Editorial

2 pages, 169 KiB

Open AccessEditorial

Dynamical Processes in Heterogeneous and Discrete Media

by Igor V. Andrianov

Symmetry 2023, 15(1), 101; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15010101 - 30 Dec 2022

Viewed by 541

There is no need to talk about the wide spreading of dynamic processes in nature, or about the infrastructure created by man [...] Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

Research

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9 pages, 331 KiB

Open AccessArticle

Decay Conditions for Antiplane Shear of a High-Contrast Multi-Layered Semi-Infinite Elastic Strip

by Ludmila Prikazchikova

Symmetry 2022, 14(8), 1697; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081697 - 16 Aug 2022

Cited by 5 | Viewed by 919

Abstract

The antiplane shear of a semi-infinite multi-layered elastic strip with traction free faces and edges subject to prescribed stress is studied. A high contrast is assumed in the stiffnesses of two types of homogeneous isotropic layers. Explicit conditions on the edge load are [...] Read more.

The antiplane shear of a semi-infinite multi-layered elastic strip with traction free faces and edges subject to prescribed stress is studied. A high contrast is assumed in the stiffnesses of two types of homogeneous isotropic layers. Explicit conditions on the edge load are derived, ensuring the decay of stress components at the distance of order strip thickness. One of these conditions corresponds to the canonical Saint-Venant’s principle, manifesting the self-equilibrium of the load. The rest of the decay conditions consider the presence of high contrast and are of an asymptotic nature, in contrast to the exact former condition. The number of asymptotic conditions is the same as that of soft layers. An example of the implementation of the proposed decay conditions for calculating the solution for the interior (outside of a boundary layer zone) domain of a three-layered semi-strip, considering geometric asymmetry, is presented. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

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18 pages, 4374 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Analysis of Drill-String Nonlinear Dynamics Using the Lumped-Parameter Method

by Lelya A. Khajiyeva, Igor V. Andrianov, Yuliya F. Sabirova and Askar K. Kudaibergenov

Symmetry 2022, 14(7), 1495; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071495 - 21 Jul 2022

Cited by 5 | Viewed by 1600

Abstract

This work aims at studying the nonlinear dynamics of drill strings using the lumped-parameter method (LPM). The study is based on the good consistency of the results of the test problem where the model of the longitudinal vibrations of a horizontal drill string [...] Read more.

This work aims at studying the nonlinear dynamics of drill strings using the lumped-parameter method (LPM). The study is based on the good consistency of the results of the test problem where the model of the longitudinal vibrations of a horizontal drill string with a static compressive load at the left end is considered. In this paper, this method is applied to discretize linear and nonlinear models of the lateral vibrations of a vertical drill string under the effect of a supersonic gas flow. The obtained results are verified with the previously published data. The optimal number of the drill-string partitions is determined using the developed application, which allows us to estimate the accuracy of the loaded data. The numerical solution of the model is obtained using the fourth-order Runge–Kutta method. The optimization of the numerical algorithm using parallel-programming tools is carried out, and the efficiency of the method is analyzed. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

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13 pages, 491 KiB

Open AccessArticle

Analytically Solvable Models and Physically Realizable Solutions to Some Problems in Nonlinear Wave Dynamics of Cylindrical Shells

by Andrey Bochkarev, Aleksandr Zemlyanukhin, Vladimir Erofeev and Aleksandr Ratushny

Symmetry 2021, 13(11), 2227; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112227 - 21 Nov 2021

Cited by 2 | Viewed by 1208

Abstract

The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. [...] Read more.

The axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is herein studied. With the help of asymptotic integration, two analytically solvable models were obtained that have no physically realizable solitary wave solutions. The possibility for the real existence of exact solutions, in the form of traveling periodic waves of the nonlinear inhomogeneous Klein–Gordon equation, was established. Two cases were identified, which enabled the development of the modulation instability of periodic traveling waves: (1) a shell preliminarily compressed along a generatrix, surrounded by an elastic medium with hard nonlinearity, and (2) a preliminarily stretched shell interacting with an elastic medium with soft nonlinearity. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

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9 pages, 262 KiB

Open AccessArticle

New Oscillation Results of Even-Order Emden–Fowler Neutral Differential Equations

by Saeed Althubiti, Ibtisam Aldawish, Jan Awrejcewicz and Omar Bazighifan

Symmetry 2021, 13(11), 2177; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112177 - 15 Nov 2021

Cited by 4 | Viewed by 1130

Abstract

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term [...] Read more.

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term

{(r (ı) {({(y (ı) + m (ı) y (g (ı)))}^{(n - 1)})}^{γ})}^{'} + \sum_{i = 1}^{j} q_{i} (ı) y^{γ} (μ_{i} (ı)) = 0

. We use Riccati transformation and the comparison with first-order differential inequalities to obtain theses criteria. Moreover, the presented oscillation conditions essentially simplify and extend known criteria in the literature. To show the importance of our results, we provide some examples. Symmetry plays an essential role in determining the correct methods for solutions to differential equations. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

10 pages, 5045 KiB

Open AccessArticle

Design of Energy Absorbing Metamaterials Using Stochastic Soft-Wall Billiards

by Valery Pilipchuk

Symmetry 2021, 13(10), 1798; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101798 - 27 Sep 2021

Cited by 1 | Viewed by 1223

Abstract

Physical principles for designing cellwise artificial materials with energy-absorbing/harvesting and wave guiding properties are discussed in the present work. We analyzed the evolution of waves in a one-dimensional lattice of 3D massive potential wells with light particles inside. The potential wells were coupled [...] Read more.

Physical principles for designing cellwise artificial materials with energy-absorbing/harvesting and wave guiding properties are discussed in the present work. We analyzed the evolution of waves in a one-dimensional lattice of 3D massive potential wells with light particles inside. The potential wells were coupled with elastic springs and represented soft-wall versions of the so-called stochastic billiards. A billiard could switch from repelling to the stadium type as the parameter of shape changed its sign from positive to negative. We found that certain shapes of the potential wells/containers provided a one-directional trend of the energy flow from the chain of containers into the chaotically moving light inclusions by increasing their total kinetic energy. As a result, propagating waves became trapped by giving rise to standing waves with chaotic mode shapes with decaying amplitudes. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

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15 pages, 3515 KiB

Open AccessArticle

Transition from Discrete to Continuous Media: The Impact of Symmetry Changes on Asymptotic Behavior of Waves

by Igor Andrianov, Steve Koblik and Galina Starushenko

Symmetry 2021, 13(6), 1008; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061008 - 04 Jun 2021

Cited by 7 | Viewed by 1524

Abstract

This paper is devoted to comparing the asymptotics of a solution, describing the wave motion of a discrete lattice and its continuous approximations. The transition from a discrete medium to a continuous one changes the symmetry of the system. The influence of this [...] Read more.

This paper is devoted to comparing the asymptotics of a solution, describing the wave motion of a discrete lattice and its continuous approximations. The transition from a discrete medium to a continuous one changes the symmetry of the system. The influence of this change on the asymptotic behavior of waves is of great interest. For the discrete case, Schrödinger’s analytical solution of the initial-value problem for the Lagrange lattice is used. Various continuous approximations are proposed to approximate the lattice. They are based on Debye’s concept of quasicontinuum. The asymptotics of the initial motion and the behavior of the systems in the vicinity of the quasifront and at large times are compared. The approximations of phase and group velocities is analyzed. The merits and limitations of the described approaches are discussed. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

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21 pages, 311 KiB

Open AccessArticle

Optimized Factor Approximants and Critical Index

by Simon Gluzman

Symmetry 2021, 13(5), 903; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050903 - 19 May 2021

Cited by 6 | Viewed by 2541

Abstract

Based on expansions with only two coefficients and known critical points, we consider a minimal model of critical phenomena. The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations. Factor [...] Read more.

Based on expansions with only two coefficients and known critical points, we consider a minimal model of critical phenomena. The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations. Factor approximants are applied together with various natural optimization conditions of non-perturbative nature. The role of control parameter is played by the critical index by itself. The minimal derivative condition imposed on critical amplitude appears to bring the most reasonable, uniquely defined results. The minimal difference condition also imposed on amplitudes produces upper and lower bound on the critical index. While one of the bounds is close to the result from the minimal difference condition, the second bound is determined by the non-optimized factor approximant. One would expect that for the minimal derivative condition to work well, the bounds determined by the minimal difference condition should be not too wide. In this sense the technique of optimization presented above is self-consistent, since it automatically supplies the solution and the bounds. In the case of effective viscosity of passive suspensions the bounds could be found that are too wide to make any sense from either of the solutions. Other optimization conditions imposed on the factor approximants, lead to better estimates for the critical index for the effective viscosity. The optimization is based on equating two explicit expressions following from two different definitions of the critical index, while optimization parameter is introduced as the trial third-order coefficient in the expansion. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

Review

Jump to: Editorial, Research

13 pages, 1599 KiB

Open AccessReview

On Aspects of Gradient Elasticity: Green’s Functions and Concentrated Forces

by Igor V. Andrianov, Steve G. Koblik, Galina A. Starushenko and Askat K. Kudaibergenov

Symmetry 2022, 14(2), 188; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020188 - 19 Jan 2022

Cited by 1 | Viewed by 1404

Abstract

In the first part of our review paper, we consider the problem of approximating the Green’s function of the Lagrange chain by continuous analogs. It is shown that the use of continuous equations based on the two-point Padé approximants gives good results. In [...] Read more.

In the first part of our review paper, we consider the problem of approximating the Green’s function of the Lagrange chain by continuous analogs. It is shown that the use of continuous equations based on the two-point Padé approximants gives good results. In the second part of the paper, the problem of singularities arising in the classical theory of elasticity with affecting concentrated loadings is considered. To overcome this problem, instead of a transition to the gradient theory of elasticity, it is proposed to change the concept of concentrated effort. Namely, the Dirac delta function is replaced by the Whittaker–Shannon–Kotel’nikov interpolating function. The only additional parameter that characterizes the microheterogeneity of the medium is used. An analog of the Flamant problem is considered as an example. The found solution does not contain singularities and tends to the classical one when the microheterogeneity parameter approaches zero. The derived formulas have a simpler form compared to those obtained by the gradient theory of elasticity. Full article

(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)

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