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

${\langle {\boldsymbol {P}}\rangle ={\cfrac {1}{V_{0}}}\int _{\Omega _{0}}{\boldsymbol {P}}~{\text{dV}}~.}$

Recall the relation (see Appendix)

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

In the above equation, let the volume integral be over $\Omega _{0}$ and let the surface integral be over $\partial {\Omega }_{0}$. Let the unit outward normal to $\partial {\Omega }_{0}$ be $\mathbf {N}$. Let the gradient and divergence operations be with respect to the reference configuration. Also, let $\mathbf {v} \rightarrow \mathbf {X}$ and let ${\boldsymbol {S}}\rightarrow {\boldsymbol {P}}$. Then we have

$\int _{\partial {\Omega }_{0}}\mathbf {X} \otimes ({\boldsymbol {P}}^{T}\bullet \mathbf {N} )~{\text{dA}}=\int _{\Omega _{0}}[{\boldsymbol {\nabla }}_{0}~\mathbf {X} \cdot {\boldsymbol {P}}+\mathbf {X} \otimes ({\boldsymbol {\nabla }}_{0}\bullet {\boldsymbol {P}}^{T})]~{\text{dV}}=\int _{\Omega _{0}}[{\boldsymbol {\mathit {1}}}\cdot {\boldsymbol {P}}+\mathbf {X} \otimes ({\boldsymbol {\nabla }}_{0}\bullet {\boldsymbol {P}}^{T})]~{\text{dV}}=\int _{\Omega _{0}}[{\boldsymbol {P}}+\mathbf {X} \otimes ({\boldsymbol {\nabla }}_{0}\bullet {\boldsymbol {P}}^{T})]~{\text{dV}}~.$

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

$\int _{\partial {\Omega }_{0}}\mathbf {X} \otimes ({\boldsymbol {P}}^{T}\bullet \mathbf {N} )~{\text{dA}}=\int _{\Omega _{0}}{\boldsymbol {P}}~{\text{dV}}=V_{0}~\langle {\boldsymbol {P}}\rangle ~.$

Let ${\bar {\mathbf {T} }}$ 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 ${\bar {\mathbf {T} }}={\boldsymbol {P}}^{T}\cdot \mathbf {N}$ on $\partial {\Omega }_{0}$. Hence we get

${\langle {\boldsymbol {P}}\rangle ={\cfrac {1}{V_{0}}}\int _{\partial {\Omega }_{0}}\mathbf {X} \otimes {\bar {\mathbf {T} }}~{\text{dA}}~.}$

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

${\langle {\overline {\boldsymbol {\sigma }}}\rangle :={\cfrac {1}{\det \langle {\boldsymbol {F}}\rangle }}~\langle {\boldsymbol {F}}\rangle \cdot \langle {\boldsymbol {P}}\rangle ~.}$

Note that, in general, $\langle {\overline {\boldsymbol {\sigma }}}\rangle \neq \langle {\boldsymbol {\sigma }}\rangle$.

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

${\langle {\overline {\boldsymbol {\tau }}}\rangle :=\det \langle {\boldsymbol {F}}\rangle ~\langle {\overline {\boldsymbol {\sigma }}}\rangle =\langle {\boldsymbol {F}}\rangle \cdot \langle {\boldsymbol {P}}\rangle ~.}$

In general, $\langle {\overline {\boldsymbol {\tau }}}\rangle \neq \langle {\boldsymbol {\tau }}\rangle$.

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Micromechanics of composites

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