Document Type

Thesis - Open Access

Award Date


Degree Name

Master of Science (MS)


Mechanical Engineering


It is a formidable task to find exact solutions to elastoplastic problems even for plane strain or plane stress and yet more so for three dimensional problems. Some investigators tried to analyze the problem of growth of the plastic zone from the tips of cracks or notches by introducing certain models such as those of Dugdale, Barenblatt and Wnuk. Dugdale, led by an ingenious intuition, proposed that the plastic deformation at the crack tip is entirely confined to a narrow-tapered zone extending in the crack plane. Barenblatt has introduced a modulus called "cohesive modulus" in order to exp lain the quasi-ductile fracture. Wnuk proposed a simple "cut end cigar" crack to describe the effect of plasticity and time on the fracture. The scale of yielding is required to be small. Since some progress could be made in obtaining the exact solutions for cracks under longitudinal stress, it was Hult and McClintock who first succeeded in obtaining the exact solution for a particular case. We would like to mention here the formulation of crack problems proposed by Bilby, Cottrell and Swinden which are essentially based on the same assumption used by Dugdale and Field but differs in the use of dislocation theory terminology. In many papers including this thesis, the idea of "quasi-brittle" fracture is used. It agrees well with the experiment when the plastic deformations which occur before the fracture starts, are confined to a narrow layer, ahead of the crack tip. The work expended in plastic deformation exceeds many times the energy lost in the separation of two surface s in a perfectly brittle fracture process. It was Kraft, later Srawley and Brown and simultaneously McClintock and Irwin who recognized the possibility of a stable crack growth induced in a ductile metal under a monotonically increasing load. McClintock argued that in a ductile so lid sub-initial growth would not be possible without continued increase of load. He showed that after the crack progresses by an infinitesimal amount, the redistribution of stresses and strains lowers the strain at the crack-root and thus further load increase is necessary to elevate root strain to the critical value. But conclusions obtained by McClintock and Rice came from tedious numerical or graphical calculations which pertained to a solution of a non-linear integral equation. A more straight forward approach but based on a simplified Dugdale model, was devised by Wnuk who considered an opening mode of fracture and reduced the problem to a differential equation governing the sub-critical growth. A similar approach is used in this thesis to derive the solution for sub-critical growth of a crack under longitudinal stress.

Library of Congress Subject Headings

Fracture mechanics
Shear (Mechanics)



Number of Pages



South Dakota State University