Heterogeneity-dominated discrete phase transitions in multistable systems: A unified bistable chain framework

Multistable systems, characterized by their ability to undergo discrete phase transitions, underpin a broad range of phenomena and applications from snapping metamaterials to protein unfolding. However, the foundational assumption of homogeneity in conventional bistable chain models restricts them to predicting only uniformly propagating phase transitions, thereby overlooking the sequential, path-dependent behaviors that are characteristic of real-world heterogeneous systems. To address this limitation, we develop a semi-analytical heterogeneous bistable chain model composed of dissimilar bistable elements. Each element is described by trilinear force-displacement relation with distinct phase transition thresholds. The two limiting pathways, the minimum energy (thermodynamic equilibrium) and maximum hysteresis (athermal) paths, are generalized to account for heterogeneity. They provide the theoretical envelope that contains all possible mechanical responses. Furthermore, through heterogeneous bistable chain model, we analytically reveal the mechanism of coupled phase transitions: a cooperative phenomenon unique to heterogeneous multistable systems where the phase transition of one element can induce phase transitions in others. The predictive capability of the proposed framework is validated through the design of gradient multistable metamaterials, where theoretical predictions show excellent agreement with finite element simulations and experimental measurements. This work provides both a fundamental understanding of discrete phase transitions in heterogeneous systems and an efficient reduced-order modeling tool for structural design with programmable phase transition pathways.