Programmable symmetry-breaking in folded elastic ribbons via tunable pitchfork bifurcations

Controlling symmetry-breaking transitions is central to the design of programmable morphing structures and functional meta-devices. Here, we investigate the stability and bifurcation of elastic ribbons featuring localized folds under combined longitudinal compression and transverse shear. Through a combination of experimental observations, discrete differential geometry simulations, and theoretical analysis, we demonstrate that a localized fold acts as a geometric “tuner” that fundamentally reshapes the stability topology of the system. While a pristine continuous ribbon undergoes spontaneous loss of configurational rotational symmetry through a supercritical pitchfork bifurcation, the introduction of a localized fold transforms the response into an imperfect bifurcation, where a primary stable branch deterministically dictates the deformation path; beyond a critical tuner strength, the configurational symmetry is fully preserved throughout loading. By mapping the global stability landscape through eigenvalue analysis, we identify a critical boundary in the parameter space separating symmetry-breaking from symmetry-preserving regimes. This tunability is further enhanced in dual-fold configurations, where synergistic fold interactions enable a more efficient suppression of symmetry-breaking at significantly smaller fold angles. Finally, a low-dimensional toy model is developed to distill the underlying energy competition, revealing that the programmable response originates from the nonlinear coupling between localized geometric singularities and global buckling modes. Our findings provide a predictive framework for encoding complex, deterministic morphological responses into slender structures through simple geometric motifs.