Michael Marder

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.

Origin of crack tip instabilities

Abstract This paper demonstrates that rapid fracture of ideal brittle lattices naturally involves phenomena long seen in experiment, but which have been hard to understand from a continuum point of view. These idealized models do not mimic realistic microstructure, but can be solved exactly and understood completely. First it is shown that constant velocity crack solutions do not exist at all for a range of velocities starting at zero and ranging up to about one quarter of the shear wave speed. Next it is shown that above this speed cracks are by and large linearly stable, but that at sufficiently high velocity they become unstable with respect to a nonlinear microcracking instability. The way this instability works itself out is related to the scenario known as intermittency, and the basic time scale which governs it is the inverse of the amount of dissipation in the model. Finally, we compare the theoretical framework with some new experiments in Plexiglas, and show that all qualitative features of the theory are mirrored in our experimental results.

Dynamic Fracture in Single Crystal Silicon

We have measured the velocity of a running crack in brittle single crystal silicon as a function of energy flow to the crack tip. The experiments are designed to permit direct comparison with molecular dynamics simulations; therefore the experiments provide an indirect but sensitive test of interatomic potentials. Performing molecular dynamics simulations of brittle crack motion at the atomic scale we find that experiments and simulations disagree showing that interatomic potentials are not yet well understood.

Gas production in the Barnett Shale obeys a simple scaling theory

Significance Ten years ago, US natural gas cost 50% more than that from Russia. Now, it is threefold less. US gas prices plummeted because of the shale gas revolution. However, a key question remains: At what rate will the new hydrofractured horizontal wells in shales continue to produce gas? We analyze the simplest model of gas production consistent with basic physics of the extraction process. Its exact solution produces a nearly universal scaling law for gas wells in each shale play, where production first declines as 1 over the square root of time and then exponentially. The result is a surprisingly accurate description of gas extraction from thousands of wells in the United States’ oldest shale play, the Barnett Shale. Natural gas from tight shale formations will provide the United States with a major source of energy over the next several decades. Estimates of gas production from these formations have mainly relied on formulas designed for wells with a different geometry. We consider the simplest model of gas production consistent with the basic physics and geometry of the extraction process. In principle, solutions of the model depend upon many parameters, but in practice and within a given gas field, all but two can be fixed at typical values, leading to a nonlinear diffusion problem we solve exactly with a scaling curve. The scaling curve production rate declines as 1 over the square root of time early on, and it later declines exponentially. This simple model provides a surprisingly accurate description of gas extraction from 8,294 wells in the United States’ oldest shale play, the Barnett Shale. There is good agreement with the scaling theory for 2,057 horizontal wells in which production started to decline exponentially in less than 10 y. The remaining 6,237 horizontal wells in our analysis are too young for us to predict when exponential decline will set in, but the model can nevertheless be used to establish lower and upper bounds on well lifetime. Finally, we obtain upper and lower bounds on the gas that will be produced by the wells in our sample, individually and in total. The estimated ultimate recovery from our sample of 8,294 wells is between 10 and 20 trillion standard cubic feet.

Friction and fracture

Consider a block placed on a table and pushed sideways until it begins to slide. Amontons and Coulomb found that the force required to initiate sliding is proportional to the weight of the block (the constant of proportionality being the static coefficient of friction), but independent of the area of contact. This is commonly explained by asserting that, owing to the presence of asperities on the two surfaces, the actual area in physical contact is much smaller than it seems, and grows in proportion to the applied compressive force. Here we present an alternative picture of the static friction coefficient, which starts with an atomic description of surfaces in contact and then employs a multiscale analysis technique to describe how sliding occurs for large objects. We demonstrate the existence of self-healing cracks that have been postulated to solve geophysical paradoxes about heat generated by earthquakes, and we show that, when such cracks are present at the atomic scale, they result in solids that slip in accord with Coulombs law of friction. We expect that this mechanism for friction will be found to operate at many length scales, and that our approach for connecting atomic and continuum descriptions will enable more realistic first-principles calculations of friction coefficients.

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.

Mechanics: Buckling cascades in free sheets

The edge of a torn plastic sheet forms a complex three-dimensional fractal shape. We have found that the shape results from a simple elongation of the sheet in the direction along its edge. Natural growth processes in some leaves, flowers and vesicles could lead to a similar elongation and hence to the generation of characteristic wavy shapes.

Leaves, flowers and garbage bags: Making waves

the world’s most durable mysteries. Some patterns—clouds, snowflakes— form in space. Others—the ebb and flow of tides, seasonal wet and dry spells—are patterns that form in time. Natural patterns are mysterious because they are complex, organized and interconnected, even though the laws of physics on which they rest—Newton’s classical laws of motion—are simple. The living world presents the ultimate examples of pattern formation. The patterns in biological systems are the most stunningly complex of any we encounter. Consider: In order to form a complex organism from an initial featureless collection of identical cells, a system must undergo myriad transitions that break its spatial symmetries and trigger the differentiation of cells at selected sites. How are these sites selected? How complex and controlled must a growth process be to direct that particular things happen in sequence and at the right sites? It is difficult to imagine how the impersonal interactions of atoms can lead to the growth of a plant or an animal from inanimate matter. Yet in fact this is what happens with the birth and development of every living thing. Some of the simplest features of biological shapes can be explained by basic physical laws. We will describe here an elegant example: the edges of flowers and leaves, where complex rippled shapes give the impressions of ruffles and frills. We suspected that very simple growth processes might provide the mechanism that shapes thin membranes and sheets into complex shapes in space, and indeed we have found that they do. By themselves, these processes do not break

Cracks and Atoms

Many materials scientists and engineers are, with some justification, suspicious of theoretical and numerical studies ascending from the atomic scale on the mechanical response of materials. On the one hand, there is a reluctance to believe that the invisible atomic scale is important for macroscopic mechanical deformation. Out of sight, out of mind. On the other hand, many large scale computer simulations that produce brightly colored pictures with gobs of toy atoms, and sometimes even impressive statistics on processing efficiency, seem simply to avoid questions on how to compare computation with either theory or experiment. For in fact, a calculation involving ten billion atoms, necessarily with questionable effective atomic interactions, would exceed the powers of the worlds largest computers, and yet describe only a cube of matter no more than half a micrometer along each side. And even when computers become large enough to store and manipulate the coordinates of this many particles, it will not be possible to follow their behavior for much more than a nanosecond, thus making comparison with experiment seem as remote as a manned flight to Pluto. These simple observations lie behind the dominance of continuum mechanics in most studies of mechanical behavior of materials. Obviously, so the argument goes, it is an enormous waste of effort to calculate the motion of every atom when all information of interest is contained in continuous fields that are most sensibly studied by other means. Hence the feeling, widely held but seldom expressed, that areal materials are not made of atomso. The point of this article is to show that this feeling is wrong. Materials constantly betray their atomic underpinnings. When this happens, it should come as no surprise that the continuum theory breaks down, since it requires a great deal of cleverness indeed to apply continuum elastic theory to phenomena that are neither continuous nor elastic. We will discuss properties of materials for which atomic features are essential to even a qualitative understanding, and show how to design studies at the atomic scale in an efficient manner, studies which permit direct comparison with experiment. The mechanical response of materials is an enormous and varied subject. We will therefore focus on one particular case that makes it possible to examine the relationship between atomic and macroscopic scales in detail: the process of brittle fracture. Fracture is important because it determines the ultimate strength of a wide range of materials. Fracture fundamentally has to do with the severing of inter-atomic bonds: this points theoretical investigations toward atomic-scale studies. As gem-cutters know, cracks tend to run along crystal planes, showing that the process is sensitive to atomic detail. Nevertheless, most fracture research is carried out in the context of continuum elasticity through an elegant framework that bypasses most of the questions arising at the atomic scale. Our aim is to identify the questions that the continuum approach cannot address, and to show how a combination of theoretical insight and numerical computation can be employed to answer them. The ability to compare directly with experiment will then provide a strong test of the correctness of the underlying interatomic potentials used in simulations.

Rippling of graphene

We show that ripples observed in free-standing graphene sheets can be explained as a consequence of adsorbed OH molecules sitting on random sites. The adsorbates cause the bonds between carbon atoms to lengthen slightly. Static buckles then result from a mechanism like the one that leads to buckling of leaves. Buckles caused by roughly 20% coverage of adsorbates are consistent with experimental observations.