Irwin crack model

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... 

There are two principal theories, or models, that attempt to describe what happens during brittle fracture, the Griffith fracture theory and the Irwin model. Both assume that fracture takes place through the presence of preexisting cracks or flaws in the polymer and are concerned with what happens near such a crack when a load is applied. Each leads to the definition of a fracture-toughness parameter and the two parameters are closely related to each other. The Griffith theory is concerned with the elastically stored energy near the crack, whereas the Irwin model is concerned with the distribution of stresses near the crack. Both theories apply strictly only for materials that are perfectly elastic for small strains and are therefore said to describe linear fracture mechanics. 

Figure 8.7 shows how the stress increases near the tips of a crack in a sheet to which a uniform tensile stress is applied normal to the crack. Near the tip of the crack the tensile component of stress must be higher than the applied stress because the stress within the crack is zero and the stress a long way from it must be equal to the applied stress. Near the crack the other components of stress are not all zero and, in the model due to Irwin, it is shown that the whole system of stresses near the tip of a crack loaded in this way can be written in terms of the applied stress, geometrical factors and a single further quantity K called the stress-intensity factor. For an infinite sheet with a central crack of length I, it can further be shown that... 

The elastic stresses in the outer zone can be calculated from the Irwin model by assigning a fictitious length a -I- Ty to the crack. 

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

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

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