S.B. Dong

Single-mode optical waveguides

An efficient and powerful technique has been developed to treat the problem of wave propagation along arbitrarily shaped single-mode dielectric waveguides with inhomogeneous index variations in the cross-sectional plane. This technique is based on a modified finite-element method. Illustrative examples were given for the following guides: (a) the triangular fiber guide; (b) the elliptical fiber guide; (c) the single material fiber guide; (d) the rectangular fiber guide; guide; (g) the optical stripline guide.

On a hierarchy of conforming timoshenko beam elements

Abstract Presented herein is a hierarchy of beam elements which include the effects of transverse shear deformation and rotary inertia. Interdependent interpolations are employed for the deflection and bending rotation fields to obtain a family of virgin elements. By imposing a continuous shear constraint condition on each member of the virgin element family, a series of constrained elements can be generated. Stiffness and consistent mass matrices and consistent load vectors are formulated by Gaussian quadrature using formulas for exact integration. These elements exhibit excellent modeling capabilities and suffer no deficiencies in the range of small thickness/length ratios. One important feature of certain constrained elements is that it is possible to use them to explain the essence of the “reduced integration” stiffness matrices that have appeared in the literature.

Vibrations and waves in laminated orthotropic circular cylinders

Abstract A method is presented for studying the free vibrations of a circular cylinder composed of an arbitrary number of bonded elastic, cylindrically orthotropic layers. The analysis is carried out within the framework of the complete three-dimensional theory of elasticity. In this paper, all displacements are taken in the form of trigonometric functions in both the circumferential angle and axial variables, while their radial behavior is modeled by an approximate displacement field characterized by a discrete number of coordinates. The discretized model leads to an algebraic eigenvalue problem which is solved by an efficient numerical technique. The solutions give the natural frequencies and associated modal patterns, the lowest ten of which are determined. Stresses are directly calculated from the eigenvectors to give complete physical information. Several homogeneous isotropic cylinders are studied to ascertain the accuracy and effectiveness of the method and an orthotropic cylinder is compared with known approximate results. Additional examples of homogeneous and laminated orthotropic cylinders are given to furnish some insight into their physical behavior.

Free Vibration of Laminated Orthotropic Cylindrical Shells

Vibration characteristics of thin laminated orthotropic cylindrical shells are investigated. The theory, based on the Kirchhoffean hypothesis regarding deformation, can accommodate shells composed of an arbitrary number of bonded layers, each with a different thickness and different elastic orthotropic properties. Donnell‐type equations expressed in terms of the reference surface displacements are employed, and the effects of initial extensional forces in the shell are included. The frequency equation for a simply supported cylinder is derived. Next, an iterative procedure for determining the natural frequencies of a shell under an arbitrary set of homogeneous boundary conditions is described in detail. Specialization of present results to laminated orthotropic plates is indicated.

Propagating waves and edge vibrations in anisotropic composite cylinders

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f(r) exp i(ez + nθ + ωt) with the radial dependence f(r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number e or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of e. For e specified as real, an eigenvalue problem results for the natural frequencies ωi for waves propagating along the z-direction of a cylinder of infinite extent. When e is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding e as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for e which may be real, purely imaginary or complex. Complex es correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields e as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venants principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.

Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides

Using the finite element technique, a numerical method is devloped so that one may obtain the propagation characteristics of optical waves along guiding structures whose cores may be of arbitrary cross‐sectional shape and whose material media may be inhomogeneous in more than one transverse direction. Several specific examples are given and the results are compared with those obtained by other exact or approximate methods. Very close agreement was found. The method developed here can easily be applied to many important problems dealing with practical optical fiber or integrated optical waveguides whose cross‐sectional index of refraction distribution may be quite arbitrary.

Wave reflection from the free end of a cylinder with an arbitrary cross-section

Elastic wave scattering at the free end of a cylinder due to an incident monochromatic wave is investigated. The cross-section may have an arbitrary geometry with any number of distinct elastic rectilinear anisotropic materials comprising its planar profile. The governing equations are based on a semi-analytical finite element method in which the cross-sectional behavior is modeled by general two-dimensional finite elements with the axial dependence and time left unspecified at the outset. First, all the modal data for the cylinder are established. Two eigenproblems are posed for this purpose, that are obtained by inserting a wavelike solution form into the governing equations. These eigenproblems allow all propagating waves and end modes for the cylinder to be determined. Propagating modes are traveling waves with energy transport capabilities, while the end modes are standing vibrations which, in contrast, do not transport any energy into the interior of the cylinder. These eigendata are the basis for representing the wave reflection phenomenon at the free end. The amplitudes of the traveling waves and end modes that satisfy traction-free end conditions may be determined by least-squares minimization or by a virtual work method. Four cylinders with different cross-sectional geometries were considered to illustrate the analysis procedure and reveal some physical insight into the frequency dependent wave reflection phenomena in them.

Edge Effects in Laminated Composite Plates

The attenuation of self-equilibrated edge stress states into the interior of a laminated plate composed of an arbitrary number of bonded, elastic, anisotropic layers is investigated in the context of Saint-Venant’s principle using the exponential decay results of Toupin, Knowles, and Horgan. To model the plate’s behavior, a semianalytical method is used with finite element interpolations over the thickness and exponential decay into the plate’s interior. The formulation leads to a second-order algebraic eigensystem whose eigenvalues are the characteristic inverse decay lengths, and corresponding right eigenvectors depict the displacement distributions of self-equilibrated traction states. Orthogonality relations between these right and left eigenvectors of the adjoint system are established. An eigenvector expansion for representing arbitrary self-equilibrated edge tractions is then presented. This approach is useful in revealing the interlaminar effects and their decay rates in a laminated composite plate under plane strain. Two examples are provided where the interlaminar phenomena due to eigenstates of self-equilibrated edge stress are illustrated.

On Saint-Venant’s Problem for an Inhomogeneous, Anisotropic Cylinder—Part I: Methodology for Saint-Venant Solutions

In this paper, the first in a series of three, a procedure based on semi-analytical finite elements is presented for constructing Saint-Venant solutions for extension, bending, torsion, and flexure of a prismatic cylinder with inhomogeneous, anisotropic cross-sectional properties. Extension-bending-torsion involve stress fields independent of the axial coordinate and their displacements may be decomposed into two distinct parts which are called the primal field and the cross-sectional warpages herein. The primal field embodies the essence of the kinematic hypotheses of elementary bar and beam theories and that for unrestrained torsion. The cross-sectional warpages are independent of the axial coordinate and they are determined by testing the variationally derived finite element displacement equations of equilibrium with the primal field. For flexure, a restricted three-dimensional stress field is in effect where the stress can vary at most linearly along the axis. Integrating the displacement field based for extension-bending-torsion gives that for the flexure problem. The cross-sectional warpages for flexure are determined by testing the displacement equations of equilibrium with this displacement field. In the next paper, the cross-sectional properties such as the weighted-average centroid, center of twist and shear center are defined based on the Saint-Venant solutions established in the present paper and numerical examples are given. In the third paper, end effects or the quantification of Saint-Venants principle for the inhomogeneous, anisotropic cylinder is considered.

Saint-Venant solutions for prismatic anisotropic beams

Abstract An analytical model is presented for determining the displacement and stress distributions of the Saint-Venant extension, bending, torsion and flexure problems for a homogeneous prismatic beam of arbitrary section and rectilinear anisotropy. The determination of the complete displacement field requires solving a coupled two-dimensional boundary value problem for the local in-plane deformations and warping out of the section plane. The principle of minimum potential energy is applied to a discretized representation of the cross-section (Ritz method) to calculate solutions to this problem. The behavior of an anisolropic beam is studied in detail using the resulting displacement and stress solutions, where definitions are presented for the shear center, center of twist, torsion constant and a new geometric parameter: the line of extension bending centers. Two sets of numerical results are presented to illustrate how section geometry, beam length and material properties affect the behavior of a homogeneous anisotropic cantilever beam.