With the advent of scalable parallel computers, computational molecular dynamics can be a very powerful tool for providing immediate insights into the nature of fracture dynamics. We have studied two dimensional triangular solids with up to 1500 atoms on a side, or about one half of a micron in length. If we were to do three dimensions for an equivalent number of atoms, a cube would be only 130 atoms on a side! But like experiment (3-5), our interest was to study two-dimensional `mode one' loading. We were able to follow the crack propagation over sufficient time and distance intervals so that a comparison with experiment became feasible. While the laboratory experiments used amorphous materials in order to suppress the possibility that the instability is due to material defects, we used a defect-free 2D crystal: an amorphous packing is not stable for a one-component 2D system.
We refer the reader to Allen and Tildesley (10), Hoover (11) or Abraham (12) for a treatment of the molecular dynamics (MD) simulation technique, including the many procedures for implementing it. MD predicts the motion of a given number of atoms governed by their mutual interatomic interactions described by a continuous potential function and requires the numerical integration of Hamilton's classical equations of motion. We assume that the interatomic forces are described by a Lennard-Jones (LJ) 12:6 potential with a spline cutoff (13). Quantities are expressed in terms of reduced units: lengths are scaled by the parameter , the value of the interatomic separation for which the LJ potential is zero, and energies are scaled by the parameter , the depth of the minimum of the LJ potential. Reduced temperature is therefore . Our choice of the simple LJ force law is dictated by our interest to study the microscopic features of brittle fracture common to a large class of real physical systems. The LJ potential can be used to represent a generic ``brittle" material (13). For the LJ 2D solid (11), the longitudinal sound speed at zero temperature and pressure is and the transverse sound speed The Rayleigh speed is approximately equal to the transverse sound speed (11). Our parallel molecular dynamics program is implemented on the IBM PVS using 16 nodes and on the IBM SP1 using 64 nodes. For 2,027,776 atoms, the update time per time step is 0.9 seconds for the IBM SP1.
Our system is a 2D rectangular slab of atoms with atoms on a side, where = 712 for the half million atom system and 1424 for the two million atom system. The slab is initialized at a reduced temperature of 0.0001. A triangular notch of 10 and 20 lattice spacings for the two respective lengths is cut midway along the lower horizontal slab boundary, and an outward strain rate is imposed on the outer most columns of atoms defining the opposing vertical faces of the slab. A linear velocity gradient is established across the slab, and an increasing lateral strain occurs in the solid slab. This leads to eventual structural failure of the material, and this failure can take quite different forms, depending on the applied strain rate. The failure could occur by a single fracture (which is seeded by a notch), multiple internal fracture due to the occurrence of voids, as well as `necking' by slippage of atomic rows of atoms. We found that a strain rate of is sufficiently small for our size systems to prevent multiple fracture accompanying fracture at the notch. With this choice, the solid fails at the notch tip when the solid has been stretched by percent. Our system has an exaggerated critical strain for tip failure, which is about an order of magnitude larger than experiment (3,4). This is necessary to achieve a fracture dynamics that is sufficiently fast to follow by molecular dynamics. At the onset of crack motion, the imposed strain rate remains constant (experiment 1) or is set to zero (experiment 2), and the simulation is continued until the growing crack has traversed the total length of the slab.