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$$.