Fractal nature of creep-fracture surface patterns in pure Zn polycrystals

Abstract

Mandelbrot et al. [1] ®rst quantitatively evaluated the fracture surface patterns in impact-loaded and fractured steels using fractal geometry and studied the relationship between the fractal dimension of the fracture surfaces and the absorbed energy. Since then, the concept of fractal geometry has been applied to many studies in materials science. For example, the fractal nature of microstructures such as grain boundaries, has been investigated in deformed or annealed metals and alloys [2±5] and in heat-treated alloys [6]. A correlation has also been found between the fractal dimension of the grain boundaries, the fractal nature of the grain-boundary fracture surfaces and the creep-rupture properties in heat-resistant alloys [6, 7]. Ishikawa [8] recently examined the size distribution of dimples in ductile fracture surfaces of steels and found that the value of the fractal dimension is about 1.5 and is close to that of the Sierpinski gasket (about 1.585). However, fracture surface patterns are generally associated with the fracture mechanisms in materials, and it is not uncommon that both equiaxed and elongated dimples are observed on the ductile fracture surface of a single specimen. Ductile fracture surfaces with dimples can also be regarded as a network of ridges (the walls of dimples) formed by a large amount of plastic deformation between dimples. In this study, the fractal dimension of the dimple patterns, which consisted of a network of ridges on ductile fracture surfaces, was estimated by the boxcounting method on fracture surfaces with elongated or equiaxed dimples in creep-ruptured specimens of pure Zn polycrystals. According to Ishikawa [8], the fractal dimension of the dimple size distribution was also estimated on some creep-ruptured specimens to compare with that of the dimple patterns. Commercial pure Zn polycrystals of 99.99 wt % were used in this study. Specimens 16 mm in diameter and 90 mm in length were cut from forged Zn bars and were solution-heated for 1.8 ks at 473 K and then air-cooled. The average grain diameter of the heat-treated specimens was 12 im. These specimens were machined into creep-rupture specimens 5 mm in diameter and 30 mm in length. Creeprupture experiments were carried out under the initial creep stresses of 14.7, 19.6 and 29.4 MPa at 373 K. Ruptured specimens were sectioned normally to the tensile axis. Fracture surfaces of the ruptured specimens were observed and photographed in a scanning electron microscope at magni®cations of 200 or 10003. The direction of microscopic observation is parallel with the tensile axis of specimens. The dimple size, x, is the diameter of a dimple for equiaxed dimples, but the value of x is the mean diameter estimated by x ˆ (d1d2) for elongated dimples, where d1 is the length of semimajor axis and d2 is that of semi-minor axis of an elongated dimple. The cumulative number of dimples, N, whose size is equal to or larger than x, is related to the value of x by a power law with the fractal dimension of the dimple size distribution, D9, as [8, 9]: N / xyD9 (1)