Wave dispersion in heterogeneous waveguides: methods of solution (a review). part ii | Mekhanika | kompozitsionnykh | materialov i konstruktsii

Wave dispersion in heterogeneous waveguides: methods of solution (a review). part ii


A brief review of modern methods of solution of problems of dispersion of normal waves in functionally graded and laminated elastic waveguides as well as of some ways of their improvement is presented. The early published Part I of review presented some typical functionally graded materials and appropriate constitutive relations; the methods of transfer matrix, reverberation, global matrix and the Peano series method were briefly described as well as possible approximations of functionally graded waveguides by the laminated structures with properties being constant or variable across the thickness. The main ways to improve the numerical stability of matrix methods were also mentioned. In the presented below Part II, the main attention is paid to methods of semi-analytical solution of dispersion problems based on the approximation of a waveguide by an equivalent system with a finite number of degrees of freedom, i. e. to power series, generalized Fourier series, semi-analytical finite elements, as well as methods based on higher-order theories of plates and shells. The basics of power series method are stated, the appropriate recursive relations for a plane layer and a hollow cylindrical waveguide with sectorial cross-section are presented. The main alternative to power series could be based on the expansion of the unknowns into generalized Fourier series on orthogonal polynomials of the normal coordinate; contrarily to power series the so-called “orthogonal polynomial approach” does not require the solution of transcendental equation and results in the statement of the generalized eigenvalue problem, moreover the known recursive properties of orthogonal polynomials allow one to obtain the equations coefficients analytically. The formulation of the Fourier series method in terms of state-vector formalism is presented, and its application to the study of evanescent wave modes is considered. The semi-analytical finite element method is briefly described. Finally, one variant of higher-order shell theory based on the Lagrangian formalism of the analytical dynamics of continua with constraints and biorthogonal expansion of unknown functions is discussed. It is shown that both orthogonal polynomial approach and semi-analytical finite element method follows from this kind of shell theory as particular cases generated by choice of different base functions of the normal coordinate on the background of the unified variational formalism. Accounting for constraints following from boundary conditions on shells’ faces allows one to satisfy the reflection conditions for the waveguide model based on the shell model of arbitrary order.