Independent derivations are given for the failure criteria of the purely dilatational stress state involving voids nucleation failure as well as for the purely distortional stress state involving shear bands failure. The results are consistent with those from a recently derived failure theory and they further substantiate the failure theory. The voids nucleation mechanism is compared with the ideal theoretical strength of isotropic materials as derived by density functional theory and two other atomic-scale methods. It is found that a cross-over occurs from the voids nucleation failure mechanism to the ideal strength limitation as the tensile to compressive strengths ratio, T / C , increases toward a value of unity. All the results are consistent with the failure modes transition results from the general failure theory.