T Stresses for Edge Cracks and Vanishing Ligaments

Abstract

An edgecracked halfplane 0 < x < A and a halfplane x > 0 with a semiinfinite crack x > a perpendicular to the edge are examined in this paper. Uniform crackface 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,â€‌ Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding Tstress is obtained. For the additional cases of (i) nonuniform edgecrack crackface loading دƒ(x/A)k (â„œ(k)>1), (ii) concentrated loading at the edgecrack crack mouth, the Wiener–Hopf solutions and analytical Tstress expressions are provided, and tables of Tstress results for دƒ(x/A)k and دƒ(1x/A)k are presented. A Green's function for the edgecrack Tstress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,â€‌ Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,â€‌ Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic HalfPlane,â€‌ Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edgecracked halfplane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal Tstress expression for edgecracks is derived. Finally, the case of a vanishing uncracked ligament in a halfplane is examined, and the associated Wiener–Hopf solution and analytical Tstress expression are again provided. Several limiting cases are examined.