Irwin plastic zone

Although Equation (12.18a) is more empirically based than Equation (12.18) and is not formally equivalent, it has been shown to model fracture results very well. Moreover, in this formulation w relates to the size of the so-called Irwin plastic zone Ty, which can be defined simply on the basis of Equation (12.3) by assuming that a point Vy the stress reaches the yield stress Oy. Hence... 

Figure 10.18 [27] presents the empirical basis of the Bates and Clark finding. The exponent of two that appears in Eq. (10.23) induces to believe that striation spacing is somehow related to the Irwin plastic zone size (10.6) or to the crack tip opening displacement or CTOD that, as the plastic zone radius, is proportional to the square of the stress intensity factor K ... 

At the crack tip, there is a stress concentration effect but, in LEFM, stress is limited to plastic zone. The dimensions of this plastic zone can be calculated using Irwin s theory and is ... 

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... 

Fig. 3. (i) Normal stress towards the fracture plane in mode 1 for an elastic material (ii) redistribution of stress by development of a plastic zone in front of a crack tip according to Irwin. 

In the Irwin s approach, plasticity is assumed to develop when the material exceeds locally its yield stress, Cy. The stress in front of the crack tip is therefore truncated at Oy as highlighted in Fig. 3(ii). In order to take into account the stored energy in the elastic part of the material, a plastic zone of radius fp, must be added to the crack length, a, which becomes ... 

A useful approximation to these plasticity results was originally suggested by Irwin and assumes that, for small plastic zones, the crack length should be corrected by the addition of a length equal to rp/2. The result is then precisely the same as the linear case, except that cq + p/2 replaces Oq and we have ... 

An estimate of the plastic zone correction factor was made by Irwin (see [9]). He postulated that stresses ahead of the crack tip, away from a small plastic zone, can be approximated by those given by the solutions of linear elasticity, provided that an effective crack length Ug is used, where,... 

Equations (4.3) and (4.4) are written to expressly reflect Irwin s plastic zone correction factor ry (see Eqn. (3.49)). By normalizing with respect to ry, the yield loci, or... 

Figure 4.3. Schematic diagram showing the typical variation in fracture toughness with material thickness (B), or thickness relative to Irwin s plastic zone correction factor (B/ry).

One of the pioneers of fracture mechanics studies. Irwin [112], estimated the extent of plastic zone development at crack tip. The radius r. of this plastic zone is given by... 

Analysis of the stress field and deformation behavior in front of the tip of sharp notch or crack allows to calculate principal stresses and estimates the size of the plastic zone ahead of the crack tip. Under assumption that the material near the crack tip is an elastic-plastic continuum, the radius of the plastic zone on the crack plane can be expressed, according to Irwin (1964) ... 

Fig. 13. Irwin model Small plastic zone surrounded by an elastic field.

Figure 3.464. Model of plastic zone (Irwin and Dugdale) from crack region [1295].

Irwin [7] suggested that if a material is considered which is elastic up to the uniaxial yield stress, cTay, and then becomes plastic, then the extent of crack tip plasticity for a crack in bulk material is, to a first approximation, as shown in Fig. 7.5. Here 8t is the crack tip opening displacement and ry is the radius of the circular plastic zone at the tip of a notional crack. The tip of this notional crack is therefore at the centre of the plastic zone so that its length is (a H- ry). The elastic stress field ahead of this notional crack may therefore be regarded... 

Figure 7.5 Irwin model of the plastic zone at a crack tip.

One of the purposes of this paper is to differentiate between surface free energy and fracture surface energy. Thus, several basic principles of fracture mechanics are discussed with reference to the Griffith s energy-b J nce concept and the Irwin-Orowan s plastic-zone concept. We also define several frequently misused terms, e.g., effective fracture surface energy, fracture energy and fracture toughness. Actual values of related parameters are presented to illustrate the applicability of fracture mechanics to the selection of polymers for structural materials. 

The second physical quantity of interest is, r t = 90 pm, the critical crack tip stress field dimension. Irwin s analysis of the crack tip process zone dimension for an elastic-perfectly plastic material began with the perfectly elastic crack tip stress field solution of Eq. 1 and allowed for stress redistribution to account for the fact that the near crack tip field would be limited to Oj . The net result of this analysis is that the crack tip inelastic zone was nearly twice that predicted by Eq. 3, such that... 

Linear elastic fracture mechanics (including the Irwin model of confined plasticity, the line-zone or Dugdale model and viscoelastic effects)... 

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