Time-causal material modeling in the simulation of guided waves in circular viscoelastic waveguides

For the description of linear viscoelasticity, the fractional Zener model may be used. Based on the spectral decomposition of the elasticity matrix as proposed by Theocaris, we generalize the one-dimensional analysis of the material model into three dimensions and discuss appropriate simplifications to reduce the amount of unknowns for the material description. Then, a decomposition approach that considers the real valued frequency dependence of the viscoelastic moduli and the real valued frequency dependence of their attenuation separately is proposed. The Scaled Boundary Finite Element Method is used for the efficient computation of the phase velocity dispersion and the modal wave fields given a frequency dependent but real valued viscoelasticity matrix. Utilizing the modal expansion approach, the transmitting and receiving transducer are taken into account to compute the modal amplitudes. Combining these modal amplitudes, the phase velocity dispersion and re-introducing the viscoelastic attenuation results in a transfer function of the viscoelastic waveguide including excitation and receiving conditions. The performance of the proposed simulation model is shown by comparison to measurements taken on a polypropylene sample.