Morphological evolution and “local circularization” energy mechanism of surface tension–driven post-buckling in polymer rings

Surface tension can induce instability in slender ring structures, leading to complex post-buckling configurations; however, the mechanism underlying the evolution of these configurations, along with their energy characteristics, is not fully understood. This study investigates the deformation behavior of polymer rings under surface tension and clarifies the energy-driven mechanism underlying post-buckling evolution.

Polymer rings with rectangular cross-sections were fabricated using polyethylene terephthalate rods, and double-layer soap films were applied to generate uniform surface tension loading. The deformation process was recorded using a high-resolution imaging system, and geometric parameters, including the ring radius and cross-sectional properties, were extracted from the experimental images. A support vector machine model was established to classify the buckling states based on these parameters. Finite element simulations were performed using Abaqus to reproduce the deformation process and analyze the stress and deformation distributions. A three-dimensional reconstruction method based on image processing was also employed to obtain the spatial configurations of the post-buckled shapes. The total energy of the system was calculated by combining the bending strain energy, which was obtained through curvature integration along the ring centerline, and the interfacial energy, which was simplified using Young’s equation. The contact areas between the solid, liquid, and air phases were extracted from the experimental images and the simulation results.

The results show that the ring undergoes a sequence of morphological transitions with increasing radius. The shape evolves from a circular configuration to an elliptical configuration, then to a figure-eight configuration, and finally to a configuration with extended contact regions, accompanied by substantial out-of-plane deformation. The support vector machine classification results agree well with the theoretical predictions of the critical buckling condition, and the finite element results are consistent with the experimental observations in overall morphology and deformation trends. Energy analysis indicates that the total energy of the deformed configuration is lower than that of the initial circular configuration, with the largest reduction occurring at the onset of buckling. Thereafter, the total energy remains relatively stable with increasing radius. The curvature distribution of the complex configurations shows that local segments tend to approach circular arcs, thereby reducing the bending energy, and the introduction of out-of-plane deformation further decreases the system energy.

The post-buckling behavior of polymer rings is governed by energy minimization under surface tension. Buckling leads to a substantial reduction in system energy, and subsequent deformation maintains this low-energy state. Complex configurations are formed through local curvature adjustment and three-dimensional deformation. The proposed experimental and numerical methods provide an effective approach for analyzing surface tension–induced instability, and the results provide a reference for the design of flexible structures and anti-buckling systems in engineering applications.