Complex modulus estimation respecting causality: Application to viscoelastic bars

¹ Centre de Physique Théorique CNRS UMR 7644, École Polytechnique, 91128 Palaiseau, France
² Laboratoire de Mécanique des Solides, CNRS UMR 7649, École Polytechnique, 91128 Palaiseau, France
³ The Angström Laboratory, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden

Abstract

The identification of linear visco-elasticity models mostly focuses on the real and imaginary parts of the Young’s modulus. Many methods have been proposed in the past to identify these material model parameters from experiments. However, when these parameters are determined independently, they are likely to violate the principle of causality. The present work presents a method that accounts for the constraints of causality and positivity of dissipation rate. The proposed method is based on a finite set of n measured angular frequencies and complex moduli. It includes a noise reduction procedure and provides a rheological 2p + 1)-parameter model with p < gn which corresponds to a specific configuration of pairs of springs and dashpots. The poles of the complex modulus on the positive imaginary frequency axis are determined by p parameters which are obtained as the common positive zeros of a special class of rational functions, while the remaining parameters are obtained from a least squares fit. The level of refinement of the rheological model, expressed by p, is not an assumed value but a result of the method. The method is applied to an impact test with a Nylon bar specimen. In this case, data at the n = 29 lowest resonance frequencies resulted in a rheological model with 14 parameters (p = 6). The validity of the method is checked through supplementary experimental results at low frequencies.