Let
σ
{\displaystyle {\boldsymbol {\sigma }}}
be the Cauchy stress and let
∇
v
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} }
be the velocity gradient in a body
Ω
{\displaystyle \Omega }
with boundary
∂
Ω
{\displaystyle \partial {\Omega }}
. Let
n
{\displaystyle \mathbf {n} }
be the normal to the boundary. Let
V
{\displaystyle V}
be the volume of the body. If the skew-symmetric part of the velocity gradient is zero, i.e.,
∇
v
=
∇
v
T
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} ={\boldsymbol {\nabla }}\mathbf {v} ^{T}}
, or if the stress field is self equilibrated, i.e.,
⟨
σ
⟩
=
⟨
σ
⟩
T
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle =\langle {\boldsymbol {\sigma }}\rangle ^{T}}
, show that
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
=
1
V
∫
∂
Ω
[
v
−
⟨
∇
v
⟩
⋅
x
]
⋅
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
dA
.
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}[\mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\cdot [({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]~{\text{dA}}~.}
Taking the trace of each term in the identity
⟨
A
⋅
B
⟩
−
⟨
A
⟩
⋅
⟨
B
⟩
=
⟨
[
A
−
⟨
A
⟩
]
⋅
[
B
−
⟨
B
⟩
]
⟩
{\displaystyle \langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle =\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle }
the difference between the average stress power and the product of the
average stress and the average velocity gradient can be written as
(using either the symmetry of the stress or of the velocity gradient)
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
=
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
−
⟨
∇
v
⟩
:
⟨
σ
⟩
+
⟨
∇
v
⟩
:
⟨
σ
⟩
=
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
−
⟨
∇
v
⟩
:
⟨
σ
⟩
+
[
⟨
∇
v
⟩
T
⋅
⟨
σ
⟩
]
:
1
{\displaystyle {\begin{aligned}\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle &=\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :\langle {\boldsymbol {\sigma }}\rangle +\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :\langle {\boldsymbol {\sigma }}\rangle \\&=\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :\langle {\boldsymbol {\sigma }}\rangle +[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \langle {\boldsymbol {\sigma }}\rangle ]:{\boldsymbol {\mathit {1}}}\end{aligned}}}
Recall that
⟨
σ
:
∇
v
⟩
=
1
V
∫
∂
Ω
(
σ
⋅
n
)
⋅
v
dV
;
⟨
∇
v
⟩
=
1
V
∫
Ω
∇
v
dV
;
⟨
σ
⟩
=
1
V
∫
∂
Ω
x
⊗
t
¯
dA
;
1
V
∫
Ω
∇
x
dV
=
1
.
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {\sigma }}\cdot \mathbf {n} )\cdot \mathbf {v} ~{\text{dV}}~;~~\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}~;~~\langle {\boldsymbol {\sigma }}\rangle ={\cfrac {1}{V}}~\int _{\partial {\Omega }}\mathbf {x} \otimes {\bar {\mathbf {t} }}~{\text{dA}}~;~~{\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {x} ~{\text{dV}}={\boldsymbol {\mathit {1}}}~.}
Also, from the divergence theorem
∫
Ω
∇
v
dV
=
∫
∂
Ω
v
⊗
n
dA
;
∫
Ω
∇
x
dV
=
∫
∂
Ω
x
⊗
n
dA
.
{\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}=\int _{\partial {\Omega }}\mathbf {v} \otimes \mathbf {n} ~{\text{dA}}~;~~\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {x} ~{\text{dV}}=\int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {n} ~{\text{dA}}~.}
Therefore,
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
=
1
V
∫
∂
Ω
(
σ
⋅
n
)
⋅
v
dV
−
⟨
σ
⟩
:
[
1
V
∫
∂
Ω
v
⊗
n
dA
]
−
⟨
∇
v
⟩
:
[
1
V
∫
∂
Ω
x
⊗
t
¯
dA
]
+
[
⟨
∇
v
⟩
T
⋅
⟨
σ
⟩
]
:
[
1
V
∫
∂
Ω
x
⊗
n
dA
]
.
{\displaystyle {\begin{aligned}\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle &={\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {\sigma }}\cdot \mathbf {n} )\cdot \mathbf {v} ~{\text{dV}}-\langle {\boldsymbol {\sigma }}\rangle :\left[{\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {v} \otimes \mathbf {n} ~{\text{dA}}\right]-\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :\left[{\cfrac {1}{V}}~\int _{\partial {\Omega }}\mathbf {x} \otimes {\bar {\mathbf {t} }}~{\text{dA}}\right]\\&\qquad \qquad +[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \langle {\boldsymbol {\sigma }}\rangle ]:\left[{\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {n} ~{\text{dA}}\right]~.\end{aligned}}}
Since
⟨
σ
⟩
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle }
and
⟨
∇
v
⟩
{\displaystyle \langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }
are independent of
x
{\displaystyle \mathbf {x} }
, we can take
these inside the integrals to get
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
=
1
V
∫
∂
Ω
[
(
σ
⋅
n
)
⋅
v
−
⟨
σ
⟩
:
(
v
⊗
n
)
−
⟨
∇
v
⟩
:
(
x
⊗
t
¯
)
+
[
⟨
∇
v
⟩
T
⋅
⟨
σ
⟩
]
:
(
x
⊗
n
)
]
dA
{\displaystyle {\begin{aligned}\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle &={\cfrac {1}{V}}\int _{\partial {\Omega }}\left[({\boldsymbol {\sigma }}\cdot \mathbf {n} )\cdot \mathbf {v} -\langle {\boldsymbol {\sigma }}\rangle :(\mathbf {v} \otimes \mathbf {n} )-\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :(\mathbf {x} \otimes {\bar {\mathbf {t} }})+[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \langle {\boldsymbol {\sigma }}\rangle ]:(\mathbf {x} \otimes \mathbf {n} )\right]~{\text{dA}}\end{aligned}}}
Using the identity
(
A
⋅
a
)
⋅
(
B
⋅
b
)
=
(
A
T
⋅
B
)
:
(
a
⊗
b
)
{\displaystyle ({\boldsymbol {A}}\cdot \mathbf {a} )\cdot ({\boldsymbol {B}}\cdot \mathbf {b} )=({\boldsymbol {A}}^{T}\cdot {\boldsymbol {B}}):(\mathbf {a} \otimes \mathbf {b} )}
we get
[
⟨
∇
v
⟩
T
⋅
⟨
σ
⟩
]
:
(
x
⊗
n
)
=
[
⟨
∇
v
⟩
⋅
x
]
⋅
[
⟨
σ
⟩
⋅
n
]
.
{\displaystyle [\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \langle {\boldsymbol {\sigma }}\rangle ]:(\mathbf {x} \otimes \mathbf {n} )=[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\cdot [\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} ]~.}
Also, using the identity
A
:
(
a
⊗
b
)
=
(
A
⋅
b
)
⋅
a
{\displaystyle {\boldsymbol {A}}:(\mathbf {a} \otimes \mathbf {b} )=({\boldsymbol {A}}\cdot \mathbf {b} )\cdot \mathbf {a} }
we get
⟨
σ
⟩
:
(
v
⊗
n
)
=
[
⟨
σ
⟩
⋅
n
]
⋅
v
;
⟨
∇
v
⟩
:
(
x
⊗
t
¯
)
=
[
⟨
∇
v
⟩
⋅
t
¯
]
⋅
x
=
[
⟨
∇
v
⟩
T
⋅
x
]
⋅
t
¯
=
[
⟨
∇
v
⟩
T
⋅
x
]
⋅
(
σ
⋅
n
)
.
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle :(\mathbf {v} \otimes \mathbf {n} )=[\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} ]\cdot \mathbf {v} ~;~~\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :(\mathbf {x} \otimes {\bar {\mathbf {t} }})=[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot {\bar {\mathbf {t} }}]\cdot \mathbf {x} =[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \mathbf {x} ]\cdot {\bar {\mathbf {t} }}=[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \mathbf {x} ]\cdot ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~.}
Since
∇
v
T
=
∇
v
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} ^{T}={\boldsymbol {\nabla }}\mathbf {v} }
, we have
⟨
∇
v
⟩
T
=
⟨
∇
v
⟩
{\displaystyle \langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}=\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }
(we could
alternatively use the symmetry of
⟨
σ
⟩
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle }
to arrive at the following
equation). Hence,
⟨
∇
v
⟩
:
(
x
⊗
t
¯
)
=
[
⟨
∇
v
⟩
⋅
x
]
⋅
(
σ
⋅
n
)
.
{\displaystyle \langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :(\mathbf {x} \otimes {\bar {\mathbf {t} }})=[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\cdot ({\boldsymbol {\sigma }}\cdot \mathbf {n} )~.}
Plugging these back into the original equation, we have
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
=
1
V
∫
∂
Ω
{
(
σ
⋅
n
)
⋅
v
−
[
⟨
σ
⟩
⋅
n
]
⋅
v
−
[
⟨
∇
v
⟩
⋅
x
]
⋅
(
σ
⋅
n
)
+
[
⟨
∇
v
⟩
⋅
x
]
⋅
[
⟨
σ
⟩
⋅
n
]
}
dA
=
1
V
∫
∂
Ω
{
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
⋅
v
−
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
⋅
(
⟨
∇
v
⟩
⋅
x
)
}
dA
=
1
V
∫
∂
Ω
{
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
⋅
[
v
−
⟨
∇
v
⟩
⋅
x
]
}
dA
.
{\displaystyle {\begin{aligned}\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle &={\cfrac {1}{V}}\int _{\partial {\Omega }}\left\{({\boldsymbol {\sigma }}\cdot \mathbf {n} )\cdot \mathbf {v} -[\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} ]\cdot \mathbf {v} -[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\cdot ({\boldsymbol {\sigma }}\cdot \mathbf {n} )+[\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\cdot [\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} ]\right\}~{\text{dA}}\\&={\cfrac {1}{V}}\int _{\partial {\Omega }}\left\{[({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]\cdot \mathbf {v} -[({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]\cdot (\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} )\right\}~{\text{dA}}\\&={\cfrac {1}{V}}\int _{\partial {\Omega }}\left\{[({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]\cdot [\mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\right\}~{\text{dA}}~.\end{aligned}}}
Hence
⟨
σ
:
∇
v
⟩
−
⟨
σ
⟩
:
⟨
∇
v
⟩
=
1
V
∫
∂
Ω
[
[
v
−
⟨
∇
v
⟩
⋅
x
]
⋅
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
]
dA
.
◻
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}\left[[\mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ]\cdot [({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]\right]~{\text{dA}}~.}\qquad \qquad \qquad \qquad \square }