One of the useful features of stress intensity characterisation of cracks, is that the total stress intensity factor for two (or more) loadings of the same mode type (e.g. all Mode 1 or Mode 2 or Mode 3) is obtained by algebraic summation, or superposition. Perhaps one of the most common applications of this concept is in pressure vessels, where the internal pressure gives rise membrane stresses in the wall and, for the case of a surface breaking internal defect, internal pressure on the crack faces. The K values due to membrane stress in a pressure vessel and the internal pressure on the faces of a surface crack can be simply added to get the net result.
This is illustrated in the diagrams below. Case A can be shown to be the same as Case B by considering the free-body diagram for the plate and taking a section through the crack plane. Case C is an infinite plate containing a crack with no internal pressure on its faces, while Case D represents the opposite of the case we are interested in, i.e. a pressurised crack.
Hence, as an uncracked plate has a zero value of stress intensity factor, we can say that:
The case of a crack with internally pressurised faces is found by reversing the sign of the stress in Case D. Generally, therefore, one can account for internal pressure in a component by adding the internal pressure to the stress used in calculating the stress intensity factor of interest. Thus, for a cylindrical pressure vessel, where the hoop stress is given by pD/2t, the total stress intensity factor for a semi-elliptic internal surface flaw is:
Remember that Y accounts for finite specimen geometry and phi allows for crack shape effects. Thus dealing with cracks in internally pressurised components is usually straightforward.