We perform a probabilistic investigation on the effect of systematically removing imperfections on the buckling behavior of pressurized thin, elastic, hemispherical shells containing a distribution of defects. We employ finite element simulations, which were previously validated against experiments, to assess the maximum buckling pressure, as measured by the knockdown factor, of these multi-defect shells. Specifically, we remove fractions of either the least or the most severe imperfections to quantify their influence on the buckling onset. We consider shells with a random distribution of defects whose mean amplitude and standard deviation are systematically explored while, for simplicity, fixing the width of the defect to a characteristic value. Our primary finding is that the most severe imperfection of a multi-defect shell dictates its buckling onset. Notably, shells containing a single imperfection corresponding to the maximum amplitude (the most severe) defect of shells with a distribution of imperfections exhibit an identical knockdown factor to the latter case. Our results suggest a simplified approach to studying the buckling of more realistic multi-defect shells, once their most severe defect has been identified, using a well-characterized single-defect description, akin to the weakest-link setting in extreme-value probabilistic problems.

We perform numerical experiments using the finite element method (FEM) to investigate the effect of defect–defect interactions on the pressure-induced buckling of thin, elastic, spherical shells containing two dimpled imperfections. Throughout, we quantify the critical buckling pressure of these shells using their knockdown factor. We examine cases featuring either identical or different geometric defects and systematically explore the parameter space, including the angular separation between the defects, their widths and amplitudes, and the radius-to-thickness ratio of the shell. As the angular separation between the defects is increased, the buckling strength initially decreases, then increases before reaching a plateau. Our primary finding is that the onset of defect–defect interactions, as quantified by a characteristic length scale associated with the onset of the plateau, is set by the critical buckling wavelength reported in the classic shell-buckling literature. Beyond this threshold, within the plateau regime, we demonstrate that the largest defect dictates the shell buckling.

We investigate the effect of defect geometry in dictating the sensitivity of the critical buckling conditions of spherical shells under external pressure loading. Specifically, we perform a comparative study between shells containing dimpled (inward) versus bumpy (outward) Gaussian defects. The former has become the standard shape in many recent shell-buckling studies, whereas the latter has remained mostly unexplored. We employ finite-element simulations, which were validated previously against experiments, to compute the knockdown factors for the two cases while systematically exploring the parameter space of the defect geometry. For the same magnitudes of the amplitude and angular width of the defect, we find that shells containing bumpy defects consistently exhibit significantly higher knockdown factors than shells with the more classic dimpled defects. Furthermore, the relationship of the knockdown as a function of the amplitude and the width of the defect is qualitatively different between the two cases, which also exhibit distinct post-buckling behavior. A speculative interpretation of the results is provided based on the qualitative differences in the mean-curvature profiles of the two cases.

The buckling of spherical shells is plagued by a strong sensitivity to imperfections. Traditionally, imperfect shells tend to be characterized empirically by the knockdown factor, the ratio between the measured buckling strength and the corresponding classic prediction for a perfect shell. Recently, it has been demonstrated that the knockdown factor of a shell containing a single imperfection can be predicted when there is detailed a priori knowledge of the defect geometry. Still, addressing the analogous problem for a shell containing many defects remains an open question. Here, we use finite element simulations, which we validate against precision experiments, to investigate hemispherical shells containing a well defined distribution of imperfections. Our goal is to characterize the resulting knockdown factor statistics. First, we study the buckling of shells containing only two defects, uncovering non-trivial regimes of interactions that echo existing findings for cylindrical shells. Then, we construct statistical ensembles of imperfect shells, whose defect amplitudes are sampled from a lognormal distribution. We find thata 3-parameter Weibull distribution is an excellent description for the measured statistics of knockdown factors, suggesting that shell buckling can be regarded as an extreme-value statistics phenomenon.