Physica D 190 (2004) 177–189 Non-linear dynamical analysis of crack surface perturbations and their dependence on velocity and direction Dov Sherman ∗ , Ilan Be’ery Department of Materials Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 9 December 2002; received in revised form 28 July 2003; accepted 14 October 2003 Communicated by C.K.R.T. Jones Abstract The fracture surfaces of single crystal [1 0 0] silicon specimens, fractured under three-point bending (3PB) and subjected to a high strain energy upon cracking, revealed exceptional surface perturbations, generated during the unstable propagation. While macroscopically the crack is propagating on the (1 1 1) low energy cleavage plane, microscopic examination revealed small angled deviations from and fluctuations along that plane. Furthermore, while the crack is propagating at a velocity of nearly 3000 m/s in the [1 ¯ 1 0] direction, its velocity in the [1 1 ¯ 2] direction is two orders of magnitude lower, with distinctive surface perturbations. The amplitude and complexity of the perturbations increase as the normal velocity vector changes its direction and magnitude. These perturbations were recorded with a profilometer and analyzed using non-linear dynamical analysis tools. This study provides an opportunity to interpret surface phenomena of one of the most general cases of fracture and to study the effect of major variables on the nature of the perturbations involved, such as the local crack tip velocity and the crystallographic orientations. It is shown that the surface perturbations are chaotic deterministic in nature and can be described by high order non-linear differential equations; the order of the equation varying with the variations of the local velocity and direction. © 2003 Elsevier B.V. All rights reserved. PACS: 62.20.Mk; 68.35.Md; 05.70.Ln; 83.60.Uv Keywords: Non-linear dynamical analysis; Single crystal silicon; Fracture surface perturbations; Crack velocity 1. Introduction As the unbalanced energy in a brittle solid in- creases, an almost universal route of instability is ac- companied by mechanisms in the following sequence: mirror, mist, hackle, and branching. These damage mechanisms were observed not only in amorphous materials [1–7] but also in single crystals such as sapphire [8,9]. Ball and Payne [8] have reported fluc- tuations on some fracture planes of quartz crystals. ∗ Corresponding author. E-mail address: dsherman@techunix.technion.ac.il (D. Sherman). These oscillations were not stable but rather grew in amplitude until branching occurred. Wiederhorn [9] reported crack oscillations, or a hackle mechanism, in sapphire when the K I direction was parallel to the (0 0 0 1) and ( ¯ 1012) planes. No fluctuations were de- tected on the (10 ¯ 10) plane. Perturbations parallel to the crack direction were observed in association with a higher energy concentration; but the reason for their formation and the way they dissipate energy is unex- plained. Fineberg et al. [10] have measured the crack velocity in PMMA and found velocity fluctuations that coincided with the jagged structure of the fracture surface. More recent work, done in silicon by Hauch 0167-2789/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2003.10.010