Theoretical study: depth of small cracks (circa 100 microns) from photoinductive data

Sethuraman, Ananth
Major Professor
Hsiu C. Han
Committee Member
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Electrical and Computer Engineering

The reconstruction of small, tight surface-breaking cracks is one of the key problems of nondestructive evaluation. Many of the established methods of nondestructive evaluation have been defeated by the smallness and the tightness of such cracks. The photoinductive method, which combines a laser with the eddy current method, is known to have a particularly fine spatial resolution. Consequently we have investigated whether it would be well-suited for determining the size and the shape of small, tight surface-breaking cracks. A fundamental feature of the photoinductive method is that it uses a coil to induce an eddy current in the test specimen and, associated with it, an electric field. A photoinductive measurement is essentially the square of this electric field, evaluated at points on the surface of the test specimen. In this approach, the electric field interaction of the crack is described with a concept known as the current dipole density. It is shown that the square of the electric field is connected to the current dipole density through a nonlinear integral equation. Besides this, another integral equation is also set up that reflects that each point on the cross-section is (i) either in a crack, and hence blocks the flow of electric current, or (ii) is intact, and hence has a zero current dipole density. This second integral equation is also nonlinear. The result is that we can determine the shape of the crack since the current dipole density has the property that the set of points where the current dipole density is nonzero can be identified as the crack. The dissertation uses standard numerical techniques to solve the pair of nonlinear integral equations--discretizing the cross-section containing the crack into squares, modeling the current dipole density as a piecewise constant function, constant in each square, numerical integration and so on. The problem is reduced to an optimization problem simply by taking modulus squared of the difference between the left-hand and the right-hand sides as an objective function for minimization. The reconstruction method was carried out for a variety of cracks using photoinductive data. Good results were obtained for the length and depth of the crack. The reconstruction of the crack shape was somewhat more ambiguous, because of an ill-posedness inherent in this pair of nonlinear integral equations. The dissertation traces the ill-posedness to the behavior of a Green's function that appears as a kernel in one of the integral equations. This ill-posedness is essentially that which usually arises in Fredholm equations of the first kind. Cracks on the order of hundreds of microns to a few millimeters in size have been reconstructed in nonmagnetic materials. The depths of the example cracks were recovered in all the cases attempted. The shapes of the examples were also recovered, after an additional condition--that of seeking the crack with the least perimeter--was adjoined to the problem to eliminate the ill-posedness. (Abstract shortened by UMI.)