strain/stress in local material coordinates

We are doing simulations of simple bilayer composites which undergo extensive buckling and post-buckling instabilities. The simulation is being done with ABAQUS explicit to capture the highly non-linear geometries. Everything works very well, however to compare with our analytical theory we need the Green strain at each element. ABAQUS for solid elements gives only the NE or LE which is projected onto the global coordinate system. Of course the principle strains are available, however we are interested in the local rotational modes and in particular would like the strain mapped onto the local co-rotational reference coordinate system. ABAQUS clearly has access to the  deformation gradient tensor F (non-symmetric 3x3 matrix of the stretch ratios) at each integration point of an element, how can I map the unit vectors of the reference coordinate system (x=[1 0 0], y=[0 1 0], z=[0 0 1]) to the new unit vectors x', y', z' at the deformed state.x' = x . Fy' = y . Fz' = z , for each element? I have read this is possible with VUMAT subroutines, however these seem to be used only for special constitutive relations; my system is linearly elastic. Is there a way without evoking the subroutines that the above mapping can be done? Thank you for any help you can offer.