Micromechanics of composites/Average deformation gradient in a RVE

Average deformation gradient in a RVE
The average deformation gradient is defined as

{ \langle \boldsymbol{F}\rangle := \cfrac{1}{V_0}\int_{\Omega_0} \boldsymbol{F}~\text{dV} } $$ where $$V_0$$ is the volume in the reference configuration.

We can express the average deformation gradient in terms of surface quantities by using the divergence theorem. Thus,

\langle \boldsymbol{F}\rangle = \cfrac{1}{V_0}\int_{\Omega_0} \boldsymbol{F}~\text{dV} = \cfrac{1}{V_0}\int_{\Omega_0} \boldsymbol{\nabla}_0~ \mathbf{x}~\text{dV} = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} \mathbf{x}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} (\mathbf{X}+\mathbf{u})\otimes\mathbf{N}~\text{dA} $$ where $$\mathbf{N}$$ is the unit outward normal to the reference surface $$\partial{\Omega}_0$$ and $$\mathbf{u}(\mathbf{X}) = \mathbf{x} - \mathbf{X}$$ is the displacement.

The surface integral can be converted into an integral over the deformed surface using Nanson's formula for areas:

\text{d}\mathbf{a} = \det(\boldsymbol{F})~\boldsymbol{F}^{-T}~\text{d}\mathbf{A}\qquad\equiv\qquad \mathbf{n}~\text{da} = \det(\boldsymbol{F})~\boldsymbol{F}^{-T}\cdot\mathbf{N}~\text{dA} \quad \implies \quad \cfrac{1}{\det{\boldsymbol{F}}}~\boldsymbol{F}^T\cdot\mathbf{n}~\text{da} = \mathbf{N}~\text{dA} $$ where $$\text{da}$$ is an element of area on the deformed surface, $$\mathbf{n}$$ is the outward normal to the deformed surface, and $$\text{dA}$$ is an element of area on the reference surface.

The conservation of mass gives us

J := \det(\boldsymbol{F}) = \cfrac{\rho_0}{\rho} = \cfrac{V}{V_0}~. $$ Therefore,

\mathbf{x}\otimes\mathbf{N}~\text{dA} = \mathbf{x}\otimes(\mathbf{N}~\text{dA}) = \mathbf{x}\otimes\left(\cfrac{V_0}{V}~\boldsymbol{F}^T\cdot\mathbf{n}~\text{da}\right) = \left(\cfrac{V_0}{V}\right)~\mathbf{x}\otimes(\boldsymbol{F}^T\cdot\mathbf{n})~\text{da} $$ Plugging into the surface integral, we have

\langle \boldsymbol{F} \rangle = \cfrac{1}{V_0} \int_{\partial \Omega_0} \mathbf{x}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V_0} \int_{\partial \Omega} \left[ \left(\cfrac{V_0}{V}\right)~\mathbf{x}\otimes(\boldsymbol{F}^T\cdot\mathbf{n}) \right]~\text{da} = \cfrac{1}{V} \int_{\partial \Omega } \mathbf{x}\otimes(\boldsymbol{F}^T\cdot\mathbf{n})~\text{da} ~. $$ Using the identity $$\mathbf{a}\otimes(\boldsymbol{A}\cdot\mathbf{b}) = (\mathbf{a}\otimes\mathbf{b})\cdot\boldsymbol{A}^T$$ (see Appendix), we get

\langle \boldsymbol{F}\rangle = \cfrac{1}{V} \int_{\partial{\Omega}} (\mathbf{x}\otimes\mathbf{n})\cdot\boldsymbol{F}~\text{da} ~. $$ Therefore, the average deformation gradient in surface integral form can be written as

{ \langle \boldsymbol{F}\rangle = \cfrac{1}{V_0} \int_{\partial{\Omega}_0} \mathbf{x}\otimes\mathbf{N}~\text{dA} = \cfrac{1}{V} \int_{\partial{\Omega}} (\mathbf{x}\otimes\mathbf{n})\cdot\boldsymbol{F}~\text{da} ~. } $$

Note that there are three more conditions to be satisfied for the average deformation gradient to behave like a macro variable, i.e.,

\det\langle \boldsymbol{F}\rangle > 0 ~; \langle \boldsymbol{F}\rangle^{-1} = \langle \boldsymbol{F}^{-1 \rangle} ~; V = V_0 \det\langle \boldsymbol{F}\rangle ~. $$ These considerations and their detailed exploration can be found in Costanzo et al.(2005).