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含过程区裂纹的反平面问题中具有指数因子的矩阵函数的数值分解

Numerical factorization of a matrix-function with exponential factors in an anti-plane problem for a crack with process zone.

作者信息

Livasov P, Mishuris G

机构信息

Department of Mathematics, IMAPS, Aberystwyth University, Aberystwyth, Ceredigion SY23 3BZ, UK.

出版信息

Philos Trans A Math Phys Eng Sci. 2019 Oct 21;377(2156):20190109. doi: 10.1098/rsta.2019.0109. Epub 2019 Sep 2.

DOI:10.1098/rsta.2019.0109
PMID:31474200
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC6732375/
Abstract

In this paper, we consider an interface mode III crack with a process zone located in front of the fracture tip. The zone is described by imperfect transmission conditions. After application of the Fourier transform, the original problem is reduced to a vectorial Wiener-Hopf equation whose kernel contains oscillatory factors. We perform the factorization numerically using an iterative algorithm and discuss convergence of the method depending on the problem parameters. In the analysis of the solution, special attention is paid to its behaviour near both ends of the process zone. Qualitative analysis was performed to determine admissible values of the process zone's length for which equilibrium cracks exist. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

摘要

在本文中，我们考虑一个III型界面裂纹，其断裂尖端前方存在一个过程区。该区域由不完美的传输条件描述。应用傅里叶变换后，原问题简化为一个矢量维纳 - 霍普夫方程，其核包含振荡因子。我们使用迭代算法对其进行数值因式分解，并根据问题参数讨论该方法的收敛性。在解的分析中，特别关注其在过程区两端附近的行为。进行了定性分析以确定存在平衡裂纹时过程区长度的允许值。本文是主题为“结构化介质中动态现象和局部化的建模（第1部分）”的一部分。

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本文引用的文献

1
Regular approximate factorization of a class of matrix-function with an unstable set of partial indices.一类具有不稳定部分指标集的矩阵函数的正则近似分解
Proc Math Phys Eng Sci. 2018 Jan;474(2209):20170279. doi: 10.1098/rspa.2017.0279. Epub 2018 Jan 17.
2
Brittle fracture in a periodic structure with internal potential energy.具有内部势能的周期性结构中的脆性断裂。
Proc Math Phys Eng Sci. 2014 May 8;470(2165):20130821. doi: 10.1098/rspa.2013.0821.
具有损伤区域的裂纹在方形晶格上的散射。
Proc Math Phys Eng Sci. 2020 Mar;476(2235):20190686. doi: 10.1098/rspa.2019.0686. Epub 2020 Mar 18.
4
Applying an iterative method numerically to solve × matrix Wiener-Hopf equations with exponential factors.运用迭代法数值求解具有指数因子的 × 矩阵 Wiener-Hopf 方程。
Philos Trans A Math Phys Eng Sci. 2020 Jan 10;378(2162):20190241. doi: 10.1098/rsta.2019.0241. Epub 2019 Nov 25.