Micromechanics of composites/Average deformation gradient in a RVE
Average Deformation Gradient in a RVE 
The average deformation gradient is defined as
where 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,
where is the unit outward normal to the reference surface and is the displacement.
The surface integral can be converted into an integral over the deformed surface using Nanson's formula for areas:
where is an element of area on the deformed surface, is the outward normal to the deformed surface, and is an element of area on the reference surface.
The conservation of mass gives us
Plugging into the surface integral, we have
Using the identity (see Appendix), we get
Therefore, the average deformation gradient in surface integral form can be written as
Note that there are three more conditions to be satisfied for the average deformation gradient to behave like a macro variable, i.e.,
These considerations and their detailed exploration can be found in Costanzo et al.(2005).