Dugdale zone

Fig. 4 Description of a craze (a) with a Dugdale zone, and the local analysis (b) as a long strip representing the anisotropic craze structure made of main fibrils oriented along direction 2, connected with lateral cross-tie fibrils along direction 1...

One example is the stress concentration equation (Equation (12.21)). Experimental results for PMMA suggest typical values of h = 2 ftm and d = 20 nm. If a lies in the range 0.01 0.05, atip will be in the range 0.96ad-2.1od, which gives some credence to the theory in that it is remarkably consistent with values of the craze stress obtained from Dugdale zone measurements, and will account for failure in the mid-rib of the craze. 

For a closing crack a different approach is required. Schapery 13) introduces a strain intensity factor to replace K in equation (14). The length d of the Dugdale zone is still given approximately by equation (15), but the apparent work of adhesion is now given approximately by... 

A fundamental restriction in the theory developed above is that the length d of the zone at the edge of contact where the adhesive forces act (Dugdale zone) should be small... 

Creep effects are due to the bulk deformation whose size is characterized by a, the radius of the contact. Adhesive interactions result in large stresses that are concentrated near the contact edge. These can be characterized by the width d of the Dugdale zone over which the adhesive force with characteristic range ho is important or, from the fracture mechanics point-of-view, by the length L of the crack at the contact edge. If the rate at which the contact size changes is v =da/dt, two characteristic times can be defined ... 

The characteristic time t is related to the time, d/v, that it takes for the stress field to pass through the Dugdale zone. The actual value of t is a third of this time [28], giving the following expression for / ... 

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

In the traditional Dugdale model [56], a = Oy and the familiar result is obtained, Gic = cTySc- In the EPZ model, cr exceeds critical crack opening displacement <5c is proportional to the maximum stresses cr in the deformation zone... 

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... 

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

The deformation zones were calculated for the polymers of Table 5.1 and Table 6.1 according to the Dugdale-Barenblatt-model. Yield stress ay from tensile tests was used instead of the cohesive stress ctc since a reasonable agreement of ay and ctc... 

Fig. 7.3. Deformation zone as calculated from the Dugdale-Barenblatt-model (Eq. 7.2). In order to magnify the displacements, E/cry = 7 was assumed for the diagram, whereas, in reality the ratio is about 38 (Tables 5.1 and 6.1)...

Fig. 10-13. The links between the cycling of C, N, and O2 are indicated. Total primary production is composed of two parts. The production driven by new nutrient input to the euphotic zone is called new production (Dugdale and Goering, 1967). New production is mainly in the form of the upward flux of nitrate from below but river and atmospheric input and nitrogen fixation (Karl et al, 1997) are other possible sources. Other forms of nitrogen such as nitrite, ammonia, and urea may also be important under certain situations. The "new" nitrate is used to produce plankton protoplasm and oxygen according to the RKR equation. Some of the plant material produced is respired in the euphotic zone due to the combined efforts...

Another mathematical approach to modeling cohesive zones is to consider the crack tip fully shielded, that is Keff = 0, rather than partially shielded as in the case considered above. In this case, a cohesive zone lies in front of a traction free crack (Zone 1). This is the classical Barenblatt-Dugdale model in which the stress-intensity factor at the end of the cohesive zone is now zero that is, stress singularities are completely removed by the cohesive forces.29 The requirement of complete shielding results in a cusp-shaped cohesive zone or bridging zone profile. This approach has advantages, particularly for the elevated temperature case, in that the cohesive zone can... 

In this section we present a model that applies to a fully shielded crack tip model (Dugdale-Barenblatt-type cohesive zone). We consider the problem of a fully shielded crack (Keff = 0) under small-scale creep/bridging and... 

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...

From observations of interference patterns and interferometric measurements of the refractive index, the profile of a craze preceeding a crack in polystyrene has been determined. The strain along the length of the craze was calculated and found to increase toward the crack tip. The opening displacements across the craze boundaries show a deviation from those predicted for the simple Dugdale yield zone. 

In the present work the variation in stretch ratio or strain along a craze in polystyrene has been determined experimentally using methods related to those recently described by Brown and Ward (5) for poly-(methyl methacrylate), and the elastic displacements across the craze zone were compared with the values predicted for the Dugdale model (7). 

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