A finite line crack in a pressurized spherical shell

Abstract

The deformation of a thin sheet having initial spherical curvature is shown to be associated with that of an initially flat plate resting upon an elastic foundation. Using an integral formulation the coupled Reissner equations for a shell with a crack of length 2c are solved for the in-plane and Kirchhoff bending stresses, and, among other things, it is found that the explicit nature of the stresses near the crack point depends upon the inverse half power of the non-dimensional distance from the point e. The character of the combined extension-bending stress field near the crack tip is investigated in detail for the special case of a radial crack in a spherical cap which is subjected to a uniform internal pressure qo and is clamped at the boundary $$\overline {\text{R}} = \overline {\text{R}} _{\text{o}}$$ . Pending a complete study of the solution, approximate results for the combined surface stresses near the crack tip normal and along the line of crack prolongation are respectively of the form $$\sigma _y (\varepsilon ,0)|_{\nu = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \approx 0.45\sqrt {{1 \mathord{\left/ {\vphantom {1 \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }} \tfrac{{q_O R}}{h} + ...$$ $$\lambda = 0.98$$ $${\text{c = 0}}{\text{.23 in}}$$ $$\overline {\text{R}} _{\text{o}} = 4.25 {\text{in}}$$ and similarly $$\sigma _x (\varepsilon ,0)|_{\nu = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \approx 0.45\sqrt {{1 \mathord{\left/ {\vphantom {1 \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon }} \tfrac{{q_O R}}{h} + ...$$ $$\lambda = 0.98$$ $${\text{c = 0}}{\text{.23 in}}$$ $$\overline {\text{R}} _{\text{o}} = 4.25 {\text{in}}$$ It is interesting to note that the stress σx and σy, along the line of crack prolongation, for this geometry are equal. In general, they will be of the same sign and will differ only slightly in magnitude due to the bending component. Finally, the experimental and theoretical results for ey, along the line of crack prolongation, compare very well.