On the fractal nature of crack branching in MgF 2
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On the fractal nature of crack branching in MgF2 J. J. Mecholsky, Jr., Richard Linhart, and Brian D. Kwitkin University of Florida, Gainesville, Florida 32611
Roy W. Rice 5411 Hopark Drive, Alexandria, Virginia 22310-1109 (Received 5 September 1997; accepted 14 January 1998)
Nineteen disks of IR window grade, hot pressed magnesium fluoride (,0% porosity, grain size ,1 mm) previously loaded in ring-on-ring flexure tests were used to analyze the crack branching patterns. Fractal geometry was used to determine the crack branching fractal dimension which was named the crack branching coefficient or CBC. The failure stress was proportional to the CBC and the number of pieces generated during the fracture. Thus, the number of pieces was proportional to the crack branching coefficient. The crack branching coefficient is distinct from the fractal dimension obtained from the onset of mist and hackle on the fracture surface. The fractal dimension of the fracture surface is, in most cases for brittle materials, a constant and related to the crack tip stress field. The crack branching fractal dimension is a function of the stress at fracture and the far-field stress distribution, or in other words, related to both the type and magnitude of loading. The findings in this work have strong implications for many commercial processes such as comminution, attrition, grinding, and basic studies in crack branching.
I. INTRODUCTION
Fracture mechanics1,2 clearly shows that the propagation of an existing crack in a brittle material such as a ceramic commences when the stress intensity at a point on the crack tip reaches a critical value, i.e., the critical stress intensity factor at a point on the crack equals the fracture toughness. In the mechanical failure of many components and test specimens of interest in this paper, fracture typically initiates from the propagation of a single crack. This single crack is selected from many cracks present because it has the combination of the size and location in a high local stress field to yield the highest stress intensity. Even in its initial propagation, the stress intensity required is greater than that expected from the equilibrium surface energy of the material as originally proposed by Griffith.3 The difference results from other energy dissipative processes such as incomplete relaxation of broken bonds, various acoustic, electromagnetic, and particulate emissions which are involved in the propagation,4 and deviations from a perfectly smooth fracture surface. Beyond the initiation of propagation, crack surfaces and paths become progressively rougher in stages. Key aspects of these changes are discontinuous crack fronts5 and progressive scaling of fracture surface features during branching.6 First, crack branching occurs on the microscale observed as fracture surface topography, then on a macroscale observed as bifurcation, and then, followed by rebranching depending on specimen size, failure stress, and material properties.7–9 Much has been J
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