Elastic Waves Originating at the Surface of a Spherical Opening in Nonhomogeneous Isotropic Media

1972 ◽  
Vol 50 (19) ◽  
pp. 2359-2367 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Formal Method ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Radial Distance ◽  
Dimensional Case ◽  
Linearized Equations ◽  
General Law ◽  
Isotropic Media ◽  
Displacement Potentials ◽  
The One

This paper is concerned with the propagation of three-dimensional waves in nonhomogeneous isotropic elastic media. Herein we develop a formal method for the solution of the linearized equations of elasticity for nonhomogeneous media whose properties depend continuously on the radial distance from a point. The method is developed for the equations of motion formulated in terms of displacements, velocities, and stresses. It is based on a method developed by Karal and Keller for the asymptotic solution of the linearized equations of elasticity formulated in terms of displacements and displacement potentials, and is a generalization of results obtained for the one-dimensional case by Cooper and for the two-dimensional case by the author.The method is applied to the propagation of dilatational waves generated by a radial input velocity in the form of a unit function applied to the surface of the hole in a medium whose properties depend on the radial distance from the center of the hole according to a general law. Also, we apply the method to the propagation of purely distortional waves generated by a stress applied at the hole and corresponding to a tangential impulse symmetrical about a diameter of the hole.

On the theory of Langmuir solitons

1977 ◽  
Vol 17 (2) ◽  
pp. 153-170 ◽  
Author(s):  
J. Gibbons ◽  
S. G. Thornhill ◽  
M. J. Wardrop ◽  
D. Ter Haar
Keyword(s):  
Conservation Laws ◽  
Numerical Experiments ◽  
Lagrangian Density ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Turbulent Plasma ◽  
Dimensional Case ◽  
One Dimensional ◽  
The One ◽  
Langmuir Solitons

We find a Lagrangian density from which the equations of motion for the Lang-muir solitons follow in the usual way. We show how this Lagrangian leads to the usual conservation laws. For the one-dimensional case we discuss how a consideration of these conservation laws can help us to understand some of the results obtained in numerical experiments on the behaviour of a strongly turbulent plasma. We point out that the situation in the three-dimensional case may be fundamentally different, and we discuss near-sonic perturbations and Karpman's treatment of these.

Constitutive Inequalities for the Damage of Elastic-Brittle Materials

2011 ◽  
Vol 675-677 ◽  
pp. 891-899
Author(s):  
Qi Chang He ◽  
J.Z. Zhou
Keyword(s):  
Three Dimensional ◽  
Brittle Materials ◽  
Dimensional Case ◽  
Continuum Damage ◽  
One Dimensional ◽  
Damage Models ◽  
Constitutive Inequalities ◽  
The One

Starting from the requirement that the principle of determinism be satisfied, two constitutive inequalities are derived for one-dimensional strain- and stress-based continuum damage models. The one-dimensional constitutive inequality corresponding to the strain-based formulation turns out to be much less restrictive than the one associated to the stress-based formulation and is extended to the three-dimensional case. This extension gives a general constitutive inequality for the damage of elastic-brittle materials.

scholarly journals Damage Mechanics of Carbon Nanotubes

2020 ◽  
Vol 03 (02) ◽  
pp. 1-1
Author(s):  
George Z. Voyiadjis ◽  
◽  
Peter I. Kattan ◽  
Keyword(s):  
Carbon Nanotubes ◽  
Damage Mechanics ◽  
Three Dimensional ◽  
Damage Variable ◽  
Dimensional Case ◽  
Third Order ◽  
Cross Sectional ◽  
Energy Equivalence ◽  
The One

A robust mathematical method for the characterization of damage in carbon nanotubes is presented the presentation here is limited to elasticity. In this regard, the second and third order elastic stiffnesses are employed. All this is based on damage mechanics. The hypotheses of elastic strain equivalence and elastic energy equivalence are utilized. A new damage variable is proposed that is defined in terms of the surface area. This is in contrast to the classical damage variable which is defined in terms of the cross-sectional area. In the presentation, both the one-dimensional case (scalars) and the three-dimensional case (tensors) are illustrated.

scholarly journals Stress field formulation of linear electro-magneto-elastic materials

2019 ◽  
Vol 24 (12) ◽  
pp. 3806-3822
Author(s):  
A Amiri-Hezaveh ◽  
P Karimi ◽  
M Ostoja-Starzewski
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Governing Equations ◽  
Field Equations ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Sufficient Set

A stress-based approach to the analysis of linear electro-magneto-elastic materials is proposed. Firstly, field equations for linear electro-magneto-elastic solids are given in detail. Next, as a counterpart of coupled governing equations in terms of the displacement field, generalized stress equations of motion for the analysis of three-dimensional (3D) problems Are obtained – they supply a more convenient basis when mechanical boundary conditions are entirely tractions. Then, a sufficient set of conditions for the corresponding solution of generalized stress equations of motion to be unique are detailed in a uniqueness theorem. A numerical passage to obtain the solution of such equations is then given by generalizing a reciprocity theorem in terms of stress for such materials. Finally, as particular cases of the general 3D form, the stress equations of motion for planar problems (plane strain and Generalized plane stress) for transversely isotropic media are formulated.

GENERALIZED FIRST-FIT ALGORITHMS IN TWO AND THREE DIMENSIONS

1990 ◽  
Vol 01 (02) ◽  
pp. 131-150 ◽  
Author(s):  
KEQIN LI ◽  
KAM-HOI CHENG
Keyword(s):  
Bin Packing ◽  
Three Dimensional ◽  
Packing Problem ◽  
Three Dimensions ◽  
Performance Ratio ◽  
Dimensional Case ◽  
Performance Bound ◽  
Worst Case ◽  
Packing Problems ◽  
The One

We investigate the two and three dimensional bin packing problems, i.e., packing a list of rectangles (boxes) into unit square (cube) bins so that the number of bins used is a minimum. A simple on-line packing algorithm for the one dimensional bin packing problem, the First-Fit algorithm, is generalized to two and three dimensions. We first give an algorithm for the two dimensional case and show that its asymptotic worse case performance ratio is [Formula: see text]. The algorithm is then generalized to the three dimensional case and its performance ratio [Formula: see text]. The second algorithm takes a parameter and we prove that by choosing the parameter properly, it has an asymptotic worst case performance bound which can be made as close as desired to 1.72=2.89 and 1.73=4.913 respectively in two and three dimensions.

scholarly journals The Lagrangian and Hamiltonian for the Two-Dimensional Mathews-Lakshmanan Oscillator

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Wang Guangbao ◽  
Ding Guangtao
Keyword(s):  
Lagrange Function ◽  
Three Dimensional ◽  
Hamiltonian Function ◽  
First Integral ◽  
Dimensional Case ◽  
Analytical Mechanics ◽  
Two Dimensional ◽  
Rectangular Coordinates ◽  
Polar Coordinate ◽  
The One

The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata

1997 ◽  
Vol 07 (07) ◽  
pp. 1451-1496 ◽  
Author(s):  
André Barbé
Keyword(s):  
Cellular Automata ◽  
Three Dimensional ◽  
Coarse Graining ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Complex Order ◽  
Simple Order ◽  
Problems And Solutions ◽  
The One

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).

Cylindrical and Spherical Shear Waves in Nonhomogeneous Isotropic Viscoelastic Media

1973 ◽  
Vol 51 (10) ◽  
pp. 1091-1097 ◽  
Author(s):  
T. Bryant Moodie
Keyword(s):  
Equations Of Motion ◽  
Shear Waves ◽  
Three Dimensions ◽  
Radial Coordinate ◽  
Shear Stresses ◽  
Progressive Wave ◽  
Elastic Solids ◽  
Linearized Equations ◽  
Viscoelastic Media ◽  
Isotropic Media

This paper is concerned with the propagation of viscoelastic shear waves in nonhomogeneous isotropic media. Herein we develop formal methods of solving the linearized equations of viscoelastodynamics in two and three dimensions for nonhomogeneous Maxwell solids whose properties depend continuously on a single radial coordinate. These methods are developed for the linearized equations of motion formulated in terms of shear stresses, and are based on Cooper's and Reiss' extension to linear homogeneous viscoelastic media of the Karal–Keller technique. Shearing stesses are applied to the boundaries of cylindrical and spherical openings in the viscoelastic media, and formal asymptotic wave front expansions of the solutions are obtained. In both cases a modulated progressive wave that propagates with variable velocity is obtained. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends only on the modulus of rigidity. When the viscosity parameter in our Maxwell element tends to infinity, the results reduce to the known results for nonhomogeneous elastic solids.

CORRELATION AND SPECTRAL PROPERTIES OF MULTIDIMENSIONAL THUE–MORSE SEQUENCES

2007 ◽  
Vol 17 (04) ◽  
pp. 1265-1303 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Spectral Properties ◽  
Three Dimensional ◽  
Number System ◽  
Dimensional Case ◽  
Binary Number ◽  
Singular Continuous Spectrum ◽  
One Dimensional ◽  
Number Systems ◽  
Higher Dimensional ◽  
The One

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue–Morse sequence on ℕ. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in ℤm instead of in ℕ. It is shown that the corresponding ±1-valued Thue–Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.

3-D Exact Free Vibration Analysis of Transversely Isotropic Cylindrical Panels

1998 ◽  
Vol 120 (4) ◽  
pp. 982-986 ◽  
Author(s):  
Chen Weiqiu ◽  
Cai Jinbiao ◽  
Ye Guiru ◽  
Ding Haojiang
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Transversely Isotropic ◽  
Function Solution ◽  
Transversely Isotropic Media ◽  
Cylindrical Panels ◽  
Isotropic Media ◽  
Three Dimensional Elasticity

This paper presents an exact analysis of the free vibration of simply supported, transversely isotropic cylindrical panels. Based on the three dimensional elasticity for transversely isotropic media, three displacement functions are introduced so that the equations of motion are uncoupled and simplified. After expanding these functions with orthogonal series, the equations of free vibration problems are further reduced to three second order ordinary differential equations. A modified Bessel function solution with complex arguments is then directly used for the case of complex eigenvalues, which, to the authors’ knowledge, has never been reported before. To clarify the correctness and effectiveness of the developed method, numerical examples are presented and compared to the results of existent papers.

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