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L.I. Slepyan, "On Discrete Models in Fracture Mechanics," Mech. Solids. 45 (6), 803-814 (2010)
Year 2010 Volume 45 Number 6 Pages 803-814
DOI 10.3103/S0025654410060051
Title On Discrete Models in Fracture Mechanics
Author(s) L.I. Slepyan (Tel Aviv University, P.O. Box 39040, Tel Aviv, 69978 Israel, leonid@eng.tau.ac.il)
Abstract In two papers published in 1969, where the foundations of the theory of brittle fracture were laid, Valentin Valentinovich Novozhilov introduced a necessary and sufficient criterion for estimating the strength of an elastic body weakened by a cut and formulated the concept of brittle fracture. The criterion is based on taking account of the body discrete structure, and the fracture process is treated as loss of stability of elastic equilibrium. These two facts, fracture discreteness and loss of stability in the deformation of the breaking bond, underlie numerous phenomena, which could not be discovered in the framework of continuum mechanics. In the present paper, the following effects of taking these facts into account are briefly discussed: an unstable leg of the stress-strain diagram and the capture of part of the fracture energy by the lattice, the role of the lattice dimension, radiation in quasi-static and dynamic crack propagation including the radiation structure, the role of the dynamic factor, irregularities in the crack growth, crack propagation under the action of high-frequency waves including steady-state modes with periodically varying speed, and cracks in continuum discrete models.
Keywords continuity and discreteness, instability, continuum limit, radiation, irregularities in fracture, dynamic factor, wave action, periodic oscillations of speed, continuum discrete models
References
1.  V. V. Novozhilov, "On a Necessary and Sufficient Criterion for Brittle Strength," Prikl. Mat. Mekh. 33 (2), 212-222 (1969) [J. Appl. Math. Mech. (Engl. Transl.) 33 (2), 201-210 (1969)].
2.  V. V. Novozhilov, "On the Foundations of a Theory of Equilibrium Cracks in Elastic Solids," Prikl. Mat. Mekh. 33 (5), 797-812 (1969) [J. Appl. Math. Mech. (Engl. Transl.) 33 (5), 777-790 (1969)].
3.  V. V. Novozhilov, Theory of Elasticity (Sudpromgiz, Leningrad, 1958) [in Russian].
4.  L. I. Slepyan, "Dynamics of a crack in a Lattice," Dokl. Akad. Nauk SSSR 253 (3), 561-564 (1981) [Sov. Phys. Dokl. (Engl. Transl.) 26 (5), 538-540 (1981)].
5.  G. S. Mishuris, A. V. Movchan, and L. I. Slepyan, "Localized Knife Waves in a Structured Interface," J. Mech. Phys. Solids 57, 1958-1979 (2009).
6.  L. I. Slepyan, "Wave Radiation in Lattice Fracture," Acoust. Phys. 56 (6), 962-971 (2010).
7.  R. Thomposon, C. Hsieh, and V. Rana, "Lattice Trapping of Fracture Cracks," J. Appl. Phys. 42 (8), 3154-3160 (1971).
8.  L. I. Slepyan, Models and Phenomena in Fracture Mechanics (Springer, Berlin, 2002).
9.  V. V. Novozhilov and L. I. Slepyan, "On Saint-Venant's Principle in the Dynamics of Beams," Prikl. Mat. Mekh. 29 (2), 261-281 (1965) [J. Appl. Math. Mech. (Engl. Transl.) 29 (2), 293-315 (1965)].
10.  L. I. Slepyan, "Dynamic Factor in Impact Phase Transition and Fracture," J. Mech. Phys. Solids 48 (5), 927-960 (2000).
11.  M. Marder and S. Gross, "Origin of Crack Tip Instabilities," J. Mech. Phys. Solids 43 (1), 1-48 (1995).
12.  L. I. Slepyan, M. V. Ayzenberg-Stepanenko, and J. P. Dempsey, "A Lattice Model for Viscoelastic Fracture," Mech. Time-Depend. Mat., 3 (2), 159-203 (1999).
13.  L. I. Slepyan and M. V. Ayzenberg-Stepanenko, "Some Surprising Phenomena in Weak-Bond Fracture of a Triangular Lattice," J. Mech. Phys. Solids 50 (8), 1591-1625 (2002).
14.  L. I. Slepyan, "Crack Propagation in High-frequency Lattice Vibration," Dokl. Akad. Nauk SSSR 260 (3), 566-569 (1981) [Sov. Phys. Dokl. (Engl. Transl.), 26, 900-902 (1981)].
15.  L. I. Slepyan, G. S. Mishuris, A. V. Movchan, "Crack in a Lattice Waveguide," Int. J. Fract. 16 (1-2), 91-106 (2009).
16.  M. Ryvkin and L. Slepyan, "Crack in a 2D Bending Beam Lattice: Analytical Solutions for Two Fracture Modes," J. Mech. Phys. Solids 58 (6), 902-917 (2010).
17.  L. Truskinovsky, "Fracture as a Phase Transition," in Contemporary Research in the Mechanics and Mathematics of materials. A Collection of Papers Dedicated to the 69th Birthday of J. Erickson. Ed. by R. C. Batra and M. F. Beatty (CIMNE, Barcelona, 1996), pp. 322-332.
18.  L. I. Slepyan and L. V. Troyankina, "Fracture Wave in a Chain Structure," Zh. Prikl. Mekh. Tekhn. Fiz. 25 (6), 128-134 (1984) [J. Appl. Mech. Tech. Phys. (Engl. Transl.) 25 (6), 921-927 (1984)].
19.  O. Kresse and L. Truskinowsky, "Mobility of Lattice Defects: Discrete and Continuum Approaches," J. Mech. Phys. Solids 51 (7), 1305-1332 (2003).
20.  A. Vainchtein, "The Role of Spinodal Region in the Kinetics of Lattice Phase Transitions," J. Mech. Phys. Solids 58, 227-240 (2010).
Received 21 June 2010
Link to Fulltext http://www.springerlink.com/content/9682l6084q1200hm/
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