Micromechanics of composites/Average deformation gradient in a RVE

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Average deformation gradient in a RVE[edit | edit source]

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).