Axisymmetric dynamic response of the multi-layered transversely isotropic medium

By virtue of the precise integration method (PIM) and the technique of mixed variable formulations, solutions for the dynamic response of the multi-layered transversely isotropic medium subjected to the axisymmetric time-harmonic forces are presented. The planes of cross anisotropy are assumed to be parallel to the horizontal surface of the stratified media. Four kinds of vertically acting axisymmetric loads are prescribed either at the external surface or in the interior of the soil system. Thicknesses and number of the medium strata are not limited. Employing the Hankel integral transform in cylindrical coordinate, the axisymmetric governing equations in terms of displacements of the multi-layered media are uncoupled. Applying mixed variable formulations, more concise first-order ordinary differential matrix equations from the uncoupled motion equations can be obtained. Solutions of the ordinary differential matrix equations in the transformed domain are acquired by utilizing the approach of PIM. Since PIM is highly accurate to solve the sets of first-order ordinary differential equations, any desired accuracy of the solutions can be achieved. All calculations are based on the corresponding algebraic operations and computational efforts can be reduced to a great extent. Comparisons with the existing numerical solutions are made to confirm the accuracy of the present solutions proposed by this procedure. Several examples are illustrated to explore the influences of the type and degree of material anisotropy, the frequency of excitation and loading positions on the dynamic response of the stratified medium.     

Analytical layer-element solution to axisymmetric consolidation of multilayered soils

After the application of a Laplace–Hankel transform, the governing equations of Biot’s consolidation are solved analytically by using the eigenvalue approach. Then the analytical layer-element of a single soil layer can be obtained in the transformed domain by synthesizing the generalized displacements and stresses, which are both expressed by six arbitrary constants. The elements of the analytical layer-element only contain negative exponential functions, which leads to a considerable improvement in computation efficiency and stability. The global stiffness matrix equation of multilayered soils is further obtained by assembling the interrelated layer-elements, and the actual solution is achieved by numerical inversion of the Laplace–Hankel transform after the solution in the transformed domain is obtained. Numerical examples are given to demonstrate the accuracy of this method and to study the influence of the layered soil properties and time history on the consolidation behavior.     

Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space

Based on the analytical layer-element method, an analytical solution is proposed to determine the dynamic interaction between the elastic circular plate and transversely isotropic multilayered half-space. The dynamic response of the elastic circular plate is governed by the classical thin-plate theory with the assumption that the contact surface between the plate and soil is frictionless. The total stiffness matrix of the transversely isotropic multilayered half-space is acquired by assembling the analytical layer-element of each soil layer with the aid of the continuity conditions between adjacent layers. According to the displacement condition of coordination between the plate and soil, the dynamic interaction problem is reduced to that of multilayered transversely isotropic half-space subjected to axisymmetric harmonic vertical loading. Some numerical examples are given to study the vertical vibration of the plate, and the results indicate that the dynamic response of elastic circular plate depends strongly on the material properties of the soils, the rigidity of the plate, the frequency of excitation and the external load form.     

Third order wave force on axisymmetric bodies

The third order triple-frequency wave load on fixed axisymmetric bodies by monochromatic waves is considered within the frame of potential theory. Waves are assumed to be weak non-linearity and a perturbation method is used to expand velocity potentials and wave loadings into series according to a wave steepness of kA. Integral equation method is used to compute velocity potentials up to second order in wave steepness. The third order triple-frequency wave loads are computed by an indirect method and an efficient method is applied to form the third order forcing term on the free surface quickly. The method can be used to compute third order triple-frequency surge force, heave force and pitch moment on any revolution bodies with vertical axes. The comparison with Malenica and Molin's results is made on surge force on a uniform cylinder, and comparison with experimental results is made on third order surge force, heave force and pitch moment on a truncated cylinder. More numerical computations are carried out for third order forces and moments on a uniform cylinder, truncated cylinders and a hemisphere.     

Analytical layer-element solution to axisymmetric dynamic response of transversely isotropic multilayered half-space

Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the corresponding algebraic operations and the Hankel transform, the analytical layer-elements of a finite layer and a half-space are obtained in the transformed domain. According to the continuity conditions between adjacent layers, the global stiffness matrix equation is obtained by assembling the analytical layer-element of each single layer. The solutions in the transformed domain are acquired by introducing the boundary conditions into the global stiffness matrix equation, and thus, the corresponding solutions in frequency domain are achieved by taking the inversion of Hankel transform. Finally, some numerical examples are given to illustrate the accuracy of the proposed method, and to study the influence of properties and the frequency of excitation on the dynamic response of the medium.     

Linear stability analysis for the differentially heated rotating annulus

We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier–Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations.

A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis.

A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.