The mechanical properties of materials are ultimately determined by events occurring on the atomic scale. In the case of brittle fracture, this connection is obvious, since the crack in a perfectly brittle material must be atomically sharp at its tip. The crack moves by breaking individual bonds between atoms and can therefore be regarded as a macroscopic probe for the atomic bonding. Nevertheless, traditional analysis of brittle-fracture processes resorts to the treatment of Griffith,1 which implies thermodynamic equilibrium. The Griffith criterion for the mechanical stability of a crack can be formulated as a balance of the crack driving force, the energyrelease rate G, and the surface energy ɣs of the two freshly exposed fracture surfaces: G = 2ɣs. The crack driving force can be obtained from elasticity theory. Within linear elasticity, the crack is characterized by a singularity in the stress field that decays as the inverse square root of the distance R from the crack. The strength of the singularity is characterized by the stressintensity factor K, the square of which directly gives access to the energy-release rate (G = K2/E′, where E′ is an appropriate elastic modulus). While this linear elastic description of the material is not disputed for brittle materials, except for a few atomic bonds around the crack, the assumption that the resistance of the material to crack propagation will only be characterized by the surface energy of the fracture surfaces is certainly worth some further consideration. Such considerations should range from examining atomic details at the tip of a single brittle crack to the relevance of more complex fracture events involving additional irreversible processes and complex crack geometries.

That fracture is governed by processes occurring over a wide range of length scales has been recognized since the earliest developments of modern fracture mechanics. Griffitha's study of the strength of cracked solids 1,2 is perhaps the earliest example of such multiscale thinking, predating by several decades the first attempts to apply atomistically grounded traction-separation laws to fracture (e.g., the Orowan-Gilman model3,4). Griffith recognized the critical condition for crack extension to be a statement of thermodynamic equilibrium of a cracked solid, representing a balance between the mechanical energy decrease upon crack extension and the corresponding increase in energy due to the newly created crack surface. Griffith determined the elastic strain energy of the cracked body using the continuum solution of the stress field about an ellipse5 and recognized that the potential energy associated with the cleavage surfaces of the crack was directly proportional to the surface energy, the latter deriving from the cohesive molecular forces of the solid. The Irwin-Orowan extension of Griffith mechanics to include plastic dissipation,6-9 which is known to occur on the mesoscopic length scale (~1–100 μm), provides yet a further example of multiscale thinking in the early community of fracture researchers. In fact, the interaction of length scales is of central importance in most problems of fracture.

Our understanding of materials phenomena is based on a hierarchy of physical descriptions spanning the space-time regimes of electrons, atoms, and matter and given by the theories of quantum mechanics, statistical mechanics, and continuum mechanics. The pioneering work of Clementi and co-workers provides a lucid example of the traditional approach to incorporating multiscale phenomena associated with these three mechanics. Using quantum mechanics, they evaluated the interactions of several water molecules. From this data base, they created an empirical potential for use in atomistic mechanics and evaluated the viscosity of water. From this computed viscosity, they performed a fluid-dynamics simulation to predict the tidal circulation in Buzzard's Bay. This is a powerful example of the sequential coupling of length and time scales: a series of calculations is used as input to the next rung up the length/time-scale ladder.

One of the most interesting unsolved problems in fracture mechanics is the precise understanding of the energy-dissipation mechanisms that occur as a crack advances. In most cases, this energy-release rate is many times the surface energy created. One of the main reasons for this difference is the fact that plastic deformation can occur in the crack-tip region as dislocations nucleate and are emitted from the crack tip. Experimental studies provide little insight into the precise mechanisms for this process because they cannot reach the atomistic scale. For example, a crack that may seem experimentally sharp, and therefore indicative of brittle fracture, may not be sharp at the atomic level. Continuum mechanics has a similar limitation, since the assumptions of elasticity theory usually break down in the crack-tip region. Atomistic simulation studies provide researchers an opportunity to obtain precise atomic configurations in the crack-tip region under various loading conditions and to observe the basic energy-dissipation mechanisms.

Fracture dynamics remains a challenging research topic in materials science, mechanical engineering, mathematics, and nonequilibrium physics. Despite nearly a century of intense investigation, several basic problems remain unsolved. In particular, there is still no fundamental understanding of the distinction between brittle and ductile failure; there is still no definitive explanation of how breaking stresses can be transmitted through plastic zones near crack tips, nor is there an adequate understanding of why the energy-release rate even in brittle fracture is often orders of magnitude larger than the rate at which surface energy is created. These difficulties seem to stem primarily from the lack of an adequate theory of deformation near crack tips, where stresses and strain rates are large.

While it is possible to carry out fracture experiments in single crystals, in everyday experience fracture occurs in heterogeneous and often disordered materials. Tear a piece of paper, and the resulting ragged edge shows evidence of the local variation in the paper's mechanical properties. One might expect the same behavior in fracture of other disordered materials such as polycrystalline solids, fiber composites, and concrete.