The nanomechanical properties of polymethyl methacrylate and indium phosphide were measured with an atomic force microscope and a nanoindentation system. The elastic moduli measured with the atomic force microscope are in good agreement with the values obtained with the nanoindentation system. The hardness is shown to be affected by the tip radius used in our experiments. The cantilever vertical and lateral movements were independently analyzed during nanoindentation, and the tip torsion can be attributed to a change from elastic to plastic deformation regimes of materials during force microscopy nanoindentation. An analysis of the lateral movement of the laser beam associated with the cantilever torsion was used to determine the material yield stress.