Interaction of a photoelastic transducer with an elastic-hereditary medium

Abstract

Photoelastic stress transducers are simple, convenient devices for measuring stresses in rocks. They can react not only to elastic deformation of the solid rock, but also to creep deformation [1]. In this article we shall discuss the theory of interaction of a transducer with a rock mass possessing elastic-hereditary properties.In stress measurements in the field, it is possible to operate a photoelastic transducer under conditions of plane deformation, when the transducer is fixed in a deep borehole at a sufficiently great distance from its end, or in conditions of a plane state of stress, when the transducer is fixed near the surface of a mine working at a depth of 3-10 cm. In this connection the problem of the interaction of a photoelastic transducer with the rock mass must be regarded as a general case of the plane problem.In an elastic-hereditary isotropic rock mass acted on by forces p and q, suppose that we instantaneously drill a borehole of radius R and immediately install a photoelastic transducer and glue it to the walls of the borehole (Fig. 1). We shall assume that the elastic deformations in the rock due to the cutting of the borehole are completed before the transducer is installed. The internal radius of the transducer, which is a cylinder with an axial hole, is R l, and its external radius is R which is equal to that of the borehole. Let G I and G be the moduli of elasticity of the second kind of the materials of the transducer and the rock; let u I and u be Poisson's ratios for the materials of the transducer and rock; and let a and 8 be the creep parameters of the rock for a power-type creep kernel of the typewhere t is the time.In this problem we must determine the stress components in the transducer. Let orl, o01, and r tO1 be the radial, tangential, and peripheral stresses in the transducer; let Ctr, o O, and rro be the same for the rock near the transducer; let u I and v I be the radial and tangential displacements in the transducer; let u and v be the same in the rock; and let u c and v c be the displacements in the rock due to creep deformation of the rock near the transducer. In this problem we must have the initial conditions tt c=v c. :0when t=:0; = "c~0t = 0when r :--Rr;and the junction conditions (at r-R) Fig. 1. Scheme of calculations.Institute of Mining, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk.