Micromechanics of composites/Average deformation gradient in a RVE

Average deformation gradient in a RVE

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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

 

Therefore,

 

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).