Solutions of plane crack problems by mapping technique

In an earlier paper it was suggested that a knowledge of the elastic-stress variation in the neighborhood of an angular corner of an infinite plate would perhaps be of value in analyzing the stress distribution at the base of a V-notch. As a part of a more general study, the specific case of a zero-angle notch, or crack, was carried out to supplement results obtained by other investigators. This paper includes remarks upon the antisymmetric, as well as symmetric, stress distribution, and the circumferential distribution of distortion strain-energy density. For the case of a symmetrical loading about the crack, it is shown that the energy density is not a maximum along the direction of the crack but is one third higher at an angle ± cos−1 (1/3); i.e., approximately ±70 deg. It is shown that at the base of the crack in the direction of its prolongation, the principal stresses are equal, thus tending toward a state of (two-dimensional) hydrostatic tension. As the distance from the point of the crack increases, the distortion strain energy increases, suggesting the possibility of yielding ahead of the crack as well as ±70 deg to the sides. The maximum principal tension stress occurs on ±60 deg rays. For the antisymmetrical stress distribution the distortion strain energy is a relative maximum along the crack and 60 per cent lower ± 85 deg to the sides.