Spatially Resolved Measurement of the Stress Tensor in Thin Membranes Using Bending Waves

Abstract

The mode shape of bending waves in thin silicon and silicon carbide membranes is measured as a function of space and time, using a phase shift interferometer with stroboscopic light. The mode shapes hold information about all the relevant mechanical parameters of the samples, including the spatial distribution of static prestress. We present a simple algorithm to obtain a map of the lateral tensor components of the prestress, with a spatial resolution much better than the wavelength of the bending waves. The method is not limited to measuring the stress of bending waves. It is applicable in almost any situation, where the fields determining the state of the system can be measured as a function of space and time. I. INTRODUCTION Mechanical waves are a powerful tool to study material properties over a broad range of length scales. The applications range from seismic-wave methods to studying the structure of the Earth [1], SONAR for underwater location and navigation [2], ultrasonic testing of materials in engineering [3,4] and medical ultrasonography [3] ,d own to methods applicable to micro- and nanostructures like acoustic microscopy [3], Brillouin scattering [5], and picosecond pump-probe spectroscopy [6,7]. For all these methods, the length scale of the inhomogeneities of the material properties is supposed to be larger than the wavelength. In a previous work [8], we show how to extract the dispersion relation of bending waves of prestressed thin plates from measured mode shapes. Stress and bending stiffness of the membrane are determined as fit parameters using a theoretical model of the dispersion relation. Since the existence of a well-defined dispersion relation requires a homogeneous system, this method has the same shortcomings as the methods listed above. Here, we present an approach using full knowledge of the wave function to determine the mechanical properties of the sample. We measure the shape of a bending-wave mode in direct space and compare it to the equation of motion. The parameters found in the equation are the components of the stress tensor divided by the density of the membrane σij=ρ ,a s well as a constant proportional to the bending stiffness. With our simple algorithm, these parameters are obtained as a function of space using a linear fit. Since the full mode shape is available and we are not limited to far-field information, the wavelength of the bending waves is no limit for the spatial resolution. For the measurement of the stress-tensor components of semiconductors, all optical methods can be applied with spatial resolution in the order of a few micrometers [9,10] to roughly 100 μm [11], depending on the respective methods. However, these methods make use of particular optical properties and cannot straightforwardly be generalized to insulators or metals.