I am not sure what is causing your element distortions. Can you try with a single element and just apply a uniaxial load?
Abaqus/Explicit expects the stresses to be returned in a co-rotational coordinate system. You can achieve this by either calculating the Cauchy stress based on the deformation gradient and then rotating the stress, or you can achieve the same effect (but more efficiently) if you simply calculate the stress using the left Cauchy-Green tensor U.
-Jorgen