Recovery of elastic parameters of cellular materials by inversion of vibrational data

Abstract

This investigation addresses the inverse problem of the retrieval of the macroscopic Young's modulus and Poisson ratio of light fluid-saturated cellular materials from the measured resonance frequencies of their vibrational response. The latter is shown to be generally nonlinear for all except very small excitations. In order to remain in the linear regime, and thus avoid ambiguities in the characterization of the material, we employ lightweight piezoelectric transducers and sensors and a low-noise amplifier for the acquisition of dynamic response data. The cellular materials under investigation (polyurethane, melamine) are shown to behave as isotropic, (equivalent) homogeneous elastic solids with Rayleigh damping (equivalent to a rheological model having the same dependence of attenuation on frequency) in the frequency band of interest. Our viscoelastic model is validated by comparing its predictions of the dynamic response of the specimens to the predictions of the more complete Biot interaction model. The computations with both of these models are carried out by means of 3D finite element (FE) codes for a variety of specimen shapes and sizes. Once the simpler viscoelastic model is validated, it is then employed to compute the trial resonance frequencies (which are the solutions of an eigenvalue problem associated with this model) during the inversion process. The cost function, which is a measure of the discrepancy between the trial and measured resonance frequencies for a number of vibrational modes, is constructed and then minimized by the Levenberg–Marquardt optimization algorithm in order to recover the sought-for moduli. Finally, a discussion is presented concerning the influence of the size and shape of the specimens, as well as the number of employed modes, on the uniqueness and quality of the inversion.