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Discussion of fracture paper #39 - Dynamic Fracture on a Molecular Level

ESIS's picture

Dynamic fracture is a never-ending story. In 1951, EH Yoffe obtained an analytical solution for a crack of constant length travelling at constant speed along a plane. She used a Galilean transformation to get a solution for arbitrary speeds. The situation seems strange with a crack tip where the material breaks and a lagging tip where the material heals. However, there are applications. One that I encountered was several mode II cracks that travel in the contact plane between a brake pad and a brake disc. The moving cracks were blamed for the causing squeaking noise when braking.

A step towards a more versatile solution was given by KG Broberg in 1960 for crack growth with both tips moving apart from each other at the same speed. The self-similar state leads to simplifications while all coordinates are scaled with speed and time. The solution suffers from the crack growth rate instability that is expected as long as the crack growth rate is less than the speed of the Rayleigh wave, i.e. always as long as you don't push the crack tip with something. I can recall an epoxy experiment and a high-power laser that gave the crack tip a push.

Finally in 1972 LB Freund solved the problem of a semi-infinite crack and an arbitrary crack growth rate which more or less culminated the entire subject. Another milestone is the incorporation of plastic strain rate dependencies, dealt with LB Freund and JW Hutchinson in 1989, that explained the observed sudden arrest from a finite crack growth speed.

These are a few milestones as regards the mathematical physics of dynamic fracture. Maybe a new milestone is reached with the article,

"From macro fracture energy to micro bond breaking mechanisms - Shorter is tougher" by Merena Shaaeen-Mualim, Guy Kovel, Fouad Atrash, Liron Ben-Bashat-Bergman, Anna Gleizer, Lingyue Ma and Dov Sherman in Engineering Fracture Mechanics, vol. 289, 2023,,

that begins with a brilliant and detailed review of three successful models for the initiation of crack growth in brittle materials. The first is AA Griffith's theory from 1920, for the initiation of crack growth, which has served us well for decades. The limitation is that it only applies to stable quasistatic crack growth. The second model is due to LB Freund and his work from 1972. This certainly put the subject and us on a different level. The third model is based on molecular dynamics. Here the present paper gives much insight and could be a springboard for initiates.

Observations and results are based on data from fracture mechanical testing of brittle single-crystal silicon samples. The focus is on the relationship between the energy balance and the crack tip speed. Also, the details of the crack front contribute. Analysis of fracture processes on a microscopic scale enabled the development of an interesting model for the required binding energy. The involved mechanisms are the low-energy migration and high-energy kink nucleation along the crack front.

In particular, the energy release rate at crack initiation, and its derivative with respect to the crack length, play an important role. The macroscale experiments, the microscale atomistic model and the energy release rate gradient lead to several conclusions. A primary conclusion is that the energy release rate required for breaking the bonding in the crack plane is not constant. Instead, it is limited by the classical Griffith energy and an upper limit, related to the lattice trapping at as much as 3 times the Griffith energy. Also an interesting transition phase between breaking up from the virgin crack and growth. During this phase, the sequence of bond-breaking mechanisms varies, leading to an increase in the cleavage energy.

Also, the variation of the cleavage energy shows that shorter cracks require higher energy to grow, and are stronger than what is predicted by the classic Griffith's surface energy.

I consider this to be an interesting and important paper. This is why I only have two questions. First, were there ever any dislocations that were nucleated and thrown out from the crack tip? Second, there is of course no such thing as a pure mode for a kink but is the kink propagating under mixed mode closer to mode I or possibly closer to mode II?

It would be interesting to hear from the author or anyone else who would like to discuss or provide comments or thoughts, regarding the subject, the method, or anything related. If you do not have an iMechanica account and fail to register, please email me at and I will post your comments in your name. The paper will be open-access in a couple of days. 

Per Ståhle


ESIS's picture

Dear Per,

First and foremost, thank you for selecting our paper to appear in the blog.

A reply to your questions: 

  1. Dislocation emission near a crack front in silicon crystal usually occurs first at about 900K. There is no dislocation emission at room temperature since the activation energy for dislocation emission at room temperature is much higher than the cleavage energy. The covalent bonds are responsible for the high activation energy for dislocation emission. This is regardless of the suggested kinks along the crack front. The kinks are always there, even when the theory of kinking is quite new. At room temperature, silicon crystal may be considered as an ideally brittle material.
  2. To answer this question (is the kink propagating under mixed mode closer to mode I or possibly closer to mode II?) we emphasize that kink motion, both kink advance and kink formation are planar entities in a 3D coordinate system. They propagate along the cleavage system (plane of propagation and direction of the crack front velocity, e.g., (110)[110] or (111)[112]); kink advances advancing parallel to the crack front, kink formations are generated normal to the crack front (by thermally activated processes). The boundary value problem is fully symmetric to the x-axis, the cleavage system is in the middle of the specimen and parallel to the specimen edges, where the deformation vector is normal to the edges. While the fracture surface is smooth at the scale of several nanometers (confocal optical microscope), scanning tunneling microscope (STM) scans show atomistic scale misalignment steps due to unavoidable misalignment between the maximum KI plane and the cleavage plane of the material. And yet no significant shear stresses exist to activate Mode II deformation.

    Dov Sherman

ESIS's picture

Dear Dov and co-authors, thank you for your comments. 

The quantum mechanical probability-based physics gives the probability of tunnelling based on a barrier's height and thickness. A low and thin barrier increases the possibility that wave properties interfere with the classical physics comprehension of matter. I don't know what levels are needed for the wave properties of matter to set in. However, I do know that the probability of tunnelling is increasing exponentially with decreasing height and thickness. With the barrier height, that is what it is, perhaps even a thickness on the atomic scale, which is what one normally connects to quantum mechanics, is still too big and far away from being close to allowing tunnelling. 

Regarding the kinks, thanks for the hint that the kinks are confined to the crack plane. I got that part now. I found the supplementary movie with the crack front and the interesting behaviour during growth. I can see that the crack front propagates by kinks that nucleate in pairs that repel each other and propagate along the crack front, towards opposite traction-free body surfaces. 

A comparison with sliding dislocations that we often see as a two-dimensional object is interesting. Also for dislocations, the motion is not a coordinated process but is instead repeated jumps that emerge from a single atom that takes the jump and then the closest neighbours follow. The dislocations are often compared with sidewinder snakes' way of motion. A difference is that the snake's modus operandi is that the kinks are running along the body from head to tail. The same could be possible also for cracks and dislocations if the kinks are initiated where the crack or dislocation meets a body surface. So many interesting details, thanks for an indeed interesting paper / Per

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