Research Papers

T-Stresses for Edge Cracks and Vanishing Ligaments

[+] Author and Article Information
John P. Dempsey

Clarkson University,
Potsdam, NY 13699-5710
e-mail: jdempsey@clarkson.edu

Manuscript received October 12, 2012; final manuscript received November 4, 2012; accepted manuscript posted November 20, 2012; published online May 31, 2013. Assoc. Editor: Pradeep Sharma.

J. Appl. Mech 80(4), 041035 (May 31, 2013) (9 pages) Paper No: JAM-12-1473; doi: 10.1115/1.4023033 History: Received October 12, 2012; Revised November 04, 2012; Accepted November 20, 2012

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 ((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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Grahic Jump Location
Fig. 1

Half-plane with edge-crack

Grahic Jump Location
Fig. 2

T-stress for concentrated loads acting at x=x*=Aρ*

Grahic Jump Location
Fig. 3

Half-plane with finite ligament




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