Micromechanics of composites/Proof 14

Question

Let $𝑷$ be the first Piola-Kirchhoff stress and let $\dot{𝑭}$ be the time rate of the deformation gradient in a body whose reference configuration is ${\mathrm{\Omega }}_0$ with boundary $\partial {\mathrm{\Omega }}_0$. Let $𝐍$ be the normal to the boundary. Let $V_0$ be the volume of the body. Let $𝐗$ represent the position of points in the reference configuration. Let $\dot{𝐱}$ be the material time derivative of $𝐱$. Let $⟨𝑨⟩$ represent the unweighted volume average of a quantity $𝑨$. Show that

$\begin{array}{cc}⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes \{[𝑷-⟨𝑷⟩{]}^T\cdot 𝐍\}\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes \{[𝑷-⟨𝑷⟩{]}^T\cdot 𝐍\}\text{dA}.\end{array}$

Proof

Recall the identity

$⟨𝑨\cdot 𝑩⟩-⟨𝑨⟩\cdot ⟨𝑩⟩=⟨[𝑨-⟨𝑨⟩]\cdot [𝑩-⟨𝑩⟩]⟩.$

Therefore,

$\begin{array}{cc}⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩& =⟨[\dot{𝑭}-⟨\dot{𝑭}⟩]\cdot [𝑷-⟨𝑷⟩]⟩\\ & =\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }[\dot{𝑭}-⟨\dot{𝑭}⟩]\cdot [𝑷-⟨𝑷⟩]\text{dV}\\ & =\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\cdot 𝑷\text{dV}-\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\cdot ⟨𝑷⟩\text{dV}-\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }⟨\dot{𝑭}⟩\cdot 𝑷\text{dV}+\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩\text{dV}\\ & =\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\cdot 𝑷\text{dV}-\left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\text{dV}\right)\cdot ⟨𝑷⟩-⟨\dot{𝑭}⟩\cdot \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }𝑷\text{dV}\right)+⟨\dot{𝑭}⟩\cdot \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }1\text{dV}\right)\cdot ⟨𝑷⟩.\end{array}$

We want express the volume integrals above in terms of surface integrals. To do that, recall that

$\begin{array}{cc}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\cdot 𝑷\text{dV}& =\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\\ \underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\text{dV}& =\underset{{\mathrm{\Omega }}_0}{\int }{\mathrm{\nabla }}_0\dot{𝐱}\text{dV}=\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes 𝐍\text{dA}\\ \underset{{\mathrm{\Omega }}_0}{\int }𝑷\text{dV}& =\underset{\partial {\mathrm{\Omega }}_0}{\int }𝐗\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\\ \underset{{\mathrm{\Omega }}_0}{\int }1\text{dV}& =\underset{{\mathrm{\Omega }}_0}{\int }{\mathrm{\nabla }}_0𝐗\text{dV}=\underset{\partial {\mathrm{\Omega }}_0}{\int }𝐗\otimes 𝐍\text{dA}.\end{array}$

Therefore,

$\begin{array}{cc}\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\cdot 𝑷\text{dV}& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\\ \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\text{dV}\right)\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }(\dot{𝐱}\otimes 𝐍)\cdot ⟨𝑷⟩\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes [⟨𝑷{\mathit{⟩}}^{𝑻}\mathit{\cdot }𝐍\mathit{]}\text{dA}\\ ⟨\dot{𝑭}⟩\cdot \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }𝑷\text{dV}\right)& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }⟨\dot{𝑭}⟩\cdot [𝐗\otimes ({𝑷}^T\cdot 𝐍)]\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\\ ⟨\dot{𝑭}⟩\cdot \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }1\text{dV}\right)\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }⟨\dot{𝑭}⟩\cdot (𝐗\otimes 𝐍)\cdot ⟨𝑷⟩\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [⟨𝑷{⟩}^T\cdot 𝐍]\text{dA}.\end{array}$

Collecting the terms, we have

$\begin{array}{cc}⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\{\dot{𝐱}\otimes ({𝑷}^T\cdot 𝐍)-\dot{𝐱}\otimes [⟨𝑷{\mathit{⟩}}^{𝑻}\mathit{\cdot }𝐍\mathit{]}-[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)+[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [⟨𝑷{⟩}^T\cdot 𝐍]\}\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\{\dot{𝐱}\otimes [{𝑷}^T\cdot 𝐍-⟨𝑷{\mathit{⟩}}^{𝑻}\mathit{\cdot }𝐍\mathit{]}-[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [{𝑷}^T\cdot 𝐍-⟨𝑷{⟩}^T\cdot 𝐍]\}\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [{𝑷}^T\cdot 𝐍-⟨𝑷{\mathit{⟩}}^{𝑻}\mathit{\cdot }𝐍\mathit{]}\text{dA}.\end{array}$

Therefore,

$⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes \{[𝑷-⟨𝑷⟩{]}^T\cdot 𝐍\}\text{dA}.$

From the above, clearly

$\left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }\dot{𝑭}\text{dV}\right)\cdot ⟨𝑷⟩=⟨\dot{𝑭}⟩\cdot \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }𝑷\text{dV}\right)=⟨\dot{𝑭}⟩\cdot \left(\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }1\text{dV}\right)\cdot ⟨𝑷⟩.$

Therefore,

$\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes [⟨𝑷{\mathit{⟩}}^{𝑻}\mathit{\cdot }𝐍\mathit{]}\text{dA}=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)\text{dA}=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [⟨𝑷{⟩}^T\cdot 𝐍]\text{dA}.$

Thus we can alternatively write the expression for the difference as

$\begin{array}{cc}⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\{\dot{𝐱}\otimes ({𝑷}^T\cdot 𝐍)-[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)-[\dot{𝐱}\otimes [⟨{𝑷}^T⟩\cdot 𝐍]-[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [⟨{𝑷}^T⟩\cdot 𝐍]]\}\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\end{array}$

or,

$\begin{array}{cc}⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\{\dot{𝐱}\otimes ({𝑷}^T\cdot 𝐍)-\dot{𝐱}\otimes [⟨{𝑷}^T⟩\cdot 𝐍]-[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)-[⟨\dot{𝑭}⟩\cdot 𝐗]\otimes [⟨{𝑷}^T⟩\cdot 𝐍]]\}\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes [{𝑷}^T\cdot 𝐍-⟨{𝑷}^T⟩\cdot 𝐍]\text{dA}.\end{array}$

Hence,

$\begin{array}{cc}⟨\dot{𝑭}\cdot 𝑷⟩-⟨\dot{𝑭}⟩\cdot ⟨𝑷⟩& =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes \{[𝑷-⟨𝑷⟩{]}^T\cdot 𝐍\}\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }[\dot{𝐱}-⟨\dot{𝑭}⟩\cdot 𝐗]\otimes ({𝑷}^T\cdot 𝐍)\text{dA}\\ & =\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }\dot{𝐱}\otimes \{[𝑷-⟨𝑷⟩{]}^T\cdot 𝐍\}\text{dA}.\end{array}$

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