Cracks and Atoms

Abstract

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 world's 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.