A Brief Review of Elasticity

Abstract

This is a very brief review of the elasticity theory needed to understand the principles of stress, strain, and flexure in Geodynamics [Turcotte and Schubert, 2002]. This review assumes that you have already taken a class in continuum mechanics. One difference from T&S is that we follow the sign convention used by seismologists and engineers where extensional strain and stress is positive. Stress Stress is a force acting on an area is measured in Newtons per meter squared (N m –2) which corresponds to a Pascal unit (Pa). The following diagram shows a cube of solid material. Each face of the cube has three components of stress so there are 9 possible components of the stress tensor. We will consider only the symmetric part of the stress tensor so only 6 of these components are independent. The antisymmetric part of the tensor represents a torque. In Cartesian coordinates the stress tensor is given by σ ij = σ xx σ xy σ xz σ xy σ yy σ yz σ xz σ yz σ zz ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ where index notation is the shorthand for dealing with tensors and vectors; a variable with a single subscript is a vector  a = a i , a variable with two subscripts is a tensor σ = σ ij , and a repeated index indicates summation over the spatial coordinates. For example the pressure is given by P = −σ ii / 3. In addition, a comma preceding a subscript means differentiation with respect to that variable ∇  a = a i, j or for example a x, y = ∂a x ∂y .