Cohesive zone

There are two different scales of deformation in any adhesive contact (1) the bulk scale of deformation which is characterized by the radius a of contact area over which the compressive forces are significant and (2) the zone of action of surface forces or the cohesive zone at the edge of the contact, characterized by the length d over which the tensile forces are dominant. When the contact boundary is moving with a speed u, the two scales of deformation translate into two time scales, one on the order of a/ v) and the other of the order of (d/v). 

Johnson [109] for linear viscoelastic effects inside the cohesive zone. For growing cracks... 

The existence of a wedge-shaped cavitated or fibrillar deformation zone or craze, ahead of the crack-tip in mode I crack opening, has led to widespread use of models based on a planar cohesive zone in the crack plane [39, 40, 41, 42]. The applicability of such models to time-dependent failure in PE is the focus of considerable attention at present [43, 44, 45, 46, 47]. However, given the parallels with glassy polymers, a recent static model for craze breakdown developed for these latter, but which may to some extent be generalised to polyolefins [19, 48, 49], will first be introduced. This helps establish important links between microscopic quantities and macroscopic fracture, to be referred to later. 

Currently, mathematical tools are available only for the modeling of cohesive or bridging zones for cracks in linear-elastic solids, although the closure pressure function p(u) or p(u,t) can itself be nonlinear. We first review some basic approaches for the modeling of cohesive zones, beginning with time-independent bridging and then discuss the relationship between cohesive zones and crack growth at elevated temperature primarily based on some recent or just-completed studies.29,30,32,33... 

If the crack tip is considered to be located at the end of the zone of bridging or cohesion (see Fig. 10.6), then the cohesive forces exist in the wake of the crack. Under these circumstances, the cohesive forces essentially reduce the effective value of the stress intensity factor at the crack tip, Keff. Accordingly, the crack tip in this case is shielded. When the shielding is only partial, Keff is finite, whereas for complete shielding, Keff = 0. Both cases are valid fracture mechanics representations for cracks that include cohesive zones. We consider partial shielding first. 

Thus, the magnitude of the reduction in the applied stress intensity factor, Kt, due to the cohesive zone is... 

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... 

The second case in this category of creep crack growth with a cohesive zone associated with viscous interfaces is the work of Nair and Gwo.32,53 Nair and Gwo32 considered intermittent crack growth whereas continuous crack growth was considered in the latter study. The principal assumption in these studies was that the stress field in the creep zone follows the ffR-field or RR-field (see Section 10.2) with applied K replaced by Keff. This is only a first approximation, because, as mentioned earlier, these crack tip fields were originally derived for traction-free crack surfaces. 

Cohesive Zone with Power-Law Creep and Damage... 

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... 

Fig. 10.13 Schematic of a cohesive zone ahead of a crack. Note the origin of the coordinate system.

The model is formulated in terms of an integral equation which is solved with the condition at the boundary of the open crack and the bridging zone (.x — 0) that 6(0) = 5C. It is interesting to note that the structure of the rate-dependent problem is such that, aside from the material parameters, the solution is completely determined for a given crack velocity. For a given velocity, the value of the applied stress intensity factor, K, and the length of the cohesive zone, L, that maintains this condition is determined. Selected results are presented below. 

Fig. 10.14 The normalized stress in the cohesive zone,

Therefore, the condition for craze breakdown is incorporated in the cohesive zone description by adopting a critical thickness Acr that is just material dependent. We will briefly explore the influence of a temperature-dependent critical thickness in some nonisothermal calculations (Sect. 5.3) by letting the value of Acr increase by a factor of two from room temperature to Tg. 

In conclusion, the cohesive surface description presented in the foregoing sections appears suitable for capturing a ductile-to-brittle transition with increasing loading rate, and for predicting a toughening effect when the bulk is essentially elastic. These trends are reported experimentally and a calibration of the parameters used in the cohesive zone description is presented in [64],... 

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