Mathematics

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27 pages, 16743 KiB

Open AccessFeature PaperArticle

In-Plane Vibrations of Elastic Lattice Plates and Their Continuous Approximations

by Noël Challamel, Huu Phu Nguyen, Chien Ming Wang and Giuseppe Ruta

Mathematics 2024, 12(15), 2312; https://doi.org/10.3390/math12152312 - 24 Jul 2024

Viewed by 734

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding [...] Read more.

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson’s ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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28 pages, 1431 KiB

Open AccessArticle

Modal Analysis of a Multi-Supported Beam: Macroscopic Models and Boundary Conditions

by Antoine Rallu and Claude Boutin

Mathematics 2024, 12(12), 1844; https://doi.org/10.3390/math12121844 - 13 Jun 2024

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Abstract

This paper deals with the long-wavelength behaviour of a Euler beam periodically supported by co-located rotation and compression springs. An asymptotic homogenization method is applied to derive the several macroscopic models according to the stiffness contrasts between the elastic supports and the beam. [...] Read more.

This paper deals with the long-wavelength behaviour of a Euler beam periodically supported by co-located rotation and compression springs. An asymptotic homogenization method is applied to derive the several macroscopic models according to the stiffness contrasts between the elastic supports and the beam. Effective models of differential order two or four are obtained, which can be merged into a single unified model whose dispersion relations at long and medium wavelengths fit those derived by Floquet-Bloch. Moreover, the essential role of rotation supports is clearly evidenced. A mixed “discrete/continuous” approach to the boundary conditions is proposed, which allows the boundary conditions actually applied at the local scale to be expressed in terms of Robin-type boundary conditions on macroscopic variables. This approach can be applied to both dominant-order and higher-order models. The modal analysis performed with these boundary conditions and the homogenised models gives results in good agreement with a full finite element calculation, with great economy of numerical resources. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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19 pages, 1628 KiB

Open AccessFeature PaperArticle

A Multiscale Statistical Analysis of Rough Surfaces and Applications to Tribology

by Feodor M. Borodich, Andrey Pepelyshev and Xiaoqing Jin

Mathematics 2024, 12(12), 1804; https://doi.org/10.3390/math12121804 - 10 Jun 2024

Cited by 1 | Viewed by 1091

Abstract

Mathematical modelling of surface roughness is of significant interest for a variety of modern applications, including, but not limited to, tribology and optics. The most popular approaches to modelling rough surfaces are reviewed and critically examined. By providing counterexamples, it is shown that [...] Read more.

Mathematical modelling of surface roughness is of significant interest for a variety of modern applications, including, but not limited to, tribology and optics. The most popular approaches to modelling rough surfaces are reviewed and critically examined. By providing counterexamples, it is shown that approaches based solely on the use of the fractal geometry or power spectral density have many drawbacks. It is recommended to avoid these approaches. It is argued that the surfaces that cannot be distinguished from the original rough surfaces can be synthesised by employing the concept of the representative elementary pattern of roughness (REPR), i.e., the smallest interval (or area) of a rough surface that statistically represents the whole surface. The REPR may be extracted from surface measurement data by the use of the “moving window” technique in combination with the Kolmogorov–Smirnov statistic. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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14 pages, 400 KiB

Open AccessArticle

Asymptotic Analysis of an Elastic Layer under Light Fluid Loading

by Sheeru Shamsi and Ludmila Prikazchikova

Mathematics 2024, 12(10), 1465; https://doi.org/10.3390/math12101465 - 9 May 2024

Viewed by 653

Abstract

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding [...] Read more.

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding to long-wave, low-frequency, antisymmetric motions. The asymptotic plate models, including the equations of motion and the impenetrability condition, motivated by the aforementioned shortened dispersion equations, were derived for a plane-strain setup. The key findings included, in particular, the necessity of taking into account transverse plate inertia at the leading order, which is not the case for heavy fluid loading. In addition, the transverse shear deformation, rotation inertia, and a number of other corrections appeared at the next order, contrary to the previous asymptotic developments for fluid-loaded plates not assuming a light fluid loading scenario. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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13 pages, 1842 KiB

Open AccessArticle

On Aspects of Continuous Approximation of Diatomic Lattice

by Igor V. Andrianov, Lelya A. Khajiyeva, Askar K. Kudaibergenov and Galina A. Starushenko

Mathematics 2024, 12(10), 1456; https://doi.org/10.3390/math12101456 - 8 May 2024

Viewed by 666

Abstract

This paper is devoted to the continualization of a diatomic lattice, taking into account natural intervals of wavenumber changes. Continualization refers to the replacement of the original pseudo-differential equations by a system of PDEs that provides a good approximation of the dispersion relations. [...] Read more.

This paper is devoted to the continualization of a diatomic lattice, taking into account natural intervals of wavenumber changes. Continualization refers to the replacement of the original pseudo-differential equations by a system of PDEs that provides a good approximation of the dispersion relations. In this regard, the Padé approximants based on the conditions for matching the values of the dispersion relations of the discrete and continuous models at several characteristic points are utilized. As a result, a sixth-order unconditionally stable system with modified inertia is obtained. Appropriate boundary conditions are formulated. The obtained continuous approximation accurately describes the amplitude ratios of neighboring masses. It is also shown that the resulting continuous system provides a good approximation for the natural frequencies. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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21 pages, 397 KiB

Open AccessFeature PaperArticle

Norm-Resolvent Convergence for Neumann Laplacians on Manifold Thinning to Graphs

by Kirill D. Cherednichenko, Yulia Yu. Ershova and Alexander V. Kiselev

Mathematics 2024, 12(8), 1161; https://doi.org/10.3390/math12081161 - 12 Apr 2024

Viewed by 990

Abstract

Norm-resolvent convergence with an order-sharp error estimate is established for Neumann Laplacians on thin domains in

R^{d},

d \geq 2,

converging to metric graphs in the limit of vanishing thickness parameter in the “resonant” case. The vertex matching conditions [...] Read more.

Norm-resolvent convergence with an order-sharp error estimate is established for Neumann Laplacians on thin domains in

R^{d},

d \geq 2,

converging to metric graphs in the limit of vanishing thickness parameter in the “resonant” case. The vertex matching conditions of the limiting quantum graph are revealed as being closely related to those of the

δ^{'}

type. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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14 pages, 7271 KiB

Open AccessArticle

Utilization of a Genetic Algorithm to Identify Optimal Geometric Shapes for a Seismic Protective Barrier

by Vladimir Bratov, Andrey Murachev and Sergey V. Kuznetsov

Mathematics 2024, 12(3), 492; https://doi.org/10.3390/math12030492 - 4 Feb 2024

Viewed by 1053

Abstract

The utilization of seismic barriers for protection against the hazardous impact of natural or technogenic waves is an extremely promising emerging technology to secure buildings, structures and entire areas against earthquake-generated seismic waves, high-speed-transport-induced vibrations, etc. The current research is targeted at studying [...] Read more.

The utilization of seismic barriers for protection against the hazardous impact of natural or technogenic waves is an extremely promising emerging technology to secure buildings, structures and entire areas against earthquake-generated seismic waves, high-speed-transport-induced vibrations, etc. The current research is targeted at studying the effect of seismic-barrier shape on the reduction of seismic-wave magnitudes within the protected region. The analytical solution of Lamb’s problem was used to verify the adopted numerical approach. It was demonstrated that the addition of complementary geometric features to a simple barrier shape provides the possibility of significantly increasing the resulting seismic protection. A simple genetic algorithm was employed to evaluate the nontrivial but extremely effective geometry of the seismic barrier. The developed approach can be used in various problems requiring optimization of non-parameterizable geometric shapes. The applicability of genetic algorithms and other generative algorithms to discover optimal (or close to optimal) geometric configurations for the essentially multiscale problems of the interaction of mechanical waves with inclusions is discussed. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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25 pages, 1406 KiB

Open AccessArticle

A General Case of a Line Contact Lubricated by a Non-Newtonian Giesekus Fluid

by Ilya I. Kudish and Sergei S. Volkov

Mathematics 2023, 11(22), 4679; https://doi.org/10.3390/math11224679 - 17 Nov 2023

Viewed by 750

Abstract

A steady plane hydrodynamic problem of lubrication of a lightly loaded contact of two parallel cylinders lubricated by a non-Newtonian fluid with Giesekus rheology is considered. The advantage of this non-Newtonian rheology is its ability to properly describe the real behavior of formulated [...] Read more.

A steady plane hydrodynamic problem of lubrication of a lightly loaded contact of two parallel cylinders lubricated by a non-Newtonian fluid with Giesekus rheology is considered. The advantage of this non-Newtonian rheology is its ability to properly describe the real behavior of formulated lubricants at high and low shear stresses. The problem is solved by using a modification of the regular perturbation method with respect to the small parameter

α

, characterizing the degree to which the polymeric molecules of the additive to the lubricant follow the streamlines of the lubricant flow. It is assumed that the lubricant relaxation time and the value of

α

are of the order of the magnitude of the ratio of the characteristic gap between the contact surfaces and the contact length. The obtained analytical solution of the problem is analyzed numerically for the dependencies of the problem characteristics such as contact pressure, fluid flux, lubrication film thickness, friction force, energy loss in the lubricated contact, etc., on the problem input parameters. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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11 pages, 425 KiB

Open AccessArticle

Degenerated Boundary Layers and Long-Wave Low-Frequency Motion in High-Contrast Elastic Laminates

by Lenser A. Aghalovyan, Lusine G. Ghulghazaryan, Julius Kaplunov and Danila Prikazchikov

Mathematics 2023, 11(18), 3905; https://doi.org/10.3390/math11183905 - 14 Sep 2023

Cited by 1 | Viewed by 896

Abstract

The effect of high contrast on the multiscale behaviour of elastic laminates is studied. Mathematical modelling in this area is of significant interest for a variety of modern applications, including but not limited to advanced sandwich structures and photovoltaic panels. As an example, [...] Read more.

The effect of high contrast on the multiscale behaviour of elastic laminates is studied. Mathematical modelling in this area is of significant interest for a variety of modern applications, including but not limited to advanced sandwich structures and photovoltaic panels. As an example, the antiplane shear of a symmetric, three-layered plate is considered. The problem parameters expressing relative thickness, stiffness and density are assumed to be independent. The high contrast may generally support extra length and time scales corresponding to degenerated boundary layers and propagating long-wave low-frequency vibration modes. The main focus is on the relation between these two phenomena. The developed multiparametric approach demonstrates that those do not always appear simultaneously. The associated explicit estimates on contrast parameters are established. In addition, the recent asymptotic extension of the classical Saint-Venant’s principle is adapted for calculating the contribution of the degenerate boundary layer or long-wave low-frequency propagation mode. The peculiarity of the limiting absorption principle in application to layered media is also addressed. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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