The ideal strength and mechanical hardness of solids

Relationships between intrinsic mechanical hardness and atomic-scale properties are reviewed. Hardness scales closely and linearly with shear modulus for a given class of material (covalent, ionic or metallic). A two-parameter fit and a Peierls-stress model produce a more universal scaling relationship, but no model can explain differences in hardness between the transition metal carbides and nitrides. Calculations of "ideal strength" (defined by the limit of elastic stability of a perfect crystal) are proposed.; The ideal shear strengths of fcc aluminum and copper are calculated using ab initio techniques and allowing for structural relaxation of all five strain components other than the imposed strain. The strengths of Al and Cu are similar (8--9% of the shear modulus), but the geometry of the relaxations in Al and Cu is very different. The relaxations are consistent with experimentally measured third-order elastic constants.; The general thermodynamic conditions of elastic stability that set the upper limits of mechanical strength are derived. The conditions of stability are shown for cubic (hydrostatic), tetragonal (tensile) and monoclinic (shear) distortions of a cubic crystal. The implications of this stability analysis to first-principles calculations of ideal strength are discussed, and a method to detect instabilities orthogonal to the direction of the applied stress is identified.; The relaxed ideal shear and tensile strengths of bcc tungsten are also calculated using ab initio techniques and are favorably compared to recent nano-indentation measurements. The {lcub}100{rcub} tensile strength (29.5 GPa) is governed by the Bain instability. The shear strengths in the weak directions on {lcub}110{rcub}, {lcub}112{rcub}, and {lcub}123{rcub} planes are very nearly equal (≈18 GPa) and occur at approximately the same strain (17--18%). This isotropy is a function of the linear elastic isotropy for shear in directions containing ⟨111⟩ in bcc and of the atomic configurations of energetic saddle points reached during shear. This isotropy may also explain the prevalence of the pencil glide of dislocations in bcc metals.; A final chapter presents some recent ideal strength calculations of TiC and TiN and discusses future directions for research.