Deformation gradient tensor and FEA question

Hi,

Whenever I read explainations of the deformation gradient tensor (DGT) it is always with reguard to infinitesimal line segments.

Could anybody explain how this idea is expanded to obtain the DGT of a finite element?

For instance how would one go about constructing the DGT for a shell element? How would the principle stretches be obtained?

Thanks for any help with this (even if its just speculation).

B