Contact interaction of the borders of edge cracks in a flexible half plane

This paper provides, an elastic solution for multiple curved edge cracks emanating from the boundary of the half-plane. After placing the distributed dislocations at the prospective sites of cracks in an infinite plate, the principal part of the complex potentials is obtained. By using the concept of the modified complex potentials, the complementary part of the complex potentials can be derived. The whole complex potentials satisfy the traction free condition along the boundary of half-plane automatically. This is a particular advantage of the suggested method. This concept or method of the modified complex potentials is a counterpart of the Green's function method, which is universal in mathematical physics. The direct usage of this method cannot provide a solution in detail. Comparing with the line edge crack case, the following points are significant in the presented study. The relevant kernels in the integral equation are more complicated than in the line edge crack case and the relevant integrations in the problem should be completed on curves. This paper solves a rather complicated problem, the multiple curved edge crack problem, and gives the final solution. A singular integral equation is formulated with the dislocation distribution being unknown function and the traction being the right hand term. The singular integral equation is solved by using the curve length method in conjunction with the semiopening quadrature rule. Periodic curved edge crack problem is also addressed. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

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T-Stresses for Edge Cracks and Vanishing Ligaments

Journal of Applied Mechanics ◽

10.1115/1.4023033 ◽

2013 ◽

Vol 80 (4) ◽

Cited By ~ 1

Author(s):

John P. Dempsey

Keyword(s):

Half Plane ◽

Edge Crack ◽

Flexural Rigidity ◽

Quantitative Relationship ◽

Crack Mouth ◽

Crack Face ◽

T Stress ◽

Concentrated Loading ◽

Edge Cracks ◽

Uncracked Ligament

An edge-cracked half-plane 0 < x < A and a half-plane x > 0 with a semi-infinite crack x > a perpendicular to the edge are examined in this paper. Uniform crack-face loading is thoroughly examined, with a thorough exposition of the Koiter Wiener–Hopf approach (Koiter, 1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding T-stress is obtained. For the additional cases of (i) nonuniform edge-crack crack-face loading σ(x/A)k (ℜ(k)>-1), (ii) concentrated loading at the edge-crack crack mouth, the Wiener–Hopf solutions and analytical T-stress expressions are provided, and tables of T-stress results for σ(x/A)k and σ(1-x/A)k are presented. A Green's function for the edge-crack T-stress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,” Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic Half-Plane,” Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edge-cracked half-plane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal T-stress expression for edge-cracks is derived. Finally, the case of a vanishing uncracked ligament in a half-plane is examined, and the associated Wiener–Hopf solution and analytical T-stress expression are again provided. Several limiting cases are examined.

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Response of an orthotropic half-plane subjected to transient anti-plane loading with multiple edge cracks

Journal of Physics Conference Series ◽

10.1088/1742-6596/843/1/012015 ◽

2017 ◽

Vol 843 ◽

pp. 012015

Author(s):

M. Ayatollahi

Keyword(s):

Half Plane ◽

Multiple Edge ◽

Edge Cracks

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