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{align} \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{align} $$

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{align} \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{align} $$ We want express the volume integrals above in terms of surface integrals. To do that, recall that

\begin{align} \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{align} $$ Therefore,

\begin{align} \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{align} $$ Collecting the terms, we have

\begin{align} \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{align} $$ 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{align} \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{align} $$ or,

\begin{align} \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{align} $$ Hence,

{ \begin{align} \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{align} } $$