Cracks cleave crystals

– The problem of finding what direction cracks should move is not completely solved. A commonly accepted way to predict crack directions is by computing the density of elastic potential energy stored well away from the crack tip, and finding a direction of crack motion to maximize the consumption of this energy. I provide in this letter a specific case where this rule fails. The example is of a crack in a crystal. It fractures along a crystal plane, rather than in the direction normally predicted to release the most energy. Thus, a correct equation of motion for brittle cracks must take into account both energy flows that are described in conventional continuum theories and details of the environment near the tip that are not. Introduction. – Much of the continuum theory of fracture concerns itself with the initiation of cracks, and their speed in response to varying loads [1]. Cracks also choose a direction in which to move. The continuum theory of fracture has no law to describe unambiguously which way they choose. However, there is a rule that is widely employed in practice to calculate the direction of crack motion. This rule is the principle of local symmetry. It says that cracks advance in the direction such that shear stresses on the faces of the crack vanish near the tip; the stresses are purely tensile, and pull the crack faces apart. Equivalently, cracks move in a direction that maximizes the consumption of energy stored in linear elastic fields in front of the tip. The rule was first proposed for slowly moving cracks by Goldstein and Salganik [2], generalized to rapidly moving cracks by Adda-Bedia, Arias, Ben Amar and Lund [3], and has recently been derived carefully from a variational principle by Oleaga [4]. Experimental checks are not numerous, but they confirm the principle of local symmetry. They have been carried out in amorphous materials such as glass, where alternatives are difficult to imagine [5–9]. Experiments in crystals, and the art of cutting gems [10], find a tendency of cracks to travel along special atomic planes [11,12]. However, the crystals where these experiments have been performed are macroscopically anisotropic. The preference of cracks for certain directions can be attributed to the lack of isotropy in the continuum theory. Thus it has been reasonable to believe that cracks in a macroscopically homogeneous and isotropic material should always move in accord with the dictates of local symmetry. I will provide in this letter a specific system where the principle of local symmetry is not obeyed, despite the fact that macroscopically the system is homogeneous and isotropic. The demonstration comes from combined analytical and numerical work. The numerical computations involve small numbers of atoms (40000). However, by combining the computations with scaling theory, one can predict the outcome of experiments with arbitrarily large numbers of atoms, and over arbitrarily large time intervals [13].