Asymptotics of Flexural Waves in Isotropic Elastic Plates

1997 ◽  
Vol 64 (2) ◽  
pp. 336-342 ◽  
Author(s):  
N. A. Losin
Keyword(s):  
Material Parameter ◽  
Velocity Dispersion ◽  
Three Dimensional ◽  
Analytical Form ◽  
Short Wave ◽  
Frequency Equation ◽  
Elastic Plates ◽  
Flexural Waves ◽  
Explicit Formulas ◽  
Longitudinal Waves

The long and short-wave asymptotics of order O(k6h6) for the flexural vibrations of an infinite, isotropic, elastic plate are studied. The differential equation for the flexural motion is derived from the system of three-dimensional dynamic equations of linear elasticity. All coefficients of the differential operator are presented as explicit functions of the material parameter γ = cs2/cL2, the ratio of the velocities squared of the flexural (shear) and extensional (longitudinal) waves. Relatively simple frequency and velocity dispersion equations for the flexural waves are deduced in analytical form from the three-dimensional analog of Rayleigh-Lamb frequency equation for plates. The explicit formulas for the group velocity are also presented. Variations of the velocity and frequency spectrums depending on Poisson’s ratio are illustrated graphically. The results are discussed and compared to those obtained and summarized by R. D. Mindlin (1951, 1960), Tolstoy and Usdin (1953, 1957), and Achenbach (1973).

Asymptotics of Extensional Waves in Isotropic Elastic Plates

1998 ◽  
Vol 65 (4) ◽  
pp. 1042-1047 ◽  
Author(s):  
N. A. Losin
Keyword(s):  
Wave Motion ◽  
Material Parameter ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Analytical Form ◽  
Short Wave ◽  
Asymptotic Model ◽  
Correction Factors ◽  
Elastic Plates ◽  
Frequency Spectra

The long and short-wave asymptotics of order O(η6),η=kh, for free extensional vibrations of an infinite isotropic elastic plate are studied. The asymptotic model for flexural and extensional wave motion applicable for both long and short-wave approximations and for any materials is developed. The velocity and frequency dispersion relations for extensional waves are derived in analytical form from the system of three-dimensional dynamic equations of linear elasticity. All dispersion equations and the group velocity formula are presented as explicit functions in material parameter γ=cs2/cL2 (the ratio of the velocities squared of the flexural and extensional waves) without any correction factors as in the Reissner-Mindlin theory. Variations of the velocity and frequency spectra depending on Poisson’s ratio ν are illustrated graphically. The results are discussed and compared to those obtained and summarized by Mindlin (1960), Mindlin and Medick (1959), Tolstoy and Usdin (1953, 1957), Achenbach (1973), and Graff (1991).

On the Equivalence of Dispersion Relations Resulting from Rayleigh-Lamb Frequency Equation and the Operator Plate Model

2001 ◽  
Vol 123 (4) ◽  
pp. 417-420 ◽  
Author(s):  
Nikolay A. Losin
Keyword(s):  
Velocity Dispersion ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Dispersion Relations ◽  
Plate Model ◽  
Elastic Plates ◽  
Flexural Waves ◽  
Frequency Spectra ◽  
Infinite Power ◽  
Antisymmetric Vibrations

The Rayleigh-Lamb frequency equations for the free vibrations of an infinite isotropic elastic plate are expanded into the infinite power series and reduced to the polynomial frequency and velocity dispersion relations. The latter are compared to those of the operator plate model developed in [Losin, N. A., 1997, “Asymptotics of Flexural Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 64, No. 2, pp. 336–342; Losin, N. A., 1998, “Asymptotics of Extensional Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 65, No. 4, pp. 1042–1047] for both symmetric and antisymmetric vibrations. As a result of comparative analysis, the equivalence of the corresponding dispersion polynomials is established. The frequency spectra, generated by Rayleigh-Lamb equations, are illustrated graphically and briefly discussed with reference to those published in [Potter, D. S., and Leedham, C. D., 1967, “Normalized Numerical Solution for Rayleigh’s Frequency Equation,” J. Acoust. Soc. Am., 41, No. 1, pp. 148–153].

On the Nonlocal Theory of Wave Propagation in Elastic Plates

1984 ◽  
Vol 51 (3) ◽  
pp. 608-613 ◽  
Author(s):  
J. L. Nowinski
Keyword(s):  
Lattice Dynamics ◽  
Wave Speed ◽  
Nonlocal Theory ◽  
Short Wave ◽  
Elastic Plates ◽  
Field Equations ◽  
Atomic Lattice ◽  
Longitudinal Waves ◽  
Standard Manner ◽  
The Fourier Transform

Propagation of longitudinal waves in isotropic homogeneous elastic plates is studied in the context of the linear theory of nonlocal continuum mechanics. To determine the nonlocal moduli, the dispersion equation obtained for the plane longitudinal waves in an infinite medium is matched with the parallel equation derived in the theory of atomic lattice dynamics. Using the integroalgebraic representation of the stress tensor and the Fourier transform, the system of two coupled differential field equations is solved in the standard manner giving the frequency equations for the symmetric and antisymmetric wave modes. It is found that the short wave speed in the Poisson medium differs by about 13 percent from the speed established in the classical theory. A numerical example is given.

Dispersion of Flexural Waves in an Elastic, Circular Cylinder

1960 ◽  
Vol 27 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Yih-Hsing Pao ◽  
R. D. Mindlin
Keyword(s):  
Circular Cylinder ◽  
Phase Velocity ◽  
Propagation Constant ◽  
Direct Determination ◽  
Short Wave ◽  
Frequency Equation ◽  
Flexural Waves ◽  
Asymptotic Equations ◽  
Frequency Phase

In this paper it is shown how the branches of Pochhammer’s frequency equation for flexural waves in a circular cylinder may be constructed approximately with the aid of a grid of simpler curves and asymptotic equations for long and short wave lengths. With very little computation, in comparison with that required in the direct determination of the roots of Pochhammer’s equation, a qualitative view is obtained of the relations between frequency, phase velocity, group velocity, and propagation constant, for any branch, as well as some information as to the shapes of the modes.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

2019 ◽  
Vol 968 ◽  
pp. 496-510
Author(s):  
Anatoly Grigorievich Zelensky
Keyword(s):  
Mathematical Theory ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Theory Of Elasticity ◽  
Elastic Plates ◽  
Dimensional Problem ◽  
Boundary Problems ◽  
Complex Boundary ◽  
Plates And Shells ◽  
Basic Equations

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

Kinematical evolution of Globular Clusters

2019 ◽  
Vol 14 (S351) ◽  
pp. 524-527
Author(s):  
Maria A. Tiongco ◽  
Enrico Vesperini ◽  
Anna Lisa Varri
Keyword(s):  
Angular Momentum ◽  
Structural Properties ◽  
Globular Clusters ◽  
Velocity Dispersion ◽  
Three Dimensional ◽  
Stellar Populations ◽  
Velocity Space ◽  
Fundamental Understanding

AbstractWe present several results of the study of the evolution of globular clusters’ internal kinematics, as driven by two-body relaxation and the interplay between internal angular momentum and the external Galactic tidal field. Via a large suite of N-body simulations, we explored the three-dimensional velocity space of tidally perturbed clusters, by characterizing their degree of velocity dispersion anisotropy and their rotational properties. These studies have shown that a cluster’s kinematical properties contain distinct imprints of the cluster’s initial structural properties, dynamical history, and tidal environment. Building on this fundamental understanding, we then studied the dynamics of multiple stellar populations in globular clusters, with attention to the largely unexplored role of angular momentum.

Thermal Stresses in Moderately Thick Elastic Plates

1959 ◽  
Vol 26 (3) ◽  
pp. 432-436
Author(s):  
B. E. Gatewood
Keyword(s):  
Temperature Gradient ◽  
Plane Stress ◽  
Thermal Stresses ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Elastic Plates ◽  
Plate Length ◽  
Moderately Thick Plates ◽  
Thick Elastic Plates ◽  
Thickness Variable

Abstract The three-dimensional stresses in the plate are investigated without using the plane-stress or plane-strain assumptions, the thickness of the plate being limited so that the normal stress in the thickness direction can be taken as a polynomial in the thickness variable. The temperature is taken as a polynomial in the thickness variable but with relatively large, though restricted, gradients with respect to the co-ordinates of the plane of the plate. For the case of the temperature constant in thickness variable, the stresses in the plane of the plate are presented as the plane-stress solution plus correcting terms due to the plate thickness, where the correcting terms involve the product of the temperature gradient and the ratio of the plate thickness to the plate length in the direction of the temperature gradient. In many cases the corrections are small even for moderately thick plates.

Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization

2021 ◽  
pp. 1-35
Author(s):  
Chunlin Wu ◽  
Liangliang Zhang ◽  
Huiming Yin
Keyword(s):  
Taylor Series ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Elastic Field ◽  
Analytical Form ◽  
Tetrahedral Elements ◽  
Equivalent Inclusion Method ◽  
Equivalent Inclusion ◽  
Eshelby’S Tensor ◽  
Domain Discretization

Abstract The paper extends the recent work (JAM, 88, 061002, 2021) of the Eshelby's tensors for polynomial eigenstrains from a two dimensional (2D) to three dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity by using Eshelby's equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby's tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the 𝐶0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of the Eshelby's tensor, the elastic analysis is robust, stable and efficient.

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