A bistable chain on elastic foundation

Arrays of bistable elements have been studied extensively in the last two decades due to their relevance to a wide range of physical phenomena and engineering applications, from rate-independent hysteresis to multi-stable metamaterials and soft robotics. Here, we study, theoretically and experimentally, an important extension of the bistable-chain model that has been largely overlooked, namely a discrete chain of bistable elements that is supported by a linear-elastic foundation. Focus is put on equilibrium configurations and their stability, from which the sequence of phase-transition events and the overall force-displacement relation are obtained. In addition, we study the influence of each of the bistable parameters and the stiffness of the elastic foundation on the overall behavior. Closed-form analytical expressions are derived by approximating the bistable behavior with a trilinear force-displacement relation. These are later validated numerically and experimentally. Our analysis shows that the sequence of phase transition may involve two fundamentally different scenarios, depending on the system parameters. The first scenario is characterized by the propagation of a single phase boundary associated with an ordered sequence of phase transitions, while the second involves the formation of multiple phase boundaries and a disordered sequence of transition events. Also, by identifying that the displacements of the chain are related through a linear recursive sequence, we show that, in some particular cases, the relevant expressions can be conveniently reduced to formulas associated with the celebrated Lucas or Fibonacci sequences, and the physical interpretation of these solutions is discussed.