Dugdale model

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... 

The cohesive stress ac is assumed to be constant (Dugdale model) as in Eq. (7.5). Chan, Donald and Kramer [87] found a good agreement between the critical energy release rate GIC, as estimated by the Dugdale model and G)C as computed from the actual stress and displacement profiles in their experiments. 

A deep insight into the problem of contact mechanics involved in a conductivity measurement using an SFM tip can be found in the paper by Lantz et al. [441]. In this article the contact area was derived for the case of ohmic contacts using the Maugis-Dugdale model [104] (see Sect. 2.1). However, the uncertainty is still related to the problem of the conductivity of the tip apex. If a sharp tip is not absolutely necessary, a possible solution to this problem is to add electro-chemically a copper layer to the chromium sub-layer (Fig. 33e,f). 

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... 

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...

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). 

Figure 9. Normalized opening displacement for the Dugdale model (upper curve) and for polystyrene craze (lower curve)...

Fig. 3. Elastic-plastic boundary as calculated from the Dugdale model...

Using a modified Dugdale model with a variable craze stress along the craze zone this effect has quahtatively been interpreted At positions where the constant stress Dugdale model gives displacements higher than the measured ones the actual craze stress must be higher. In the case of PC a closer inspection reveals a stress peak at the crack and craze tip. Kambour predicted just such a stress distribution in a craze from the analysis of the stress distribution around a craze (without a crack)... 

Fig. 8. Measured craze zone at the crack tip and fit by the Dugdale model (—) for different polymers r + PES, o plasticized PMMA x PC , A plasticized PVC ...

Fig. 12 a and b. Material data of the micro region at the crack tip as derived by the application of the Dugdale model to measured craze sizes (Fig. 11) a craze stress 0 b creep modulus E... 

From the reported craze dimensions, the tensile creep moduli E and craze stress have been derived by the aid of the Dugdale model. For PMMA the thus evaluated creep moduli are shown as a function of temperature T in Fig. 18 together... 

Fig. 18. Influence of temperature T on modulus E determined by interference fringe pattern and Dugdale model low ( ) and high (O) molecular weight material and compliance...

Williams modelled this behavior using the Dugdale model and derived a growth law of the form ... 

Craze growth at the crack tip has been qualitatively interpreted as a cooperative effect between the inhomogeneous stress field at the crack tip and the viscoelastic material behavior of PMMA, the latter leading to a decrease of creep modulus and yield stress with loading time. If a constant stress on the whole craze is assumed then time dependent material parameters can be derived by the aid of the Dugdale model. An averaged curve of the creep modulus E(t) is shown in Fig. 13 as a function of time, whilst the craze stress is shown in Fig. 24. 

For craze zones at the tips of static and of moving cracks under quasistatic loading conditions it has been shown in Sect. 3 that the normal stress acting on the craze zone and the modulus E can be derived from the measured craze dimensions using the Dugdale model. 

This is confirmed by the experimental data and especially the extended analysis of the measured craze contours in high molecular weight PMMA. In this material an excellent fit is achieved by using a uniform craze stress in the Dugdale model... 

In this context it should be mentioned that the optical interference results on PVC indicate that the simple Dugdale model with a constant stress is not appropriate... 

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... 

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