Atomistic Aspects of Brittle Fracture
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Peter Gumbsch and Rowland M. Cannon Introduction 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 s of the two freshly exposed fracture surfaces: G 2s. The crack driving force can be obtained from elasticity theory.2 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 K 2/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. From an atomistic point of view, resistance to crack propagation should be characterized by the forces needed to separate bonds successively, as schematically illustrated in Figure 1. Indeed, the first atomistic studies of fracture, accomplished two decades ago using simple pairwise inter-
MRS BULLETIN/MAY 2000
actions, showed that the discreteness of the lattice manifests itself in the so-called lattice-trapping effect.3 Lattice trapping causes the crack to remain stable and not to advance until loads K that are somewhat larger than the Griffith load KG are reached. Similarly, the crack will not heal until a load K KG is reached.4 The critical loads K, are called the upper and lower trapping limits. The lattice trapping of the crack can be viewed as the equivalent
Figure 1. (a) Schematic drawing of the atomic configuration of the tip of a brittle crack with force f. (b) Different force laws (force f versus distance from the crack, R) for the crack-tip bond give the same surface energy, but lead to very different lattice trapping. Illustrated are the linear-elastic “snapping-spring” law (heavy dashed line) and short-range (solid line) and long-range (dotted line) nonlinear interactions.
of the Peierls barrier, which impedes the motion
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