\\{\boldsymbol {\sigma }}&=\rho ~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {E}}}}\cdot {\boldsymbol {F}}^{T}\\{\boldsymbol {\tau }}&=\rho _{0}~{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {E}}}}\cdot {\boldsymbol {F}}^{T}\end{aligned}}}
The stored energy function
W
{\displaystyle W}
Show this.
↑ C. Truesdell and W. Noll, 1992,
The Nonlinear Field Theories of Mechanics:2nd ed. , Springer-Verlag, Berlin
![{\displaystyle {\boldsymbol {N}}^{T}(\mathbf {X} ,t)={\boldsymbol {P}}(\mathbf {X} ,t)=\rho _{0}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\left[\mathbf {X} ,{\boldsymbol {F}}(\mathbf {X} ,t)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d1edd1eea6840b8113617de431230fc4e8355e)
![{\displaystyle {\boldsymbol {P}}=\rho _{0}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\left[\mathbf {X} ,{\boldsymbol {F}}(\mathbf {X} ,t)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c292d13628035f797ff1b69a23c463e90451e35)
![{\displaystyle {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:({\boldsymbol {T}}^{T}\cdot {\boldsymbol {F}})=\left[{\boldsymbol {F}}\cdot \left({\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right)^{T}\right]:{\boldsymbol {T}}\quad {\text{and}}\quad {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}:({\boldsymbol {F}}^{T}\cdot {\boldsymbol {T}})=\left[{\boldsymbol {F}}\cdot {\frac {\partial {\hat {W}}}{\partial {\boldsymbol {C}}}}\right]:{\boldsymbol {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f08a2da1577dc1038a1316b6fa2a8280533dcb3)
