Lorentz force-induced stresses in long current-carrying conductors

Abstract

Technique for calculation of Lorentz force-induced stresses in long straight conductors with the uniform current density in the presence of an external field is proposed. Reduction of the plane elastic problem for the conductor of a narrow rectangular cross section embedded in an arbitrary external field to the known homogeneous problems is considered. As an example stress analysis of an elliptical conductor with the stress-free lateral surface under the interaction of the current with its self field is performed and particular solutions of the problem of the interaction with the external field produced by the current flowing in an analogous conductor are given. Calculation of Lorentz force-induced stresses in long current-carrying conductors in the presence of high transverse fields is of interest. In practice, boundary conditions, distributions of the field and hence the Lorentz body force over the conductor cross section are, as a rule, complex and the latter can be of various shapes. Therefore, solving the corresponding plane elastic problem often requires invoking numerical methods. Routine analytical method is based on finding a particular solution of the non-homogeneous equations and reducing them to the homogeneous ones. There are no difficulties of principle in obtaining such solution since it is represented in the form of the integral taken over the cross-sectional area (1). But in the case of the complex distribution of the field there exist calculational difficulties even for the canonical cross-sectional shapes. Neither is suitable here the approach based on finding two functions satisfying non-homogeneous biharmonic equations in which the right-hand sides are the components of the body force (2). However, for the conductor having the relative permeability close to 1 and carrying the uniform current density in the presence of any permissible field distribution a simple technique for getting particular solutions can be proposed (3). These solutions are a sum of magnetic terms of the Maxwell stress tensor of opposite sign and any stresses the Airy's function of which satisfies the biharmonic equation with the constant right-hand side. Below such technique is developed; it is shown that when an external field is available a modified Maxwell stress tensor can be used. At the beginning the general case of both arbitrary field and the conductor cross section is considered; the problem is reduced to the homogeneous one and the methods widely used for its solving are discussed. Then the reduction of