Micromechanics of composites/Average stress in a RVE with finite strain

Average stress in a RVE

The average nominal (first Piola-Kirchhoff ) stress is defined as

$⟨𝑷⟩=\frac{1}{V_0}\underset{{\mathrm{\Omega }}_0}{\int }𝑷\text{dV}.$

Recall the relation (see Appendix)

$\underset{\partial \mathrm{\Omega }}{\int }𝐯\otimes ({𝑺}^T\bullet 𝐧)\text{dA}=\underset{\mathrm{\Omega }}{\int }[\mathrm{\nabla }𝐯\cdot 𝑺+𝐯\otimes (\mathrm{\nabla }\bullet {𝑺}^T)]\text{dV}.$

In the above equation, let the volume integral be over ${\mathrm{\Omega }}_0$ and let the surface integral be over $\partial {\mathrm{\Omega }}_0$. Let the unit outward normal to $\partial {\mathrm{\Omega }}_0$ be $𝐍$. Let the gradient and divergence operations be with respect to the reference configuration. Also, let $𝐯\to 𝐗$ and let $𝑺\to 𝑷$. Then we have

$\underset{\partial {\mathrm{\Omega }}_0}{\int }𝐗\otimes ({𝑷}^T\bullet 𝐍)\text{dA}=\underset{{\mathrm{\Omega }}_0}{\int }[{\mathrm{\nabla }}_0𝐗\cdot 𝑷+𝐗\otimes ({\mathrm{\nabla }}_0\bullet {𝑷}^T)]\text{dV}=\underset{{\mathrm{\Omega }}_0}{\int }[1\cdot 𝑷+𝐗\otimes ({\mathrm{\nabla }}_0\bullet {𝑷}^T)]\text{dV}=\underset{{\mathrm{\Omega }}_0}{\int }[𝑷+𝐗\otimes ({\mathrm{\nabla }}_0\bullet {𝑷}^T)]\text{dV}.$

If we assume that there are no inertial forces or body forces, then ${\mathrm{\nabla }}_0\bullet {𝑷}^T=0$ (from the conservation of linear momentum), and we have

$\underset{\partial {\mathrm{\Omega }}_0}{\int }𝐗\otimes ({𝑷}^T\bullet 𝐍)\text{dA}=\underset{{\mathrm{\Omega }}_0}{\int }𝑷\text{dV}=V_0⟨𝑷⟩.$

Let $\overline{𝐓}$ be a self equilibrating traction that is applied to the RVE, i.e., it does not lead to any inertial forces. Then, Cauchy's law states that $\overline{𝐓}={𝑷}^T\cdot 𝐍$ on $\partial {\mathrm{\Omega }}_0$. Hence we get

$⟨𝑷⟩=\frac{1}{V_0}\underset{\partial {\mathrm{\Omega }}_0}{\int }𝐗\otimes \overline{𝐓}\text{dA}.$

Given the above, the average Cauchy stress in the RVE is defined as

$⟨\bar{\mathit{\sigma }}⟩:=\frac{1}{\det ⟨𝑭⟩}⟨𝑭⟩\cdot ⟨𝑷⟩.$

Note that, in general, $⟨\bar{\mathit{\sigma }}⟩\ne ⟨\mathit{\sigma }⟩$.

The Kirchhoff stress is defined as $\mathit{\tau }:=\det 𝑭\mathit{\sigma }$. The average Kirchhoff stress in the RVE is defined as

$⟨\bar{\mathit{\tau }}⟩:=\det ⟨𝑭⟩⟨\bar{\mathit{\sigma }}⟩=⟨𝑭⟩\cdot ⟨𝑷⟩.$

In general, $⟨\bar{\mathit{\tau }}⟩\ne ⟨\mathit{\tau }⟩$.

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