Vibrational Technique for Stress Measurement in Films: I, Ideal Membrane Behavior

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Abstract
A vibrational technique for the measurement of stress in films has been developed, based on the theory of membranes. Specimens for vibrational analysis are prepared by removing a circular or square piece of the substrate. It is shown that the error introduced due to substrate removal is negligible except for thin metallic films. The resulting specimen, a film stretched across a hole in the substrate, behaves as a membrane. Since its natural frequencies increase with the square root of in‐plane stress, these can be measured to compute the in‐plane stress. The only material parameter needed is the density of the film. Because a number of natural frequencies can be measured, each of which can be used to compute the stress, this method has a strong check for internal consistency. Two different techniques have been used to measure natural frequencies: (1) sinusoidal excitation and holographic imaging of the vibration modes, and (2) random excitation and Fourier analysis of measured vibrations. In this, the first part of the study, we discuss cases when ideal membrane behavior is obtained. Three applications of the method are presented: measurement of stress in a filled‐glass layer bonded to an aluminum substrate, a study of the effect of temperature and humidity on stress in a thermoset polymer coating, and stress in a silver film during heat treatment. The second part of the study1 deals with extensions of the technique when complicating effects, such as air loading, bending stiffness, and an aniso‐tropic state of stress, are important. In the third part of the study,2 the method is used to identify the orthotropic axes and all nine elasticity coefficients of a commercially made film.