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, $\langle \overline{\boldsymbol{\sigma}} \rangle \ne \langle \boldsymbol{\sigma} \rangle$.

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, $\langle \overline{\boldsymbol{\tau}} \rangle \ne \langle \boldsymbol{\tau} \rangle$.