Geometry of a typical spherical shell with a Gaussian defect at the pole, clamped at its free boundary. Meridian cross section of the shell, defining all the relevant geometric parameters. Three-dimensional visualization of a partial shell in the spherical coordinate system; the color map represents the radial deviation, , from a perfect sphere. Parameterized Gaussian defect profile according to Eq. (3), for . The thick horizontal line represents angular width, , associated with the theoretical buckling wavelength (see text) of the axisymmetric mode for this particular shell.
Geometry of a typical spherical shell with a Gaussian defect at the pole, clamped at its free boundary. Meridian cross section of the shell, defining all the relevant geometric parameters. Three-dimensional visualization of a partial shell in the spherical coordinate system; the color map represents the radial deviation, , from a perfect sphere. Parameterized Gaussian defect profile according to Eq. (3), for . The thick horizontal line represents angular width, , associated with the theoretical buckling wavelength (see text) of the axisymmetric mode for this particular shell.
Abstract
We present the results from a numerical investigation using the finite element method to study the buckling strength of near-perfect spherical shells containing a single, localized, Gaussian-dimple defect whose profile is systematically varied toward the limit of vanishing amplitude. In this limit, our simulations reveal distinct buckling behaviors for hemispheres, full spheres, and partial spherical caps. Hemispherical shells exhibit boundary-dominated buckling modes, resulting in a knockdown factor of 0.8. By contrast, full spherical shells display localized buckling at their pole with knockdown factors near unity. Furthermore, for partial spherical shells, we observed a transition from boundary modes to these localized buckling modes as a function of the cap angle. We characterize these behaviors by systematically examining the effects of the discretization level, solver parameters, and radius-to-thickness ratio on knockdown factors. Specifically, we identify the conditions under which knockdown factors converge across shell configurations. Our findings highlight the critical importance of carefully controlled numerical parameters in shell-buckling simulations in the near-perfect limit, demonstrating how precise choices in discretization and solver parameters are essential for accurately predicting the distinct buckling modes across different shell geometries.
