The average deformation gradient is defined as
⟨
F
⟩
:=
1
V
0
∫
Ω
0
F
dV
{\displaystyle {\langle {\boldsymbol {F}}\rangle :={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {F}}~{\text{dV}}}}
where
V
0
{\displaystyle V_{0}}
We can express the average deformation gradient in terms of surface
quantities by using the divergence theorem. Thus,
⟨
F
⟩
=
1
V
0
∫
Ω
0
F
dV
=
1
V
0
∫
Ω
0
∇
0
x
dV
=
1
V
0
∫
∂
Ω
0
x
⊗
N
dA
=
1
V
0
∫
∂
Ω
0
(
X
+
u
)
⊗
N
dA
{\displaystyle \langle {\boldsymbol {F}}\rangle ={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {F}}~{\text{dV}}={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {\nabla }}_{0}~\mathbf {x} ~{\text{dV}}={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\mathbf {x} \otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}(\mathbf {X} +\mathbf {u} )\otimes \mathbf {N} ~{\text{dA}}}
where
N
{\displaystyle \mathbf {N} }
∂
Ω
0
{\displaystyle \partial {\Omega }_{0}}
u
(
X
)
=
x
−
X
{\displaystyle \mathbf {u} (\mathbf {X} )=\mathbf {x} -\mathbf {X} }
The surface integral can be converted into an integral over the deformed
surface using Nanson's formula for areas:
d
a
=
det
(
F
)
F
−
T
d
A
≡
n
da
=
det
(
F
)
F
−
T
⋅
N
dA
⟹
1
det
F
F
T
⋅
n
da
=
N
dA
{\displaystyle {\text{d}}\mathbf {a} =\det({\boldsymbol {F}})~{\boldsymbol {F}}^{-T}~{\text{d}}\mathbf {A} \qquad \equiv \qquad \mathbf {n} ~{\text{da}}=\det({\boldsymbol {F}})~{\boldsymbol {F}}^{-T}\cdot \mathbf {N} ~{\text{dA}}\quad \implies \quad {\cfrac {1}{\det {\boldsymbol {F}}}}~{\boldsymbol {F}}^{T}\cdot \mathbf {n} ~{\text{da}}=\mathbf {N} ~{\text{dA}}}
where
da
{\displaystyle {\text{da}}}
n
{\displaystyle \mathbf {n} }
dA
{\displaystyle {\text{dA}}}
The conservation of mass gives us
J
:=
det
(
F
)
=
ρ
0
ρ
=
V
V
0
.
{\displaystyle J:=\det({\boldsymbol {F}})={\cfrac {\rho _{0}}{\rho }}={\cfrac {V}{V_{0}}}~.}
Therefore,
x
⊗
N
dA
=
x
⊗
(
N
dA
)
=
x
⊗
(
V
0
V
F
T
⋅
n
da
)
=
(
V
0
V
)
x
⊗
(
F
T
⋅
n
)
da
{\displaystyle \mathbf {x} \otimes \mathbf {N} ~{\text{dA}}=\mathbf {x} \otimes (\mathbf {N} ~{\text{dA}})=\mathbf {x} \otimes \left({\cfrac {V_{0}}{V}}~{\boldsymbol {F}}^{T}\cdot \mathbf {n} ~{\text{da}}\right)=\left({\cfrac {V_{0}}{V}}\right)~\mathbf {x} \otimes ({\boldsymbol {F}}^{T}\cdot \mathbf {n} )~{\text{da}}}
Plugging into the surface integral, we have
⟨
F
⟩
=
1
V
0
∫
∂
Ω
0
x
⊗
N
dA
=
1
V
0
∫
∂
Ω
[
(
V
0
V
)
x
⊗
(
F
T
⋅
n
)
]
da
=
1
V
∫
∂
Ω
x
⊗
(
F
T
⋅
n
)
da
.
{\displaystyle \langle {\boldsymbol {F}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial \Omega _{0}}\mathbf {x} \otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V_{0}}}\int _{\partial \Omega }\left[\left({\cfrac {V_{0}}{V}}\right)~\mathbf {x} \otimes ({\boldsymbol {F}}^{T}\cdot \mathbf {n} )\right]~{\text{da}}={\cfrac {1}{V}}\int _{\partial \Omega }\mathbf {x} \otimes ({\boldsymbol {F}}^{T}\cdot \mathbf {n} )~{\text{da}}~.}
Using the identity
a
⊗
(
A
⋅
b
)
=
(
a
⊗
b
)
⋅
A
T
{\displaystyle \mathbf {a} \otimes ({\boldsymbol {A}}\cdot \mathbf {b} )=(\mathbf {a} \otimes \mathbf {b} )\cdot {\boldsymbol {A}}^{T}}
⟨
F
⟩
=
1
V
∫
∂
Ω
(
x
⊗
n
)
⋅
F
da
.
{\displaystyle \langle {\boldsymbol {F}}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}(\mathbf {x} \otimes \mathbf {n} )\cdot {\boldsymbol {F}}~{\text{da}}~.}
Therefore, the average deformation gradient in surface integral form
can be written as
⟨
F
⟩
=
1
V
0
∫
∂
Ω
0
x
⊗
N
dA
=
1
V
∫
∂
Ω
(
x
⊗
n
)
⋅
F
da
.
{\displaystyle {\langle {\boldsymbol {F}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\mathbf {x} \otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V}}\int _{\partial {\Omega }}(\mathbf {x} \otimes \mathbf {n} )\cdot {\boldsymbol {F}}~{\text{da}}~.}}
Note that there are three more conditions to be satisfied for the average
deformation gradient to behave like a macro variable, i.e.,
det
⟨
F
⟩
>
0
;
⟨
F
⟩
−
1
=
⟨
F
−
1
⟩
;
V
=
V
0
det
⟨
F
⟩
.
{\displaystyle \det \langle {\boldsymbol {F}}\rangle >0~;~~\langle {\boldsymbol {F}}\rangle ^{-1}=\langle {\boldsymbol {F}}^{-1\rangle }~;~~V=V_{0}\det \langle {\boldsymbol {F}}\rangle ~.}
These considerations and their detailed exploration can be found
in Costanzo et al.(2005).
![{\displaystyle \langle {\boldsymbol {F}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial \Omega _{0}}\mathbf {x} \otimes \mathbf {N} ~{\text{dA}}={\cfrac {1}{V_{0}}}\int _{\partial \Omega }\left[\left({\cfrac {V_{0}}{V}}\right)~\mathbf {x} \otimes ({\boldsymbol {F}}^{T}\cdot \mathbf {n} )\right]~{\text{da}}={\cfrac {1}{V}}\int _{\partial \Omega }\mathbf {x} \otimes ({\boldsymbol {F}}^{T}\cdot \mathbf {n} )~{\text{da}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/089e775d437b4d7dd883fe2d2df3c0893865a067)
