Lecture on mineral physics part 2: linear elasticity

Abstract

This article explores linear elasticity to understand the elastic properties of minerals. First, strain is introduced. When a body is displaced by some function, the translation of the gravitational center of the body should be subtracted to understand the deformation in the body. In addition, the rotation of the body is not allowed also subtracted. We define the strain as a partial differential of the displacement with respect to the position in the body. Strains perpendicular and parallel to the coordinates are referred to as normal and shear strains, respectively. The strains are second-rank symmetric tensors. Because of the symmetry, the set of strains can be viewed as a combination of normal strains perpendicular to each other. Stress is a force per unit area in a given plane on the surface or inside the body. The stresses perpendicular and parallel to the plane of interest are referred to as normal and shear stresses, respectively. The stresses are also second-rank symmetric tensors. The set of stresses can also be viewed as a combination of normal stresses perpendicular to each other. Hook's law states the linearity of an applied force to a displacement of the body. Hook's law is generalized using the strain and stress tensors. The proportional constants of strains to stresses are referred to as the elastic moduli. The elastic moduli are fourth-rank tensors. Due to symmetry, the number of independent elastic moduli is 21.