Fractal fracture of single crystal silicon
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The quantitative description of surfaces that are created during the fracture process is one of the fundamental issues in materials science. In this study, single crystal silicon was selected as a model material in which to study the correlation of fracture surface features as characterized by their fractal dimension for two different orientations of fracture with the fracture toughness of the material as measured using the strength-indentation and fracture surface analysis techniques. The fracture toughness on the {110} fracture plane of single crystal silicon was determined to be 1.19 MPa m1/2 for the {100} tensile surface and 1.05 MPa m1/2 for the {110} tensile surface using the indentation-strength three-point bending method. The fracture surface features of these two orientations are correspondingly different. Within our limitations of measurements (1-100 /zm), the fractal dimension appeared different in different regions of the fracture surface. It has a higher value in the branching region and a lower value in the pre-branching and post-branching regions. The fractal dimensions are about the same in the pre-branching regions and post-branching region for these two orientations (D = 1.01 ± 0.01), i.e., nearly Euclidean (smooth); but the fractal dimensions are higher in the branching region for these two orientations. The fractal dimension is 1.10 ±0.4 for the {100} tensile surface and is 1.04 ±0.3 for the {110} tensile surface. If we select the highest dimension on a surface to represent the dimensionality of the surface, then a material with a higher fracture toughness has a higher fractal dimension in the branching region.
I. INTRODUCTION The strength of brittle materials is determined by the size of the crack which causes fracture and the value of the fracture toughness.1 Fracture surface topography is related to the size of the fracture initiating crack. However, quantitative characterization of the fracture surface has been limited. 2 " 4 Techniques to determine fracture toughness of ceramic materials include cleavage techniques,5 double cantilever beam (DCB),6 crackindentation,7 strength-indentation,8 and fracture surface analysis1 (FSA). FSA is important because it can be applied to a fractured part rather than a representative material. Fracture markings on glasses, glass ceramics, single crystals, and polycrystalline brittle materials, known as mirror, mist, and hackle, and which are considered precursors to macroscopic crack branching,9 can be used to describe the stress state1 and characteristics of crack propagation.10 These markings have been observed for more than 100 years and were related quantitatively to the stress at fracture in the 1950's. 11 More recently, the repetition of these features was observed and quantitatively related to stress intensity. Ravi-Chandar and Knauss9 noted that mist and hackle are self-similar; i.e., they appear to be physically similar and produced in the same fashion. However, their description did not emphasize the self-similar nature of the features.2 The dista
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