Lal Sharma Basant, Mishuris Gennady
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India.
Department of Mathematics, Aberystwyth University, Aberystwyth, UK.
Proc Math Phys Eng Sci. 2020 Mar;476(2235):20190686. doi: 10.1098/rspa.2019.0686. Epub 2020 Mar 18.
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 , , 1171-1192 (doi:10.1137/140985093); Sharma 2015 , 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 × equations, where 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.
无限方形晶格中的半无限裂纹受到来自无穷远处的波的作用，从而导致裂纹面产生散射。裂纹尖端前方的部分损伤区域通过损伤链节的任意分布刚度来建模。虽然晶格中具有原子尖锐裂纹尖端的开口裂纹已借助标量维纳 - 霍普夫公式以封闭形式求解（沙玛2015年，第1171 - 1192页（doi:10.1137/140985093）；沙玛2015年，第1915 - 1940页。（doi:10.1137/15M1010646）），但此处考虑的问题根据损伤链节的性质变得非常复杂。例如，在部分桥接有限区域的情况下，它涉及一个具有强大类别的2×2矩阵核。但是使用一种原始技术，该问题，包括任意损伤链节的一般情况，除了涉及求解一个×方程的辅助线性系统（其中定义了损伤区域的长度）外，被简化为一个标量问题。所提出的方法确实有效地允许构建精确解。还给出了数值示例以及远离裂纹尖端的散射场的渐近近似。
