Buckling of planar curved beams with finite prebuckling deformation

The serpentine structure with a sufficiently thick cross section has recently been proposed as an important design concept in stretchable electronics, which features mechanically stable in-plane deformation mechanism and very low electrical resistance, bringing unique advantages for devices compared with the traditional thin ribbon layout. However, unduly increasing the thickness is well known to sacrifice the overall flexibility and functionality of devices. Such a contradiction leads to challenges in structural stability, as a relatively thick but insufficient serpentine structure may eventually undergo the out-of-plane buckling after significant in-plane prebuckling deformation and appreciable alterations in initial configuration, which is ignored by most conventional buckling theories (CBTs) and linear buckling analysis in commercial finite element analysis software, producing intolerable errors when predicting the critical loads. In this paper, a systematic and straightforward theory considering the finite prebuckling deformation (FPD buckling theory) is established to investigate the underlying mechanism. Two sets of governing equations related to the prebuckling and FPD buckling behavior are obtained. Four representative examples, including two classical problems of planar curved beams and two typical loading conditions of serpentine structures, have been carefully studied. Comparisons with the accurate geometrically-nonlinear-analysis-based (GNAB) buckling analysis have amply demonstrated the validity of our theory in predicting the reinforcement effect of prebuckling deformation on the buckling resistance of structures. Key dimensionless geometric parameters that govern this effect have also been identified, providing direct and effective guidance for the design and optimization of stretchable electronic devices.