Because there is no breaking of bonds, the material springs back when the force is released. This type of deformation is modeled using a linear relationship between stress and strain, i.e. a Hookean relationship. Where σ denotes stress, denotes strain, and E is the elastic modulus, which is material dependent. The elastic modulus is, in effect, a measure of the strength of atomic bonds. Plastic deformation refers to non-reversible deformation. The relationship between stress and strain for permanent deformation is nonlinear. Stress has caused permanent change of shape in the material by involving the breaking of bonds. One mechanism of plastic deformation is the movement of dislocations by an applied stress. Because rocks are essentially aggregates of minerals, we can think of them as poly-crystalline materials. Dislocations are a type of crystallographic defect which consists of an extra or missing half plane of atoms in the periodic array of atoms that make up a crystal lattice. Dislocations are present in all real crystallographic materials. Hardness Hardness is difficult to quantify. It is a measure of resistance to deformation, specifically permanent deformation. There is precedent for hardness as a surface quality, a measure of the abrasiveness or surface-scratching resistance of a material. If the material being tested, however, is uniform in composition and structure, then the surface of the material is only a few atomic layers thick, and measurements are of the bulk material. Thus, simple surface measurements yield information about the bulk properties. Ways to measure hardness include: Mohs Scale Dorry abrasion test Deval abrasion test Indentation hardness Indentation hardness is used often in metallurgy and materials science and can be thought of as resistance to penetration by an indenter. Toughness Toughness can be described best by a material's resistance to cracking. During plastic deformation, a material absorbs energy until fracture occurs. The area under the stress-strain curve is the work required to fracture the material. The toughness modulus is defined as: Where is the ultimate tensile strength, and is the strain at failure. The modulus is the maximum amount of energy per unit volume a material can absorb without fracturing. From the equation for modulus, for large toughness, high strength and high ductility are needed. These two properties are usually mutually exclusive. Brittle materials have low toughness because low plastic deformation decreases the strain (low ductility). Ways to measure toughness include: Page impact machine and Charpy impact test Resilience