Comparative analysis of mechanical efficiency and robustness of different spherical cap shaping strategies in elastic sheets

The transformation of flat elastic or viscoelastic sheets into three-dimensional structures is central to biological processes and the design of smart materials. The existence of a degeneracy where multiple physical mechanisms are capable of achieving the same final geometry, raises the question of how a specific path may be optimal in nature or engineering. We present a theoretical and computational comparative analysis of distinct mechanisms that induce flat to spherical cap transitions, focusing on two main classes: bi-layer area mismatches and patterning intrinsic curvature via spontaneous strains. We systematically quantify and compare the performance of representatives of these classes across several metrics: kinetics, energetics, robustness to noise, and the information content required to encode the target shape. In so doing, we elucidate fundamental trade-offs inherent to shape morphing. We anticipate that by quantifying these differences, this work will resolve the observed shape degeneracy by revealing a direct link between the mechanism's performance and its context or constraints. These findings will ultimately offer new insights into biological shape transitions and provide design principles for engineered morphing materials.