The stresses in a spherical shell containing a crack

Abstract

NOTE: There is a 3-page Notation Eqivalency Chart preceding the Introduction. NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Introduction is included in .pdf document. Chapter I Introduction One of the problems in fracture mechanics which apparently has not received extensive theoretical treatment is that concerning the effect of initial curvature upon the stress distribution in a thin sheet containing a crack. Considerable work has been carried out on initially flat sheets subjected to either extensional or bending stresses, and for small deformations the superposition of these separate effects [1] is permissible. On the other hand, if a thin sheet is initially curved, a bending load will generally produce both bending and extensional stresses, and similarly a stretching load will also induce both bending and extensional stresses. The subject of eventual concern therefore is that of the simultaneous stress fields produced in an initially curved sheet containing a crack. Two geometries immediately come to mind: a spherical shell, and a cylindrical shell. In the latter case one of the principal radii of curvature is infinite and the other constant. It might appear therefore that this geometric simplicity leads to a rather straightforward analytical solution. However, the fact that the curvature varies between zero and a constant as one considers different angular positions - say around the point of a crack which is aligned parallel to the cylinder axis - more than obviates the initial geometric simplification. For this reason a spherical section of a large radius of curvature constant in all directions is chosen for consideration. It is of some practical value to be able to correlate flat sheet behavior with that of initially curved specimens. In experimental work, for example, considerable time could be saved if a reliable prediction of curved sheet response behavior could be made from flat sheet tests. For this reason an exploratory study was undertaken to assess analytically how the two problems might be related. Although it is recognized that elastic analysis is not directly applicable to fracture prediction because of the plastic flow near the crack tip, considerable information can be obtained. Chapter II lists the basic assumptions and equations of shallow spherical shells. Then the complementary problem of a cracked spherical shell is formulated in terms of Reissner's shallow shell equations in Chapter III where the problem is separated into two parts, symmetric and antisymmetric. In Chapters IV and V the solutions to the symmetric and antisymmetric parts respectively are expressed in integral form. They are then reduced to the solution of a pair of coupled singular integral equations, which are solved by successive approximations for small values of the characteristic shell parameter [...]. No effort was made to convert the pair of singular integral equations to a corresponding Fredholm type, however Appendix II shows that the two methods are equivalent. The particular example of a clamped segment of a thin shallow spherical shell is considered in Chapter VI which serves to illustrate how the local solution may be combined in a particular case. Then in Chapter VII, Griffith's criterion is extended to the local region of an initially spherical curved sheet and an expression for its critical crack length is obtained. Finally, Chapter VIII compares the experimental and theoretical results for the particular problem described in Chapter VI.