Dugdale plastic-zone model

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

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... 

These measurements show that the shape of the craze is well described by the Dugdale plastic zone model originally proposed for metals. The tip of the craze is approximated by a straight line perpendicular to the direction of advance of the crack and the thickness profile can be deduced from the assumption that there is a constant craze stress normal to the plane of the craze. Provided that cr cTcr, where a is the applied tensile stress causing the crack, the length R of the craze ahead of the crack tip is given by... 

Figure 12.8 The Dugdale plastic zone model for a craze...

These extracted chains, when bundled in fibrils, form the crazes observed in fracture of bulk polymers.(81) As the fibrils in the craze sustain an almost constant stress (Jq, the craze zone is the most perfect example of the Dugdale plastic zone at a crack tip, as first suggested by Marshall et and the term v/L can be seen as the term of the Dugdale model. 

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

The measure of the craze shape ahead of a propagating crack by Brown and Ward [38] appears consistent with the geometry of the plastic zone according to a Dugdale [33] model of a craze. For a precracked specimen under the remote load (Fig. 3), the craze is represented by a plastic zone similar to a strip at the tip of the crack. The profile of the plastic zone varies from zero at the location (a + Ac) to the value Acr at the crack tip. 

Fig. 3 Schematic of the Dugdale model. The plastic zone is modeled by a strip subjected to a constant normal stress ac. The length of the crack is 2a and the size of the plastic zone ahead of the crack tip is Ac...

On the other hand, in glassy polymers the zone of crack tip yielding is often found to be a thin wedge rather than a circle. It is now well documented that a good description of the shape and size of this yielded zone at the crack tip can be provided by the plastic zone size model proposed by Dugdale and by the cohesive force model of Barenblatt " . Similar solutions and further developments have been contributed by other authors... 

The outstanding importance of the Dugdale-Barenblatt-model lies in the fact that yielding occurs by tensile stresses (normal yielding). This becomes obvious if, in the limit of the ellipse becoming a crack, the stresses at the boundary of the plastic zone are considered they are oriented normal to the crack plane. Now, it was shown by Sternstein et coworkers that crazing is a phenomen which essentially occurs under the action of normal stresses. The fibrils in the craze are oriented under the action of the maximum principal tensile stress component and the propagation direction of the craze zone is perpendicular to it and parallel to the axis of the minor principal tensile stress. 

In this context it has to be pointed out that in the original Dugdale model the material behavior is assumed to be linearly elastic and perfectly plastic the latter assumption leads to a uniform stress distribution in the plastic zone. This may be a simplified situation for many materials to model, however, the material behavior in the crack tip region where high inhomogeneous stresses and strains are acting is a rather complex task if nonlinear, rate-dependent effects in the continuum... 

Precise determination of CTOD is often difficult. Furthermore, these measurements are unsuitable for use in design. For these reasons, models that enable CTOD to be calculated in terms of stresses have been developed [Dugdale, I960]. In a wide plate with a central crack of length 2a with a narrow planar plastic zone of length L, extending from each of the crack tips, the apphed stress a is given by [BucknaU, 1978] ... 

Figure 3.464. Model of plastic zone (Irwin and Dugdale) from crack region [1295].

In the Dugdale model the stresses in the cohesion zone are assumed to have a constant value CTq over a length d, so that at equilibrium G = where bj is the crack opening displacement (COD) beyond which interactions disappear. This model is used for elastoplastic materials where CTq is the yield stress and d the plastic zone length, and for polymers where ag is the craze stress and d the craze length it can also be applied to liquid bridges where (Jq is the Laplace pressure (see Section 6). 

Despite the fact, that HDPE testing samples have nominally a sharp notch, it is well-known [37], that local plasticity zone availability results to a sharp notch blunting and as the first approximation the value R can be accepted equal to notch critical opening 8 [37], determined according to Dugdale model [4] ... 

FIGURE 10.7 The schematic picture of crack and local plasticity zone (which is shaded) within the frameworks of the Dugdale-Barenblatt model [22],... 

Using the Dugdale (1960) model, one may estimate the size A of the plastic zone at the crack tip to be... 

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