# 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).