Stress intensity factor superposition

One of the simplest techniques to determine in a complex configuration is to use superposition to build up the solution from a set of simpler and known solutions. Clearly, the precision with which the superposed geometries replicate the final, more complex, structure will impact the accuracy of the final solution. Consider the situation shown in Fig. 8.24, in which cracks emanating from a circular hole is subjected to a biaxial stress. This solution can be broken down into two uniaxial stress solutions, and K. Thus, the total stress intensity factor is found by superposition, K=K, +K. A somewhat more complex configuration is shown in Fig. 8.25. The problem again involves a cracked circular hole but, in this case, it is being loaded along a semi-circular portion of the hole. The problem is asymmetric but, as shown, it can be found from the superposition of two symmetric solutions, i.e., K = K +K H. 

Superposition of K solutions is subjected to the same restrictions as those used for stresses and displacements. For example, the stress intensity factors must be associated with a single loading mode, often mode I, and the body geometry should be the same. An additional restriction is that the crack surfaces must be separated along their entire length in the final configuration. This can be a problem if one of the basic solutions involves compressive stresses that push the crack surfaces together. 

As shown in Fig. 8.26 and discussed earlier, if the stress distribution is known in an uncracked body, one can determine AT by superposition. If the crack length is small, then, to a first approximation, the stress could be considered constant. Three possible choices for this stress, denoted as a, would be a) the maximum stress, b) the stress at the crack tip location and c) the mean stress. Consider the problem shown in Fig. 8.29. If the surface tractions can be considered constant, the stress intensity factor for a small crack is given as... 

Part (b) of the Figure 11.48 shows the superposition of the behaviors of mechanical fatigue and stress corrosion cracking. In general, the influence of the latter shows up mostly at low strain rates (d /dy = 10 - 10 s ). In the graph that represents Aa/AN as a function of log K, a step appears at the point where the stress intensity factor that corresponds to the maximum value of the applied cyclic stress, reaches the value iscc. the threshold stress intensity for stress corrosion cracking. 

Figure 3.23. Stress components at a point near a hollow microsphere in the vicinity of the crack (a) before superposition, (b) after superposition, and (c) effective stress intensity factor Kj) [31]...

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