Elastic-plastic boundary

Partially Plastic Thick-Walled Cylinders. As the internal pressure is increased above the yield pressure, P, plastic deformation penetrates the wad of the cylinder so that the inner layers are stressed plasticady while the outer ones remain elastic. A rigorous analysis of the stresses and strains in a partiady plastic thick-waded cylinder made of a material which work hardens is very compHcated. However, if it is assumed that the material yields at a constant value of the yield shear stress (Fig. 4a), that the elastic—plastic boundary is cylindrical and concentric with the bore of the cylinder (Fig. 4b), and that the axial stress is the mean of the tangential and radial stresses, then it may be shown (10) that the internal pressure, needed to take the boundary to any radius r such that is given by... 

Assume pressure, needed to take the elastic—plastic boundary to radius r corresponds to point B (see Fig. 3). Then provided the cylinder unloads elasticady when the internal pressure is removed, ie, unloading path BE is paradel to OA, the residual shear stress distribution is as fodows. 

An analytical expression for the displacements 2v of the elastic-plastic boundaries in front of and behind the crack tip was derived by Goodier and Field... 

In Fig. 3 the dependence of the displacements on the distance from the crack tip is shown using a normalized scale and assuming ct = 0.10. At a distance behind the crack tip the elastic-plastic boundary has an almost parabolic form, whilst directly at the crack tip — in contrast to the ideal elastic solution (Eq. (4)) — there is a certain displacement which is also the maximum width of the plastic zone. Here, and in the following text, the width of the plastic zone at the crack tip will be denoted by 2v without any coordinate parameter. In fracture mechanics terms it can be expressed as... 

Fig. 3. Elastic-plastic boundary as calculated from the Dugdale model...

Fig. 54. Schematic of indentation damage for (a) Vickers and (b) spherical indentations in zirconia-based ceramics. Dark areas comespond to higher microcrack density. Dashed lines show the elastic-plastic boundary. After Reference [251].

The median crack is a single, penny-shaped crack nucleated beneath the apex of the plastic zone created by the indentor. The diameter of the median crack is comparable with the indent size, and a median crack is not visible in a polycrystalline, opaque material. The driving force for nucleation of the median crack is the elastic tensile stress developed normal to the indentation direction at the elastic-plastic boundary when the external load is relaxed. Nucleation of a median crack depends on the presence of a suitable flaw. Once nucleated, the median crack will propagate spontaneously to a stable flaw size. The critical flaw size for growth is ... 

Also referred to as median vent cracks, these are caused to pop-in by exceeding a critical indenter load. It is the pop-in phenomenon that is important to the development of this subject in ceramic science because the halfpenny crack has the surface trace which allows opaque materials to be analyzed by recording the radial crack size as a function of increasing load. There is, however, the implication that the surface must be prepared carefully by polishing to an optical finish in order to see the radial cracks. If necessary, samples must be annealed to remove polishing stresses. Radial cracks are the result of surface tensile stresses, (Xyy in equation (1.29). Such stresses are at a maximum at the elastic-plastic boundary. 

Analytic solutions of the integrals and calculation of the stresses at various points for indents of varying sizes and shapes lead to several common features the peak tensions are at the elastic-plastic boundary there is a rapid change to compression within the plastic zone and a linear decrease in tension into the elastic zone and, for indenters with large the residual stresses are greater than the peak stress at loading, so that these stresses are important as the load is removed. These predictions are consistent with observations made on many ceramic crystals. 

Lateral cracks are nucleated along the flow lines at the elastic-plastic boundary. 

The two plots in Fig. 15.3 are significant because they show the redistribution of the stress pattern as the inner region of the cylinder becomes plastic. Also note the reduction of the stress at the inner surface and the increase of the stress at the elastic-plastic boundary as the pressure is increased beyond P. ... 

This relation is established balancing the radial forces at the elastic-plastic boundary (Kramer et al, 1999 Figure H.6) ... 

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