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Research Articles

Stresses in a Long Cylindrical Conductor Moving Axially Through a Pair of Electrode Plates Under Stationary Conditions

[+] Author and Article Information
L. Westerling

Swedish Defense Research Agency (FOI),
SE-164 90 Stockholm, Sweden

B. Lundberg

The Angstrom Laboratory,
Uppsala University,
SE-751 21 Uppsala, Sweden

Manuscript received March 2, 2012; final manuscript received June 8, 2012; accepted manuscript posted July 27, 2012; published online January 22, 2013. Assoc. Editor: Bo S. G. Janzon.

J. Appl. Mech 80(2), 021013 (Jan 22, 2013) (11 pages) Paper No: JAM-12-1088; doi: 10.1115/1.4007221 History: Received March 02, 2012; Revised June 08, 2012; Accepted July 27, 2012

In a conductor carrying electric current, the Lorentz force gives rise to mechanical stresses. Here, we study a long elastic cylindrical conductor that moves axially with constant velocity through two electrode plates. The aims are to explore how the stresses in the conductor depend on the velocity in the stationary case of constant current and to assess the validity of the analytic method used. The diffusion equation for the magnetic flux density is solved by use of Fourier transform, and the current density is determined. The stresses, due to the Lorentz force, are found by use of an analytic method combining the solutions of a quasi-static radial problem of plane deformation and a dynamic axial problem of uniaxial stress. They are also determined through FE analysis. Radial field profiles between the plates indicate a velocity skin effect signifying that the current and the magnetic field are concentrated near the cylindrical surface up-stream and are more uniformly distributed downstream. The radial and hoop stresses are compressive, while the axial stress is tensile. The von Mises effective stress increases towards the symmetry axis, in the downstream direction, and with velocity. There are circumstances under which a large current can produce an effective stress in a copper conductor of the order of the yield stress without causing a significant temperature rise. The stresses obtained with the two methods agree well, even relatively near the electrode plates. The analytical method should be useful in similar cases as well as for the provision of test cases for more general simulation tools.

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Figures

Grahic Jump Location
Fig. 1

Cylindrical conductor moving through a pair of electrode plates. The function f(z) is used in the Fourier analysis.

Grahic Jump Location
Fig. 2

Contour plots for normalized magnetic flux density B(r,z) for velocities corresponding to magnetic Reynolds number Rm= (a) 0, (b) 90, and (c) 150. The contours represent B= 0.1, 0.2,… and 0.9 in the order of increasing radius.

Grahic Jump Location
Fig. 3

Radial profiles for normalized (a) current density Jz, (b) magnetic flux density B, (c) Lorentz force Fr and stresses (d) τrr, (e) τϕϕ, (f) τzz at the up-stream cross-section z=5 for five magnetic Reynolds numbers Rm (velocities)

Grahic Jump Location
Fig. 4

Radial profiles for normalized (a) current density Jz, (b) magnetic flux density B, (c) Lorentz force Fr and stresses (d) τrr, (e) τϕϕ, (f) τzz at the mid cross-section z=15 for five magnetic Reynolds numbers Rm (velocities)

Grahic Jump Location
Fig. 5

Radial profiles for normalized (a) current density Jz, (b) magnetic flux density B, (c) Lorentz force Fr and stresses (d) τrr, (e) τϕϕ, (f) τzz at the down-stream cross-section z=25 for five magnetic Reynolds numbers Rm (velocities)

Grahic Jump Location
Fig. 6

Radial profiles for von Mises effective stress τeff for five magnetic Reynolds numbers Rm (velocities) at (a) up-stream cross-section z=5, (b) mid cross-section z=15, and (c) down-stream cross-section z=25. Comparison between stresses obtained with analytic method (solid curves) and FE method (symbols).

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