Vibration modeling and testing of bilayer beams for determination of film elastic modulus

Abstract

Analysis of the main parameters affecting the fundamental vibrating frequency of film/substrate bilayer beams of rectangular cross-section is discussed based on modeling and testing. Initially, the limits of validity of two analytical models to obtain the fundamental frequency of perfectly-bonded bilayer beams in cantilever configuration are determined by comparing the predicted frequencies to a finite element model developed herein. Using a selected analytical formulation, a modeling-assisted methodology is employed to investigate the parameters that are most influential on the determination of the elastic modulus of the film using a vibratory technique. Modeling suggests the use of thin compliant substrates for extracting the modulus of stiff (metallic) films. If the substrate is stiffer than the film, a thicker film is required to yield measurable shifts in the resonant frequency. The elastic modulus of a millimeter-thick thermosetting polymer extracted by this method agrees with the results obtained from conventional tensile testing of the polymer. Measurements carried out on a gold (100 nm)/polysulfone (130 µm) system yield an average elastic modulus of the gold film similar to the values reported in the literature.