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See an animation of results (2MB -- 5 mins at 56k) in this window:
Seeded Crack or a
No Seed (pure shear) Crack.
Or in a new window:
Seeded Crack or a
No Seed (pure shear) Crack.
No javascript:
Seeded Crack or a
No Seed (pure shear) Crack.
(click to stop)
Why are we interested in self-healing cracks? Well, it is a dirty secret of physics that we still don't fully understand friction. Dry friction between solid rigid objects is described by an extremely simple rule: F=uN. This states that the friction force sideways between two objects is proportional to the external force pushing them together. It also states that the friction force depends on nothing else -- for example, the total area and the speed of slip do not come into it. At the macroscopic level, friction is as simple as physics gets.
At a closer level it gets more complicated. All surfaces are rough to some extent. While sandpaper has bumps that are about 1 mm in size, even glass has bumps that are . A machined, lapped and polished surface will still have bumps of xxx and smaller.
These asperities (bumps) exist at all scales from microns down to nanometers (the surface is mathematically described as a self-affine fractal). Two surfaces in contact will have a complicated interface of interlocking asperities pushing on and deforming into each other. At some level it seems that these interlocking aperities must give rise to macroscopic friction.
To understand the contribution of numerous interlocking asperities we should understand the behavior at one such asperity. The microscopic picture, however, is spirit-crushingly complex. We like to think that two bumps pushing on each other is a mechanical interation, when it is in fact due to electrical interactions, which are in fact quantum mechanical interactions. The scale is too small to neglect quantum mechanics, entropy, and the like, yet too large to be able to isolate from the bulk and surface behavior. There is too much to still understand about the interaction of two bumps pushing on each other at the atomic level to profitably proceed to the macroscopic level. Even with insight into the atomic picture, there are oxide layers, surface contaminants, and a host of other complications going on under the hood.
The mystery, then, is how such remarkable simplicity can come from such mind-blowing complexity.
Dr. Marder's idea (and my project) is to disregard the obvious intermediate model of interlocking asperities and instead pretend that the surfaces interact over a small area that is perfectly flat and free of complications. The two surfaces could slip if a self-healing crack formed at the interface and traveled along its length (like pushing a bump in a rug from one side of the room to the other). Now, it is a known result that cracks propagating through materials have a range of forbidden speeds -- a crack can move faster than a certain minimum average speed, but it can't move at less than that -- it would just sit still and not progress. The minimum speed depends on the external forces imposed; in our case, on the strain (normal force) and shear (friction force). This range of forbidden speeds, and the stick-slip dynamics it produces, gives rise to a coupling between the normal and friction forces and finally to the F=uN law. Woohoo! A direct connection from the microscopic to the macroscopic!
Now we have to see if this picture is correct... We need to understand traveling self-healing cracks (TraSH Cracks?) and determine whether they can arise spontaneously, how long and how fast they travel, and what magnitude of external forces are involved. It would be nice to know how such a crack forms as well as its characteristics once underway. All this will let us know if the leap from micro to macro is valid. There is also the serious problem of neglecting the physical picture of two rough surfaces meeting at widely scattered points. My judgement is that if the numbers work for our picture then it must be right at some level -- that in some mysterious way, a traveling crack along our one interface mirrors the behavior for the aggregate interface of all the interlocked asperities. Simulations may help point to a connection between the micro and meso pictures. Finally, we would like to find physical signatures of this behaviour. If this model is correct then a thorough understanding should allow us to make experimental predictions that would have been otherwise unacheivable.
The animations at the top of the page depict a molecular dynamics (MD) simulation of a traveling, self-healing crack. Our "crystal" is a two-dimensional, 46x175 atom triangular lattice. Each atom exerts a force on nearby atoms as if they were connected by a "snapping" linear spring -- as two atoms move apart or together the force will increase, up to a certain cutoff where the force drops to zero. Each atom is colored according to the total force on it; blue means it is near equilibrium, while yellow and red indicate strong forces on the atom. High forces occur near the crack tip and in traveling sound waves caused by the crack. In the above simulation, the force cutoff is equal to about 1½ layers, slightly less in the interface region. As the top surface reconnects in the seeded crack animation, you can briefly see the atoms turn yellow when they pass through the cutoff.
We start with an externally strained, sheared crystal. The bottom layer is held fixed and the top boundary is rigidly displaced to impose the external force. The rest of the crystal just evolves naturally. The arrow on the top layer indicates the magnitude and direction of the total strain (vertical) and shear (horizontal). We also either insert a seed crack in the crystal before beginning simulation (the first frame shows an unseeded crystal) or turn the interfac cutoff down so that the crak starts along the boundary.
Due to the external force, the crack starts advancing. As it does so, we decrease the external strain but continue to increase the shear. After 7000 or so steps, the top face of the crystal reconnects with the bottom forming a self-healing crack. Notice that the two faces have moved by 5 atomic spacings where they rejoin.
When the crack tip nears the borders of the simulation we chop off the left side and paste fresh crystal on the right side. [Unfortunately, I'm still working on the paste code so they don't match up that well right now. This makes the crystal unhappy, and you can see a big red scar on the crystal as it figures out how to join up correctly].
Read a good (technical) account of the conventional model of friction (by Persson) or a non-technical article about work on the microscopic picture of friction. Persson gives a sanguine view of our current understanding of friction; for another opinion see this New Scientist article.
Non-technical articles on the research by Dr. Marder and Eric Gerde have appeared in Scientific American; Physics Web; Science News (9/22/01 160:12 p181); and others. The original research paper was published in Nature (Requires registration to view). Dr. Marder's site has much more information.
