Statistical-Thermodynamic Study of Nonequilibrium Phenomena in Three-Dimensional Anharmonic Crystal Lattices: I. Microscopic Basic Equations

2003 ◽  
Vol 72 (3) ◽  
pp. 545-550 ◽  
Author(s):  
Masaru Sugiyama ◽  
Kuniaki Goto
Keyword(s):  
Three Dimensional ◽  
Thermodynamic Study ◽  
Statistical Thermodynamic ◽  
Crystal Lattices ◽  
Anharmonic Crystal ◽  
Nonequilibrium Phenomena ◽  
Basic Equations

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.

scholarly journals Basic Solutions of Three Dimensional Elasticity

2021 ◽  
Vol 236 ◽  
pp. 05039
Author(s):  
Wx Zhang
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Practical Calculation ◽  
Basic Solution ◽  
Final Solution ◽  
Theoretical Derivation ◽  
Basic Solutions ◽  
Special Solutions ◽  
Basic Equations ◽  
Three Dimensional Elasticity

Elastic calculation method is an important research content of computational mechanics. The problems of elasticity include basic equations and boundary conditions. Therefore, the final solution consists of the general solutions of the basic equations and the special solutions satisfying the boundary conditions. Numerical method is often used in practical calculation, but the analytical solution is also an important subject for researchers. In this paper, the basic solution of three-dimensional elastic materials is given by theoretical derivation.

Band Structure Theory for Extended Systems

2020 ◽  
pp. 246-278
Author(s):  
Jochen Autschbach
Keyword(s):  
Electronic Structure ◽  
Band Structure ◽  
Electron Gas ◽  
Three Dimensional ◽  
Structure Theory ◽  
Crystal Lattices ◽  
Hückel Theory ◽  
Periodic Linear ◽  
Crystal Orbitals ◽  
Dimensional Crystal

The electronic structure of infinite periodic systems (crystals) is treated with band structure theory, replacing molecular orbitals by crystal orbitals. The chapter starts out by introducing the electron gas and definitions of the Fermi momentum, the Fermi energy, and the density of states (DOS). A periodic linear combination of atomic orbitals (LCAO) type treatment of an infinite periodic system is facilitated by the construction of Bloch functions. The notions of energy band and band gap are discussed with band structure concepts, using the approximations made in Huckel theory (chapter 12). One, two, and three-dimensional crystal lattices and the associated reciprocal lattices are introduced. The band structures of sodium metal, boron nitride, silicon, and graphite, are discussed as examples of metals, insulators, semi-conductors, and semi-metals, respectively. The chapter concludes with a brief discussion of the projected DOS and measures to determine bonding or antibonding interactions between atoms in a crystal.

Chaotic and regular behavior in two-dimensional anharmonic crystal lattices

1993 ◽  
Vol 48 (21) ◽  
pp. 15732-15739 ◽  
Author(s):  
M. L. A. Nip ◽  
J. A. Tuszyn´ski ◽  
Z. W. Gortel ◽  
T. A. Riauka
Keyword(s):  
Two Dimensional ◽  
Crystal Lattices ◽  
Anharmonic Crystal ◽  
Regular Behavior

Application of a Vortex Method to Fluid Dynamics in Sports Science

2007 ◽  
Author(s):  
Kyoji Kamemoto ◽  
Akira Ojima
Keyword(s):  
Fluid Dynamics ◽  
Three Dimensional ◽  
Vortex Method ◽  
Mathematical Treatment ◽  
Sports Science ◽  
Lagrangian Simulation ◽  
Eddy Simulation ◽  
Flow Features ◽  
Vortex Element ◽  
Basic Equations

This paper describes a pioneering work of practical application of an advanced vortex method in the field of fluid dynamics in sports science. The vortex method developed by the present authors is one of vortex element methods based on the Biot-Savart law, and it is known that the method provides a Lagrangian simulation of unsteady and vortical flows. In this study, in order to examine the applicability of the vortex method, three-dimensional, complex and unsteady flows around an isolated 100 m runner and a ski-jumper were calculated. Basic equations and mathematical treatment of the method are explained in this paper, and calculation conditions and panel data of deforming configuration of the athletes are described. As results of the present study, vortical and unsteady flow features around a runner and a ski-jumper are understood, and unsteady variation of aerodynamic forces corresponding to deformation of body configuration due to athletic motion are calculated. And, it is confirmed that the advanced vortex element method is a promising way to a grid-free Lagrangian large eddy simulation of unsteady and complex flows around dynamic bodies of athletes.

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