Jay Fineberg

Instability in dynamic fracture

Abstract The fracture of brittle amorphous materials is an especially challenging problem, because the way a large object shatters is intimately tied to details of cohesion at microscopic scales. This subject has been plagued by conceptual puzzles, and to make matters worse, experiments seemed to contradict the most firmly established theories. In this review, we will show that the theory and experiments fit within a coherent picture where dynamic instabilities of a crack tip play a crucial role. To accomplish this task, we first summarize the central results of linear elastic dynamic fracture mechanics, an elegant and powerful description of crack motion from the continuum perspective. We point out that this theory is unable to make predictions without additional input, information that must come either from experiment, or from other types of theories. We then proceed to discuss some of the most important experimental observations, and the methods that were used to obtain the them. Once the flux of energy to a crack tip passes a critical value, the crack becomes unstable, and it propagates in increasingly complicated ways. As a result, the crack cannot travel as quickly as theory had supposed, fracture surfaces become rough, it begins to branch and radiate sound, and the energy cost for crack motion increases considerably. All these phenomena are perfectly consistent with the continuum theory, but are not described by it. Therefore, we close the review with an account of theoretical and numerical work that attempts to explain the instabilities. Currently, the experimental understanding of crack tip instabilities in brittle amorphous materials is fairly detailed. We also have a detailed theoretical understanding of crack tip instabilities in crystals, reproducing qualitatively many features of the experiments, while numerical work is beginning to make the missing connections between experiment and theory.

Detachment fronts and the onset of dynamic friction.

The dynamics of friction have been studied for hundreds of years, yet many aspects of these everyday processes are not understood. One such aspect is the onset of frictional motion (slip). First described more than 200 years ago as the transition from static to dynamic friction, the onset of slip is central to fields as diverse as physics, tribology, mechanics of earthquakes and fracture. Here we show that the onset of frictional slip is governed by three different types of coherent crack-like fronts: these are observed by real-time visualization of the net contact area that forms the interface separating two blocks of like material. Two of these fronts, which propagate at subsonic and intersonic velocities, have been the subject of intensive recent interest. We show that a third type of front, which propagates an order of magnitude more slowly, is the dominant mechanism for the rupture of the interface. No overall motion (sliding) of the blocks occurs until either of the slower two fronts traverses the entire interface.

Confirming the continuum theory of dynamic brittle fracture for fast cracks

Crack propagation is the basic mechanism of materials failure. Experiments on dynamic fracture in brittle amorphous materials have produced results that agree with theoretical predictions for single-crack motion at very low velocities. But numerous apparent discrepancies with theory have been observed at higher velocities. In particular, the maximum crack velocities attained in amorphous materials are far slower than the predicted asymptotic value, vR (ref. 3). Beyond a critical velocity, vc ≈ 0.4vR, an intrinsic instability has been observed in which a multiple-crack state is formed by repetitive, frustrated micro-branching events. These cause velocity oscillations and may explain the apparent anomaly. Here we report measurements of dynamic fracture in a brittle, amorphous material that are in quantitative agreement with the theoretical single-crack equation of motion, from the initial stages of propagation up to vc. Beyond vc, agreement breaks down owing to the appearance of the multiple-crack ensemble. But in this regime, the micro-branching process can momentarily produce a single-crack state which instantaneously attains its predicted single-crack velocity, for velocities up to 0.9vR. Our results therefore confirm the validity of the single-crack continuum theory of elastic brittle fracture even in the dynamical regime where the crack morphology is complex.

The Dynamics of the Onset of Frictional Slip

Slip Sliding Away The initiation of frictional motion, or slip, along ideal surfaces typically behaves as predicted by rupture models. When stress heterogeneities—similar to irregularities in fault zones in Earths crust—are introduced, rupture propagation speeds are not as well constrained by models. To improve understanding of slip behavior, Ben-David et al. (p. 211; see the Perspective by Zapperi) measured rupture speeds and stress profiles along an extended frictional interface between two polymer blocks. The experiments revealed a slow mode of slip that occurs when the local ratio of shear to normal stress is sufficiently low. The selection and arrest of three distinct modes of rupture depended on the value of the local stress ratio. The selection and arrest of three distinct modes of stress-induced rupture depend on the value of the local stress ratio. The way in which a frictional interface fails is critical to our fundamental understanding of failure processes in fields ranging from engineering to the study of earthquakes. Frictional motion is initiated by rupture fronts that propagate within the thin interface that separates two sheared bodies. By measuring the shear and normal stresses along the interface, together with the subsequent rapid real-contact-area dynamics, we find that the ratio of shear stress to normal stress can locally far exceed the static-friction coefficient without precipitating slip. Moreover, different modes of rupture selected by the system correspond to distinct regimes of the local stress ratio. These results indicate the key role of nonuniformity to frictional stability and dynamics with implications for the prediction, selection, and arrest of different modes of earthquakes.

Slip-stick and the evolution of frictional strength

The evolution of frictional strength has great fundamental and practical importance. Applications range from earthquake dynamics to hard-drive read/write cycles. Frictional strength is governed by the resistance to shear of the large ensemble of discrete contacts that forms the interface that separates two sliding bodies. An interface’s overall strength is determined by both the real contact area and the contacts’ shear strength. Whereas the average motion of large, slowly sliding bodies is well-described by empirical friction laws, interface strength is a dynamic entity that is inherently related to both fast processes such as detachment/re-attachment and the slow process of contact area rejuvenation. Here we show how frictional strength evolves from extremely short to long timescales, by continuous measurements of the concurrent local evolution of the real contact area and the corresponding interface motion (slip) from the first microseconds when contact detachment occurs to large (100-second) timescales. We identify four distinct and inter-related phases of evolution. First, all of the local contact area reduction occurs within a few microseconds, on the passage of a crack-like front. This is followed by the onset of rapid slip over a characteristic time, the value of which suggests a fracture-induced reduction of contact strength before any slip occurs. This rapid slip phase culminates with a sharp transition to slip at velocities an order of magnitude slower. At slip arrest, ‘ageing’ immediately commences as contact area increases on a characteristic timescale determined by the system’s local memory of its effective contact time before slip arrest. We show how the singular logarithmic behaviour generally associated with ageing is cut off at short times. These results provide a comprehensive picture of how frictional strength evolves from the short times and rapid slip velocities at the onset of motion to ageing at the long times following slip arrest.

Singularity dynamics in curvature collapse and jet eruption on a fluid surface

Finite-time singularities—local divergences in the amplitude or gradient of a physical observable at a particular time—occur in a diverse range of physical systems. Examples include singularities capable of damaging optical fibres and lasers in nonlinear optical systems, and gravitational singularities associated with black holes. In fluid systems, the formation of finite-time singularities cause spray and air-bubble entrainment, processes which influence air–sea interaction on a global scale. Singularities driven by surface tension have been studied in the break-up of pendant drops and liquid sheets. Here we report a theoretical and experimental study of the generation of a singularity by inertial focusing, in which no break-up of the fluid surface occurs. Inertial forces cause a collapse of the surface that leads to jet formation; our analysis, which includes surface tension effects, predicts that the surface profiles should be describable by a single universal exponent. These theoretical predictions correlate closely with our experimental measurements of a collapsing surface singularity. The solution can be generalized to apply to a broad class of singular phenomena.

How Things Break

Galileo Galilei was almost seventy years old, his life nearly shattered by a trial for heresy before the Inquisition, when he retired in 1633 to his villa near Florence to construct the Dialogues Concerning ‘Two New Sciences. His first science was the study of the forces that hold objects together and the conditions that cause them to fall apart—the dialogue taking place in a shipyard, triggered by observations of craftsmen building the Venetian fleet. His second science concerned local motions—laws governing the movement of projectiles.

Static friction coefficient is not a material constant.

The static friction coefficient between two materials is considered to be a material constant. We present experiments demonstrating that the ratio of shear to normal force needed to move contacting blocks can, instead, vary systematically with controllable changes in the external loading configuration. Large variations in both the friction coefficient and consequent stress drop are tightly linked to changes in the rupture dynamics of the rough interface separating the two blocks.

The Near-Tip Fields of Fast Cracks

Slightly Cracked While there are detailed theories to explain the propagation of a crack in the bulk of a material, our understanding of cracking breaks down near the tip of the crack. Experimentally, it is very hard to observe the propagation of a crack at the tip region because it tends to move very quickly. Livne et al. (p. 1359) approached this problem by working with a polyacrylamide gel in which cracks progress slowly enough to monitor them. A hierarchy of linear and nonlinear regions was observed through which energy is transported before being dissipated by the growing crack. How stresses are distributed during cracking will determine whether the resulting failure will be brittle or ductile. The linear and nonlinear elastic responses near a growing crack tip can reveal how materials fail. In a stressed body, crack propagation is the main vehicle for material failure. Cracks create large stress amplification at their tips, leading to large material deformation. The material response within this highly deformed region will determine its mode of failure. Despite its great importance, we have only a limited knowledge of the structure of this region, because it is generally experimentally intractable. By using a brittle neo-Hookean material, we overcame this barrier and performed direct and precise measurements of the near-tip structure of rapid cracks. These experiments reveal a hierarchy of linear and nonlinear elastic zones through which energy is transported before being dissipated at a crack’s tip. This result provides a comprehensive picture of how remotely applied forces drive material failure in the most fundamental of fracture states: straight, rapidly moving cracks.

Propagating solitary waves along a rapidly moving crack front

A rapidly moving crack in a brittle material is often idealized as a one-dimensional object with a singular tip, moving through a two-dimensional material. However, in real three-dimensional materials, tensile cracks form a planar surface whose edge is a rapidly moving one-dimensional singular front. The dynamics of these fronts under repetitive interaction with material inhomogeneities (asperities) and the morphology of the fracture surface that they create are not yet understood. Here we show that perturbations to a crack front in a brittle material result in long-lived and highly localized waves, which we call ‘front waves’. These waves exhibit a unique characteristic shape and propagate along the crack front at approximately the Rayleigh wave speed (the speed of sound along a free surface). Following interaction, counter-propagating front waves retain both their shape and amplitude. They create characteristic traces along the fracture surface, providing cracks with both inertia and a new mode of dissipation. Front waves are intrinsically three-dimensional, and cannot exist in conventional two-dimensional theories of fracture. Because front waves can transport and distribute asperity-induced energy fluctuations throughout the crack front, they may help to explain how cracks remain a single coherent entity, despite repeated interactions with randomly dispersed asperities.