The average displacement in a RVE may be defined as
⟨
u
⟩
:=
1
V
∫
Ω
u
(
x
)
dV
.
{\displaystyle {\langle \mathbf {u} \rangle :={\cfrac {1}{V}}\int _{\Omega }\mathbf {u} (\mathbf {x} )~{\text{dV}}~.}}
We would like to find the relation between the average displacement in a RVE and the applied displacements at the boundary of the RVE. To do that, recall the identity
∇
∙
(
v
⊗
w
)
=
v
⋅
(
∇
∙
w
)
+
(
∇
v
)
⋅
w
{\displaystyle {\boldsymbol {\nabla }}\bullet (\mathbf {v} \otimes {\boldsymbol {w}})=\mathbf {v} \cdot ({\boldsymbol {\nabla }}\bullet {\boldsymbol {w}})+({\boldsymbol {\nabla }}\mathbf {v} )\cdot {\boldsymbol {w}}}
where
v
{\displaystyle \mathbf {v} }
w
{\displaystyle {\boldsymbol {w}}}
If we choose
v
{\displaystyle \mathbf {v} }
∇
v
=
1
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} ={\boldsymbol {\mathit {1}}}}
w
{\displaystyle {\boldsymbol {w}}}
∇
∙
(
v
⊗
w
)
=
v
⋅
(
∇
∙
w
)
+
1
⋅
w
=
v
⋅
(
∇
∙
w
)
+
w
.
{\displaystyle {\boldsymbol {\nabla }}\bullet (\mathbf {v} \otimes {\boldsymbol {w}})=\mathbf {v} \cdot ({\boldsymbol {\nabla }}\bullet {\boldsymbol {w}})+{\boldsymbol {\mathit {1}}}\cdot {\boldsymbol {w}}=\mathbf {v} \cdot ({\boldsymbol {\nabla }}\bullet {\boldsymbol {w}})+{\boldsymbol {w}}~.}
Now,
∇
v
=
1
{\displaystyle {\boldsymbol {\nabla }}\mathbf {v} ={\boldsymbol {\mathit {1}}}}
v
=
x
{\displaystyle \mathbf {v} =\mathbf {x} }
∇
∙
(
x
⊗
w
)
=
x
⋅
(
∇
∙
w
)
+
w
⟹
w
=
∇
∙
(
x
⊗
w
)
−
x
⋅
(
∇
∙
w
)
.
{\displaystyle {\boldsymbol {\nabla }}\bullet (\mathbf {x} \otimes {\boldsymbol {w}})=\mathbf {x} \cdot ({\boldsymbol {\nabla }}\bullet {\boldsymbol {w}})+{\boldsymbol {w}}\qquad \implies \qquad {\boldsymbol {w}}={\boldsymbol {\nabla }}\bullet (\mathbf {x} \otimes {\boldsymbol {w}})-\mathbf {x} \cdot ({\boldsymbol {\nabla }}\bullet {\boldsymbol {w}})~.}
Using the above in the expression for the average displacement, we have
⟨
u
⟩
=
1
V
∫
Ω
[
∇
∙
(
x
⊗
u
)
−
x
⋅
(
∇
∙
u
)
]
dV
.
{\displaystyle \langle \mathbf {u} \rangle ={\cfrac {1}{V}}\int _{\Omega }[{\boldsymbol {\nabla }}\bullet (\mathbf {x} \otimes \mathbf {u} )-\mathbf {x} \cdot ({\boldsymbol {\nabla }}\bullet \mathbf {u} )]~{\text{dV}}~.}
Applying the divergence theorem to the first term on the right, we get
⟨
u
⟩
=
1
V
∫
∂
Ω
(
x
⊗
u
)
⋅
n
dV
−
1
V
∫
Ω
x
⋅
(
∇
∙
u
)
dV
.
{\displaystyle {\langle \mathbf {u} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}(\mathbf {x} \otimes \mathbf {u} )\cdot \mathbf {n} ~{\text{dV}}-{\cfrac {1}{V}}\int _{\Omega }\mathbf {x} \cdot ({\boldsymbol {\nabla }}\bullet \mathbf {u} )~{\text{dV}}~.}}
There are two terms in the above expression: the first is a boundary term while the second requires information from the interior of the body. Hence, in general, the average displacement of a RVE cannot be determined solely on the basis of boundary displacements.
In the material is incompressible, the balance of mass gives us
∇
∙
u
=
0
.
{\displaystyle {\boldsymbol {\nabla }}\bullet \mathbf {u} =0~.}
In that case,
⟨
u
⟩
=
1
V
∫
∂
Ω
(
x
⊗
u
)
⋅
n
dV
.
{\displaystyle \langle \mathbf {u} \rangle ={\cfrac {1}{V}}\int _{\partial {\Omega }}(\mathbf {x} \otimes \mathbf {u} )\cdot \mathbf {n} ~{\text{dV}}~.}
It's only in this special case that the average displacement in the RVE can be expressed in terms of boundary displacements.
![{\displaystyle \langle \mathbf {u} \rangle ={\cfrac {1}{V}}\int _{\Omega }[{\boldsymbol {\nabla }}\bullet (\mathbf {x} \otimes \mathbf {u} )-\mathbf {x} \cdot ({\boldsymbol {\nabla }}\bullet \mathbf {u} )]~{\text{dV}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d82f2d0f96430e4faec906c1187ecb45332a8e7)
