Issue 43

L.C.H. Ricardo, Frattura ed Integrità Strutturale, 43 (2018) 57-78; DOI: 10.3221/IGF-ESIS.43.04 59 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 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 as a correction parameter that accounts for plasticity effects adjacent to the crack tip is vital in determining 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. A small plastic zone, ( r << a ) 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 may be determined in order to set an approximate limit for both LEFM and EPFM theories. Fig. 1.b shows schematic plastic zones for plane stress (thin plate) and plane strain (thick plate) conditions [16]. Plane strain: 1. Large thickness B, and ε z ~ 0 on in an internal region and. σ z = υ(σ x + σ y ). This means that the material is constrained in the z-direction due to a sufficiently large thickness and the absence of strain in this axis. In fact, the stress in the z- direction develops due to the Poisson’s effect as explicitly included in the equation that defines σ z . 2. Yielding is suppressed due to the kinematics constrain from the surrounding elastic material. 3. Plastic deformation is associated with the hinge mechanism (internal necking) Fig. 1.a) 4. The plastic zone size is small in the midsection of the plate (Fig. 1.a).This condition implies that the plastic zone must be smaller than the crack length Plane stress: 1. The thickness B is small, σ z = 0 and ε z ≠ 0 on the surface (external region) and through the whole thickness. This means that the stresses normal to the free surface are absent and therefore, σ z = 0 through the thickness. Consequently, a biaxial state of stress results. 2. If σ y ≥ σ x >0 (Tresca Criterion), then yielding occurs by a cumulative slip mechanism (Fig. 1.b). 3. The height of the yielded zone is limited due to the slip mechanism. 4. The total motion has a necking effect in front of the crack as it opens.            2 1 2 I p ys K r Plane Stress B ≤ 2.5( K IC /σ ys ) 2 (2.4) Irwin [6] has shown that the effect on the plastic zone is to artificially extend the crack by a distance r 1 (Fig. 2) known as Irwin’s plastic zone correction. The elastic stress distribution shown in Fig. 2 indicates that as σ y → ∞. Actually, σ y is limited to σ ys as shown by the elastic-plastic stress distribution. This means that σ y → ∞ occurs mathematically, not physically. In order to account for the changes due to the artificial crack extension or virtual crack length and to visualize the plastic zone as r → 0 a cylinder, the crack length a can be replaced by a e in eqs. (2.4 and 2.5). Moreover, the virtual