a material's hardness but this no longer remains the dominant school of thought. For example, some alkali and noble metals (Pd, Ag) have anomalously high ratio of the bulk modulus to the Vickers of Brinell hardness. In the early 2000s, a direct relationship between bulk modulus and valence electron density was found as the more electrons were present the greater the repulsions within the structure were. Bulk modulus is still used as a preliminary measure of a material as superhard but it is now known that other properties must be taken into account. In contrast to bulk modulus, shear modulus measures the resistance to shape change at a constant volume, taking into account the crystalline plane and direction of shear. The shear modulus G is defined as ratio of shear stress to shear strain: G = stress/strain = F·L/(A·dx), where F is the applied force, A is the area upon which the force acts, dx is the resulting displacement and L is the initial length. The larger the shear modulus, the greater the ability for a material to resist shearing forces. Therefore, the shear modulus is a measure of rigidity. Shear modulus is related to bulk modulus as , where v is the Poisson's ratio, which is typically ~0.1 in covalent materials. If a material contains highly directional bonds, the shear modulus will increase and give a low Poisson ratio. A material is also considered hard if it resists plastic deformation. If a material has short covalent bonds, atomic dislocations that lead to plastic deformation are less likely to occur than in materials with longer, delocalized bonds. If a material contains many delocalized bonds it is likely to be soft. Somewhat related to hardness is another mechanical property fracture toughness, which is a material's ability to resist breakage from forceful impact (note that this concept is distinct from the notion of toughness). A superhard material is not necessarily "supertough". For example, the fracture toughness of diamond is about 7–10 MPa·m1/2, which is high compared to other gemstones and ceramic materials, but poor compared to many metals and alloys – common steels and aluminium alloys have the toughness values at least 5 times higher. Several properties must be taken into account when evaluating a material as (super)hard. While hard materials have high bulk moduli, a high bulk modulus does not mean a material is hard. Inelastic characteristics must be considered as