Some boundary value problems in elasticity with and without magnetic field

Abstract

In our paper, in order to solve the problem we need the knowledge of three analytic functions 4>(z), and f^(z).. In the case of plane magneto-elasticity we arrive at a homogeneous biharmonic equation, as is the case with classical plane elasticity theory, and it is here where lies the difference between the techniques of Paria [3 ] and those of ours in discussing the same problem of magnetoelastic deformation of a thin conducting plate with a hole. The complex variable technique is made use of in the usual way to solve the problem afterwards. We have first presented the case when there is an arbitrary hole in an infinite■plate which is subjected to uniform magnetic field in the plane of the deformation, the plate being deformed by a uniaxial tension. V/e next proceed for the case when the arbitrary hole is replaced by a circular hole of radius unit3r. The technique of conformal mapping is found very helpful in solving such problems. The exterior region of the circular hole of unit radius has been mapped onto the interior of a circular hole of unit radius. ¥e have followed the method of Sokolnikoff [7] to determine the Kolosov- Muskhelishvilli potentials <j)(z) and 4^(2) by means of a decent application of Schwarz's formula of the theory of functions of complex variables. It has been observed that in most of the problems that the author has come across sufficient care has not been taken to -determine the stresses at infinity when the magnetic field is present in the plane of the deformation. In our paper we have presented the analysis in such a way that no flaw exists in it and the .infinity conditions have been properly taken care off. It is interesting to note that we arrive at the same biharmonic equation, as in the case of pure elasticity, but the stresses here are the sum of the elastic and the Maxwell stress tensor. The same conclusion can be drawn from Chandrasekhariah's [4 ] paper when the thermal field is absent.