# Micromechanics of composites/Proof 14
## Question

Let ${\boldsymbol {P}}$ be the first Piola-Kirchhoff stress and let ${\dot {\boldsymbol {F}}}$ be the time rate of the deformation gradient in a body whose reference configuration is $\Omega _{0}$ with boundary $\partial {\Omega }_{0}$. Let $\mathbf {N}$ be the normal to the boundary. Let $V_{0}$ be the volume of the body. Let $\mathbf {X}$ represent the position of points in the reference configuration. Let ${\dot {\mathbf {x} }}$ be the material time derivative of $\mathbf {x}$. Let $\langle {\boldsymbol {A}}\rangle$ represent the unweighted volume average of a quantity ${\boldsymbol {A}}$. Show that
    ${\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}~.\end{aligned}}$

## Proof

Recall the identity
    $\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle =\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle ~.$
Therefore,
    ${\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &=\langle [{\dot {\boldsymbol {F}}}-\langle {\dot {\boldsymbol {F}}}\rangle ]\cdot [{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]\rangle \\&={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}[{\dot {\boldsymbol {F}}}-\langle {\dot {\boldsymbol {F}}}\rangle ]\cdot [{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]~{\text{dV}}\\&={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}~{\text{dV}}-{\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}\cdot \langle {\boldsymbol {P}}\rangle ~{\text{dV}}-{\cfrac {1}{V_{0}}}\int _{\Omega _{0}}\langle {\dot {\boldsymbol {F}}}\rangle \cdot {\boldsymbol {P}}~{\text{dV}}+{\cfrac {1}{V_{0}}}\int _{\Omega _{0}}\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle ~{\text{dV}}\\&={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}~{\text{dV}}-\left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {P}}~{\text{dV}}\right)+\langle {\dot {\boldsymbol {F}}}\rangle \cdot \left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {\mathit {1}}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {P}}\rangle ~.\end{aligned}}$
We want express the volume integrals above in terms of surface integrals. To do that, recall that
    ${\begin{aligned}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}~{\text{dV}}&=\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}&=\int _{\Omega _{0}}{\boldsymbol {\nabla }}_{0}~{\dot {\mathbf {x} }}~{\text{dV}}=\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \mathbf {N} ~{\text{dA}}\\\int _{\Omega _{0}}{\boldsymbol {P}}~{\text{dV}}&=\int _{\partial {\Omega }_{0}}\mathbf {X} \otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\\int _{\Omega _{0}}{\boldsymbol {\mathit {1}}}~{\text{dV}}&=\int _{\Omega _{0}}{\boldsymbol {\nabla }}_{0}~\mathbf {X} ~{\text{dV}}=\int _{\partial {\Omega }_{0}}\mathbf {X} \otimes \mathbf {N} ~{\text{dA}}~.\end{aligned}}$
Therefore,
    ${\begin{aligned}{\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}~{\text{dV}}&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\\left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}({\dot {\mathbf {x} }}\otimes \mathbf {N} )\cdot \langle {\boldsymbol {P}}\rangle ~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes [\langle {\boldsymbol {P\rangle ^{T}\cdot \mathbf {N} ]}}~{\text{dA}}\\\langle {\dot {\boldsymbol {F}}}\rangle \cdot \left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {P}}~{\text{dV}}\right)&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\langle {\dot {\boldsymbol {F}}}\rangle \cdot [\mathbf {X} \otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )]~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\\langle {\dot {\boldsymbol {F}}}\rangle \cdot \left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {\mathit {1}}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\langle {\dot {\boldsymbol {F}}}\rangle \cdot (\mathbf {X} \otimes \mathbf {N} )\cdot \langle {\boldsymbol {P}}\rangle ~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [\langle {\boldsymbol {P}}\rangle ^{T}\cdot \mathbf {N} ]~{\text{dA}}~.\end{aligned}}$
Collecting the terms, we have
    ${\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\left\{{\dot {\mathbf {x} }}\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )-{\dot {\mathbf {x} }}\otimes [\langle {\boldsymbol {P\rangle ^{T}\cdot \mathbf {N} ]}}-[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )+[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [\langle {\boldsymbol {P}}\rangle ^{T}\cdot \mathbf {N} ]\right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\left\{{\dot {\mathbf {x} }}\otimes [{\boldsymbol {P}}^{T}\cdot \mathbf {N} -\langle {\boldsymbol {P\rangle ^{T}\cdot \mathbf {N} ]}}-[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [{\boldsymbol {P}}^{T}\cdot \mathbf {N} -\langle {\boldsymbol {P}}\rangle ^{T}\cdot \mathbf {N} ]\right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [{\boldsymbol {P}}^{T}\cdot \mathbf {N} -\langle {\boldsymbol {P\rangle ^{T}\cdot \mathbf {N} ]}}~{\text{dA}}~.\end{aligned}}$
Therefore,
    ${\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}~.}$
From the above, clearly
    $\left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\dot {\boldsymbol {F}}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {P}}\rangle =\langle {\dot {\boldsymbol {F}}}\rangle \cdot \left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {P}}~{\text{dV}}\right)=\langle {\dot {\boldsymbol {F}}}\rangle \cdot \left({\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {\mathit {1}}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {P}}\rangle ~.$
Therefore,
    ${\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes [\langle {\boldsymbol {P\rangle ^{T}\cdot \mathbf {N} ]}}~{\text{dA}}={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [\langle {\boldsymbol {P}}\rangle ^{T}\cdot \mathbf {N} ]~{\text{dA}}~.$
Thus we can alternatively write the expression for the difference as
    ${\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\left\{{\dot {\mathbf {x} }}\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )-[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )-\left[{\dot {\mathbf {x} }}\otimes [\langle {\boldsymbol {P}}^{T}\rangle \cdot \mathbf {N} ]-[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [\langle {\boldsymbol {P}}^{T}\rangle \cdot \mathbf {N} ]\right]\right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\end{aligned}}$
or,
    ${\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\left\{{\dot {\mathbf {x} }}\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )-{\dot {\mathbf {x} }}\otimes [\langle {\boldsymbol {P}}^{T}\rangle \cdot \mathbf {N} ]-\left[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )-[\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes [\langle {\boldsymbol {P}}^{T}\rangle \cdot \mathbf {N} ]\right]\right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes [{\boldsymbol {P}}^{T}\cdot \mathbf {N} -\langle {\boldsymbol {P}}^{T}\rangle \cdot \mathbf {N} ]~{\text{dA}}~.\end{aligned}}$
Hence,
    ${\begin{aligned}\langle {\dot {\boldsymbol {F}}}\cdot {\boldsymbol {P}}\rangle -\langle {\dot {\boldsymbol {F}}}\rangle \cdot \langle {\boldsymbol {P}}\rangle &={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}[{\dot {\mathbf {x} }}-\langle {\dot {\boldsymbol {F}}}\rangle \cdot \mathbf {X} ]\otimes ({\boldsymbol {P}}^{T}\cdot \mathbf {N} )~{\text{dA}}\\&={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}{\dot {\mathbf {x} }}\otimes \left\{[{\boldsymbol {P}}-\langle {\boldsymbol {P}}\rangle ]^{T}\cdot \mathbf {N} \right\}~{\text{dA}}~.\end{aligned}}$