Abstract
A semi-infinite crack in an infinite square lattice is subjected to a wave coming from infinity, thereby leading to its scattering by the crack surfaces. A partially damaged zone ahead of the crack tip is modelled by an arbitrarily distributed stiffness of the damaged links. While an open crack, with an atomically sharp crack tip, in the lattice has been solved in closed form with the help of the scalar Wiener–Hopf formulation (Sharma 2015 SIAM J. Appl. Math., 75, 1171–1192 (doi:10.1137/140985093); Sharma 2015 SIAM J. Appl. Math. 75, 1915–1940. (doi:10.1137/15M1010646)), the problem considered here becomes very intricate depending on the nature of the damaged links. For instance, in the case of a partially bridged finite zone it involves a 2 × 2 matrix kernel of formidable class. But using an original technique, the problem, including the general case of arbitrarily damaged links, is reduced to a scalar one with the exception that it involves solving an auxiliary linear system of N × N equations, where N defines the length of the damage zone. The proposed method does allow, effectively, the construction of an exact solution. Numerical examples and the asymptotic approximation of the scattered field far away from the crack tip are also presented.
Highlights
Among other distinguished as well as popular works [1], Peter Chadwick made several contributions to the wave propagation problems in anisotropic models with different kinds of symmetries as well as those applicable to the theory of lattice defects [2,3,4,5,6,7,8]
In the context of the matrix kernel (4.17), with the distinguished presence of the off-diagonal factors z−2M and z2M, the reduction to the linear algebraic equation obtained above is reminiscent of that proposed for the Wiener–Hopf kernel with exponential phase factors that appear in several continuum scattering problems in fluid mechanics and fracture mechanics [52,53,54,55], and their discrete analogues in the form of scattering due to a pair of staggered cracks and rigid constraints [34,56,78]; both of these are based on an exact solution of the corresponding staggerless case [35,79,80,81]
We have shown how the scattering problem in a square lattice with an infinite crack with a damage zone near the crack tip of arbitrary properties can be effectively solved by We were able to reduce it to a scalar Wiener–Hopf method
Summary
Among other distinguished as well as popular works [1], Peter Chadwick made several contributions to the wave propagation problems in anisotropic models with different kinds of symmetries as well as those applicable to the theory of lattice defects [2,3,4,5,6,7,8]. The problem considered in this paper, becomes much more intractable when compared with the scattering due to an atomically sharp crack tip that has been solved in [12,13] using the scalar Wiener–Hopf factorization [11,51]. It is shown that a reduction to a scalar problem is possible with the additional clause that it involves solving an auxiliary linear system of N × N equations, where N represents the size of the cohesive zone Such a reduction resembles the one proposed for the Wiener–Hopf kernel with exponential phase factors in the continuum case [52,53,54,55], and its recently investigated discrete analogue of scattering due to a pair of staggered crack tips [34,56]. For the issues dealing with the difficult cases of the matrix Wiener–Hopf problems, the reader is referred to [57,67,68,69,70,71,72,73]
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