Determination of Stresses from the Stress Trajectory Pattern in a Plane Elastic Domain

Abstract

Identification of stresses acting in a plane (homogeneous and isotropic) elastic domain is performed based on the analysis of principal stress trajectories. This problem, originated in photoelasticity, is now of great importance in geodynamics. Given stress trajectories, in photoelasticity stresses are found by solving a certain boundary value problem. We propose the solution of the problem without appealing to boundary conditions, which is advantageous to geodynamics where boundary stresses are poorly constrained. The analysis of the given stress trajectory pattern is equivalently reduced to the investigation of the argument, α, of the complex-valued bi-holomorphic function, D, which represents the stress deviator of the 2D stress tensor. Necessary and sufficient conditions for the stress trajectory pattern to be admissible in elasticity are established. A procedure to obtain a particular solution, D ₁, is presented. The general solution for D derived from D ₁ depends on four (if α is a harmonic function) or one (otherwise) arbitrary real constants. The procedure is illustrated by model examples for West European and Australian platforms.