Three-Dimensional Photoelasticity

Abstract

The classical Sommerfeld-Runge method for deriving the laws of geometric optics in isotropic inhomogeneous media from Maxwell's equations is applied to problems in ``weakly'' anisotropic and inhomogeneous media. This leads to a generalized Hamilton-Jacobi partial differential equation for the phases of the two waves (eikonals) which can propagate with finite velocity in such a medium. If one restricts oneself to a limited class of stress distributions, one derives a very powerful method of solving axially and spherically symmetric problems which would be very difficult to treat by the direct solution of the classical equations of elasticity. The Maxwell-Neumann law is derived in variational form so that, in principle, one can estimate in any given case the error involved in the conventional assumption that the paths are straight lines instead of the actual Fermat paths. The last section treats the so-called general problem of photoelasticity in which the principal stress directions in the plane of the wave front rotate arbitrarily as the latter propagates through the medium. Again the Sommerfeld-Runge method gives one a solution which claims that if the rate of rotation is small in a wavelength one can ignore the contributions of the rotation to the integrated phase. An experiment is proposed to check this claim.