Fractal analysis of the surface cracks on continuously cast steel slabs
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I.
INTRODUCTION
THE continuous casting of steel slabs has recently undergone significant development and is now widely used. Problems associated with the formation of surface cracks, which were experienced during the early stages of development, have now been largely overcome and the casting speed has been increased. Many investigations have studied the operating conditions and fundamental mathematical modeling of crack formation.[1–6] Little has, however, been written about the distribution frequency of the surface cracks. Recently, the author was involved in an analysis of practical data relating to the frequency of cracks formed on the surface of continuously cast steel slabs. The data reminded the author of the relation between the maximum amplitude and the frequency of earthquakes.[7] Although the subjects are very different, both involve the brittle fracture of solids, which is amenable to fractal analysis. It is the purpose of this article to analyze the continuous casting data in this light. II.
FRACTAL DISTRIBUTION
Japan suffers many earthquakes, and, as a consequence, an extensive study has been made of this subject. Early workers in this field related the frequency of the earthquakes to the inverse of the square of the maximum amplitude.[7] Gutenberg and Richter[8] replaced the maximum amplitude by the magnitude of the earthquake, while Tsu-
HIROSHI KAMETANI, Lecturer, is with the Department of Chemistry, Faculty of Science, Science University of Tokyo, 1-3 Kagurazaka, Shinjukuku, Tokyo, Japan. Manuscript submitted September 26, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
boi[9] assumed that the earthquake energy per unit volume was constant and gave the fractal dimension of 2 for the statistical distribution of the cumulative frequency as a function of the length of earthquake dislocation. These distributions and others were recently reviewed and treated as fractal phenomena.[10] Either the interval frequency or the cumulative frequency could be used to represent the crack frequency as a function of the length (L) of the surface cracks. In problems of this kind, the interval frequency is usually used, where the interval between events is uniform and the data set is large. In the present instance, however, the intervals are not uniform and the data, especially for the long cracks, are few in number. Accordingly, the cumulative frequency, which is the sum of the interval frequency, in order, from the longest to shortest L, similar to the fractal distribution for earthquakes, was preferred in the present analysis. In order to clarify the essential features of plots using methods of representation, a simplified model was used for the simulation. The model includes the following. (1) The interval frequency (Ni), which is represented by the following equation, i.e., the fractal distribution: Ni 5 ki L2n
[1]
where ki is a constant, 3.162 3 104 (Ni 5 1 at L 5 103 in this simulation only), and n is the fractal dimension, 1.5. (2) The range of L is from the upper limit, Lu 5 103, to L 5 1, and this r
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