Theory of dynamic processes in mechanical systems and materials of regular structure

Abstract

The theory of vibrations and waves in natural and synthesized materials of regular structure is analyzed. Models based on different averaging and continualization methods are outlined. Emphasis is on periodically inhomogeneous structures. The exact solutions are obtained and analyzed using the closed-form solution of infinite algebraic systems, representing equations in Hamiltonian operator form and solving them based on the theory of differential equations with periodic coefficients, mode selection rule, and methods of drawing wave shapes at limit and arbitrary frequencies Keywords: vibrations and waves, averaging, structural model, exact solutions, Hamiltonian formalism, mode selection rule, wave shape 1. Introduction. General Characteristic. Natural and artificial regular structures are widely used and there is an extensive scientific and technical literature on their properties. A qualified analysis of all aspects of achievements in this field is an unreal task. The present review is a logical continuation and generalization of the reviews [64,65]. The goal is to describe the most important aspects in the development of the theory of vibrations and waves in regular structures, at least in the author's field of research.It is of fundamental and applied importance to develop methods for analysis of and to analyze the dynamic processes in regular structures. To analyze processes in inhomogeneous structures, two types of models are developed and used: continuous and structural (piecewise-inhomogeneous).Continuum theories model an inhomogeneous regular (or quasiregular) structure by a homogeneous system with certain (effective) properties. The transition from an inhomogeneous structure to a continuous model is called averaging. Averaging methods are best developed at the physical level of rigor and were first used in solid-state physics. Voigt determined [88] the elastic moduli c ijmn of polycrystals by averaging the properties of crystallites. Reiss proposed [88] to average the compliance moduli of s ijmn . Voigt proceeded from the assumption that strains are homogeneous, Reiss from the assumption that so are stresses. Voigt's averaging method was later named the mixing rule. Later, Hill [88] showed that Voigt's and Reiss' methods give upper-and lower-bound estimates, respectively, for the elastic characteristics of a polycrystal. The rapid development of methods for determining the average (effective) elastic characteristics of structurally inhomogeneous materials is characteristic of composite mechanics. The results obtained in this field for layered, fibrous, particulate, and other materials are elucidated in the monographs [35, 37, etc.].The continuum theory of the first order or the theory of effective moduli (the terms "first-order theory" and "second-order theory" were introduced in [62]), which is the best developed and has a clear physical interpretation, is predominant among the continuum theories of composites. This theory, however, can describe only long-wave processes, which do not display the geomet...

Application of the hamilton formalism in a timoshenko-type theory of vibrations of plates

“…In many successive works analyzed in survey works [2,[7][8][9] and other publications, the Hamilton formalism in a form developed in [6] was extended to the equations and the problems of elasticity theory, electroelasticity, and magnetoelasticity. Due to these achievements, the properties of characteristic equations and the general structure of solutions of wave problems for periodic structures were revealed.…”