How do things break? The fracture of materials is part of our everyday experience, and yet the process is not well understood. A study of crack propagation at microscopic scales shows the devil in the details.
The effects of material failure are obvious on macroscopic scales, and yet the dynamics of fracture that lead to such failure are governed entirely by the material's behaviour at the smallest scales. This fact is underscored by work presented by Buehler et al.1 on page 141 of this issue. Ninety years after Inglis pointed out that cracks in a material play a key role as stress enhancers, Buehler et al. show that the dynamics of a crack change unexpectedly as the huge (putatively infinite) stresses generated around the crack's tip alter the material properties across a microscopic region.
Once a crack starts to develop in a material, what determines its dynamics? You might think that the dynamics of fracture are self-evident: huge stresses at the tip of a crack cause molecular bonds to rupture, and the crack lengthens ever more rapidly. But that is only the beginning of the story of how different materials cope with the mathematical singularity at the tip of a crack that, as a crack becomes infinitely sharp, gives rise to infinite stresses.
The theoretical framework for the formation of the stress singularity is based on the assumption that a material under stress follows the rules of linear elasticity — that, like a common spring, the amount that a material stretches is proportional to how hard it is pulled. In nearly all cases, this assumption is perfectly valid if a material is only slightly deformed from its initial state. Fracture, on the other hand, is a nonlinear process and the model of spring-like behaviour breaks down around the tip of a crack when the stresses approach the mathematical singularity predicted by Inglis. The way that a particular material acts to 'blunt' this singularity ultimately determines its strength, its mode of failure and the dynamics that describe it. Both the rapid failure that is typical of low-strength, brittle materials (such as glass) and the gradual failure of high-strength, ductile materials (such as steel) are governed by the microscopic response of the material around crack tips.
There have been numerous attempts2,3,4,5,6 to model this near-tip zone, but progress has been hampered by lack of knowledge of the detailed physical processes at these quasi-infinite stress levels. Experimental access to this region is extremely difficult — the zone is small, sometimes approaching atomic dimensions, and fracture can occur very rapidly (brittle fracture can happen at speeds approaching the speed of sound in the material). Computational advances7,8,9 now offer a way in.
Buehler et al.1 ran simulations of millions of interacting atoms to analyse the fine details of the fracture process in two qualitatively different types of (artificial) material. These materials are conceptually simple, but their properties are designed to elucidate the very real effects of the nonlinear material response to the large stresses present around a crack tip. The force between neighbouring atoms in both materials is allowed to behave in a linear (or elastic) way until their separation exceeds a critical value. Beyond that value, a 'hyperelastic' force law applies: in one material, the effective spring constant between the neighbouring atoms is doubled (called hyperelastic stiffening); in the other material it is halved (hyperelastic softening). Any pair of atoms separated by more than a second critical value are allowed to detach, thereby enabling fracture to occur.
The calculations show how significant the effects of hyperelasticity can be, even when the hyperelastic zone is only a few hundred atoms in size and the remainder of the material is behaving elastically. Buehler et al.1 suggest that, under the right conditions, the properties of this tiny hyperelastic region entirely control the fracture process. Supersonic propagation of cracks — far surpassing the limiting velocity allowed by linear elastic theory10 — becomes possible, and as a result shock fronts may be emitted from the hyperelastic region.
As every crack is surrounded by a nonlinear region, hyperelastic effects should be commonplace. On the other hand, laboratory experiments11 have indicated that linear elastic theory provides an excellent description of the dynamics of rapidly moving cracks. So when would the more exotic behaviour induced by hyperelastic effects be seen? Buehler et al.1 predict that the key to this question is the ratio of the size of the hyperelastic region marking the onset of hyperelasticity, to an 'energy' length scale, defined as the size of the region around the tip that encompasses enough energy to drive the fracture process. When the hyperelastic region is much smaller than the energy length scale, hyperelastic effects are negligible; but if the two are similar, these effects could dominate the fracture process. We might then expect to observe the effects of hyperelastic behaviour when fracture occurs in a highly strained material, or if a material suffers a high rate of strain, such as caused by the impact of a projectile.
The process of rapid fracture is strongly influenced by the interplay of physical effects on many different scales. Simulations, such as these by Buehler et al.1, should enable us to bridge the gap between the laboratory scales where fracture is observed and the near-atomic scale where fracture germinates. Describing the detailed motion of millions of atoms involved in the fracture process is an impressive technical feat, but it is still important to be able to see the wood for the trees — to identify the key features relating the delicate interplay between the myriad dancing atoms at a crack tip and the macroscopic effects that they generate. This work, together with other studies7,8,9, is a step towards both a fundamental understanding of these processes and, possibly, a powerful tool for the design of new materials.
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International Journal of Computational Methods (2019)
AIMS Materials Science (2017)
Earth and Planetary Science Letters (2012)
Acta Mechanica Solida Sinica (2008)
Journal of Applied Physics (2007)