Spectral Properties of Hexagonal Lattices with The -R Coupling

This thesis constitutes an exposition of the work carried out by the author whilst examining several physical problems under the broad theme of the dynamic response of structures. An outline of the thesis is provided in chapter one. Chapter two introduces some notation and preliminary results on general lattice equations. Chapter three examines the dispersive behaviour of non-classical discrete elastic lattice systems. In particular, the effect of distributing the inertial properties of the lattice over the elastic rods, in addition to at the junctions, is considered. It is demonstrated that the material properties in the long wavelength limit are not what one would expect from the static response of the lattice. The effect of various interactions on the dispersive properties of the triangular cell lattice is considered, including so-called truss, frame, and micro-polar interactions. Compact analytical estimates for the band widths are presented, allowing the design of structures possessing pass and/or stop bands at specific frequencies and in specified directions. The finite frequency response of several lattice structures is considered in chapter four. In particular, the dynamic anisotropy of both scalar and elastic lattices is examined. The resulting strongly anisotropic material response is linked, explicitly, to the dispersive properties of the lattice. A novel application of dynamic anisotropy to the focusing, shielding, and negative refraction of elastic waves using a flat discrete metamaterial lens'' is presented. Chapter five is devoted to the analysis, using the dynamic Green's function, of a finite rectilinear inclusion in an infinite square lattice. Several representations of the Green's function are presented, including expression in terms of hypergeometric functions, which are employed in deriving band edge expansions. It is shown that localised defect modes, characterised by displacements which decay rapidly away from the defect, can be initiated by reducing the mass of one or more lattice nodes, whilst ensuring that the mass of the nodes remains positive. For one- and three-dimensional multi-atomic lattices, there exists a bound on the contrast in mass between the defect and ambient lattice such that localised defect modes exist. However, it is shown that for the two-dimensional lattice, no such bound exists, provided that the masses remain positive. The analysis of a finite-sized defect region is accompanied by the waveguide modes that may exist in a lattice containing an infinite chain of point defects. A numerical simulation illustrates that the solution of the problem for an infinite chain can be used to predict the range of eigenfrequencies of localised modes for a finite but, sufficiently long, array of masses representing a rectilinear defect in a square lattice. Continuing with the theme of defects, chapter six examines response of a triangular thermoelastic lattice, with an edge crack under mode I loading. The response of the triangular lattice is compared with that of the corresponding continuum. The model is related to the phenomenon of thermal striping, which occurs when a structure is exposed to periodic variations in temperature. In the thermal striping regime, crack propagation is a fatiguing processes with the rate of crack growth being proportional to some power of the peak-to-peak amplitude of the stress intensity factor. An effective stress intensity factor'' for the lattice is introduced and it is demonstrated that, in the homogenised limit, the effective stress intensity factor'' is lower than the stress intensity factor of the continuum for sufficiently long cracks and low frequencies. Finally, chapter seven presents a detailed analysis of a non-singular square cloak for acoustic, out-of-plane shear elastic, and electromagnetic waves. The propagation of waves through the cloak is examined analytically and is complemented with a range of numerical illustrations. The efficacy of the regularised cloak is demonstrated and an objective numerical measure of the quality of the cloaking effect is introduced. The results presented show that the cloaking effect persists over a sufficiently wide range of frequencies. To illustrate further the effectiveness of the regularised cloak, a Young's double slit experiment is presented. The stability of the interference pattern is examined when a cloaked and uncloaked obstacle are successively placed in front of one of the apertures. A significant advantage of this particular regularised square cloak is the straightforward connection with a discrete lattice. It is shown that an approximate cloak can be constructed using a discrete lattice structure. The efficiency of such a lattice cloak is analysed and several illustrative simulations are presented. It is demonstrated that cloaking can be achieved by using a relatively simple lattice, particularly in the low frequency regime. This discrete lattice structure provides a possible avenue toward the physical realisation of invisibility cloaks.

On the elastic properties of three-dimensional honeycomb lattices

Modal properties of a cyclic symmetric hexagon lattice

This article is devoted to the study of auxetic properties of Cosserat hexagonal lattices composed of finite‐sized particles with complex connections. The description of complex connections is given; their mathematical model is elaborated and the properties are studied. The introduction of complex connections enables one varying their structure and component parameters. Due to that there arise possibilities for both simulation of nonchiral lattices with symmetrical bonds and with a chiral microstructure and construction of lattices with desired properties. The discrete and micropolar equations of the lattice are obtained. As a result, the macroparameters are expressed in terms of the lattice microparameters. The dependence of the Poisson's ratio on the lattice microparameters is obtained. It allows finding and analyzing parameters, for which the lattice possesses auxetic properties. The importance of rotational degrees of freedom of particles and chirality for the appearance of auxetic properties is shown. For verification, the results of the calculation of the Poisson's ratio obtained on the basis of theoretically obtained relations are compared with the results of numerical simulation of the stretching of the lattice.

By means of exact numerical calculations, the properties of small lattices with mixed exchange interactions (or bonds) are studied. A particular bond can be either ferromagnetic (F) or antiferromagnetic (AF). We assume equal magnitudes and equal concentrations for each kind of bonds. An important independent variable is the number of spins (N) or size of the lattice. We consider here two different two-dimensional geometries: triangular lattices (TL) and honeycomb lattices (HL).1 The maximum size is 42 for TL and 64 for HL. The distribution of these spins considers all possible rectangular and square arrays. The shape is an interesting independent variable. Each distribution of bonds for a given array is a sample. Once the sample is generated no mutation or migration of bonds is allowed. Thus for each sample the distribution of bonds is fixed. Periodic boundary conditions are used to keep the coordination number constant through the lattice. To get a statistical representation of these systems, 500 samples are considered for each array. Our main interest is to study the properties of the ground level of these lattices and their dependence with size and shape. In the present article we report the following properties for TL and HL: average energy per bond, remnant entropy, and order parameters p and h.2 The results agree well with the expected behavior toward the thermodynamic limit. A comparison with similar results reported for square lattices (SL) will be also performed. A deeper discussion is carried out for the recently defined order parameters p and h.2,3 Open image in new window Figure 1 Distribution of sites for a triangular lattice (filled circles) and for a honeycomb lattice (open circles).

The properties of lattices are strongly influenced by their nodal connectivity; yet, previous studies have focused mainly on topologies with a single vertex configuration. This work investigates the potential of demi-regular lattices, with two vertex configurations, to outperform existing topologies, such as triangular and kagome lattices. We used finite element simulations to predict the fracture toughness of three elastic-brittle demi-regular lattices under modes I, II, and mixed-mode loading. The fracture toughness of two demi-regular lattices scales linearly with relative density ρ¯, and outperforms a triangular lattice by 15% under mode I and 30% under mode II. The third demi-regular lattice has a fracture toughness KIc that scales with ρ¯ and matches the remarkable toughness of a kagome lattice. Finally, a kinematic matrix analysis revealed that topologies with KIc∝ρ¯ have periodic mechanisms and this may be a key feature explaining their high fracture toughness.

Lattice materials are extremely efficient in combining high stiffness and strength at low densities. Their architecture is a periodic assembly of bars, which, in most cases, all have the same length and cross-section. This is, however, suboptimal since the level of stress is not the same in all bars. To take these variations into account, we propose to design prismatic lattices with two different bar thicknesses. The ratio of these two thicknesses introduces a new parameter in the design of lattices. Analytical expressions are developed to capture the effect of this new parameter on the elastic modulus, failure mode and compressive strength of hexagonal and triangular lattices. This analytical work is then validated by finite element simulations and experiments performed on polymer lattices fabricated by additive manufacturing. This new parameter offers two advantages in the design of prismatic lattices. First, the thickness ratio can be used to vary the properties of a lattice without changing its relative density. Second, it allows to stiffen and strengthen the lattice along a specific loading direction and therefore, controls the degree of anisotropy. This work opens new possibilities to tailor the mechanical properties of prismatic lattices, and facilitates the creation of new materials by design.

Order-disorder in hexagonal lattices

The specific topology of the line centered square lattice (known also as the Lieb lattice) induces remarkable spectral properties as the macroscopically degenerated zero energy flat band, the Dirac cone in the low energy spectrum, and the peculiar Hofstadter-type spectrum in magnetic field. We study here the properties of the finite Lieb lattice with periodic and vanishing boundary conditions. We find out the behavior of the flat band induced by disorder and external magnetic and electric fields. We show that in the confined Lieb plaquette threaded by a perpendicular magnetic flux there are edge states with nontrivial behavior. The specific class of twisted edge states, which have alternating chirality, are sensitive to disorder and do not support IQHE, but contribute to the longitudinal resistance. The symmetry of the transmittance matrix in the energy range where these states are located is revealed. The diamagnetic moments of the bulk and edge states in the Dirac-Landau domain, and also of the flat states in crossed magnetic and electric fields are shown.

We explore the effect of antidot size on electronic and magnetic properties of graphene antidot lattices from first-principles calculations. The spin-polarized density of states, band gap, formation energy and the total magnetization of two different equilateral triangular and right triangular antidots with zigzag and mixed zigzag-armchair edges are studied. We find that although the values of band gap, formation energy and the total magnetization of both structures are different, these values may increase when the number of zigzag edges is increased. The armchair edges have no contribution in the total magnetization of right triangular antidots. The induced magnetic moments are mainly localized on the edge atoms with a maximum value at the center of each side of the triangles. We show that a spin-dependent band gap opens up in bilayer graphene as a result of antidot pattern in only one layer of the structure. Such periodic arrays of triangular antidots that cause a spin-dependent band gap around the Fermi energy can be utilized for turning graphene from a diamagnetic semimetal into a magnetic semiconductor.

The thermoelectric properties of graphene-based antidot lattices are theoretically investigated. A third nearest-neighbor tight-binding model and a fourth nearest-neighbor force constant model are employed to study the electronic and phononic band structures of graphene antidot lattices with circular, rectangular, hexagonal, and triangular antidot shapes. Ballistic transport models are used to evaluate transport coefficients. Methods to reduce the thermal conductance and to increase the thermoelectric power factor of such structures are studied. Our results indicate that triangular antidot lattices have the smallest thermal conductance due to longer boundaries and the smallest distance between the neighboring antidots. Among them, iso-triangular antidot lattices have also a large power factor and as a result a large figure of merit.

We investigate the local and global optimality of the triangular, square, simple cubic, face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices and the hexagonal-close-packing (hcp) structure for a potential energy per point generated by a Morse potential with parameters (α, r0). In dimension 2 and for α large enough, the optimality of the triangular lattice is shown at fixed densities belonging to an explicit interval, using a method based on lattice theta function properties. Furthermore, this energy per point is numerically studied among all two-dimensional Bravais lattices with respect to their density. The behavior of the minimizer, when the density varies, matches with the one that has been already observed for the Lennard-Jones potential, confirming a conjecture we have previously stated for differences of completely monotone functions. Furthermore, in dimension 3, the local minimality of the cubic, fcc, and bcc lattices is checked, showing several interesting similarities with the Lennard-Jones potential case. We also show that the square, triangular, cubic, fcc, and bcc lattices are the only Bravais lattices in dimensions 2 and 3 being critical points of a large class of lattice energies (including the one studied in this paper) in some open intervals of densities as we observe for the Lennard-Jones and the Morse potential lattice energies. More surprisingly, in the Morse potential case, we numerically found a transition of the global minimizer from bcc, fcc to hcp, as α increases, that we partially and heuristically explain from the lattice theta function properties. Thus, it allows us to state a conjecture about the global minimizer of the Morse lattice energy with respect to the value of α. Finally, we compare the values of α found experimentally for metals and rare-gas crystals with the expected lattice ground-state structure given by our numerical investigation/conjecture. Only in a few cases does the known ground-state crystal structure match the minimizer we find for the expected value of α. Our conclusion is that the pairwise interaction model with Morse potential and fixed α is not adapted to describe metals and rare-gas crystals if we want to take into consideration that the lattice structure we find in nature is the ground-state of the associated potential energy.

Interest in the elastic properties of regular lattices constructed from domain walls has recently been motivated by cosmological applications as solid dark energy. This work investigates the particularly simple examples of triangular, hexagonal, and square lattices in two dimensions and a variety of more complicated lattices in three dimensions which have cubic symmetry. The relevant rigidity coefficients are computed taking into account nonaffine perturbations where necessary, and these are used to evaluate the propagation velocity for any macroscopic scale perturbation mode. Using this information we assess the stability of the various configurations. It is found that triangular lattices are isotropic and stable, whereas hexagonal lattices are unstable. It is argued that the simple orthonormal cases of a square in two dimensions and the cube in three are stable, except to perturbations of infinite extent. We also find that the more complicated case of a rhombic dodecahedral lattice is stable, except to the existence of transverse modes in certain directions, whereas a lattice formed from truncated octahedra is unstable.

Low-density series expansions for the backbone properties of two-dimensional bond percolation clusters are derived and analysed. Expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice. Earlier series work was confined to three properties of the square lattice and was to order . The fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest Monte Carlo results. The union, intersection and length of the longest self-avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for . On the other hand, the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection, the triangular and square lattices are found to have sigificantly different dimensions. The fractal dimension of the shortest path also appears to be non-universal and we find for the square lattice and for the triangular lattice. Critical amplitude ratios are considered and found to be in agreement with theoretical inequalities.

Small Ising lattices with both ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) and increasing numbers of spins are studied by means of two independent methods: computational solutions to the Hamiltonian problem and topological counting of frustration paths. Equal magnitudes and concentrations are assumed for both types of bonds. Two different geometries are considered: square lattices (SL's) with coordination number 4 and triangular lattices (TL's) with coordination number 6. Two-dimensional samples with a total number of spins N between 4 and 64 are considered for SL's, while N is varied between 4 and 44 for TL's. They are distributed in two-dimensional arrays where periodic boundary conditions are imposed. After an array is selected, bond distributions (samples) are independently and randomly generated in fixed positions. The physical parameters are then calculated exactly for each sample. The emphasis here is on the ground-state properties and their dependence with size and shape for the two kinds of lattices. All magnitudes correspond to a basic statistics over a large number of samples for each array. The following magnitudes are reported: ground-state energy per bond, frustration segment, abundance of first excited states, remnant entropy, low-temperature specific heat, and site order parameters q, p, and h. Parameters p and h are introduced here, showing advantages over other similar magnitudes. The results are in good correspondence with analytic studies for the thermodynamic limit. This means that the spin site correlation (p) tends to vanish as N grows. However, we have found that the shape dependence modulates the behavior of these systems toward the thermodynamic limit. There is no tendency to vanish for the bond correlation parameter (h). For both kinds of lattices h might be a constant independent of size and shape.