In FEA, the nodes (grid points) move, but the properties do not change. Properties include area for 1D elements (CROD, CBAR) and thicknesses for 2D elements (CQUAD4/CTRIA3). 3D elements do not have section properties, because the "width" and "thickness" come from the node locations.
If you modeled a simple tension test with 1D CROD elements, you would get the axial extension, but not area change. If you modeled a flat tensile specimen with CQUAD4 elements, you would be the stretch, but you would also get the transverse contraction, because the nodes will move in that direction. But you will not get the contraction of thickness. If you truly want to see the contraction, both laterally and through the thickness, you would need to model your tension specimen with 3D solid (CHEXA) elements. But those changes in width and thickness come from the nodes moving, not from changes in the element section properties.
1D and 2D elements are engineering approximations, which are entirely adequate for solving many real engineering problems. I am not suggesting you need 3D solids for everything, it is just that you should be aware of the limitations of different types of modeling.
I would suggest another test case for SOL 106: a flat plate under uniform pressure. Make a square plate with all 4 sides simply supported. Apply a uniform normal pressure (PLOAD). Run it both linear (SOL 101) and nonlinear (SOL 106, large displacements but linear material). Analytical solutions are available for both cases, say in Timoshenko or Roark. At small loads, the models should give similar results. At higher loads, the nonlinear model should deflect much less than the linear one. This is because the linear model does not include the effect of membrane stiffening that occurs after the plate deflects more than about 1/2 the thickness.
Note that in a nonlinear solution, you really want to track the behavior, not just get to the end point all at once. You want to see how linear or nonlinear it really is, and if you are nearing a point of instability. Therefore you should get output at several increments of load along the way, not just at the final load. There are different ways to do this (see the documentation). In my example above, one thing you could track would be the deflection of the node at the center of the plate. Then you would have something to compare linear to nonlinear results. If the normal pressure was 10 psi, get results at 1, 2, 5, 10 psi to trace the curve.
P.S. It is always a good idea to run your models linearly first to debug them. Only when they are running correctly and giving expected results should you move on to the nonlinear runs.