Issue 36

L. C. H. Ricardo et alii, Frattura ed Integrità Strutturale, 36 (2016) 201-215; DOI: 10.3221/IGF-ESIS.36.20 203 max min K K K    (2.1) where K max and K min are the maximum and the minimum stress intensity factors, respectively. In the crack propagation curve, the linear part represents the Paris - Erdogan law, when plotting the values of  K vs da/dN in logarithmic scale, as can be seen in Fig. 1. Fatigue crack initiation and growth under cyclic loading conditions is controlled by the plastic zones that result from the applied stresses and exist in the vicinity (ahead) of a propagating crack and in its wake or flanks of the adjoining surfaces. For example, the fatigue characteristics of a cracked specimen or component under a single overload or variable amplitude loading situations are significantly influenced by these plastic zones. In modelling the fatigue crack growth rate this is accounted by the incorporation of accumulative damage cycle after cycle and should include plasticity effects. During the crack propagation the plastic zone should grown and the plastic wake will have compressive plastic zones that can help to keep the crack close. Hairman & Provan [9] discuss the problems pertaining to fatigue loading of engineering structures under single overload and variable amplitude loading involving the estimation of plasticity affected zones ahead of the crack tip. Crack tip plasticity Most solid materials develop plastic strains when the yield strength is exceeded in the region near a crack tip. Thus, the amount of plastic deformation is restricted by the surrounding material, which remains elastic during loading. Theoretically, linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. In fact, inelastic deformation, such as plasticity in metals and crazing in polymers, leads to relaxation of crack tip stresses caused by the yielding phenomenon at the crack tip. As a result, a plastic zone is formed containing microstructural defects such dislocations and voids. Consequently, the local stresses are limited to the yield strength of the material. This implies that the elastic stress analysis becomes increasingly inaccurate as the inelastic region at the crack tip becomes sufficiently large and, so, linear elastic fracture mechanics (LEFM) is no longer useful for predicting the field equations. The size of the plastic zone can be estimated when moderate crack tip yielding occurs. Thus, the introduction of the plastic zone size ( r ) as a correction parameter, which accounts for plasticity effects adjacent to the crack tip, is vital to determine the effective stress intensity factor ( K eff ) or a corrected stress intensity factor. The plastic zone is also determined for plane conditions; that is, plane strain for maximum constraint on relatively thick components and plane stress for variable constraint due to thickness effects of thin solid bodies. Moreover, the plastic zone develops in most common in materials subjected to an increase in the tensile stress that causes local yielding at the crack tip. Most engineering metallic materials are subjected to an irreversible plastic deformation. If plastic deformation occurs, then the elastic stresses are limited by yielding since stress singularity cannot occur, but stress relaxation takes place within the plastic zone. This plastic deformation occurs in a small region and it is called the crack-tip plastic zone ( r ). A small plastic zone, ( r << a ) being a crack length of the structure or specimen, is referred to as small-scale yielding. On the other hand, a large-scale yielding corresponds to a large plastic zone, which occurs in ductile materials in which r >> a . This suggests that the stress intensity factors within and outside the boundary of the plastic zone are different in magnitude so that K I (plastic) > K I (elastic). In fact, K I (plastic) must be defined in terms of plastic stresses and displacements in order to characterize crack growth, and subsequently ductile fracture. As a consequence of plastic deformation ahead of the crack tip, the linear elastic fracture mechanics (LEFM) theory is limited to r << a ; otherwise, elastic-plastic fracture mechanics (EPFM) theory controls the fracture process due to a large plastic zone size ( r ≥ a ). This argument implies that r should be determined in order to set an approximate limit for both LEFM and EPFM theories [10]. Fig. 2 shows schematic plastic zones for plane stress (thin plate) and plane strain (thick plate) conditions Even if in the interior of a plate a condition of plane strain exists, there will always be plane stress at the surface. Stresses perpendicular to the outer surface are non- existent, and hence σ z = σ 3 = 0 at the surface. If plane strain prevails in the interior of the plate, the stress σ 3 gradually increases from zero (at the surface) to the plane strain value in the interior [11]. Consequently, the plastic zone gradually decreases from the plane stress size at the surface, to the plane strain size in the interior of the plate, illustrated schematically in Fig. 2. Dugdale [12] proposed a strip yield model for the plastic zone under plane stress conditions. Consider Fig. 3, which shows the plastic zones in the form of narrow strips extending a distance r each, and carrying the yield stress σ ys . In the case of cyclic stress, and as the crack grows, behind the new crack front there is a region with compressive (residual) stresses. The phenomenon of crack closure is caused by these internal (compressive) stresses since they tend to close the crack in the region where a < x < c.