Electromechanical buckling of periodic patterns on stiff film bonded to a compliant substrate - Analytical and numerical postbuckling analyses

In this work, the analytical and numerical analyses of the electromechanical buckling and post- buckling states of a planar film bonded to a compliant dielectric substrate are presented. The film is a stiff thin metal layer and forms an elastic electrode. The compliant substrate is attached to a bottom fixed and rigid electrode. The film is simultaneously subjected to in-plane compression stresses, which can be induced by the thermal expansion mismatch effect, and to out-plane electrostatic traction that is induced by applying a voltage difference between the upper elastic electrode and the lower fixed electrode. Electromechanical buckling instability stems from the coupling of mechanical buckling instability and electromechanical pull-in instability and instigates mechanical buckling. This manuscript establishes an analytical approach to study the electromechanical critical and post buckling of one-dimensional, square checkerboard, hexagonal, and herringbone periodic buckling modes. Unlike previous works, in the current study the electrostatic traction and energy contributions are derived from the analytical solution of the electric potential function. A modified version of von-Karman plate equations are formulated and used along with the upper-bound energy method to derive analytical solutions for the critical and post electromechanical buckling states. The analytical critical and postbuckling solutions and energy state of periodic patterns are validated by a rigorous numerical finite element solution carried out in COMSOL Multiphysics. It has been found that the buckling stress and periodicity wavelength of a unit cell strongly depend on the applied voltage. The ability to manipulate the value of buckling stress by voltage increases the potential of applications in microsystem technologies such as electrical on/off switching of a surface wrinkling or deformable micromirror.