Measurement of deformations and displacements by means of photoelastic sensors

Abstract

Photoelastic sensors in the form of cylinders with an axial hole, made from optically active material (usually optical glass), are used to measure stresses in rocks [1][2][3][4]. This article deals with the use of such sensors to measure deformations and displacements of rocks.Let a bore~ole of radius R be drilled in a uniform, isotropic, and elastic rock; a photoetastic sensor, the outer radius of which is equal to R and coincides with the borehole radius, and the inner radius of which is equM to R i. is glued m the borehole wails. The material of the sensor is also uniform, isotropic, and elastic. It will be assumed that the distance between the sensor and the end of the borehole is so great that the effect of the latter on the operation of the sensor may be disregarded. Hence the problem of the interaction of the sensor with the rock may be regarded as a problem of the plane theory of elasticity.We will assume that after the sensor has been placed in the borehole, some unknown uniformly distributed forces p and q, acting in the plane perpendicular to uhe sensor axis, are applied to the rock at a fairly considerable distance from the borehole with the sensor (the distance is such that, in comparison with the dimensions of the sensor. it may be taken as infinite) (see Fig. 1). Let E, G, and u be the modulus of elasticity, the modulus of shear, and Poisson's ratio, respectively, and E 1, GI, and v t the corresponding values for the material of the sensor.Forces p and q induce stresses in the rock and the sensor; the following conditions must be satisfied: the conditions at inf~ity (x = y = ~) Gz=P;Gv~-q,the boundary conditions when r = R 1 at=0;%e=O,the conjugation conditions when r = R Or= oft; Tre = %ei;ttr.-~U, rl; Oe~---eel,where o and o. are the stresses in the rock in a rectangular system of coordinates, o r and fro are respectively the x y radial and tangential stresses in the rock in a polar system of coordinates, u r and v O are respectively the radial and tangential displacements in the rock, Urz and ~-tO1 are respectively the radial and tangential stresses in the sensor, and Uri and v01 are respectively the radial and tangential displacements in the sensor.