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# Section 11.4.1. Effect of Stress Concentration

The effect of stress concentration is fairly easy to estimate for small cracks because the stress-intensity factor for an elementary crack problem can be multiplied by the elastic stress concentration factor (kt).  Example 11.4.1 illustrates this point.  For longer cracks initiating at stress concentrations, the crack will be propagating through the stress field created by the stress concentration and the influence of stress gradient should be taken into account.  Example 11.4.2 discusses an approximate method for estimating the stress intensity factor for a crack moving through a stress field generated by a stress concentration.

A geometrical description of the physical problem is provided in the figure, where a small edge crack is shown growing from the edge of a wing cutout.  The stress-intensity factor for an edge crack (small with respect to the element width) is found in Table 11.3.8, and is given by

A Small Edge Crack Located at Stress Concentration

The stress term (s) in the general equation typically represents the remote stress in the uniformly loaded edge cracked plate.  This stress is also the stress that would exist along the line of crack propagation if no crack were present.  As indicated by the figure, the stress along the line of crack propagation (assuming no crack for a moment) for the given structural configuration is the product of the remote stress and the stress concentration factor (kt) associated with the cutout, i.e., the local stress is:

For the given structural configuration, the stresses along the line of crack propagation more closely represent the type of loading that the small edge crack would experience if it were in a

In general, as long as one is dealing with small edge cracks in which the width or other geometrical effects are not important, the final equation provides a reasonable approximation to the stress-intensity factor for an edge crack in the vicinity of a stress concentration.  See Example 11.4.2 for a discussion of stress gradient effects.

EXAMPLE 11.4.2       An Edge Crack Growing from a Stress Concentration Site

One difficulty in utilizing the Example 11.4.1 final equation for cracks that emanate from a stress concentration site is that the stress concentration normally generates its own stress field.  The stress concentration stress field typically exhibits the highest stresses in the vicinity of the concentration and these high stresses decay as a function of distance from the stress concentration site.  The question that needs to be answered is:  If the stresses along the crack propagation path are not constant, as in the case of a uniformly loaded edge cracked plate, what stresses should be used to estimate the stress-intensity factor:

Distribution of Stresses Normal to the Crack Path for a Radial Crack Growing from an Uniaxially Loaded Hole in a Wide Plate

The stress distribution associated with an uncracked hole in a wide plate is shown in the figure.  As can be seen, the (normal) stress drops off rapidly as a function of distance from the edge of the hole.  An evaluation of the normal stress right at the edge of the hole, i.e., the local stress, leads one to the fact that

(which is obtained by letting R/X = 1 in the equation given in the figure).  Thus kt for the uniaxially loaded hole problem is three, i.e., kt = 3 and the stress-int ensity factor for a very small crack of length a at the edge of the hole is

One estimate of the stress-intensity factor for a longer crack would be given by the equation above; but, this estimate would be high since the stresses along the crack propagation path are noted to be dropping.