Coherent.Light-Shadow Spot of a Crack Under Mode I Loading: Theory and Experiment by S.M. Kamath and K.S. Kim ABSTRACT--The coherent-light-shadow field formed by crack-tip deformation under Mode I loading is studied theoretically and experimentally. First-order approximation of geometrical optics and higher order expansions are examined in the theoretical development. The wave-optical analysis shows that the interference fringe spacing around the caustic is approximately proportional to K -2/~~ for the ~lear-tip singular field, and, that the peak amplitude of the light-intensity distri- bution around the caustic is proportional to K 4/'~. The effects of diffraction on the measurement of the caustic diameter are examined in detail. The analysis and accompanying experi- ments show that the diffraction effect a~one could lead to K estimation errors of +_20 percent or more, if the edge of the shadow spot or if the peak-intensity point is used to determine the caustic diameter. An alternate measure of caustic diameter is therefore suggested for practical use. EIfects of the size of the nominal initial curve and crack-tip bluntness on K measure- ments are included. Finally, possible applications of the co- herent-light-shadow spot in fracture mechanics are outlined. Introduction The origins of the study of caustics dates back to at least the mid-nineteenth century with contributions from Airy ' in 1838 and by Cayley 2 in 1857. Its application to fracture studies was first suggested by Manogg ~ in 196t. Since then, the optical method of shadow spots or caustics has been widely used in experimental fracture mechanics, particularly in the determination of the stress-intensity factor, K, for rapidly propagating cracks? Its application in linear-elastic fracture mechanics for complex geometry and/or loadings is considered to be accurate and efficient.' Recently, an extension to elastic-plastic fracture of power- law hardening materials has also been investigated? The advantage of using the optical method of caustics in fracture studies is its practical simplicity in extracting a near-tip parameter that characterizes near-tip deforma- tion. The caustic formula used in fracture mechanics is generally derived by geometrical optics formulation 3-6 which leads to the result that the light-intensity along the caustic curve is infinite, and that the dimension (diameter) of the caustic is well defined. In practice, however, the actual light intensity at the caustic is bounded, and the exact position of the caustic curve is ambiguous because of diffraction effects. Consequently the exact position of the crack tip is also obscure when the method is applied to the case of fast fracture. The uncertainty becomes S.M. Kamath (SEM Member) is Graduate Research Assistant, and K.S. Kim (SEA1 Member) is Associate Professor, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801. Original manuscript submitted: Oct. 4, 1985, Final manuscript received: July 21, 1986. more pronounced if a high-magnification scheme is used as relevant to small nominal initial-curve radius. This case is encountered in the study of very-near-tip deforma- tion, such as the three-dimensional field of an elastic- fracture specimen, or, the three-dimensional and non- proportional loading region of an elastic-plastic frac- ture specimen. More recently, the wave-optics aspect of caustic analysis has been explored by several mathematicians. 7 In the present paper the wave-optical nature in caustic formation is applied to fracture studies. By doing so, the accuracy of practical evaluation of the stress-intensity factor and the crack-tip location is enhanced. This is valuable in the study of the three-dimensional as well as the dynamic near-tip-deformation fields. A formulation based on wave-optical analysis is also appropriate since coherent laser light is widely used in experimental mechanics nowadays. The analysis carried out in this paper reveals that the spacing of the interference fringes formed around the caustic by incident coherent light is approximately proportional to K 2~'~. Comparison between the analytical and experimental radial dependence of fringe spacing shows the deviation of the deformation pattern from the plane stress K-field approximation very near the crack tip. The analysis and experiment on the fringe spacing also shows that the spacing is insensitive to the crack-tip bluntness whereas the diameter and the shape of the caustic are sensitive to bluntness. The physical significance of this observation is discussed later in the paper. The weak dependence of fringe spacing on the stress-intensity factor is somewhat discouraging from a practical stand- point, if one proposes to use fringe spacing instead of caustic diameter as a measure of the stress-intensity factor. However, the fringe spacing ahead of the crack is promis- ing in studying three-dimensional deformation fields very near the crack tip. In addition to fringe-spacing analysis, the light-intensity distribution around the caustic is also evaluated in this paper. This analysis provides a method of locating accurately the position of the caustic curve. It is found that the peak intensity of the caustic is propor- tional to K ~'~, and that it is only two to four times brighter than the average background intensity for typical caustics of a fracture specimen. The applicability of these results in dynamic i~ractu're studies is also discussed. Theory Figure 1 shows an experimental configuration of caustic formation. In this configuration, a light beam emerging from a point source L passes through a transparent speci- men, X-plane, that contains a crack, and makes a shadow on a screen, x-plane. Then, the situation can be described 386 " December 1986