Micromechanics of composites/Average displacement in a RVE

Average Displacement in a RVE
The average displacement in a RVE may be defined as

{ \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

\boldsymbol{\nabla} \bullet (\mathbf{v}\otimes\boldsymbol{w}) = \mathbf{v}\cdot(\boldsymbol{\nabla} \bullet \boldsymbol{w}) + (\boldsymbol{\nabla}\mathbf{v})\cdot\boldsymbol{w} $$ where $$\mathbf{v}$$ and $$\boldsymbol{w}$$ are two vector fields.

If we choose $$\mathbf{v}$$ such that $$\boldsymbol{\nabla}\mathbf{v} = \boldsymbol{\mathit{1}}$$ in the above identity, then we can get an equation for $$\boldsymbol{w}$$, i.e.,

\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} ~. $$ Now, $$\boldsymbol{\nabla}\mathbf{v} = \boldsymbol{\mathit{1}}$$ if $$\mathbf{v} = \mathbf{x}$$. Therefore,

\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

\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

{ \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.

Incompressible materials
In the material is incompressible, the balance of mass gives us

\boldsymbol{\nabla} \bullet \mathbf{u} = 0 ~. $$ In that case,

\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.