Fractal Nature of the Brittle Fracture Surfaces of Metal
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FRACTAL NATURE OF THE BRITTLE FRACTURE SURFACES OF METAL V. V. Usov and N. M. Shkatulyak
UDC 539.4: 669.176
We have studied the fracture surfaces of low-alloy low-perlite steel after impact bending tests and of aluminum wire after fatigue failure at different temperatures. We have established that the fracture surfaces after brittle destruction are fractal surfaces. On the basis of the fractal model of a crack and the determined fractal dimensionalities of the boundaries of fracture surfaces, we have evaluated the critical sizes of brittle cracks.
According to present-day concepts, the fatigue failure of metal materials occurs by means of crack initiation and propagation [1]. The cracks and, hence, fracture surfaces formed in an inhomogeneous material are irregular and have unevennesses (cavities, indentations, etc.) of various sizes [2]. This feature is evidence of their fractal nature. A characteristic feature of the objects that are called fractals is their nonintegral dimensionality D. For example, a coastline represents a set occupying an intermediate position between a common line ( D = 1 ) and a surface ( D = 2 ), and the more indented the coastline, the greater is the quantity 1 < D < 2. The coastline length L depends on the length of scale rule l as [3] L ( l ) ∞ l1 – D ,
(1)
where D is the fractal dimensionality of an indented (fractal) line, which exceeds the topological dimension (1 < D < 2). Under actual conditions, there exists a finite range of l values where relation (1) is valid. Obviously, l has to be much smaller than the fractal itself L and, at the same time, to exceed substantially the minimum distance between the points under consideration. There exists an interval where a relation of the type (1) describes the fundamental property of fractals, namely, their identical structure on various scale levels (self-similarity), i.e., scaling invariance [4]. Even in a simple model of the structure of a sheet of material as a perfect plane lattice, the nearest points of which are connected with solid brittle bars, tension leads to the development of cracks whose configurations represent fractal curves with fractional dimensionalities D1 = 1.65 ± 0.05 for branch-shaped and D1 = 1.90 ± 0.05 for bush-shaped fractals [5]. The statistical self-similarity of the microrelief of fracture surfaces was established earlier [6]. It has been shown that one can simulate fracture surfaces by different fractal structures. In addition, a theoretical dependence of the stress concentration factor on the load, average crack size, and its fractal dimensionality was found in [7]. In the present work, we describe the fractal nature and determine the fractal dimensionality of fracture surfaces after impact bending tests of the specimens of structural steel and after fatigue failure of aluminum wire. Ushyns’kyi South-Ukrainian State Pedagogic University, Odessa. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 1, pp. 58 – 62, January – February, 2005. Original article submitted May 25, 2004. 62
1068–82
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