The phenomenon of sling-shot buckling in stacked piezoelectric structures

The behavior of stacked structures consisting of two centrally adhered beam-plates of different lengths and piezoelectric materials is studied for the case when they are subjected to a uniform electric field. A variational formulation is presented based on the properties of the individual substructures, and yields the self-consistent governing equations and constitutive relations for the piezoelectric laminate, as well as the associated boundary and matching conditions. The stability of equilibrium configurations is assessed through evaluation of the second variation of the potential energy under perturbation. The resulting stability function, together with the total potential energy, admits prediction of the behavior of the structure under a changing electric field. A closed form analytical solution is obtained, and numerical simulations based on this solution are presented which reveal the characteristic behavior of the piezoelectric laminate. A critical electric field is identified analytically, and it is shown that stacked structures comprised of certain combinations of piezoelectric materials exhibit sling-shot buckling (an instability where the deflection of the structure dynamically slings from one direction to the opposite direction) when the critical electric field is achieved. A parameter study demonstrates the dependence of the critical electric field on the geometric and material properties of the system.