Finite element based technique for modeling propagative stress waves in axisymmetric viscoelastic waveguides

The knowledge of the dispersive characteristics of waves is essential for the design of any structural health monitoring technique that uses stress waves in solids of finite dimension. Here an enhanced Semi-Analytical Finite Element (SAFE) algorithm is presented for the extraction of guided wave attenuation, energy velocity and wavestructure in straight viscoelastic axisymmetric waveguides. In SAFE algorithms the wave displacement field is approximated by coupling finite elements over the waveguide cross-section and harmonic functions along the waveguide length [1]. For axisymmetric problems, by using cylindrical coordinates, the waveguide cross-section mesh can be reduced to a line of mono-dimensional elements along a single radius. In this way, the dispersive features of a three-dimensional wave propagation problem are represented by the modal properties of a mono-dimensional mesh. In force of the correspondence principle [2], the proposed formulation accounts for linear viscoelastic materials by using complex constitutive tensors in the stress-strain relations. Based on the approximate displacement field, the weak form of the governing wave equation leads to a set of algebraic equations in the frequency and complex wavenumber unknowns. For free guided wave propagation, the solution of a twin parameters eigenproblem allows the dispersive and modal characteristics to be extracted. In particular, for a given frequency, guided waves phase velocity, attenuation and wavestructure, are obtained directly from the eigensolution [3]. Wave’s energy velocity can be calculated by averaging in time the ratio of power flow and total energy density [4]. Finally, an application for a bitumen coated pipe is presented [5]. References [1] Aalami B, Waves in prismatic guides of arbitrary cross section, Journal of Applied Mechanics 40, 1973, 1067-1072. [2] Auld BA, Acoustic Fields and Waves in Solids, Malabar FL, Kreiger, 1990. [3] Bartoli I, Marzani A, Lanza di Scalea F, Viola E, Modeling wave propagation in damped waveguides of arbitrary cross-section, Journal of Sound and Vibration, (available on line 20 March 2006). [4] Biot AM, General theorems on the equivalence of group and energy transport, The Physical Review 105(4), 1957, 1129-1137. [5] Barshinger J, Rose JL, Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 51(11), 2004, 1547-1556