The equation for the balance of energy is
ρ
e
˙
−
σ
:
(
∇
v
)
+
∇
∙
q
−
ρ
s
=
0
.
{\displaystyle \rho ~{\dot {e}}-{\boldsymbol {\sigma }}:({\boldsymbol {\nabla }}\mathbf {v} )+{\boldsymbol {\nabla }}\bullet \mathbf {q} -\rho ~s=0~.}
If the absence of heat flux or heat sources in the RVE, the equation reduces to
ρ
e
˙
=
σ
:
(
∇
v
)
.
{\displaystyle \rho ~{\dot {e}}={\boldsymbol {\sigma }}:({\boldsymbol {\nabla }}\mathbf {v} )~.}
The quantity on the right is the stress power density and is a measure of the internal energy density of the material.
The average stress power in a RVE is defined as
⟨
σ
:
∇
v
⟩
:=
1
V
∫
Ω
σ
:
∇
v
dV
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle :={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}~.}}
Note that the quantities
σ
{\displaystyle {\boldsymbol {\sigma }}}
∇
v
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} }
The average velocity gradient
⟨
∇
v
⟩
{\displaystyle \langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }
⟨
∇
v
⟩
:=
1
V
∫
Ω
∇
v
dV
.
{\displaystyle {\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle :={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}~.}}
To get an expression for the average stress power in terms of the boundary conditions, we use the identity
∇
∙
(
S
T
⋅
v
)
=
S
:
∇
v
+
(
∇
∙
S
)
⋅
v
{\displaystyle {\boldsymbol {\nabla }}\bullet ({\boldsymbol {S}}^{T}\cdot \mathbf {v} )={\boldsymbol {S}}:{\boldsymbol {\nabla }}\mathbf {v} +({\boldsymbol {\nabla }}\bullet {\boldsymbol {S}})\cdot \mathbf {v} }
to get
⟨
σ
:
∇
v
⟩
=
1
V
∫
Ω
σ
:
∇
v
dV
=
1
V
∫
Ω
[
∇
∙
(
σ
T
⋅
v
)
−
(
∇
∙
σ
)
⋅
v
]
dV
.
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}={\cfrac {1}{V}}\int _{\Omega }\left[{\boldsymbol {\nabla }}\bullet ({\boldsymbol {\sigma }}^{T}\cdot \mathbf {v} )-({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})\cdot \mathbf {v} \right]~{\text{dV}}~.}
Using the balance of linear momentum (
∇
∙
σ
=
0
{\displaystyle {\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}=0}
⟨
σ
:
∇
v
⟩
=
1
V
∫
Ω
∇
∙
(
σ
T
⋅
v
)
dV
.
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\nabla }}\bullet ({\boldsymbol {\sigma }}^{T}\cdot \mathbf {v} )~{\text{dV}}~.}
Using the divergence theorem, we have
⟨
σ
:
∇
v
⟩
=
1
V
∫
∂
Ω
(
σ
T
⋅
v
)
⋅
n
dV
=
1
V
∫
∂
Ω
(
σ
T
⋅
v
)
⋅
n
dV
=
1
V
∫
∂
Ω
(
σ
⋅
n
)
⋅
v
dV
.
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {\sigma }}^{T}\cdot \mathbf {v} )\cdot \mathbf {n} ~{\text{dV}}={\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {\sigma }}^{T}\cdot \mathbf {v} )\cdot \mathbf {n} ~{\text{dV}}={\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {\sigma }}\cdot \mathbf {n} )\cdot \mathbf {v} ~{\text{dV}}~.}
Now, the surface traction is given by
t
¯
=
σ
⋅
n
{\displaystyle {\bar {\mathbf {t} }}={\boldsymbol {\sigma }}\cdot \mathbf {n} }
⟨
σ
:
∇
v
⟩
=
1
V
∫
∂
Ω
t
¯
⋅
v
dV
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}{\bar {\mathbf {t} }}\cdot \mathbf {v} ~{\text{dV}}~.}}
{\scriptsize
}
In micromechanics, it is of interest to see how the average stress power of a RVE is related to the product of the average stress
⟨
σ
⟩
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle }
⟨
∇
v
⟩
{\displaystyle \langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }
⟨
σ
:
∇
v
⟩
=
⟨
σ
⟩
:
⟨
∇
v
⟩
.
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle =\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ~.}
However, this is not true in general. We can show that if the gradient of the velocity is a symmetric tensor (i.e., there is no spin) , then (see Appendix for proof)
⟨
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}}~.}}
We can arrive at
⟨
σ
:
∇
v
⟩
=
⟨
σ
⟩
:
⟨
∇
v
⟩
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle =\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }
∂
Ω
{\displaystyle \partial {\Omega }}
v
=
⟨
∇
v
⟩
⋅
x
{\displaystyle \mathbf {v} =\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} }
σ
⋅
n
=
⟨
σ
⟩
⋅
n
{\displaystyle {\boldsymbol {\sigma }}\cdot \mathbf {n} =\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} }
Linear boundary velocities
edit
If the prescribed velocities on
∂
Ω
{\displaystyle \partial {\Omega }}
x
{\displaystyle \mathbf {x} }
v
(
x
)
=
G
⋅
x
∀
x
∈
∂
Ω
{\displaystyle \mathbf {v} (\mathbf {x} )={\boldsymbol {G}}\cdot \mathbf {x} \qquad \qquad \forall ~\mathbf {x} \in \partial {\Omega }}
where
G
{\displaystyle {\boldsymbol {G}}}
From the divergence theorem
∫
Ω
∇
a
dV
=
∫
∂
Ω
a
⊗
n
dA
.
{\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}\mathbf {a} ~{\text{dV}}=\int _{\partial {\Omega }}\mathbf {a} \otimes \mathbf {n} ~{\text{dA}}~.}
Therefore,
⟨
∇
v
⟩
=
1
V
∫
Ω
∇
v
dV
=
1
V
∫
∂
Ω
v
⊗
n
dA
.
{\displaystyle \langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}={\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {v} \otimes \mathbf {n} ~{\text{dA}}~.}
Hence, on the boundary
v
−
⟨
∇
v
⟩
⋅
x
=
G
⋅
x
−
[
1
V
∫
∂
Ω
(
G
⋅
x
)
⊗
n
dA
]
⋅
x
{\displaystyle \mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ={\boldsymbol {G}}\cdot \mathbf {x} -\left[{\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {G}}\cdot \mathbf {x} )\otimes \mathbf {n} ~{\text{dA}}\right]\cdot \mathbf {x} }
Using the identity (see Appendix)
(
A
⋅
a
)
⊗
b
=
A
⋅
(
a
⊗
b
)
{\displaystyle ({\boldsymbol {A}}\cdot \mathbf {a} )\otimes \mathbf {b} ={\boldsymbol {A}}\cdot (\mathbf {a} \otimes \mathbf {b} )}
and since
G
{\displaystyle {\boldsymbol {G}}}
v
−
⟨
∇
v
⟩
T
⋅
x
=
G
⋅
x
−
[
G
⋅
(
1
V
∫
∂
Ω
x
⊗
n
dA
)
]
⋅
x
.
{\displaystyle \mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \mathbf {x} ={\boldsymbol {G}}\cdot \mathbf {x} -\left[{\boldsymbol {G}}\cdot \left({\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {n} ~{\text{dA}}\right)\right]\cdot \mathbf {x} ~.}
From the divergence theorem,
∫
∂
Ω
x
⊗
n
dA
=
∫
Ω
∇
x
dV
=
∫
Ω
1
dV
=
V
1
.
{\displaystyle \int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {n} ~{\text{dA}}=\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {x} ~{\text{dV}}=\int _{\Omega }{\boldsymbol {\mathit {1}}}~{\text{dV}}=V~{\boldsymbol {\mathit {1}}}~.}
Therefore,
v
−
⟨
∇
v
⟩
⋅
x
=
G
⋅
x
−
(
G
⋅
1
)
⋅
x
=
G
⋅
x
−
G
⋅
x
=
0
⟹
⟨
σ
:
∇
v
⟩
=
⟨
σ
⟩
:
⟨
∇
v
⟩
.
{\displaystyle \mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ={\boldsymbol {G}}\cdot \mathbf {x} -({\boldsymbol {G}}\cdot {\boldsymbol {\mathit {1}}})\cdot \mathbf {x} ={\boldsymbol {G}}\cdot \mathbf {x} -{\boldsymbol {G}}\cdot \mathbf {x} =\mathbf {0} \qquad \implies \qquad {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle =\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }~.}
If the prescribed tractions on the boundary
∂
Ω
{\displaystyle \partial {\Omega }}
G
{\displaystyle {\boldsymbol {G}}}
t
¯
(
x
)
=
G
⋅
n
(
x
)
∀
x
∈
∂
Ω
.
{\displaystyle {\bar {t}}(\mathbf {x} )={\boldsymbol {G}}\cdot \mathbf {n} (\mathbf {x} )\qquad \qquad \forall ~\mathbf {x} \in \partial {\Omega }~.}
The tractions are related to the stresses at the boundary of the RVE by
t
¯
=
σ
⋅
n
{\displaystyle {\bar {t}}={\boldsymbol {\sigma }}\cdot \mathbf {n} }
The average stress in the RVE is given by
⟨
σ
⟩
=
1
V
∫
∂
Ω
x
⊗
t
¯
dA
=
1
V
∫
∂
Ω
x
⊗
(
G
⋅
n
)
dA
.
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {x} \otimes {\bar {t}}~{\text{dA}}={\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {x} \otimes ({\boldsymbol {G}}\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}}
⟨
σ
⟩
=
1
V
∫
∂
Ω
(
x
⊗
n
)
⋅
G
T
dA
.
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}(\mathbf {x} \otimes \mathbf {n} )\cdot {\boldsymbol {G}}^{T}~{\text{dA}}~.}
Since
G
{\displaystyle {\boldsymbol {G}}}
⟨
σ
⟩
=
(
1
V
∫
∂
Ω
x
⊗
n
dA
)
⋅
G
.
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle =\left({\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {n} ~{\text{dA}}\right)\cdot {\boldsymbol {G}}~.}
Applying the divergence theorem,
⟨
σ
⟩
=
(
1
V
∫
Ω
∇
x
dV
)
⋅
G
=
1
⋅
G
=
G
.
{\displaystyle \langle {\boldsymbol {\sigma }}\rangle =\left({\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\nabla }}\mathbf {x} ~{\text{dV}}\right)\cdot {\boldsymbol {G}}={\boldsymbol {\mathit {1}}}\cdot {\boldsymbol {G}}={\boldsymbol {G}}~.}
Therefore,
σ
⋅
n
−
⟨
σ
⟩
⋅
n
=
t
¯
−
G
⋅
n
=
0
⟹
⟨
σ
:
∇
v
⟩
=
⟨
σ
⟩
:
⟨
∇
v
⟩
.
{\displaystyle {\boldsymbol {\sigma }}\cdot \mathbf {n} -\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} ={\bar {t}}-{\boldsymbol {G}}\cdot \mathbf {n} =\mathbf {0} \qquad \implies \qquad {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle =\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle }~.}
Recall that for small deformations, the displacement gradient
∇
u
{\displaystyle {\boldsymbol {\nabla }}\mathbf {u} }
∇
u
=
ε
+
ω
.
{\displaystyle {\boldsymbol {\nabla }}\mathbf {u} ={\boldsymbol {\varepsilon }}+{\boldsymbol {\omega }}~.}
For small deformations, the time derivative of
∇
u
{\displaystyle {\boldsymbol {\nabla }}\mathbf {u} }
∇
v
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} }
∇
v
=
ε
˙
+
ω
˙
.
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} ={\dot {\boldsymbol {\varepsilon }}}+{\dot {\boldsymbol {\omega }}}~.}
If
ω
=
0
{\displaystyle {\boldsymbol {\omega }}=0}
∇
v
=
ε
˙
.
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} ={\dot {\boldsymbol {\varepsilon }}}~.}
Hence, for small strains and in the absence of rigid body rotations, the stress power density is given by
σ
:
ε
˙
{\displaystyle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}}
⟨
σ
:
ε
˙
⟩
:=
1
V
∫
Ω
σ
:
ε
˙
dV
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}\rangle :={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}~{\text{dV}}~.}}
and the average strain rate is defined as
⟨
ε
˙
⟩
:=
1
V
∫
Ω
ε
˙
dV
.
{\displaystyle {\langle {\dot {\boldsymbol {\varepsilon }}}\rangle :={\cfrac {1}{V}}\int _{\Omega }{\dot {\boldsymbol {\varepsilon }}}~{\text{dV}}~.}}
In terms of the surface tractions and the applied boundary velocities, we have
⟨
σ
:
ε
˙
⟩
=
1
V
∫
∂
Ω
t
¯
⋅
u
˙
dV
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}{\bar {\mathbf {t} }}\cdot {\dot {\mathbf {u} }}~{\text{dV}}~.}}
For small strains and no rotation, the stress-power difference relation becomes
⟨
σ
:
ε
˙
⟩
−
⟨
σ
⟩
:
⟨
ε
˙
⟩
=
1
V
∫
∂
Ω
[
u
˙
−
⟨
∇
u
˙
⟩
⋅
x
]
⋅
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
dA
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}\rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\dot {\boldsymbol {\varepsilon }}}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}[{\dot {\mathbf {u} }}-\langle {\boldsymbol {\nabla }}{\dot {\mathbf {u} }}\rangle \cdot \mathbf {x} ]\cdot [({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]~{\text{dA}}~.}}
We can arrive at
⟨
σ
:
ε
˙
⟩
=
⟨
σ
⟩
:
⟨
ε
˙
⟩
{\displaystyle \langle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}\rangle =\langle {\boldsymbol {\sigma }}\rangle :\langle {\dot {\boldsymbol {\varepsilon }}}\rangle }
∂
Ω
{\displaystyle \partial {\Omega }}
u
˙
=
⟨
∇
u
˙
⟩
⋅
x
⟹
{\displaystyle {\dot {\mathbf {u} }}=\langle {\boldsymbol {\nabla }}{\dot {\mathbf {u} }}\rangle \cdot \mathbf {x} \qquad \implies \qquad }
σ
⋅
n
=
⟨
σ
⟩
⋅
n
⟹
{\displaystyle {\boldsymbol {\sigma }}\cdot \mathbf {n} =\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} \qquad \implies \qquad }
We can also show in an identical manner that
⟨
σ
:
ε
⟩
=
1
V
∫
∂
Ω
t
¯
⋅
u
dV
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}{\bar {\mathbf {t} }}\cdot \mathbf {u} ~{\text{dV}}~.}}
and that, when
∇
u
{\displaystyle {\boldsymbol {\nabla }}\mathbf {u} }
⟨
σ
:
ε
⟩
−
⟨
σ
⟩
:
⟨
ε
⟩
=
1
V
∫
∂
Ω
[
u
−
⟨
∇
u
⟩
⋅
x
]
⋅
[
(
σ
−
⟨
σ
⟩
)
⋅
n
]
dA
.
{\displaystyle {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\varepsilon }}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}[\mathbf {u} -\langle {\boldsymbol {\nabla }}\mathbf {u} \rangle \cdot \mathbf {x} ]\cdot [({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]~{\text{dA}}~.}}
In this case, we can arrive at the relation
⟨
σ
:
ε
⟩
=
⟨
σ
⟩
:
⟨
ε
⟩
{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\rangle =\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\varepsilon }}\rangle }
u
=
⟨
∇
u
⟩
⋅
x
⟹
{\displaystyle \mathbf {u} =\langle {\boldsymbol {\nabla }}\mathbf {u} \rangle \cdot \mathbf {x} \qquad \implies \qquad }
σ
⋅
n
=
⟨
σ
⟩
⋅
n
⟹
{\displaystyle {\boldsymbol {\sigma }}\cdot \mathbf {n} =\langle {\boldsymbol {\sigma }}\rangle \cdot \mathbf {n} \qquad \implies \qquad }
![{\displaystyle \langle {\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} \rangle ={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {\sigma }}:{\boldsymbol {\nabla }}\mathbf {v} ~{\text{dV}}={\cfrac {1}{V}}\int _{\Omega }\left[{\boldsymbol {\nabla }}\bullet ({\boldsymbol {\sigma }}^{T}\cdot \mathbf {v} )-({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }})\cdot \mathbf {v} \right]~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d22cf3137b65175518e8a533278573492809d469)
![{\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}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1440a30417998062deb25a8876f186cdfa719968)
![{\displaystyle \mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle \cdot \mathbf {x} ={\boldsymbol {G}}\cdot \mathbf {x} -\left[{\cfrac {1}{V}}\int _{\partial {\Omega }}({\boldsymbol {G}}\cdot \mathbf {x} )\otimes \mathbf {n} ~{\text{dA}}\right]\cdot \mathbf {x} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0551303921fd242b12f53cf5aedda1bb5693c4)
![{\displaystyle \mathbf {v} -\langle {\boldsymbol {\nabla }}\mathbf {v} \rangle ^{T}\cdot \mathbf {x} ={\boldsymbol {G}}\cdot \mathbf {x} -\left[{\boldsymbol {G}}\cdot \left({\cfrac {1}{V}}\int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {n} ~{\text{dA}}\right)\right]\cdot \mathbf {x} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c7fe9e1e1b7bdd7e2ca9f632d2e0fc4bc0da38)
![{\displaystyle {\langle {\boldsymbol {\sigma }}:{\dot {\boldsymbol {\varepsilon }}}\rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\dot {\boldsymbol {\varepsilon }}}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}[{\dot {\mathbf {u} }}-\langle {\boldsymbol {\nabla }}{\dot {\mathbf {u} }}\rangle \cdot \mathbf {x} ]\cdot [({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]~{\text{dA}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7aa0ed2e928f825db24d9631681911d1cbe448c)
![{\displaystyle {\langle {\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}\rangle -\langle {\boldsymbol {\sigma }}\rangle :\langle {\boldsymbol {\varepsilon }}\rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}[\mathbf {u} -\langle {\boldsymbol {\nabla }}\mathbf {u} \rangle \cdot \mathbf {x} ]\cdot [({\boldsymbol {\sigma }}-\langle {\boldsymbol {\sigma }}\rangle )\cdot \mathbf {n} ]~{\text{dA}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ae7768c9861cfedcfcbed92e15284113b4a6fc)
