Plastic Zone Modeling

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... 

Konczol [81] looked at PMMA, PC, PVC and other thermoplastic polymers. Their results support the plastic zone model. 

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...

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

An approximate model of the ISE can be developed with the aid of Figure 2.9. This figure shows a schematic cylindrical indenter with a conical tip being pushed into a specimen. The plastic zone is approximated by a segment of a sphere, and the diameter of the indent is 2r. The yield stress is Y, and the friction coefficient is a. 

The more rigorous stress/strain nonlinear material model, oflen referred to as the plastic zone method, is theoretically capable of handling any general cross section Both isotropic and kinematic hardening rules are usually available. This method is... 

The dissolution-condensation-model (Figure 7.91) adequately explains the crack growth rate and its dependence on stress, temperature and interfacial energy for short cracks. This model does not account for high crack growth rates and hence a third-party process involving the plastic zone has been involved. 

POSTER TITLE Modelling of Inconel 718 crack tip s plastic zone softening during crack propagation... 

Characterization and modelling of Inconel 718 cyclic yielding behaviour in order to include the crack tip s plastic zone softening in the Navarro and de los Rios model. 

Crack growth models in monolithic solids have been well document-ed. 1-3,36-45 These have been derived from the crack tip fields by the application of suitable fracture criteria within a creep process zone in advance of the crack tip. Generally, it is assumed that secondary failure in the crack tip process zone is initiated by a creep plastic deformation mechanism and that advance of the primary crack is controlled by such secondary fracture initiation inside the creep plastic zone. An example of such a fracture mechanism is the well-known creep-induced grain boundary void initiation, growth and coalescence inside the creep zone observed both in metals1-3 and ceramics.4-10 Such creep plastic-zone-induced failure can be described by a criterion involving both a critical plastic strain as well as a critical microstructure-dependent distance. The criterion states that advance of the primary creep crack can occur when a critical strain, ec, is exceeded over a critical distance, lc in front of the crack tip. In other words... 

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...

Equation (3) indicates that near the crack tip the stresses can exceed the yield stress cjy of real materials. Therefore, in a certain zone around the crack tip the material undergoes plastic deformations. To account for this two basically different models of plastic zones have been developed. 

In the model derived by McClintock and Irwin the shape and size of the plastic zone were calculated by a combination of the stress field at the crack tip (e.g. Eq. (2)) with a yield criterion (e.g. von Mises, Tresca). This leads to the well known dog-bone type of plastic zone showing the influence of stress state. Its form is often approximated to by a circle of radius Tp, where... 

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. 

Fig. 19a. Variation of plastic zone size in a 5 mm thick SEN-spechnen of HDPE al 0.5 mm below the surface, a2 l mm below the surface, a3 2 mm below the surface b Dog bone model of plastic zone radius, r, vs. SEN-specimen thickness (F = load)...

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