Waves in a viscoelastic cylindrical waveguide with a defect

A. O. Vatulyan ; V. O. Yurov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 3, pp. 352-367

Résumé

In this paper, we consider a direct problem on waves in a viscoelastic inhomogeneous cylindrical waveguide with annular delamination and investigate an inverse problem on the identification of the delamination parameters on the basis of the additional data on the displacement field at the outer boundary of the waveguide. In order to account rheological properties within the framework of the complex modules concept, we use a model of a standard viscoelastic body. After applying the integral Fourier transform along the axial coordinate in the transform space, the problem is reduced to solving a canonical system of first-order differential equations with two spectral parameters. The corresponding boundary-value problems are solved numerically by using the shooting method. To satisfy the boundary conditions on the delamination, a system of two hypersingular integral equations for the opening functions (radial and axial displacements jumps) are compiled and solved on the basis of the boundary element method. To construct the displacement field on the outer boundary of the waveguide, the techniques of direct numerical integration by quadrature formulas and the residue theorem are used. When using the theorem on residues, the calculations are performed considering the three smallest complex poles in the absolute value, which corresponds to the retention of three non-uniform vibration modes. We carry out a series of computational experiments allowing to construct the wave field at the waveguide’s outer boundary. We perform the analysis of the effect of the delamination width and geometric characteristics of loading on the wave fields. On the basis of the asymptotic formula for the field at the outer boundary of the waveguide and additional data on the radial and axial displacements at one given point, a system of transcendental equations is compiled to find the delamination width and distance to the loading region. A series of computational experiments on the reconstruction of the axial position of the defect and its width are also carried out. We also perform the analysis of the damping effect on the inverse problem equations and estimate the error. Finally, we reveal the area of applicability of the proposed reconstruction method.