Mechanics of Fracture

Abstract

Calculations of the theoretical strengths of crystalline solids are usually based on idealised forms of atomic force-displacement curves, in which the force is defined as the differential with respect to distance of the inter-atomic or inter-ionic energy. The energy curve is similar to that for a diatomic molecule in that it represents the resultant of inter-atomic repulsions and attractions; the nature and strength of the attractive forces depending on the bond type: ionic, covalent, metallic, or Van der Waals. Differences in character between a lattice and a molecule occur at separations of order one lattice spacing, when Friedel oscillations in the screening charge cause the long range component of the interaction potential in the lattice to undergo a damped oscillation about zero. For small displacements, the atomic force/displacement curve is linear, having a slope equivalent to Young’s modulus, E. A lattice also has shear stiffness, denoted by the shear modulus μ. Both E and μ are defined macroscopically, usually for randomly-oriented polycrystals which are assumed to be isotropic. In single crystals, both the tensile stiffness and the shear stiffness vary with the orientation of the crystal with respect to the tensile axis and these variations can be substantial: in iron, for example, the minimum value of E is in the [100]direction (132 GPa at room temperature) and the maximum value is in the [111] direction (260 GPa).