I wanted to know whether it was possible to obtain the deformation gradient F directly as an output from the Abaqus software, or whether it was necessary to post-process displacement data.
deformation gradient tensor components are not "canned" output variables, but I can suggest two ways to get it (probably more accurately than trying to calculate from displacement data):
Make an output request for log strain (If you are running with NLGEOM=yes, you will get this by default even if you merely request strain) and then either:
1. Defining custom field variables for F components by inverting the definition of log strain, or
2. Make a data report (rpt) file, import to Excel and do a spreadsheet calculation to get F as a function of log strain, again using an inversion of the definition of log strain.