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Research Papers

# A Mechanical Approach to Solve Two-Dimensional Static Electrical and Magnetic Fields: Applications to Contact Between Conductors Under Electrical Load

[+] Author and Article Information
S. Hao1

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208hao0@suhao-acii.com

Q. Wang, L. M. Keer

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208

1

Corresponding author.

J. Appl. Mech 77(3), 031013 (Feb 23, 2010) (8 pages) doi:10.1115/1.4000037 History: Received April 11, 2008; Revised September 05, 2008; Published February 23, 2010; Online February 23, 2010

## Abstract

A general approach has been developed to obtain analytical solutions to the boundary-value problems for a two-dimensional conductor under static electric and magnetic fields. This approach is based on a “congruity principle” between a solution of Maxwell’s equation and the corresponding linear elastic plane stress solution with constant mean stress or plane strain solution with constant mean in-plane stress. It also leads to a new avenue to construct analytical solutions of antiplane strain boundary-value problems using plane stress/strain solutions, or vice versa, in linear elastic theory. This approach has been applied for such engineering problems as contact between two conducting solid bodies under electrical load.

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## Figures

Figure 1

The model analyzed: (a) a rigid conductor contacts a semi-infinite elastic conducting substrate under mechanical pressure and electrical load and the corresponding solutions of Maxwell’s equations can be classified as the boundary-value problems described by (b)

Figure 2

Analytic solution of an infinitely large conductor plate containing a circular hole under a uniform electrical field E∞ at remote: (a) model analyzed, (b) contours of the electrical potential, and (c) contours of the magnetic field

Figure 3

The normalized electrical potential (a) and magnetic field (b) for the contact problem in Fig. 1

Figure 4

Electrical field: (a) the two branches of the Riemann’s surface of Eq. 45, where the branch RII corresponds to the boundary value problem in Fig. 1, and (b) Ey according to Eq. 45

Figure 5

Contours of electrical potentials for the problem of contact between two semi-infinite conductors; on the contact surface |x|≤1, y=0 the electrical potential 36 is applied with a0=1,ai≠0=0, and the value of conductivity σC in upper semi-infinite conductor is one half of that in the lower plate

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