Discussion on "the fractal effect of irregularity of crack branching on the fracture toughness of brittle materials" by Xie Heping

Abstract

As an instability phenomenon, the branching event of a running crack happens most frequently in essentially brittle fractures and high crack-propagation speeds and has been correlated with fracture surface topography and surface roughness in several investigations [1][2][3][4]. The author of [5] tried to analyze the crack branching event and correlate it with the increase of fracture toughness of brittle materials based on fractal geometry, using the fractal model as shown in Fig. 1. The present paper points out some essential defects of [5].For a co-planar fractal curve, from Richardson's relation, L/Lo = (b/Lo)l -o.(1)One can easily reach the expression of fractal dimension aswhere b is the yardstick, L 0 and L are the initiator and the total length of a fractal curve, respectively; all of which are with the Euclidean measure; D is the fractal dimension.Neglecting the specific physical background of "crack branching", the fractal dimension D in Fig. lb varies with the initiator L 0 as shown in Fig. 2: D = log(3)/lo~2 cos(-~)]

Response to discussion on "the fractal effect of irrergularity of crack branching on the fracture toughness of brittle materials"

“…If we choose L 0 = B C as discussed in [1], the final construction is a crack net as shown in Fig. 3 rather than a crack branching pattern.…”

“…It should be pointed out that in [5] the only consideration was the common loading situations for mode I, II and III with free crack faces rather than the specific cracked bodies examined in [1].…”

“…The author is grateful to Mr. Weisheng Lei and Bingsen Chen for their comments [1] which raise three questions: the determination of fractal dimension, fractal model of fracture toughness and crack branching induced increase in fracture toughness of materials. However, I believe that the comments are based on a poor understanding of fractal geometry.…”

“…For a smooth curve with D = 1, L = L0. However, the authors of [1] incorrectly consider L 0 and L to be the initiator and the total length of a fractal curve.…”

Foundations of Measurement Fractal Theory for the Fracture Mechanics

“…Therefore, for a generalized measurement there is a generalized dimension which the measurement unit converge to the determined value, M , of the measurement series, according to the extension of the measuring unit tends to zero, as shown in equations equações (9) and (10), namely: i. the first is the measure itself. Because it is actually the step that evaluates the extension of the set, summing the geometrical size of the recover units.…”