Extreme crack shapes in a shallow shell

This section is concerned with an extreme crack shape problem for a shallow shell (see Khludnev, 1997a). The shell is assumed to have a vertical crack the shape of which may change. From all admissible crack shapes with fixed tips we have to find an extreme one. This means that the shell displacements should be as close to the given functions as possible. To be more precise, we consider a functional defined on the set describing crack shapes, which, in particular, depends on the solution of the equilibrium problem for the shell. The purpose is to minimize this functional. We assume that the 

Here [ ] is a jump of a function at the crack faces, v is the unit normal vector to the crack shape, and 2h is the thickness of the shell. A similar extreme crack shape problem for a plate was considered in Section 2.4. 

Note that the problem analysed in this section can be viewed as a shape sensitivity problem (see Sokolowski, Zolesio, 1992 Pironneau, 1984 Oht-suka, 1986, 1994 Mroz, 1963 Schaeffer, 1975 Kinderlehrer, 1982 Athana-sopoulos, 1981). 

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