TY - JOUR

T1 - Scattering on a square lattice from a crack with a damage zone

AU - Sharma, Basant Lal

AU - Mishuris, Gennady

N1 - © 2020 The Authors.

PY - 2020/3/18

Y1 - 2020/3/18

N2 - 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.

AB - 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.

KW - Crack

KW - Damage zone

KW - Diffraction

KW - Wiener-hopf method

UR - http://www.scopus.com/inward/record.url?scp=85083625744&partnerID=8YFLogxK

U2 - 10.1098/rspa.2019.0686

DO - 10.1098/rspa.2019.0686

M3 - Article

C2 - 32269488

AN - SCOPUS:85083625744

SN - 1364-5021

VL - 476

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2235

M1 - 20190686

ER -