The stress field near a crack tip

A crack can be considered as an infinitely sharp notch. If we estimate the maximal stress by setting the notch radius to zero, the stress concentration 

This assumption is correct if the thickness is small compared to all other dimensions i.e., t -C 1. 

The stress field at the crack causes an ellipsoidal opening of the crack due to Hooke s law (figure 5.6). Using a coordinate system centred in the middle of the crack as shown in figure 5.6, the displacement t o of the crack surfaces is given by 

As we have neglected small terms in the derivation of equations (5.1), they are valid only near the crack tip. This can immediately be seen by the fact that the stresses in equations (5.1) approach zero for large r, whereas the true far-field stress is a. Directly at the crack tip, the equations are also not valid because they are based on continuum mechanics which is not applicable on the atomic scale (see also exercise 13). 

If we increase the load and thus the stress intensity factor K, the crack will propagate when K reaches a certain critical value, the fracture toughness K c (again, T denotes loading in mode 1 ). The fracture toughness is a material property. Consequently, the failure criterion for loading in mode one is given 

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