Integral equation methods for multiple crack problemsand related topics
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摘要: The content of thisreview consists of recent developments covering an advancedtreatment of multiple crack problems in plane elasticity. Severalelementary solutions are highlighted, which are the fundamentalsfor the formulation of the integral equations. The elementarysolutions include those initiated by point sources or by adistributed traction along the crack face. Two kinds of singularintegral equations, three kinds of Fredholm integral equations,and one kind of hypersingular integral equation are suggested forthe multiple crack problems in plane elasticity. Regularizationprocedures are also investigated. For the solution of the integralequations, the relevant quadrature rules are addressed. A varietyof methods for solving the multiple crack problems is introduced.Applications for the solution of the multiple crack problems arealso addressed. The concept of the modified complex potential(MCP) is emphasized, which will extend the solution range, forexample, from the multiple crack problem in an infinite plate tothat in a circular plate. Many multiple crack problems areaddressed. Those problems include: (i) multiple semi-infinitecrack problem, (ii) multiple crack problem with a general loading,(iii) multiple crack problem for the bonded half-planes, (iv)multiple crack problem for a finite region, (v) multiple crackproblem for a circular region, (vi) multiple crack problem inantiplane elasticity, (vii) T-stress in the multiple crackproblem, and (viii) periodic crack problem and many others. Thisreview article cites 187 references.Abstract: 综述了平面弹性力学多裂纹问题的一些近代先进解法.一些基本解被着重提出, 它们是构成积分方程的基础.这些基本解包括由点源引起和沿裂纹线分布载荷引起.关于平面弹性力学多裂纹问题, 介绍了二类奇异积分方程, 三类Fredholm积分方程和一类超奇异积分方程.文中还研究了奇异积分方程的正则化问题, 即转化为Fredholm积分方程的方法. 为了求解上述积分方程,介绍了相应求积公式,并详细介绍求解其它众多多裂纹问题的各种方法,阐明了多裂纹解的应用.本文强调了修正复位函数这一概念的重要性, 因为它扩大了求解范围.还研究了下列多裂纹问题: (1)多半无限长裂纹问题; (2)一般载荷情况下的多裂纹问题; (3)粘合半平面情况下的多裂纹问题; (4)有限区域的多裂纹问题; (5) 圆形域多裂纹问题; (6)反平面弹性情况下的多裂纹问题; (7) 多裂纹问题中的$T$应力; (8)周期裂纹问题 及其它等等. 本综述共引用了187 篇学术论文.
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