Symmetry

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11 pages, 603 KiB

Open AccessArticle

Kinetics of Nanostructuring Processes of Material Surface under Influence of Laser Radiation

by Alexei Khomenko, Olga Yushchenko and Anna Badalian

Symmetry 2020, 12(11), 1914; https://doi.org/10.3390/sym12111914 - 20 Nov 2020

Cited by 2 | Viewed by 1782

In this paper, further research is conducted for a synergetic model that describes the state of the material surface in the process of laser irradiation. Namely, the previously studied approach of mutually coordinated behavior of the relaxation field, concentration of relaxation zones, and [...] Read more.

In this paper, further research is conducted for a synergetic model that describes the state of the material surface in the process of laser irradiation. Namely, the previously studied approach of mutually coordinated behavior of the relaxation field, concentration of relaxation zones, and field of stress is supplemented with a nonlinear term. It is shown that, using this model, we can describe the behavior of different types of systems. During the analysis, five stationary states were found which correspond to different modes of formation of relaxation areas on the surface. The regions of parameters are found at which one or another mode of the system behavior is established. Phase portraits are constructed for each mode and the kinetics of the system is described. The obtained results qualitatively coincide with the experimental data. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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9 pages, 246 KiB

Open AccessArticle

On Dynamic Extension of a Local Material Symmetry Group for Micropolar Media

by Victor A. Eremeyev and Violetta Konopińska-Zmysłowska

Symmetry 2020, 12(10), 1632; https://doi.org/10.3390/sym12101632 - 3 Oct 2020

Cited by 9 | Viewed by 2082

Abstract

For micropolar media we present a new definition of the local material symmetry group considering invariant properties of the both kinetic energy and strain energy density under changes of a reference placement. Unlike simple (Cauchy) materials, micropolar media can be characterized through two [...] Read more.

For micropolar media we present a new definition of the local material symmetry group considering invariant properties of the both kinetic energy and strain energy density under changes of a reference placement. Unlike simple (Cauchy) materials, micropolar media can be characterized through two kinematically independent fields, that are translation vector and orthogonal microrotation tensor. In other words, in micropolar continua we have six degrees of freedom (DOF) that are three DOFs for translations and three DOFs for rotations. So the corresponding kinetic energy density nontrivially depends on linear and angular velocity. Here we define the local material symmetry group as a set of ordered triples of tensors which keep both kinetic energy density and strain energy density unchanged during the related change of a reference placement. The triples were obtained using transformation rules of strain measures and microinertia tensors under replacement of a reference placement. From the physical point of view, the local material symmetry group consists of such density-preserving transformations of a reference placement, that cannot be experimentally detected. So the constitutive relations become invariant under such transformations. Knowing a priori a material’s symmetry, one can establish a simplified form of constitutive relations. In particular, the number of independent arguments in constitutive relations could be significantly reduced. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

17 pages, 1225 KiB

Open AccessArticle

Steady Solitary and Periodic Waves in a Nonlinear Nonintegrable Lattice

by Igor Andrianov, Aleksandr Zemlyanukhin, Andrey Bochkarev and Vladimir Erofeev

Symmetry 2020, 12(10), 1608; https://doi.org/10.3390/sym12101608 - 27 Sep 2020

Cited by 3 | Viewed by 2741

Abstract

In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found [...] Read more.

In this paper, stationary solitary and periodic waves of a nonlinear nonintegrable lattice are numerically constructed using a two-stage approach. First, as a result of continualization, a nonintegrable generalized Boussinesq—Ostrovsky equation is obtained, for which the solitary-wave and periodic solutions are numerically found by the Petviashvili method. In the second stage, discrete analogs of the obtained solutions are used as initial conditions in the numerical simulation of the original lattice. It is shown that the initial perturbations constructed in this way propagate along the lattice without changing their shape. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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24 pages, 2154 KiB

Open AccessArticle

On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths

by Giuseppe Rosi, Nicolas Auffray and Christelle Combescure

Symmetry 2020, 12(8), 1243; https://doi.org/10.3390/sym12081243 - 28 Jul 2020

Cited by 12 | Viewed by 3166

Abstract

In this work we investigate the properties of elastic waves propagating in gyroid lattices. First, we rigorously characterize the lattice from the point of view of crystallography. Second, we use Bloch–Floquet analysis to compute the dispersion relations for elastic waves. The results for [...] Read more.

In this work we investigate the properties of elastic waves propagating in gyroid lattices. First, we rigorously characterize the lattice from the point of view of crystallography. Second, we use Bloch–Floquet analysis to compute the dispersion relations for elastic waves. The results for very long wavelengths are then compared to those given by classic elasticity for a cubic material. A discrepancy is found in terms of the polarization of waves and it is related to the noncentrosymmetry of the gyroid. The gyroid lattice results to be acoustically active, meaning that transverse waves exhibit a circular polarization when they propagate along an axis of rotational symmetry. This phenomenon is present even for very long wavelengths and is not captured by classic elasticity. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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18 pages, 333 KiB

Open AccessArticle

A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers

by Milad Shirani and David J. Steigmann

Symmetry 2020, 12(7), 1133; https://doi.org/10.3390/sym12071133 - 7 Jul 2020

Cited by 30 | Viewed by 2593

Abstract

A Cosserat theory for fiber-reinforced elastic solids developed in Steigmann (2012) is generalized to accommodate initial curvature and twist of the fibers. The basic variables of the theory are a conventional deformation field and a rotation field that describes the local fiber orientation. [...] Read more.

A Cosserat theory for fiber-reinforced elastic solids developed in Steigmann (2012) is generalized to accommodate initial curvature and twist of the fibers. The basic variables of the theory are a conventional deformation field and a rotation field that describes the local fiber orientation. Constraints on these fields are introduced to model the materiality of the fibers with respect to the underlying matrix deformation. A variational argument delivers the relevant equilibrium equations and boundary conditions and furnishes the interpretation of the Lagrange multipliers associated with the constraints as shear tractions acting on the fiber cross sections. Finally, the theory of material symmetry for such solids is developed and applied to the classification of some explicit constitutive functions. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

27 pages, 1614 KiB

Open AccessArticle

Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles

by Markus Lazar and Jakob Leck

Symmetry 2020, 12(7), 1104; https://doi.org/10.3390/sym12071104 - 2 Jul 2020

Cited by 7 | Viewed by 4557

Abstract

In this paper, we study the theory of second gradient electromagnetostatics as the static version of second gradient electrodynamics. The theory of second gradient electrodynamics is a linear generalization of higher order of classical Maxwell electrodynamics whose Lagrangian is both Lorentz and [...] Read more.

In this paper, we study the theory of second gradient electromagnetostatics as the static version of second gradient electrodynamics. The theory of second gradient electrodynamics is a linear generalization of higher order of classical Maxwell electrodynamics whose Lagrangian is both Lorentz and

U (1)

-gauge invariant. Second gradient electromagnetostatics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian. Moreover, it possesses a weak nonlocality in space and gives a regularization based on higher-order partial differential equations. From the group theoretical point of view, in second gradient electromagnetostatics the (isotropic) constitutive relations involve an invariant scalar differential operator of fourth order in addition to scalar constitutive parameters. We investigate the classical static problems of an electric point charge, and electric and magnetic dipoles in the framework of second gradient electromagnetostatics, and we show that all the electromagnetic fields (potential, field strength, interaction energy, interaction force) are singularity-free, unlike the corresponding solutions in the classical Maxwell electromagnetism and in the Bopp–Podolsky theory. The theory of second gradient electromagnetostatics delivers a singularity-free electromagnetic field theory with weak spatial nonlocality. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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17 pages, 4745 KiB

Open AccessArticle

Symmetric-in-Plane Compression of Polyamide Pantographic Fabrics—Modelling, Experiments and Numerical Exploration

by Chuong Anthony Tran, Maciej Gołaszewski and Emilio Barchiesi

Symmetry 2020, 12(5), 693; https://doi.org/10.3390/sym12050693 - 1 May 2020

Cited by 20 | Viewed by 3580

Abstract

Symmetric in-plane compression of a pantographic lattice structure is modelled and simulated, and the results are compared to previously available experimental data. Said experimental results had shown a peculiar behaviour: depending on the fiber density, the deformed shape could present either one or [...] Read more.

Symmetric in-plane compression of a pantographic lattice structure is modelled and simulated, and the results are compared to previously available experimental data. Said experimental results had shown a peculiar behaviour: depending on the fiber density, the deformed shape could present either one or two swellings under compression. The present article is a preliminary modelling attempt aiming at capturing that behaviour numerically. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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29 pages, 387 KiB

Open AccessArticle

Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law

by Houssam Abdoul-Anziz, Nicolas Auffray and Boris Desmorat

Symmetry 2020, 12(4), 674; https://doi.org/10.3390/sym12040674 - 23 Apr 2020

Cited by 4 | Viewed by 2825

Abstract

We determine the different symmetry classes of bi-dimensional flexoelectric tensors. Using the harmonic decomposition method, we show that there are six symmetry classes. We also provide the matrix representations of the flexoelectric tensor and of the complete flexoelectric law, for each symmetry class. [...] Read more.

We determine the different symmetry classes of bi-dimensional flexoelectric tensors. Using the harmonic decomposition method, we show that there are six symmetry classes. We also provide the matrix representations of the flexoelectric tensor and of the complete flexoelectric law, for each symmetry class. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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21 pages, 1945 KiB

Open AccessArticle

On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam

by Mohammad Malikan and Victor A. Eremeyev

Symmetry 2020, 12(4), 643; https://doi.org/10.3390/sym12040643 - 17 Apr 2020

Cited by 60 | Viewed by 5713

Abstract

The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical [...] Read more.

The fundamental motivation of this research is to investigate the effect of flexoelectricity on a piezoelectric nanobeam for the first time involving internal viscoelasticity. To date, the effect of flexoelectricity on the mechanical behavior of nanobeams has been investigated extensively under various physical and environmental conditions. However, this effect as an internal property of materials has not been studied when the nanobeams include an internal damping feature. To this end, a closed-circuit condition is considered taking converse piezo–flexoelectric behavior. The kinematic displacement of the classical beam using Lagrangian strains, also applying Hamilton’s principle, creates the needed frequency equation. The natural frequencies are measured in nanoscale by the available nonlocal strain gradient elasticity model. The linear Kelvin–Voigt viscoelastic model here defines the inner viscoelastic coupling. An analytical solution technique determines the values of the numerical frequencies. The best findings show that the viscoelastic coupling can directly affect the flexoelectricity property of the material. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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22 pages, 4813 KiB

Open AccessArticle

Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters

by Ali Shariati, Dong won Jung, Hamid Mohammad-Sedighi, Krzysztof Kamil Żur, Mostafa Habibi and Maryam Safa

Symmetry 2020, 12(4), 586; https://doi.org/10.3390/sym12040586 - 7 Apr 2020

Cited by 161 | Viewed by 4342

Abstract

In this article, vibration of viscoelastic axially functionally graded (AFG) moving Rayleigh and Euler–Bernoulli (EB) beams are investigated and compared, aiming at a performance improvement of translating systems. Additionally, a detailed study is performed to elucidate the influence of various factors, such as [...] Read more.

In this article, vibration of viscoelastic axially functionally graded (AFG) moving Rayleigh and Euler–Bernoulli (EB) beams are investigated and compared, aiming at a performance improvement of translating systems. Additionally, a detailed study is performed to elucidate the influence of various factors, such as the rotary inertia factor and axial gradation of material on the stability borders of the system. The material properties of the beam are distributed linearly or exponentially in the longitudinal direction. The Galerkin procedure and eigenvalue analysis are adopted to acquire the natural frequencies, dynamic configuration, and instability thresholds of the system. Furthermore, an exact analytical expression for the critical velocity of the AFG moving Rayleigh beams is presented. The stability maps and critical velocity contours for various material distributions are examined. In the case of variable density and elastic modulus, it is demonstrated that linear and exponential distributions provide a more stable system, respectively. Furthermore, the results revealed that the decrease of density gradient parameter and the increase of the elastic modulus gradient parameter enhance the natural frequencies and enlarge the instability threshold of the system. Hence, the density and elastic modulus gradients play opposite roles in the dynamic behavior of the system. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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22 pages, 1430 KiB

Open AccessArticle

Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials

by Elena F. Grekova, Alexey V. Porubov and Francesco dell’Isola

Symmetry 2020, 12(4), 521; https://doi.org/10.3390/sym12040521 - 2 Apr 2020

Cited by 16 | Viewed by 2727

Abstract

We consider the reduced constrained linear Cosserat continuum, a particular type of a Cosserat medium, for three different material behaviors or symmetries: the isotropic elastic case, a special type of elastic transversely isotropic case, and the isotropic viscoelastic case. Such continua, in which [...] Read more.

We consider the reduced constrained linear Cosserat continuum, a particular type of a Cosserat medium, for three different material behaviors or symmetries: the isotropic elastic case, a special type of elastic transversely isotropic case, and the isotropic viscoelastic case. Such continua, in which stresses do not work on rates of microrotation gradients, behave as acoustic metamaterials for the (pure) shear waves and also for one branch of the mixed wave in the considered anisotropic material case. In elastic media, those waves do not propagate for frequencies exceeding a certain threshold, whence these media exhibit a single negative acoustic metamaterial behavior in this range. In the isotropic viscoelastic case, dissipation destroys the bandgap and favors wave propagation. This curious effect is, probably, due to the fact that the bandgap is associated not with the dissipation, but with the wave localization which can be destroyed by the viscosity. The dispersion curve is now decreasing in some part of the former bandgap, above a certain frequency, whence the medium is a double negative acoustic metamaterial. We prove the existence of a boundary wavenumber in the viscoelastic case and estimate its value. Below the characteristic frequency corresponding to the boundary of the elastic bandgap, the wave attenuation (logarithmic decrement) is a growing function of the viscous dissipation parameter. Above this frequency, the attenuation decreases as the viscosity increases. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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20 pages, 6583 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

by Daria Scerrato and Ivan Giorgio

Symmetry 2019, 11(12), 1523; https://doi.org/10.3390/sym11121523 - 16 Dec 2019

Cited by 25 | Viewed by 3174

Abstract

A particular pantographic sheet, modeled as a two-dimensional elastic continuum consisting of an orthogonal lattice of continuously distributed fibers with a cycloidal texture, is introduced and investigated. These fibers conceived as embedded beams on the surface are allowed to be deformed in a [...] Read more.

A particular pantographic sheet, modeled as a two-dimensional elastic continuum consisting of an orthogonal lattice of continuously distributed fibers with a cycloidal texture, is introduced and investigated. These fibers conceived as embedded beams on the surface are allowed to be deformed in a three-dimensional space and are endowed with resistance to stretching, shearing, bending, and twisting. A finite element analysis directly derived from a variational formulation was performed for some explanatory tests to illustrate the behavior of the newly introduced material. Specifically, we considered tests on: (1) bias extension; (2) compressive; (3) shear; and (4) torsion. The numerical results are discussed to some extent. Finally, attention is drawn to a comparison with other kinds of orthogonal lattices, namely straight, parabolic, and oscillatory, to show the differences in the behavior of the samples due to the diverse arrangements of the fibers. Full article

(This article belongs to the Special Issue Recent Advances in the Study of Symmetry and Continuum Mechanics)

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