The variational equations of the extended n'Th order shell theory and its application to some problems of dynamics

One of topical problems of thick shells theory consists in an accurate modelling of high-frequency shell vibrations and wave propagation, as well as in the application of developed mathematical models to engineering problems of structural dynamics. Many observed processes of both periodic and non-steady shell dynamics cannot be adequately simulated on the groundwork of most traditional hypothesis-based shell theories. Therefore the engineering practice requires new shell models considering higher degrees of freedom besides the displacement and rotation vectors and taking into account the interference of tangential and transversal motions. Here a variant of the N'th order shell model of I.N. Vekua type is analyzed. The proposed model improves the A. Amosov's shell theory; it is based on the Lagrange's formalism of analytical mechanics extended to continuum systems, on the dimensional reduction approach, and on the biorthogonal expansion technique. The shell is finally represented by a material surface framed by a set of generalized coordinates (or field variables) and a scalar generator function - the surficial density of the Lagrangian. The dynamic equations for the shell are formulated as generalized Lagrange equations of the second kind. A propagation of normal waves in a plane elastic layer is considered, the corresponding homogeneous problem of N'th order plate theory is formulated, and its solution analysis is extended. In particular, the investigation of the second longitudinal propagating mode and its modelling by approximated plate theories are expanded. The main feature of the second mode is the “inverse wave” effect, i.e. the opposite signs of the phase and group velocities at small wavenumbers. It was shown earlier that the accurate dispersion curves for both phase frequency and group velocity can be obtained on the basis of the fourth- and fifth-order theories. Using the known solution of the spectral problem of N'th order plate theory the eigenfunctions are obtained. The second propagating mode evaluation at the smallest wavenumbers is modelled, the forms of the second propagating wave are investigated, and the approximation of the waveforms at some wavenumbers by plate theories of various orders is estimated. The accuracy of the approximation of the “inverse wave” by some applied theories of various orders based on the orthogonal expansions of displacement vector is discussed.