Near-Tip Field for Diffraction on Square Lattice by Crack

Abstract

The displacement field near a tip of a finite crack, due to the diffraction of a wave on a square lattice, is studied. The finite section method, in the theory of Toeplitz operators on $\ell_2$, is invoked as the semi-infinite crack diffraction problem is shown equivalent to the inversion of a Toeplitz operator, a truncation of which appears in the finite crack diffraction problem for the same incident wave. The existence and uniqueness of the solution in $\ell_2$ for the semi-infinite crack problem is established by an application of the well-known Krein conditions. Continuum limit of the semi-infinite crack diffraction problem is established in a discrete Sobolev space; a graphical illustration of convergence in the relevant Sobolev norm is also included. A low-frequency asymptotic approximation of the normalized shear force, in “vertical” bonds ahead of the crack tip, recovers the classical crack tip singularity. Displacement of particles in the vicinity of the crack tip, a closed form expression of which is provided, is compared graphically with that obtained by a numerical solution of the diffraction problem on a finite grid. Graphical results are also included to demonstrate that the normalized shear force in “horizontal” bonds, along the crack face, approaches the corresponding stress component in the continuum model at sufficiently low frequency. Numerical solutions indicate that the crack opening displacement of a semi-infinite crack approximates that of a finite crack of sufficiently large size, and at sufficiently high frequency of incident wave, away from the neighborhood of the other crack tip, while it differs significantly for low frequencies as a result of multiple scattering due to the two crack tips.