Contents lists available at ScienceDirect Journal of the European Ceramic Society journal homepage: www.elsevier.com/locate/jeurceramsoc Original Article Relationship of energy to geometry in brittle fracture J.J. Mecholsky Jr. a, *, N.A. Mecholsky b , D.P. DeLellis a a Materials Science and Engineering Department, University of Florida, Gainesville FL 32611 USA b Vitreous State Laboratory, The Catholic University of America, Washington, DC 20012, USA ARTICLE INFO Keywords: Fracture Fractal geometry Crack branching Fractography Fracture energy ABSTRACT Numerous investigators have noticed that there is a relationship between the energy of branching and the energy of initiation during a fracture event in materials that fail in a brittle manner. Usually, this is measured in terms of the stress intensities, i.e., K B /K c . The ratio has been reported between 3 and 4, implying a constant value. However, data suggests that it is a constant for a material, but not a universal constant. The fractal dimension of the fracture surface is related to the critical stress intensity factor. It is a measure of the tortuosity of the fracture surface. We show that the K B /K C ratio is directly related to the square root of the fractal dimensional increment, indicating a relationship between the energy of crack propagation and the tortuosity of the fracture surface. 1. Background Since the discovery of the fractal nature of fracture surfaces by Mandelbrot, Passoja and Paullay [1], there have been numerous papers on fractal fracture of materials [2–10]. The most alluring aspect of the fractal nature is the scaling of geometry on the fracture surface [11]. This scaling aspect is most intriguingly observed in materials that fail in a brittle manner, e.g., glass, glass ceramics and ceramics [12]. When a brittle material fractures, there is usually one source of the initial fracture. The crack then grows in a radial pattern and forms what has been termed the mirror, mist and hackle regions. If enough energy is available, the crack macroscopically branches. These are the branches that are visibly obvious to any observer. Many studies have identified these regions in numerous materials [13]. Further studies [14,15] have demonstrated that there is scaling relationships between the size of the fracture initiating crack and the distance to the boundaries, r j , between the mirror-mist, mist-hackle, and hackle-crack branching boundaries c/r j =C j for j=1,2,3 (1) where C j is the corresponding constant for the mirror-mist (1), mist- hackle (2), and hackle-crack branching (3) boundaries, respectively. The fractal dimensions of fracture surfaces have been measured for a number of materials including metals, polymers, and ceramics [6,9,16,17]. Here, the term “ceramics” includes inorganic glasses, glass ceramics, and single crystal ceramics. It has been shown [18] that the fractal dimension in the form of the fractal dimensional increment is directly related to the energy of fracture, γ: 2γ =a 0 ED* (2) where a 0 is a characteristic dimension of the structure, E is the elastic modulus, and D* is the fractal dimensional increment. The fractal di- mensional increment is the fractional part of the fractal dimension of the fracture surface, i.e., 2 + D*. It is known to be constant for non-R- curve materials. Similar to Eq. (2), it can be shown that D* is related to the critical stress intensity factor by the following relationship [3]: K c =a 0 1/2 E D* 1/2 . (3) It was later shown [19] that c/r B = (Y B /Y c ) 2 (a 0 /b 0 ) D* (4) where Y B is the geometric factor at branching, Y c is the geometric factor at the initiation of fracture, a 0 is the characteristic dimension in Eq. (2) and b 0 ½ is the slope of the crack branching stress intensity, K B , and elastic modulus, E, graph [20–22], i.e., K B /E=Y B A B / E = (b 0 ) 1/2 . (5) A B is the “mirror” constant at branching. Eq. (4) indicates that the scaling in the plane of fracture is related to the tortuosity of the fracture surface, i.e., the scaling perpendicular to the apparent planar fracture surface. There have been many studies attempting to relate the crack branching phenomenon to the initiation of fracture [23,24]. Some re- late this to the dynamics of the fracture process [24,25], others to the details of the stress at the tip of the propagating crack [26,27], and still others to the velocity of crack propagation [28–31]. Fractographic https://doi.org/10.1016/j.jeurceramsoc.2020.05.002 Received 26 March 2020; Received in revised form 21 April 2020; Accepted 2 May 2020 ⁎ Corresponding author. E-mail address: jmech@mse.ufl.edu (J.J. Mecholsky). Journal of the European Ceramic Society 40 (2020) 4602–4604 Available online 08 June 2020 0955-2219/ © 2020 Elsevier Ltd. All rights reserved. T