Research over the past ten years has generated an increased interest in studying elastic structural instabilities as a useful response for smart applications rather than a failure. Buckling under axial compression is a type of structural instability that can be used for rapid geometric transformations (switching) and energy harvesting applications, if the deformations arising from buckling are properly controlled. Controlling transverse deformations due to buckling in slender elements usually needs external constraints/boundaries. Short thin-walled cylinders can experience several elastic buckling events under axial compression without additional constraints. However, predicting the post-buckling response in cylinders is very challenging, particularly far in the post-buckling regime since they are highly sensitivity to initial imperfections.The concept of cylinders with non-uniform stiffness distribution (NSD) was recently proposed to localize a cylinder's buckling events in targeted zones. This notion has been proven effective for controlling the number of elastic buckling events, the sequence at which they occur, and the regions experiencing buckling. However, this information is not enough to design NSD cylinders for smart applications, which requires being able to predict the actual applied force for each buckling event, the end shortening of the cylinder for the buckling event, the drop in force, the drop in strain energy, and the post-buckling stiffness of the cylinder.Here, a semi-analytical model has been developed to predict the elastic post-buckling response of NSD cylinders under compression. The developed semi-analytical model is based on three general steps:⁰́ØSeparate the NSD cylinder into parallel segments,⁰́ØSimplify and predict the response of each segment, and⁰́ØIntegrate the response of individual segments.The first step in predicting the elastic post-buckling response of a cylindrical segment was to simplify its geometry into a cylindrical panel with uniform thickness. Linear springs are connected to the top and bottom of the uniform cylinder to match the stiffness of the simplified segment to the actual one. Based on classical shell theory, the elastic post-buckling response of a cylindrical panel is solved as a boundary value differential equation using the pseudo-arclength method. Comparing the post-buckling response of four cylinders from the proposed semi-analytical model with the response of the same cylinders from the experiment and finite element analysis showed the effectiveness of the proposed model. Results from the proposed model predict well the axial deformation and force level corresponding to buckling events more accurately than the post-buckling stiffness.The response of cylindrical panels for a large variety of dimensions is needed to design NSD cylinders for targeted post-buckling behavior. Thus, the classic differential equation of the cylindrical panels under axial compression was solved independently of the material's cylinder radius and elastic modulus. These results allowed the development of design maps for several post-buckling responses such as axial strain and stresses corresponding to the first buckling event, force, and energy drops from the buckling event, the secondary (or post-buckling) stiffness of the panel, the radial deformation at the panel center, and the maximum von Mises stress in the panel. By using genetic programming, predictive equations were developed for each design parameter to relate it to the geometry of the panels.Three cylinders were designed using the developed design maps to validate the proposed approach. One NSD cylinder was designed to undergo several buckling events under compression at pre-defined end shortenings. A second NSD cylinder was designed to feature a post-buckling force-deformation response that plateaus at a constant force level. The third cylinder was designed to experience the same force drop at each buckling event and in identical axial end shortenings after the first event. Finite element analyses of the designed cylinders verified that using the proposed design procedure using the developed design maps provides NSD cylinders with a post-buckling response that is very close to the desired one, and the ultimate design goal can be achieved by slight modifications to the geometry of the cylinder.This study advances the knowledge on the elastic buckling and post-buckling response of slender cylindrical shells under axial compression and provides an approach to analyze and design them for a desired far post-buckling response. The proposed framework, which combines the notion of decomposing NSD cylindrical segments into linear and nonlinear springs in series, a semi-analytical model for NSD equivalent panels, and design maps for several nonlinear responses provides insight for designing these elements for smart devices and structures relying on structural instabilities. This work expands the harnessing of elastic instabilities to the area of thin-shell buckling under compression, which has received less attention in comparison to other forms of structural instability.
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