Finding transition state and minimum energy path of bistable elastic continua through energy landscape explorations

Mechanical bistable structures have two stable equilibria and can transit between them under external stimuli. Due to their unique behaviors such as snap-through and substantial shape changes, bistable structures exhibit unprecedented properties compared to conventional structures and thus have found applications in various fields such as soft robots, morphing wings and logic units. To quantitatively predict the performance of bistable structures in these applications, it is desirable to acquire information about the minimum energy barrier and an energy-efficient transition path between the two stable states. However, there is still a general lack of efficient methodologies to obtain this information, particularly for elastic continua with complicated, unintuitive transition paths. To overcome this challenge, here we integrate energy landscape exploration algorithms into finite element method (FEM). We first utilize the binary image transition state search (BITSS) method to identify the saddle point and then perform nudged elastic band (NEB) calculations with an initial guess based on the BITSS results to find the minimum energy path (MEP). This integrated strategy greatly helps the convergence of MEP calculations, which are highly nonlinear. Two representative cases are studied, including bistable buckled beams and a bistable unit of mechanical metamaterials, and the numerical results agree well with the previous works. Importantly, we numerically predict the complicated MEP of an asymmetric bistable unit of mechanical metamaterials and use experiments to demonstrate that following this MEP leads to successful transition between stable states while intuitive uniaxial compression fails to do so. Our work provides an effective numerical platform for identifying the minimum energy barrier and energy-efficient transition path of a bistable continuum, which can offer valuable guidance to the design of actuators, damping structures, energy harvesters, and mechanical metamaterials.