Shear crack propagation along weak planes in solids: A finite deformation analysis incorporating the linear harmonic potential

Recent molecular dynamics simulations of dynamic crack propagation have shown that a shear crack propagating at a sub-Rayleigh speed may have a finite crack opening, but the crack opening vanishes once the crack tip velocity exceeds the shear wave speed. This observation is at odds with classical linear elastic solutions which indicate that a pure shear crack should have zero opening. To understand this discrepancy, we develop in this paper a finite deformation continuum theory incorporating the linear harmonic potential to describe the deformation of a crack in a solid with triangular lattice structure. Using the asymptotic method of Knowles [Eng. Fract. Mech. 15 (1981) 469], we show that there is indeed a finite crack opening for a dynamic, sub-Rayleigh shear crack. This opening is on the order of the lattice constant, and is attributable to the geometric nonlinearity of finite deformation near the crack tip. We also show in another paper that, once the crack tip velocity exceeds the shear wave speed, Knowles' asymptotic method gives a vanishing crack opening. These conclusions based on the continuum analysis for sub-Rayleigh and super-shear cracks agree well with the molecular dynamics simulations.