Energetic Developments in Fracture - CiteSeerX

Energetic Developments in Fracture M. Marder Department of Physics and Center for Nonlinear Dynamics The University of Texas at Austin, Austin TX 78712 August 29, 2004 What drives physicists to study cracks? There is certainly some attraction in being able to tell the children one is getting paid to break things. There is also a perverse pleasure in learning the physical laws underlying irreversible change, decay, and destruction. A paper of Eran Sharon, Steven Gross, and Jay Fineberg, “Energy Dissipation in Dynamic Fracture” which appeared on 18 March in Physical Review Letters contains an answer to an old and deceptively simple question, “How fast do things break, and why?” The first scientific attempts to answer this question go back to research of Hubert Schardin and Wolfgang Struth in 1937[1], who used sparks to take photographs of cracks in less than one ten-millionth of a second. They concluded that “the maximum velocity of propagation of glass fractures is to be considered a physical constant,” and they measured crack speeds that were approximately one quarter the speed of sound in many different glasses. Some time after these experiments came theories of crack motion, which firmly insisted that cracks should move at about twice the speed observed. The classic theory[2], developed by B. V. Kostrov, J. D. Eshelby, and L. B. Freund, left little room for uncertainty. Cracks leave two new surfaces behind them. The speed at which vibrations travel across a free surface is the Rayleigh wave speed, a speed governing the motion of sound when one raps one’s knuckles on the top of a table, or the speed at which earthquakes travel on the surface of the earth. Cracks, said the theory, should move at this Rayleigh wave speed too – but it does not happen. In Plexiglas, for example, the Rayleigh wave speed is around 1000 metres per second, but cracks never exceed 600 metres per second. This descrepancy was widely acknowledged to 1

be rather puzzling, but there was the consolation that it was probably rather unimportant. After all, if a wing is falling off an airplane, who has time to wonder whether the crack is moving at 600 or 1000 metres per second? In 1991, Jay Fineberg, Steve Gross, Harry Swinney, and others at the University of Texas developed a method to look at the motion of cracks, involving deposition of a very thin layer of aluminum on the surface of a Plexiglas or glass sample, and then monitoring the resistance of the aluminum as a crack raced through it. This technique made it possible to measure the velocity of a crack on time scales much shorter than a millionth of a second, hundreds of thousands of times in succession. With this abundance of new information came numerous clues about the crack’s stubborn refusal to obey a perfectly good theory, and accelerate to the proper speed. The most important clue was found in velocity traces, such as shown on the left hand side of Fig. . After accelerating rapidly in its youth, the crack suffers a mid-life crisis at a speed of 330 metres per second, and the smooth motion of the earlier times degenerates into very rough oscillations in velocity. A large number of different theoretical ideas was proposed to explain these experiments. However, one particular set of ideas is particularly supported by the recent experimental work of Sharon, Gross, and Fineberg. From this point of view, one regards a crack in its early stages as a thin blade, one atom wide, slicing through a piece of brittle material. The effective sharpness of the blade depends upon the power with which it presses through material; press too hard, and it blunts, presenting enormous resistance to speeding up further. Both idealized calculations[3, 4], as well as more realistic simulations[5, 6] showed how this process could occur, as indicated on the right hand side of Fig. . Last fall, Sharon, Fineberg, and Gross[7] showed that cracks in Plexiglas develop branching structures, as shown on the right hand side of Fig. . It began to seem plausible that an explanation for the sluggish behavior of cracks was in reach. However, one part of the argument was still missing: definite proof that the energy used to make the the branches could fully account for energy robbed from crack acceleration, thus reconciling experimental observations with fracture theory. This full accounting has now been carried out. Sharon, Gross, and Fineberg have carefully looked beneath the surfaces left behind a crack, have determined the total energy which would be needed to create the ramified 2

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Figure 1: A crack in Plexiglas traveling at less than the critical velocity of 340 m/sec moves calmly, and leaves smooth surface in its wake. Above this velocity, the crack bucks and plunges, leaving a ramified branching structure beneath the surface. Crack velocity versus time for two experimental runs appears on the left, and photographs of the corresponding samples appear on the right. Experimental data courtesy of Sharon, Gross, and Fineberg.

Figure 2: Simulations in an atomic model of fracture show an instability agreeably similar to experimental observations of Fig. . Velocities from two crack simulations are plotted on the left, and snapshots of corresponding atomic locations are shown on the right. 3

branching structures they found there, and compared this energy with the the total energy fracture theory claims is rushing in to the crack tip. The two agree, although the energy needed to create the many subsurface branches is nearly ten times larger than the energy needed to create a smooth single crack. A consistent picture of crack motion emerges. When small amounts of energy are delivered to a crack tip, it plunges forwards, cleanly slicing material in two, leaving perfectly flat surfaces in its wake. However, once a critical energy flux is exceeded, the crack tip can no longer handle the responsibility, and becomes unstable building increasingly ramified zones of damaged material beneath the visible fracture surface. One might wonder if understanding such details of the fracture process could have practical consequences. The prospects are not entirely so discouraging as the discussion of the airplane wing above would suggest. Indeed, airplane safety provides a case in which improved understanding of crack dynamics might be important. Airplanes can tolerate rather large holes, and still land safely. If a hole does begin to form, the crucial question is whether the cracks responsible will come to rest, or whether they will continue to run until damage is irrevocable. If the current understanding of what sets the speed of cracks can be turned into new ideas on how to make them stop, this research will work its way out of the realm of curiosity, and into new stronger and safer materials.

References [1] H. Schardin, L. Mucke, W. Struth, and W. a. Rhein, The Glass Industry 36, 133 (1955). [2] L. B. Freund, Dynamic Fracture Mechanics (Cambridge University Press, Cambridge, 1990). [3] M. Marder and X. Liu, Physical Review Letters 71, 2417 (1993). [4] M. Marder and S. Gross, JMPS 43, 1 (1995). [5] X. P. Xu and A. Needleman, JMPS 42, 1397 (1994). [6] F. F. Abraham, D. Brodbeck, R. A. Rafey, and W. E. Rudge, PRL 73, 272 (1994). 4

[7] E. Sharon, S. Gross, and J. Fineberg, PRL 74, 5096 (1995).

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