Plastic zone

Fig. 4. (a) The crack tip plastic zone and (b) the Dugdale plastic zone model. Terms are defined in text. 

The distance from the crack tip, along the X-axis, at which the von Mises equivalent stress falls below the yield stress, defines the size of the plastic zone, r. For the plane stress case of unconstrained yielding, which corresponds to the free surface of the specimen in Figure 4, this gives... 

The process zone is a measure of the yield stress or plasticity of the material in comparison to its brittleness. Yielding within the process zone may take place either plastically or by dimise microcracking, depending on the brittleness of the material. For plastic yielding, / is also referred to as the plastic zone size. 

The crack propagates when K is equal to the width of the plastic zone, r,, is then given by eqn. (14.2) with K replaced by K. Note that the zone of plasticity shrinks... 

Cracks grow in the way shown in Fig. 15.9. In a pure metal or polymer (left-hand diagram), the tensile stress produces a plastic zone (Chapter 14) which makes the crack... 

We mentioned in Chapter 14 that real engineering alloys always have little inclusions in them. Then (right-hand diagram of Fig. 15.9), within the plastic zone, holes form and link with each other, and with the crack tip. The crack now advances a little faster than before, aided by the holes. 

In metals, inelastic deformation occurs at the crack tip, yielding a plastic zone. Smith [34] has argued that the elastic stress intensity factor is adequate to describe the crack tip field condition if the inelastic zone is limited in size compared with the near crack tip field, which is then assumed to dominate the crack tip inelastic response. He suggested that the inelastic zone be 1/5 of the size of the near crack tip elastic field (a/10). This restriction is in accordance with the generally accepted limitation on the maximum size of the plastic zone allowed in a valid fracture toughness test [35,36]. For the case of crack propagation, the minimum crack size for which continuum considerations hold should be at least 50 x (r ,J. 

In the case of metals, R is mainly plastic energy associated with the formation of a crack tip plastic zone. It is obvious from Eq. 9 that, for plane stress,... 

Fig. 1. The microscopic enlanglemenl slruciure, e.g, at an interface or in the bulk, is related to the measured macroscopic fracture energy G, via the VP theory of breaking connectivity in the embedded plastic zone (EPZ) at the crack tip. The VP theory determines

Another approach to the question of resistance to crack growth is to consider the extent to which yielding occurs prior to fracture. In a ductile material it has been found that yielding occurs at the crack tip and this has the effect of blunting the crack. The extent of the plastic zone (see Fig. 2.70) is given by... 

Use Ae data in Table 2.2 to compare crack tip plastic zone sizes in acrylic, ABS and polypropylene. 

This plastic deformation is localised around the crack tip and is present in all stressed engineering materials at normal temperatures. The shape and size of this plastic zone can be calculated using Westergaards analysis. The plastic zone has a characteristic butterfly shape (Fig. 8.83). There are two sizes of plastic zone. One is associated with plane stress conditions, e.g. thin sections of materials, and the other with plane strain conditions in thick sections-this zone is smaller than found under plane stress. 

Where rj is the plastic zone size along the plane of the crack and perpendicular to the applied stress. The above two equations are only a first approximation as the plastic zone contributes to the crack size. The true plastic zone is twice the initial approximation. 

Fig. 8.83 Stressed crack tip with associated plastic zone...

There are a number of restrictions on the test for for it to be a valid measure of plane strain fracture toughness. Firstly, the plastic zone must not extend through the test piece and secondly the thickness of the material must be such that the test is conducted under plane strain conditions. 

Good agreement is reported to exist between the Dugdale plastic zone model and optical interference experiments, performed at the tip of a crack. Morgan and Ward [79], Fraser and Ward [80] and more recently and extensively Doll and... 

Konczol [81] looked at PMMA, PC, PVC and other thermoplastic polymers. Their results support the plastic zone model. 

The length s of the plastic zone for small-scale yielding can be expressed by... 

Fig. 7.2. Dugdale-Muskhelishvili-model of plastic zones at the ends of a loaded elliptical hole [81].

An approximate model of the ISE can be developed with the aid of Figure 2.9. This figure shows a schematic cylindrical indenter with a conical tip being pushed into a specimen. The plastic zone is approximated by a segment of a sphere, and the diameter of the indent is 2r. The yield stress is Y, and the friction coefficient is a. 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

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