Dugdale plastic zone

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... 

The similarity between the measured craze zone and the calculated Dugdale plastic zone is apparent when Figs. 3 and 7 are compared. Before applying the Dugdale formula to the measured craze zone some points of detail should be noted. Whilst the position of the crack tip is known fairly precisely, the location of the craze tip and hence the craze length s cannot be measured directly, but can — in a similar way to... 

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... 

Equations (12.3) imply that there is an infinite stress at the crack tip. In practice this clearly cannot be so, and there are two possibilities. First, there can be a zone where shear yielding of the polymer occurs. In principle this can occur in both thin sheets where conditions of plane stress pertain and in thick sheets where there is a plane strain. Secondly, for thick specimens under conditions of plane strain, the stress singularity at the crack tip can be released by the formation of a craze, which is a line zone, in contrast to the approximately oval (plane stress) or kidneyshaped (plane strain) shear yield zones. As indicated, its shape approximates very well to the idealized Dugdale plastic zone where the stress singularity at the crack tip is cancelled by the superposition of a second stress field in which the stresses are compressive along the length of the crack (Fig. 12.8). A constant compressive stress is assumed and is identified with the craze stress. It is not the yield stress,... 

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] ... 

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