Micromechanics of composites/Average velocity gradient in a RVE

Average velocity gradient in a RVE
The time rate of the deformation gradient is given by

\dot{\boldsymbol{F}} = \frac{\partial }{\partial t}[\boldsymbol{F}(\mathbf{X}, t)] = \frac{\partial }{\partial t}\left(\frac{\partial }{\partial \mathbf{X}}[\mathbf{x}(\mathbf{X},t)]\right) = \frac{\partial }{\partial \mathbf{X}}\left(\frac{\partial }{\partial t}[\mathbf{x}(\mathbf{X},t)]\right) = \frac{\partial \dot{\mathbf{x}}}{\partial \mathbf{X}} = \boldsymbol{\nabla}_0~ \dot{\mathbf{x}} ~. $$ The average time rate of the deformation gradient is defined as

{ \langle \dot{\boldsymbol{F}} \rangle := \cfrac{1}{V_0}\int_{\Omega_0} \dot{\boldsymbol{F}}~\text{dV} ~. } $$ Following the same procedure as in the previous section, we can show that

{ \langle \dot{\boldsymbol{F}} \rangle = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} \dot{\mathbf{x}}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V} \int_{\partial{\Omega}} (\dot{\mathbf{x}}\otimes\mathbf{n})\cdot\boldsymbol{F}~\text{da} ~. } $$ The velocity gradient ($$\boldsymbol{l}$$) is given by

\boldsymbol{l} = \boldsymbol{\nabla}\mathbf{v} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} $$ where $$\mathbf{v}(\mathbf{x})$$ is the velocity.

The average velocity gradient in a RVE is defined as

{ \overline{\boldsymbol{l}} := \langle \dot{\boldsymbol{F}} \rangle\cdot\langle \boldsymbol{F}\rangle^{-1} ~. } $$ Note that $$\overline{\boldsymbol{l}} = \langle\boldsymbol{l}\rangle = \langle\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}\rangle$$ only if $$\boldsymbol{F} = \boldsymbol{\mathit{1}}$$.