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REVIEW ARTICLES

# Integral Equation Methods for Multiple Crack Problems and Related Topics

[+] Author and Article Information
Y. Z. Chen

Division of Engineering Mechanics, Jiangsu University, Zhenjiang, Jiangsu, 212013, People’s Republic of Chinachens@ujs.edu.cn

Appl. Mech. Rev 60(4), 172-194 (Jul 01, 2007) (23 pages) doi:10.1115/1.2750671 History:

## Abstract

The content of this review consists of recent developments covering an advanced treatment of multiple crack problems in plane elasticity. Several elementary solutions are highlighted, which are the fundamentals for the formulation of the integral equations. The elementary solutions include those initiated by point sources or by a distributed traction along the crack face. Two kinds of singular integral equations, three kinds of Fredholm integral equations, and one kind of hypersingular integral equation are suggested for the multiple crack problems in plane elasticity. Regularization procedures are also investigated. For the solution of the integral equations, the relevant quadrature rules are addressed. A variety of methods for solving the multiple crack problems is introduced. Applications for the solution of the multiple crack problems are also addressed. The concept of the modified complex potential (MCP) is emphasized, which will extend the solution range, for example, from the multiple crack problem in an infinite plate to that in a circular plate. Many multiple crack problems are addressed. Those problems include: (i) multiple semi-infinite crack 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 crack problem for a circular region, (vi) multiple crack problem in antiplane elasticity, (vii) T-stress in the multiple crack problem, and (viii) periodic crack problem and many others. This review article cites 187 references.

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## Figures

Figure 1

A smooth curve L in an infinite plane

Figure 2

(a) A point dislocation applied at the point z=t, (b) concentrated forces applied at the point z=t, (c) a dislocation doublet in an infinite plate, and (d) a force doublet in an infinite plate

Figure 3

(a) A point dislocation pair at the point z=t, (b) two concentrated force pairs at the point z=t, (c) other point dislocation pair at the point z=t, and (d) other two concentrated force pairs at the point z=t

Figure 4

(a) Loadings with same magnitude and opposite direction on crack faces, and (b) loadings with same magnitude and same direction on crack faces

Figure 5

Superposition method for the multiple crack problems: (a) the original problem, (b) superposition by distributed dislocations, and (c) superposition by many cracks with the undetermined tractions

Figure 6

Superposition for two collinear cracks, (a) the original problem, (b) a single crack problem, (c) the other single crack problem

Figure 7

Various conditions for the multiple crack problem for the bonded half planes, strip, or finite region: (a) a general loading applied at upper half plane, (b) multiple crack problems for bonded half planes, (c) multiple crack problems for upper half plane with a traction free boundary, (d) multiple cracks in a strip, and (e) multiple cracks in a finite region

Figure 8

Multiple crack problems for a circular region

Figure 9

Superposition method for the periodic crack problem: (a) the original problem, (b) superposition by the distributed dislocations, and (c) superposition by many cracks with the undetermined tractions

Figure 10

(a) An infinite plate with the doubly periodic cracks and (b) the cracked rectangular cell

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