A reduced Föppl–von Kármán model for magnetic plates

Slender plate-like structural elements are widely employed in both traditional engineering structures and advanced functional devices. Their counterparts made of magnetically-active materials can provide fast and reversible shape-morphing through contactless remote actuation. Several promising applications involving magnetic plates have been recently proposed in the literature, albeit with designs that have been primarily driven by intuition. As such, there is a striking lack of formal theoretical tools to aid in the predictive and rational design of this class of magneto-elastic slender structures. Here, we develop a 2D theory for the mechanics of thin hard-magnetic plates. Following a dimensional reduction procedure akin to that of classic Föppl–von Kármán plates, we reduce the system's 3D energy functional, including the magneto-elastic coupling, into a 2D description based on the plate's mid-surface. The equilibrium equations obtained through variational methods are then employed to predict the plate deformation under a combination of mechanical and magnetic loading conditions. We also analyze a magnetically-actuated buckling problem. Our theoretical framework for the mechanics of magnetic plates is validated thoroughly using experiments and full 3D finite element modeling.