Stresses and displacements in thick layer containing an axially symmetrical Dugdale crack



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Presented in this thesis is an analytical solution to the problem of the determination of stresses and displacements in a layer of elastic-perfectly plastic material containing a penny-shaped crack. It is assumed that the zone of plastic deformations surrounding the crack is very thin so that the problem can be reduced to one, within the theory of elasticity, of determining the stresses and displacements in an elastic half-layer with proper boundary conditions. Through the application of Hankel transforms, the problem is reduced to that of solving a pair of dual-integral equations. The dual-integral equations are solved by reducing them to a Fredholm integral equation of the second kind. The integral equation is then solved by numerical methods. The width of the annulus of yielded material is determined from the condition that the stress at the crack tips must be finite. The stresses and displacements on the plane of symmetry are then computed by numerical integration of the inverse Hankel transforms. Numerical examples are tabulated and plotted for plastic zone widths, stresses, and displacements corresponding to various layer thicknesses and loading pressures. The results are discussed regarding their relation to the case of an infinite body, and regarding the applicability of the assumptions of small strains and thin yield zones.