Georgian Mathematical Journal

Wave propagation on hexagonal lattices

Publication type: Journal Article
Publication date: 2024-06-26
scimago Q2
wos Q2
SJR: 0.391
CiteScore: 1.7
Impact factor: 0.8
ISSN: 15729176, 1072947X
Abstract

We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ ( 0 , 6 ) ∖ { 2 , 3 , 2 } {k\in(0,\sqrt{6})\setminus\{\sqrt{2},\sqrt{3},2\}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ ( 0 , 2 ) ∪ ( 2 , 6 ) {k\in(0,\sqrt{2})\cup(2,\sqrt{6})} for the exterior Dirichlet problem.

Found 20
By date By citations
Elsevier
Interaction of in-plane waves with a structured penetrable line defect in an elastic lattice
Nieves M.J., Sharma B.L.
International Journal of Engineering Science scimago Q1 wos Q1 , 
2024-04-01,  citations by CoLab: 4 ,  Abstract  
We consider the scattering of in-plane waves that interact with an edge of a structured penetrable inertial line defect contained in a triangular lattice, composed of periodically placed masses interconnected by massless elastic rods. The steady state problem for a time-harmonic excitation is converted into a vector Wiener–Hopf equation using the Fourier transform. The matrix Wiener–Hopf kernel of this equation describes the dynamic phenomena engaged in the scattering process, which includes instances where localised interfacial waves can emerge along the structured defect. This information is exploited to identify the dependency of the existence of these waves on the incident wave parameters and the properties of inertial defect. Symmetry in the structure of the scattering medium allows us to convert the vectorial problem into a pair of uncoupled scalar Wiener–Hopf equations posed along the lattice row containing the defect. The solution embodies an exact representation of the scattered field, in terms of a contour integral in the complex plane, that includes the contributions of evanescent and propagating waves. The solution reveals that in the remote lattice, the reflected and transmitted components of incident field are accompanied by dynamic modes from three symmetry classes, which include localised interfacial waves. These classes correspond to tensile modes acting transverse to the defected lattice row, shear modes that act parallel to this row, and wave modes represented as a mixture of these two responses. Benchmark finite element calculations are also provided to validate the results against our semi-analytical solution which involves, in particular, numerical computation of the contour integrals. Graphical illustrations demonstrate special dynamic responses encountered during the wave scattering process, including dynamic anisotropy, negative reflection and negative refraction.
Elsevier
Forcing the silence of the Lamb waves: Uni-directional propagation in structured gyro-elastic strips and networks
Carta G., Nieves M.J., Brun M.
European Journal of Mechanics, A/Solids scimago Q1 wos Q1 , 
2023-09-01,  citations by CoLab: 6 ,  Abstract  
In this paper, we propose an innovative design of an elastic network, which is capable of channelling the energy supplied by an external source towards any of its endpoints, that can be chosen arbitrarily and in advance. This system, named Mechanical Switching Network (MSN), consists of an interconnected array of branches, each of which is represented by a lattice strip endowed with gyroscopic spinners. The latter make the system non-reciprocal and, hence, are responsible for the preferential directionality exhibited by the network. We formulate and solve the forced problem for the gyro-elastic strip in the analytical form and compare the derived solutions with the results of independent finite element simulations, showing an excellent agreement. Additionally, we carry out a parametric analysis to evaluate the influence of the main parameters of the system on the uni-directional wave propagation of Lamb waves. We envisage that the proposed model can have important implications in many engineering applications, where control and tunability of guided waves play a key role.
SAGE
Exterior diffraction problems for a triangular lattice
Kapanadze D., Pesetskaya E.
Mathematics and Mechanics of Solids scimago Q2 wos Q3 , 
2023-05-26,  citations by CoLab: 2 ,  Abstract  
Exterior Dirichlet problems for two-dimensional (2D) lattice waves on the infinite triangular lattice are considered. Namely, we study Dirichlet problems for the 2D discrete Helmholtz equation in a plane with a hole. New results are obtained for the existence and uniqueness of the solution in the case of the real wave number [Formula: see text] without passing to a complex wave number. Besides, Green’s representation formula for the solution is derived with the help of difference potentials. To demonstrate the results, we propose a method for numerical calculation.
Elsevier
A review on the mechanics of graphene nanoplatelets reinforced structures
Yee K., Ghayesh M.H.
International Journal of Engineering Science scimago Q1 wos Q1 , 
2023-05-01,  citations by CoLab: 70 ,  Abstract  
Graphene nanoplatelets (GPL) are becoming an emerging nanofiller in the field of nanocomposites due to the ease and low cost of manufacturing, which have made it a preferred reinforcing element compared to graphene and other graphene derivatives. In the recent years, an increasing amount of literature has been published on GPL reinforced materials and structures, with a considerable amount of attention focused on GPL reinforced beam, plate, and shell structures. With the purpose of consolidating recent research undertaken into the study of the mechanical behaviour of GPL reinforced composite (GPLRC), this article reviews the mechanics of GPLRC beam, plate, and shell structures. More specifically, it provides a brief introduction into graphene, GPL, functionally graded materials, porous materials, and micro/nanostructures. Then, it presents an inclusive literature review on the theoretical analysis and the numerical formulation of linear and nonlinear behaviour of GPL reinforced macro/micro/nanostructures, focusing on the bending, buckling, and dynamic characteristics. Experimental analysis performed on GPLRC are also reviewed, addressing different aspects of the reinforcement of GPL. Finally, a summary of the findings in this field along with potential future applications for research and experimentation is provided. An overview to address the methodologies in studies covered in this review paper is provided as an appendix.
Wiley
The far‐field behaviour of Green's function for a triangular lattice and radiation conditions
Kapanadze D.
Mathematical Methods in the Applied Sciences scimago Q2 wos Q1 , 
2021-06-07,  citations by CoLab: 5 ,  Abstract  
We consider the Helmholtz equation (Δd + k2)u = f on the triangular lattice, where Δd is the discrete Laplacian, f has finite support, and wave number k belongs to the pass-band. Using the limiting absorption principle, we derive the discrete analogue of the Sommerfeld radiation condition for all values of k ∈ ( 0,3 ) \ { 2 2 }. It turns out that this condition is anisotropic and depends on the value of k. We introduce the notion of a radiating solution and prove the unique solvability result.
Elsevier
Rayleigh waves in micro-structured elastic systems: Non-reciprocity and energy symmetry breaking
Nieves M.J., Carta G., Pagneux V., Brun M.
International Journal of Engineering Science scimago Q1 wos Q1 , 
2020-11-01,  citations by CoLab: 33 ,  Abstract  
Rayleigh waves are analysed in elastic lattices incorporating inertial devices that couple in-plane displacements. The vector problems of elasticity for a triangular lattice and its long-wavelength/low-frequency continuum approximation are considered. The analytical procedure for the derivation of the Rayleigh dispersion relation is fully detailed and, remarkably, explicit solutions for the Rayleigh waves for both the discrete and continuous systems are found. The dispersion at low wavenumbers and the softening induced by the presence of in-plane inertial coupling are shown. Despite the symmetry of the dispersion curves with respect to the wavenumber, the introduction of the inertial coupling breaks the symmetry of the eigenmodes and makes the system non-reciprocal. Such an uncommon effect is demonstrated in a set of numerical computations, where a point force applied on the boundary generates surface and bulk waves that do not propagate symmetrically from the source.
Springer Nature
Inverse Acoustic and Electromagnetic Scattering Theory
Colton D., Kress R.
Applied Mathematical Sciences (Switzerland) , 
2019-11-06,  citations by CoLab: 192 , 
The Royal Society
Resonant waves and localization phenomena in lattices
Abdukadirov S.A., Ayzenberg-Stepanenko M.V., Osharovich G.G.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences scimago Q1 wos Q1 , 
2019-09-02,  citations by CoLab: 4 ,  Abstract  
Transient wave processes in mass-spring lattices excited by point oscillating sources are studied. Dispersion properties of uniform periodic three-dimensional (3D) square-cell and two-dimensional (2D) hexagonal-cell lattices including revealed star-shaped localization phenomena are analysed. The resonant-like waves and localization-like patterns in non-uniform lattices possessing predetermined and randomly distributed defects are numerically examined in order to identify the sensitivity of star-shape forms to different types of defects. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.
Springer Nature
Exterior diffraction problems for two-dimensional square lattice
Kapanadze D.
Zeitschrift fur Angewandte Mathematik und Physik scimago Q1 wos Q2 , 
2018-09-11,  citations by CoLab: 7 ,  Abstract  
We study exterior Dirichlet problems for the two-dimensional discrete Helmholtz equation with the real wave number $$k\in (0,2\sqrt{2})\backslash \{2\}$$ . The investigation is carried out without passing to the complex wave number. Similarly to the continuum theory, we use the notion of radiating solution. Then, the unique solvability result and Green’s representation formula are obtained with the help of difference potentials. Finally, we proposed a method for numerical calculation. Efficiency of our approach is demonstrated on an example related to the propagation problems in the left-handed 2D inductor–capacitor metamaterial with a hole.
Springer Nature
Discrete Sommerfeld diffraction problems on hexagonal lattice with a zigzag semi-infinite crack and rigid constraint
Sharma B.L.
Zeitschrift fur Angewandte Mathematik und Physik scimago Q1 wos Q2 , 
2015-09-05,  citations by CoLab: 10 ,  Abstract  
Diffraction problems, associated with waves scattered by a semi-infinite crack and rigid constraint, in a hexagonal (honeycomb) lattice model, with nearest neighbor interactions, are solved exactly using the method of Wiener and Hopf. Asymptotic expressions for the scattered waves in far field are provided for both problems, by application of the method of stationary phase to corresponding diffraction integrals. Additionally, for the crack diffraction problem, bond lengths on the semi-infinite row complementing the crack, as well as the crack opening displacement, are provided in closed form except for the presence of concomitant Fourier coefficients of the Wiener–Hopf kernel. For the rigid constraint diffraction problem, the solution on the semi-infinite row complementing the constrained lattice sites, as well as that adjacent to the constrained row, are provided in similar closed form. The amplitude, as well as phase, of waves in far field is compared, through graphical plots, with that of a numerical solution on finite grid. Also, the analytical solution for few sites near the tip of each defect is compared with numerical solution. Both discrete Sommerfeld diffraction problems and their solutions are also relevant to numerical solution of the two-dimensional Helmholtz equation using a 4-point hexagonal grid, besides having applications inherent to the scattering of waves on a honeycomb structure.
American Association for the Advancement of Science (AAAS)
Observation of phononic helical edge states in a mechanical topological insulator
Süsstrunk R., Huber S.D.
Science scimago Q1 wos Q1 Open Access , 
2015-07-03,  citations by CoLab: 841 ,  PDF,  Abstract  
Designing mechanical complexity The quantum properties of topological insulators translate to mechanical systems governed by Newton's equations of motion. Many-body interactions and the multiple degrees of freedom available to charge carriers give electronic systems a range of exotic behaviors. Süsstrunk and Huber show that this extends to mechanical systems made up of a large lattice of coupled pendula. Mechanical excitations can be eliminated from the inner part of the lattice and confined to the edges, much like topological insulators. In addition to presenting a tractable toy system in which to study complex phenomena, the approach has potential uses in vibration isolation. Science , this issue p. 47
Elsevier
Dynamic problems for metamaterials: Review of existing models and ideas for further research
Del Vescovo D., Giorgio I.
International Journal of Engineering Science scimago Q1 wos Q1 , 
2014-07-01,  citations by CoLab: 194 ,  Abstract  
Metamaterials are materials especially engineered to have a peculiar physical behaviour, to be exploited for some well-specified technological application. In this context we focus on the conception of general micro-structured continua, with particular attention to piezoelectromechanical structures, having a strong coupling between macroscopic motion and some internal degrees of freedom, which may be electric or, more generally, related to some micro-motion . An interesting class of problems in this context regards the design of wave-guides aimed to control wave propagation. The description of the state of the art is followed by some hints addressed to describe some possible research developments and in particular to design optimal design techniques for bone reconstruction or systems which may block wave propagation in some frequency ranges, in both linear and non-linear fields.
Elsevier
Propagation of Slepyan's crack in a non-uniform elastic lattice
Nieves M.J., Movchan A.B., Jones I.S., Mishuris G.S.
Journal of the Mechanics and Physics of Solids scimago Q1 wos Q2 , 
2013-06-01,  citations by CoLab: 27 ,  Abstract  
We model and derive the solution for the problem of a Mode I semi-infinite crack propagating in a discrete triangular lattice with bonds having a contrast in stiffness in the principal lattice directions. The corresponding Green's kernel is found and from this wave dispersion dependencies are obtained in explicit form. An equation of the Wiener-Hopf type is also derived and solved along the crack face, in order to compute the stress intensity factor for the semi-infinite crack. The crack stability is analysed via the evaluation of the energy release rate for different contrasts in stiffness of the bonds.
American Physical Society (APS)
Nobel Lecture: Graphene: Materials in the Flatland
Novoselov K.S.
Reviews of Modern Physics scimago Q1 wos Q1 , 
2011-08-03,  citations by CoLab: 687 ,  Abstract  
Much like the world described in Abbott's Flatland, graphene is a two-dimensional object. And, as ``Flatland'' is ``a romance of many dimensions,'' graphene is much more than just a flat crystal. It possesses a number of unusual properties which are often unique or superior to those in other materials. In this brief lecture I would like to explain the reason for my (and many other people's) fascination with this material, and invite the reader to share some of the excitement I've experienced while researching it.
American Physical Society (APS)
The electronic properties of graphene
Castro Neto A.H., Guinea F., Peres N.M., Novoselov K.S., Geim A.K.
Reviews of Modern Physics scimago Q1 wos Q1 , 
2009-01-14,  citations by CoLab: 21099 ,  Abstract  
This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.
Found 1
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Walter de Gruyter
Propagation of waves from finite sources arranged in line segments within an infinite triangular lattice
Kapanadze D., Vashakidze Z.
Georgian Mathematical Journal scimago Q2 wos Q2 , 
2025-02-10,  citations by CoLab: 0 ,  Abstract  
Abstract This paper examines the propagation of time-harmonic waves in a two-dimensional triangular lattice with a lattice constant a = 1 {a=1} . The sources are positioned along line segments within the lattice. Specifically, we investigate the discrete Helmholtz equation with a wavenumber k ∈ ( 0 , 2 ⁢ 2 ) {k\in(0,2\sqrt{2})} , where input data is prescribed on finite rows or columns of lattice sites. We focus on two main questions: the efficacy of the numerical methods employed in evaluating the Green’s function, and the necessity of the cone condition. Consistent with a continuum theory, we employ the notion of radiating solution and establish a unique solvability result and Green’s representation formula using difference potentials. Finally, we propose a numerical computation method and demonstrate its efficiency through examples related to the propagation problems in the left-handed two-dimensional inductor-capacitor metamaterial.

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Georgian Mathematical Journal
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