Electromechanical instability in layered materials

This paper deals with instability of a semi-infinite strip of polarizable layered material which is subjected to both a boundary displacement and an externally applied electrostatic potential in a plane deformation setting. Since the material is polarizable, it contributes (here in a linear fashion) to the applied electrostatic field. The nonlinear equilibrium problem is solved through a perturbative scheme and the Euler–Lagrange equations are presented. Closed-form solutions are found for some special situations and they are checked against some established results. It is shown that the general condition which lends the instability threshold is obtained enforcing that a third degree polynomial admits a double negative real solution. This amounts to seeking the roots of the discriminant of the polynomial and to checking two conditions. The negative double root yields the perturbation frequency. In the general case, a numerical solution is called upon and an instability curve, in terms of electrostatic potential vs. boundary displacement at threshold, is found. At reaching such curve, the material suddenly superposes to a homogeneously stretched configuration a periodic undulation in both the displacement and the electrostatic fields. A parametric analysis is put forward and an interesting non-monotonic behavior is found. The frequency as well as the amplitude of both the mechanical and the electrostatic undulations are found and discussed.