Micromechanics of composites/Proof 12

Question

Let ${\boldsymbol {A}}$ and ${\boldsymbol {B}}$ be two second-order tensor fields. Let the average of any second-order tensor field (${\boldsymbol {S}}$) over the region $\Omega$ (of volume $V$) be defined as

$\langle {\boldsymbol {S}}\rangle :={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {S}}~{\text{dV}}~.$

Show that

$\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle =\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle ~.$

Proof

Expanding out the right hand side, we have

${\begin{aligned}\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle &={\cfrac {1}{V}}\int _{\Omega }[{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]~{\text{dV}}\\&={\cfrac {1}{V}}\int _{\Omega }[{\boldsymbol {A}}\cdot {\boldsymbol {B}}-\langle {\boldsymbol {A}}\rangle \cdot {\boldsymbol {B}}-{\boldsymbol {A}}\cdot \langle {\boldsymbol {B}}\rangle +\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle ]~{\text{dV}}\\&={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}\cdot {\boldsymbol {B}}~{\text{dV}}-{\cfrac {1}{V}}\int _{\Omega }\langle {\boldsymbol {A}}\rangle \cdot {\boldsymbol {B}}~{\text{dV}}-{\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}\cdot \langle {\boldsymbol {B}}\rangle ~{\text{dV}}+{\cfrac {1}{V}}\int _{\Omega }\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle ~{\text{dV}}~.\end{aligned}}$

Now $\langle {\boldsymbol {A}}\rangle$ and $\langle {\boldsymbol {B}}\rangle$ are constants with respect to the integration. Hence,

${\begin{aligned}\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle &={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}\cdot {\boldsymbol {B}}~{\text{dV}}-\langle {\boldsymbol {A}}\rangle \cdot \left({\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {B}}~{\text{dV}}\right)-\left({\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {B}}\rangle +\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle \left({\cfrac {1}{V}}\int _{\Omega }~{\text{dV}}\right)\\&=\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle +\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle \\&=\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle ~.\end{aligned}}$

Therefore,

${\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle =\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle \qquad \square }$

← Proof 11

Micromechanics of composites

Proof 13 →