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Two-dimensional waves in extended square lattice
Abstract
We consider a two-dimensional square lattice model extended by additional not closed neighboring interactions. We assume the elastic forces between the masses in the lattice to be nonlinearly dependent on the spring elongations. First, we use an analysis of the linearized discrete equations to reveal the influence of additional interactions on the properties of the dispersion relation for longitudinal and shear plane waves. Then we develop an asymptotic procedure to obtain continuum two-dimensional non-linear equations to study the transverse instability of weakly non-linear localized plane longitudinal and shear waves. We find that the additional interactions used in the model may affect the sign of the amplitude of the plane strain waves (existence of compression (minus sign) or tensile (plus sign) plane waves) and their transverse stability.
... A development of this approach is needed in two-dimensional lattice models. Recently various two-dimensional nonlinear lattices were studied in [4,5]. As a rule, an analytical study of the discrete nonlinear problem is impossible, and a continuum limit was applied in [4,5] in order to obtain the governing nonlinear partial differential equations for the strains. ...
... Recently various two-dimensional nonlinear lattices were studied in [4,5]. As a rule, an analytical study of the discrete nonlinear problem is impossible, and a continuum limit was applied in [4,5] in order to obtain the governing nonlinear partial differential equations for the strains. ...
... It was found in [4,5] that longitudinal strains obey the famous Kadomtsev-Petviashvili equation [6] whose solutions including localized ones are well known. An equation with other features has been derived for shear strains. ...
Two-dimensional localized strain wave solutions of the nonlinear equation for shear waves in two-dimensional lattices are studied. The corresponding equation does not possess an invariance in one of the spatial direction while its exact plane traveling wave solution does not reflect that. However, the numerical simulation of a two-dimensional localized wave reveals a non-symmetric evolution.
... Nonlinear discrete equations are not solved analytically as a rule [33]. That is why a continualization of the discrete model is widely used [34][35][36][37]. In this case the discrete displacements are approximated by the continuum functions that yields the obtaining of differential equations instead of the original discrete ones. ...
A new nonlinear lattice model for an influence of the hydrogen concentration on the elastic constants of the lattice model of a material is developed. A weakly nonlinear long-wavelength continuum model is considered, and a model nonlinear equation for the dynamics of concentration of hydrogen is obtained asymptotically. The model predicts a decrease in the stiffness coefficient as well as a light local moving increase due to the localized nonlinear concentration wave propagation.
... One of most common modeling approaches for the incorporation of nonlinear effects in the wave propagation characteristics of structural systems, is the so-called perturbation approach [26]. Perturbation methods have found application in the modeling of both weakly [27,28] and strongly non-linear systems [29][30][31][32]. They make use of the eigenfrequencies of the linear system in the computation of its nonlinear attributes, with the correction of the displacement and frequency fields to depend on the mode of interest [33]. ...
In the current work, we study the role of chirality and non-centrosymmetry on the nonlinear wave propagation characteristics of periodic architectured media. The considered nonlinearities arise from the higher-order inner element kinematics of the periodic media and are therefore directly related to its structural pattern. Regarding centrosymmetric designs, the frequency corrections obtained -in the context of the Lindstedt-Poincare method- suggest that chiral architectures are more sensitive to inner kinematic nonlinearities than well-known, achiral lattice designs. In particular, for hexachiral lattice designs, non-negligible frequency corrections are obtained, not only for the primal eigenmode, but also for higher-order modes, extensively modifying the linear band diagram structure. To the contrary, for achiral, triangular and square lattice designs, inner kinematic nonlinearities mainly influence the primal, lowest eigenmode, with the higher-order modes to remain practically unaffected. Non-centrosymmetric inner designs modify the linear and nonlinear wave propagation material attributes both for chiral and achiral lattice patterns. However, the frequency ranges affected are strongly lattice dependent, with hexachiral and triangular lattices to be primarily influenced in their high frequency range, contrary to square lattices, which are mainly affected in their low frequency region.
... Nonlinear modeling in lattices is mainly based on the consideration of nonlinear stiffness of the springs between the masses [10,[12][13][14][15][16][17][18]. This corresponds to the physical nonlinearity. ...
It is shown that angular stiffness in the hexagonal lattice model plays a significant role in the geometrical nonlinear terms in the equations of the continuum limit. A geometrically nonlinear discrete model is formulated for the hexagonal lattice by considering the interaction of two sublattices. An asymptotic procedure is developed in order to obtain the nonlinear coupled equations of motion in the continuum limit of the discrete model. An interaction of longitudinal and shear plane strain waves is studied by using the solutions of the obtained equations.
... This is a broad problem, closely related to the issues of dynamic loads and dissipative characteristics of structures and bases discussed above. At present, it is sometimes suggested that the applied theory of linear oscillations has completely exhausted itself [17][18][19][20]. One can hardly agree with this opinion. ...
The value of vibration, defined as the oscillatory phenomenon of a solid subjected to the action of force, appeared relatively recently. In fact, mechanical vibration became real only after the industrial revolution. Mechanical oscillatory phenomena, for example, due to working equipment and train movement, began to emerge only by the second half of the eighteenth century. The intensive use of new machines and mechanisms suddenly filled the world, until then, quiet and silent, with new intense noises and vibrations. Different ways of calculating the estimated oscillations of engineering structures, methods of reducing vibrations to the level allowed or their complete isolation, determine the purpose and main problems of the applied theory of mechanical vibrations. In engineering practice, we are almost always interested in predicting the response of a structure or mechanical system to an external effect. For example, we may need to predict the response of a bridge or high-rise building to wind loads, earthquakes, or ground motion. In mechanics and construction, resonant catastrophe describes the destruction of a building or a technical mechanism by induced vibrations at the resonant frequency of a system, which causes its oscillation. Periodic excitation optimally transfers vibration energy to the system and stores it there. Because of this repeated and additional energy input, the system sways more and more until its load limit is exceeded. Frequent reason for these disasters were periodic oscillations of bridges. Periodic fluctuations can be described as a body movement that regularly goes through an equilibrium position. Any oscillatory movement of a mechanical system relative to its equilibrium position is called vibration.
... The described procedure can be extended to the study of weakly transversely perturbed shear waves. Previous studies of lattices of another structure revealed important variations in the governing equation from that of the longitudinal waves (Porubov and Berinskii, 2016;Porubov et al., 2018). ...
A two-dimensional graphene lattice modelLattice modelgraphene with translational and angular interactions between the elements of two sublattices is considered. The nonlinearity is introduced via nonlinearly elastic translational forces between the masses in the lattice. The nonlinear strain behaviour is studied using the continuum limit of the lattice that results in the derivation of four coupled governing equations describing the dynamics of interacting sublattices. An asymptotic procedure is developed to account for weakly transversely perturbed nonlinear longitudinal plane waves giving rise to a two-dimensional nonlinear governing equation for longitudinal strains.
... The composite sample is a cylinder with an internal through hole of complex geometry (the cylinder goes into an expanding cone). For the measurements, a special design was developed using elastic elements, allowing to ensure a guaranteed preload of the contact electrode of the currentgenerating and measuring circuits as to the smooth surface of the hole (Ra ~ 1 μm), so to the outer surface of the sample, having structures of honeycomb-shaped [36][37][38][39]. ...
The paper devotes a new general experimental method for studying the anisotropy of conducting materials of various functional purposes. The difference between the presented method and the previously known one is in the high rate of anisotropy assessment, ensuring the possibility of anisotropy assessment of the properties of materials by the volume of the sample or product under investigation during their manufacture or operation. The procedure for measuring the anisotropy of electrophysical properties and the results of the study of some samples are presented.
... In the same time discrete models can describe in all details the material behaviour in the vicinity of crack tips, changes in geometry and other geometrical singularities, see, e.g. Mishuris et al. (2007Mishuris et al. ( , 2009 ;Slepyan (2002); Sharma (2015aSharma ( , 2017c; Porubov and Andrianov (2013); Porubov et al. (2018); Gorbushin et al. (2018), where the efficiency of discrete lattice models is demonstrated. As an example, we recently provided the comparison of the surface wave propagation within the Gurtin-Murdoch model and the lattice dynamics which shows similarities in dispersion relations for both models after some re-scaling, see Eremeyev and Sharma (2019). ...
Within the lattice dynamics formulation, we present an exact solution for anti-plane surface waves in a square lattice strip with a surface row of material particles of two types separated by a linear interface. The considered problem is a discrete analog of an elastic half-space with surface stresses modelled through the simplified Gurtin-Murdoch model, where we have an interfacial line separating areas with different surface elastic properties. The main attention is paid to the transmittance and the reflectance of a wave across the interface. The presented results shed a light on the influence on surface waves of surface inhomogeneity in surface elastic properties such as grain and subgrain boundaries.
... An influence of non-neighboring particles on the processes in discrete media is an important issue. The dynamic features of the discrete systems with the additional non-neighboring interactions are studied in [14][15][16]. The detailed influence of the particles of the second neighbor interaction on the heat propagation has not been studied previously. ...
An influence of the second neighbor interaction on the process of heat propagation in a one‐dimensional crystal is studied. Previously developed model of the ballistic nature of the heat transfer is used. It is shown that the initial thermal perturbation evolves into two consecutive thermal wave fronts propagating with finite and different velocities. The velocity of the first front corresponds to the maximum group velocity of the discrete crystalline model. The velocity of the second front is determined by the second group‐velocity extremum, which arises at a certain ratio between the stiffnesses of the first and second neighbor interaction in the lattice.
... An alternative approach uses the numerical technique of molecular dynamics simulations, see, e.g., Miller and Shenoy (2000); Shenoy (2005), where the surface elastic moduli are determined from direct atomistic simulations. Let us note that the lattice dynamics as described by Brillouin (1946); 20 Born and Huang (1985) provides the possibility to solve various dynamical problems involving waves with attention focussed on the influence of microstructure in as much detail as possible, including even the surface microstructure, see Slepyan (2002); Mishuris et al. (2007Mishuris et al. ( , 2009 ;Porubov and Andrianov (2013); Sharma (2017b);Porubov et al. (2018) and the references therein. ...
We present a comparison of the dispersion relations derived for anti-plane surface waves using the two distinct approaches of the surface elasticity vis-a-vis the lattice dynamics. We consider an elastic half-space with surface stresses described within the Gurtin-Murdoch model, and present a formulation of its discrete counterpart that is a square lattice half-plane with surface row of particles having mass and elastic bonds different from the ones in the bulk. As both models possess anti-plane surface waves we discuss similarities between the continuum and discrete viewpoint. In particular, in the context of the behaviour of phase velocity, we discuss the possible characterization of the surface shear modulus through the parameters involved in lattice formulation.
... An aternative is to apply a combined discrete-continuum approach when a continuum limit of the discrete equations is analyzed. In particular, we successfully used this approach to study nonlinear wave processes in several twodimensional lattices [39,40]. ...
It is shown that plane longitudinal nonlinear strain waves in a 2D graphene-type hexagonal lattice are described by a nonlinear double dispersion equation previously developed for the description of waves in an elastic rod. A procedure is developed to derive the governing equation as a continuum limit of the original lattice model. The lattice is described by an interaction of two sub-lattices and both translational and angular interactions between the lattice masses are taken into account.
We apply the asymptotic approach for the description of two plane solitons interacting under an angle to each other in 2D Toda lattices with the quadratic or triangle/hexagonal structure. We demonstrate the influence of lattice anisotropy on dispersion relations and soliton structures. With the help of the asymptotic approach, we study soliton interactions in the long-wave approximation and show that it is possible to find the spatial phase shifts experienced by each soliton due to the interaction with its counterpart. In contrast to earlier studied soliton interactions within the quasi-one-dimensional equations, here we consider a fully two-dimensional arrangement with an arbitrary angle between soliton fronts.
In this work, solitary wave solutions of particle mechanical metamaterials are studied, in which the mass-in-mass structure with local resonators is considered. The Hertzian contact theory is used to describe adjacent particles in a precompressed granular chain. The governing wave equations are decoupled, and the expressions of bright, dark, and peaked solitary waves are derived, respectively. According to the results, both the wave velocity and prestress can affect the propagation of solitary waves. The amplitudes of bright and peaked solitary waves are smaller when a larger prestress is applied, which are different from the dark solitons. Furthermore, the wave widths become larger as the prestress increases.
Mechanical properties of a granular consolidated medium depend on the geometry of the microparticles, their location, and the forces of interaction between them. One of the main goals of mathematical modeling of such media is obtaining equations of motion and equations of state, which are capable to describe a discrete nature of a medium.
The principles of the structural modeling method, the development of the theoretical foundations of which this monograph is devoted, are formulated in the first chapter. Moreover, the problem of the applicability of the classical mechanics laws to a theoretical description of media with micro- and nanostructure is discussed here.
The work is devoted to the description of unsteady thermal processes in low-dimensional structures. To obtain the relationship between the microscopic and macroscopic descriptions of solids, it is necessary to understand the heat transfer mechanism at the micro-level. At the latter, in contrast to the macro-level, analytical, numerical, and experimental studies demonstrate significant deviations from the Fourier’s law. The paper bases on the ballistic heat transfer model, according to which the heat is carried by the thermal waves. This effect can be applied, for example, to signal transmission and heat removal problems. The influence of non-nearest neighbors on processes in discrete media, as well as processes in polyatomic lattices, is investigated. To describe the evolution of the initial thermal perturbation, the dispersion characteristics and group velocities in one-dimensional crystal are analyzed for (i) a diatomic chain with variable masses or stiffnesses and for (ii) a monatomic chain with regard for interaction with second neighbors. A fundamental solution to the heat distribution problem for the corresponding crystal models is obtained and investigated. The fundamental solution allows to obtain a description of waves traveling from a point source, and can serve as the basis for constructing all other solutions. In both cases, the solution consists of two thermal fronts moving one after another with different speeds and intensities. Quantitative estimates of the intensity of the thermal wave front are given, and the dynamics of changes in the velocities and intensities of the waves depending on the parameters of the problem is analyzed. Thus, a simple method for estimating the wavefronts intensity is proposed and tested on two models. These results can be used to identify and analyze those parts of the wave processes that are of interest in terms of interpretation of the effects observed in the experiments.
We present a comparison of the dispersion relations derived for anti-plane surface waves using the two distinct approaches of the surface elasticity vis-a-vis the lattice dynamics. We consider an elastic half-space with surface stresses described within the Gurtin–Murdoch model, and present a formulation of its discrete counterpart that is a square lattice half-plane with surface row of particles having mass and elastic bonds different from the ones in the bulk. As both models possess anti-plane surface waves we discuss similarities between the continuum and discrete viewpoint. In particular, in the context of the behaviour of phase velocity, we discuss the possible characterization of the surface shear modulus through the parameters involved in lattice formulation.
The dynamics of two-dimensional lattice structures is studied. The complexity of the structure includes non-neighboring interactions between the lattice masses, consideration of both translational and rotational interactions and also their nonlinear character (physical nonlinearity). The asymptotic procedures are developed to obtain the governing nonlinear equations of motion in the continuum limit. The equations obtained are studied both analytically and numerically. Of special interest are the propagation and transverse instability of the plane solitary strain waves. It is shown that the dynamics of longitudinal and shear waves is different in various two-dimensional lattices. The relationships for the elastic constants are obtained to characterize the type of the localized strain waves (tensile or compression), their transverse instability and possible auxetic behavior. Numerical solutions are obtained that describe unstable and stable dynamics of the plane longitudinal and shear waves.
Рассматриваются нестационарные тепловые процессы в низкоразмерных структурах. Понимание теплопередачи на микроуровне необходимо для получения связи между микро- и макроскопическим описанием твердых тел. На макроскопическом уровне распространение тепла описывается законом Фурье. Однако на микроскопическом уровне аналитические, численные и экспериментальные исследования показывают существенные отклонения от этого закона. В работе используется созданная ранее модель теплопереноса на микроуровне, имеющая баллистический характер. Изучается влияние не ближайших соседей на тепловые процессы в дискретных средах, а также рассматривается распространение тепла в многоатомных решетках. Для описания эволюции начального теплового возмущения проведен анализ дисперсионных характеристик и групповых скоростей в одномерном кристалле для двухатомной цепочки с чередующимися массами или жесткостями и одноатомной цепочки с учетом взаимодействия со вторыми соседями. Получено и исследовано фундаментальное решение задачи распространения тепла для соответствующих моделей кристаллов. Фундаментальное решение позволяет получить описание волн, бегущих от точечного источника, и служит основой для построения всех остальных решений. Для обеих цепочек решение состоит из двух фронтов, двигающихся друг за другом с различными скоростями и характеристиками интенсивности. Приведены количественные оценки коэффициентов интенсивности фронта тепловой волны, проанализирована динамика изменения скоростей и интенсивностей волн в зависимости от параметров задачи. Выявлено два механизма эволюции фронта тепловых волн в одномерных дискретных системах. Представленные результаты могут быть использованы для корректной интерпретации экспериментов по нестационарному баллистическому теплопереносу в кристаллах.
Within the lattice dynamics formulation, we present an exact solution for anti-plane surface waves in a square lattice strip with a surface row of material particles of two types separated by a linear interface. The considered problem is a discrete analog of an elastic half-space with surface stresses modelled through the simplified Gurtin-Murdoch model, where we have an interfacial line separating areas with different surface elastic properties. The main attention is paid to the transmittance and the reflectance of a wave across the interface. The presented results shed a light on the influence on surface waves of surface inhomogeneity in surface elastic properties such as grain and subgrain boundaries.
This chapter analyzes some aspects of the remarkably rich dynamics of a “fractional” random walk strategy, which referred to as “fractional random walk” (FRW). The FRW appears to be particularly interesting when applied as random walk based search strategy due to its relationship to anomalous transport and diffusion phenomena and Levy flights. The chapter explains the FRW on periodic and infinite d‐dimensional cubic primitive lattices with special focus on recurrence/transience behavior emerging in the infinite lattice limit. It describes basic features of Markovian walks on regular networks. The chapter deals with asymptotic fractal scaling in the transient regime of the set of distinct nodes visited by the fractional walker in an infinitely long walk. It shows that the set of distinct nodes visited constitutes a stochastic fractal of isotropic (spherical) symmetry with respect to the departure node.
This work focuses on investigation of structural (phase) transformations in crystal lattices from
continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the
sense of the Cauchy–Born energy, is constructed using long-wave approximation, and its strong ellipticity
domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular
and square lattices, and the number of the domains depends on the interaction potential parameters.
Non-convex energy profiles and stress–strain diagrams, which are typical for materials allowing twinning
and phase transformations, are obtained on the straining paths which connect the domains and cross nonellipticity
zones. The procedures of the lattice stability examinations and estimation of energy relaxation by
means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope
of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The
MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase
structures.
Linear elastic deformation of the two-dimensional triangular lattice with multiple vacancies is considered. Closed-form analytical expressions for displacement field in the lattice with doubly periodic system of vacancies are derived. Effective elastic moduli are calculated. The results are compared with the ones obtained by molecular dynamics simulations of a lattice with random distribution of vacancies. At low vacancy concentrations, less than $4\%$, random and periodic distributions of vacancies produce the same effect on elastic moduli. One of the main goals is to examine the possibilities and limitations of modeling of the lattice with vacancies by an elastic continuum with holes. It is found that the effective elastic properties are modeled adequately, provided the shape of the holes is chosen appropriately. On the contrary, the strain field, in particular, strain concentration differs significantly.
In the last years, the rapid increase in the technical capability to control and design materials at the nanoscale has pushed toward an intensive exploitation of new possibilities concerning optical, chemical, thermoelectrical and electronic applications. As a result, new materials have been developed to obtain specific physical properties and performances. In this general picture, it was natural that the attention toward mechanical characterization of the new structures was left, in a sense, behind. Anyway, once the theoretically designed objects proceed toward concrete manufacturing and applications, an accurate and general description of their mechanical properties becomes more and more scientifically relevant. The aim of the paper is therefore to discuss new methods and techniques for modeling the behavior of nanostructured materials considering surface/interface properties, which are responsible for the main differences between nano- and macroscale, and to determine their actual material properties at the macroscale. Our approach is intended to study the mechanical properties of materials taking into account surface properties including possible complex inner microstructure of surface coatings. We use the Gurtin–Murdoch model of surface elasticity. We consider the inner regular and irregular surface thin coatings (i.e., ordered or disordered nanofibers arrays) and present few examples of averaged 2D properties of them. Since the actual 2D properties depend not only on the mechanical properties of fibers or other elements of a coating, but also on the interaction forces between them, the analysis also includes information on the geometry of the microstructure of the coating, on mechanical properties of elements and on interaction forces. Further we use the obtained 2D properties to derive the effective properties of solids and structures at the macroscale, such as the bending stiffness or Young’s modulus.
Various continuum limits of the original discrete hexagonal lattice model are used to obtain transverse weakly nonlinear equations for longitudinal waves. It is shown, that the long wavelength continuum limit gives rise to the Kadomtsev–Petviashvili equation, while another continuum limit results in obtaining two-dimensional generalization of the nonlinear Schrödinger equation.
This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy–Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular and square lattices, and the number of the domains depends on the interaction potential parameters. Non-convex energy profiles and stress–strain diagrams, which are typical for materials allowing twinning and phase transformations, are obtained on the straining paths which connect the domains and cross non-ellipticity zones. The procedures of the lattice stability examinations and estimation of energy relaxation by means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase structures.
An asymptotic procedure is developed to obtain governing two-dimensional nonlinear equations as a result of the continuum limits of the original discrete hexagonal lattice model. Possible continualization is analyzed on the basis of linearized discrete equations. New weakly nonlinear continuum equations for plane and weakly transverse disturbed shear waves in a hexagonal lattice are obtained using different continuum limits. It is shown that nonlinear plane shear waves are described different from the model of isotropic continuum.
Thermal expansion of a classical chain with pair interactions performing longitudinal and transverse vibrations is investigated. Corresponding equations of state are derived analytically using series expansions of pressure and thermal energy with respect to deformations of the bonds caused by thermal motion. In the first approximation the equation of state has Mie–Grüneisen form. The dependence of Grüneisen parameter on deformation of the chain is obtained. For Lennard-Jones-like potential Grüneisen parameter varies with deformation from minus infinity (at zero stretching) to plus infinity (at the breakage point). Necessary and sufficient condition for negative thermal expansion at low thermal energies is formulated. Using this condition the potential giving rise to negative thermal expansion in the given range of deformations can be designed. It is shown that at small deformations and finite thermal energies Mie–Grüneisen equation of state strongly overestimates the absolute value of pressure. More accurate nonlinear equation of state is derived for this case. The equation implies that thermal pressure in the unstretched chain is proportional to square root of thermal energy. In the vicinity of the deformation, corresponding to zero Grüneisen parameter, the chain demonstrates negative thermal expansion at low temperatures and positive thermal expansion at higher temperatures. This phenomenon is qualitatively described by the nonlinear equation of state derived in the present paper. The theoretical findings are supported by the results of molecular dynamics simulations.
We analyze one-dimensional discrete and quasi-continuous linear chains of
$N>>1$ equidistant and identical mass points with periodic boundary conditions
and generalized nonlocal interparticle interactions in the harmonic
approximation. We introduce elastic potentials which define by Hamilton's
principle discrete "Laplacian operators" ("Laplacian matrices") which are
operator functions ($N\times N$-matrix functions) of the Laplacian of the
Born-von-Karman linear chain with next neighbor interactions. The non-locality
of the constitutive law of the present model is a natural consequence of the
{\it non-diagonality} of these Laplacian matrix functions in the $N$
dimensional vector space of particle displacement fields where the periodic
boundary conditions (cyclic boundary conditions) and as a consequence the
(Bloch-) eigenvectors of the linear chain are maintained. In the
quasi-continuum limit (long-wave limit) the Laplacian matrices yield "Laplacian
convolution kernels" (and the related elastic modulus kernels) of the non-local
constitutive law. The elastic stability is guaranteed by the positiveness of
the elastic potentials. We establish criteria for "weak" and "strong"
nonlocality of the constitutive behavior which can be controlled by scaling
behavior of material constants in the continuum limit when the interparticle
spacing $h\rightarrow 0$. The approach provides a general method to generate
physically admissible (elastically stable) {\it non-local constitutive laws} by
means of "simple" Laplacian matrix functions. The model can be generalized to
model non-locality in $n=2,3,..$ dimensions of the physical space.
This review presents the potential that lattice or spring network models hold for micromechan-ics applications. The models have their origin in the atomistic representations of matter on one hand, and in the truss-type systems in engineering on the other. The paper evolves by first giving a rather detailed presentation of one-dimensional and planar lattice models for classical continua. This is followed by a section on applications in mechanics of composites and key computational aspects. We then return to planar lattice models made of beams, which are a discrete counterpart of non-classical continua. The final two sections of the paper are devoted to issues of connectivity and rigidity of networks, and lattices of disordered rather than peri-odic topology. Spring network models offer an attractive alternative to finite element analyses of planar systems ranging from metals, composites, ceramics and polymers to functionally graded and granular materials, whereby a fiber network model of paper is treated in consider-able detail. This review article contains 81 references.
The paper focuses on continuous models derived from a discrete microstructure. Various continualization procedures that take into account the nonlocal interaction between variables of the discrete media are analysed.
Interaction of two long-crested shallow water waves is analysed in the framework of the two-soliton solution of the Kadomtsev-Petviashvili equation. The wave system is decomposed into the incoming waves and the interaction soliton that represents the particularly high wave hump in the crossing area of the waves. Shown is that extreme surface elevations up to four times exceeding the amplitude of the incoming waves typically cover a very small area but in the near-resonance case they may have considerable extension. An application of the proposed mechanism to fast ferries wash is discussed.
Elastic networks model is used to model a class of two–dimensional cellular structures including re–entrant and regular honeycombs is considered. Two mechanical parameters (longitudinal stiffness and torsional stiffness) are used to describe the interaction between the nodes in a lattice for a given geometry. Effective properties with emphasis on the auxeticy of solid materials corresponding to the considered microstructures are investigated. A method based on comparison of the strain energies at macro- and microscale is proposed to determine the effective properties of continuum and can be applied to other cellular materials. All components of stiffness tensor are determined as functions of interaction parameters. Thus, the general parametric study of the honeycomb structures is made possible. Re–entrant honeycomb structures demonstrate the auxetic properties at any combination of parameters whereas the regular honeycombs can have both positive and negative Poisson's ratios. Comparison with other model shows that the 2-parameter elastic networks can successfully describe isotropic materials. However in cases of higher anisotropy this model can be used within certain limitations.
We present a new mechanical model of interatomic bonds, which can be used to describe the elastic properties of the carbon allotropes, such as graphite, diamond, fullerene, and carbon nanotubes. The interatomic bond is modeled by a hyperboloid–shape truss structure. The elastic characteristics of this bond are determined. Previous known structural models also used elastic elements (beams, trusses) to simulate a carbon bond. However unlike them our model satisfies to the correct ratio of the longitudinal and lateral stiffness, observed from the previous experimental and theoretical results. Parameters of the bond in application for graphene and diamond were determined.
A two-dimensional model of an anisotropic crystalline material with cubic symmetry is considered. This model consists of a square lattice of round rigid particles, each possessing two translational and one rotational degree of freedom. Differential equations that describe propagation of elastic and rotational waves in such a medium are derived. A relationship between three groups of parameters is found: second-order elastic constants, acoustic wave velocities, and microstructure parameters. Values of the microstructure parameters of the considered anisotropic material at which its Poisson’s ratios become negative are found.
Single-degree-of-freedom systems -- Autonomous two-degree-of-freedom
symmetric cubic systems with close natural frequencies -- Non-autonomous
two-degree-of-freedom cubic systems with close natural frequencies --
Nonlinear flexural free and forced oscillations of a circular ring --
Localized normal modes in a chain of nonlinear coupled oscillators --
Nonlinear dynamics of coupled oscillatory chains -- Nonlinear dynamics
of strongly non-homogeneous chains with symmetric characteristics --
Transversal dynamics of one-dimensional chain on nonlinear asymmetric
substrate.
In the analysis of dynamics of an ideal system as well as a system with point defects, the role of interaction is considered not only for the nearest neighbors. The Green’s function is constructed for steady-state vibrations of a chain at all possible frequencies. It is shown that, if the interaction with the next-to-nearest neighbors is taken into account, the Green’s function inevitably becomes double partial, the nature of its two components depending significantly on its eigenfrequency. It is found that the Green’s function for frequencies of the continuous spectrum of small vibrations has one component of the plane wave type, while the other component is localized near the source of perturbations. Such a Green’s function describes the so-called quasilocal vibrations. At certain discrete frequencies falling in the continuous spectrum, the quasilocal vibration is transformed into local vibration (that does not propagate to infinity). The conditions of applicability of differential equations with fourth spatial derivative are analyzed for describing the longwave vibrations of the atomic chain. Relations between parameters of atomic interactions permitting the use of such equations are formulated. Asymptotic forms of soliton fields in a nonlinear medium with spatial dispersion are discussed. It is shown that most of the soliton parameters are determined by the dispersion relation for the linearized equation.
This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.
1. Introduction.- 2. The Binary Collision Model.- 2.1 Laboratory System.- 2.2 Centre-of-Mass System.- 2.3 Relations Between Laboratory and Centre-of-Mass Systems.- 2.4 Energy Transfer.- 2.5 Classical Scattering Theory.- 2.6 Asymptotic Trajectories.- 2.7 Determination of the Scattering Angle and the Time Integral.- 2.8 Limitations of the Binary Collision Approximation.- 2.9 Limitations of the Classical Mechanics Treatment.- 3. Classical Dynamics Model.- 3.1 Newton's Equations.- 3.2 Integration of Newton's Equations.- 3.2.1 Central Difference Scheme.- 3.2.2 Average Force Method.- 3.2.3 Euler-Cauchy Scheme.- 3.2.4 Predictor-Corrector Scheme.- 3.2.5 The Verlet Scheme.- 3.2.6 Nordsieck Method.- 3.3 The Time Step, Bookkeeping.- 4. Interaction Potentials.- 4.1 Screened Coulomb Potentials.- 4.2 The Born-Mayer Potential.- 4.3 Attractive Potentials.- 4.4 Combined Potentials.- 4.5 Empirical Potentials.- 4.6 Embedded Atom Method.- 4.7 Analytical Methods.- 4.8 Comparison of Potentials.- 5. Inelastic Energy Loss.- 5.1 Local Electronic Energy Loss.- 5.2 Continuous Electronic Energy Loss.- 5.3 Comparison.- 6. Thermal Vibrations and Specific Energies.- 6.1 Thermal Vibrations.- 6.2 Specific Energies.- 6.2.1 Cutoff Energy.- 6.2.2 Displacement Energy.- 6.2.3 Bulk Binding Energy.- 6.2.4 Surface Binding Energy.- 7. Programs Based on the BCA Model.- 7.1 Random Target Structures.- 7.2 Monte Carlo Programs.- 7.3 Crystalline Targets.- 7.4 Lattice Programs.- 7.5 TRIM.SP and TRIDYN.- 7.5.1 TRIM.SP.- 7.5.2 TRIDYN.- 7.6 MARLOWE.- 8. Programs Based on the Classical Dynamics Model.- 8.1 Stable, Metastable and Quasi-Stable Programs.- 8.2 Classical Dynamics Programs.- 9. Trajectories.- 10. Ranges.- 10.1 Definitions.- 10.2 Literature.- 10.3 Examples.- 11. Backscattering.- 11.1 Definitions.- 11.2 Literature.- 11.3 Examples.- 12. Sputtering.- 12.1 Definitions.- 12.2 Negative Binomial Distribution.- 12.3 Literature.- 12.4 Examples.- 13. Radiation Damage.- 13.1 Definitions.- 13.2 Component Analysis.- 13.3 Fuzzy Clustering.- 13.4 Literature.- 13.5 Examples.- Abbreviations Used in the Tables.- Constants.- References.- Author Index.
The results of numerical simulations are presented for propagation of solitary waves along an auxetic infinite plate. An analysis of amplitudes and velocities of the waves is described for various initial pulses for both common and auxetic materials.
Elastic properties of a rotator phase of hard homonuclear dumbbells are determined by constant pressure Monte Carlo simulations. Simple approximations expressing the influence of molecular anisotropy on the elastic constants and Poisson’s ratio are found. The maximum density at which the dumbbells with their mass centers fixed at the lattice sites can freely rotate is observed to be strongly correlated with the density at which C12=C44. It is also shown that Poisson’s ratio measured along any direction and averaged with respect to directions transverse to it is always positive. At any fixed pressure, the averaged Poisson ratio increases with increasing dumbbell anisotropy.
Poisson's ratio is, for specified directions, the ratio of a lateral contraction to the longitudinal extension during the stretching of a material. Although a negative Poisson's ratio (that is, a lateral extension in response to stretching) is not forbidden by thermodynamics, this property is generally believed to be rare in crystalline solids. In contrast to this belief, 69% of the cubic elemental metals have a negative Poisson's ratio when stretched along the [110] direction. For these metals, we find that correlations exist between the work function and the external values of Poisson's ratio for this stretch direction, which we explain using a simple electron-gas model. Moreover, these negative Poisson's ratios permit the existence, in the orthogonal lateral direction, of positive Poisson's ratios up to the stability limit of 2 for cubic crystals. Such metals having negative Poisson's ratios may find application as electrodes that amplify the response of piezo-electric sensors.
The directional behaviour of Young's modulus, shear modulus, and Poisson's ratio are expressed for a number of crystallographic planes for cubic materials. Their behaviour as a function of direction on a particular plane is shown to be a simple sum of sines and cosines of twice and sometimes of four times the angle. A general expression gives the coefficients for the sines and cosines and a table gives the coefficients for 19 planes on the standard triangle of the stereographic projection. Superposed on the stereographic projection are shown polar plots of the shear modulus Gprime12 (1prime direction normal to the plane) for silicon. Examples are also given showing G12prime for copper for some planes of zones [1bar above 00], [1bar above 10] and [2bar above 10].
The global maximum and global minimum Poisson's ratio (PR) surfaces, regions of different auxetic behaviour and the domains of the different extreme directions of cubic materials are shown and discussed in terms of the elastic moduli ratios, , where K is the bulk modulus and are shear moduli. A straightforward way is given to classify any cubic material as being auxetic, nonauxetic and partially auxetic, as well as calculating its extreme PR values. A decomposition of the XY plane with the elastic constant ratio is introduced to facilitate the interpretation of cubic materials under hydrostatic pressure, , and isotropic tension, , which are two qualitatively different situations. Using this representation it is demonstrated that cubic materials with can be auxetic only under ‘negative’ pressure (). It is shown that the influence of pressure on the auxetic behaviour is different for the positive and negative cases. In particular, a cubic material with may be auxetic at negative as well as positive pressure. The work demonstrates the crucial role of P in obtaining desired auxetic behaviour. Microscopic mechanisms which tune the cubic system to targeted regions in the XY plane are investigated with a fcc crystal of particles interacting via the pairwise spherically symmetric potential, . It is found that all fcc static models in which the range of interaction is only between nearest neighbour particles are placed on a universal curve in the XY plane. It is shown that auxetic behaviours can be achieved with simple static analytic forms. The influence of the next neighbour interactions on the universal curve is also considered. In order to assess the role of thermal fluctuations, molecular dynamic simulations for two different (the Lennard–Jones (LJ) and the tethered particle potentials) were performed. At low temperatures the studied systems are well represented by the universal curve and on increasing temperature a systematic departure from it is observed.
Nonlinear oscillations of a single particle in the potential well and for one-dimensional chain of interacting particles are considered. The law of interaction is of Lennard-Jones type, mimicking interaction in atomic systems. Similarities in average behaviour of the systems with one and many degrees of freedom are shown. Time averaging for the random oscillations is used to obtain thermodynamic characteristics such as pressure, specific volume, and thermal energy. Second order equation of state is obtained, which is valid in the conditions of strong extension, where fails the widely used Mie–Gr€ u uneisen equation of state.
The dependence of the elastic moduli of a nanocrystal on its size is investigated theoretically with reference to a two-dimensional
single-crystal strip. It is shown that the uncertainty (of a fundamental nature) in the size of a nanocrystal causes the determination
of many of its mechanical characteristics to be ambiguous. It is found that the Cauchy-Green relations are modified and the
elastic-constant tensor ceases to be symmetric; the size and shape of a nanocrystal render its mechanical properties more
anisotropic. For a single-crystal strip, the Poisson ratio decreases and the Young modulus increases with decreasing thickness
of the strip; in the case of a very thin crystal film (two atomic layers thick), these elastic moduli can differ from their
macroscopic values by a factor of two. The size effects which make the continuum elasticity theory inapplicable to nanocrystals
are estimated. The size effects that occur when the molecular dynamics method is applied for modeling macroscopic objects
are also discussed.
It is shown that generation of the rogue waves in the ocean may be described in framework of non-linear two-dimensional shallow water theory where the simplest two-dimensional long wave non-linear model corresponds to the Kadomtsev–Petviashvili (KP) equation. Numerical solution of the KP equation is obtained to account for the formation of localized abnormally high amplitude wave due to a resonant superposition of two incidentally non-interacting long-crested waves. Peculiarities of the solution allow to explain rare and unexpected appearance of the rogue waves. However, our solution differs from the exact two-solitary wave solution of the KP equation used before for the rogue waves description.
The influence of an external medium on the evolution of two-dimensional long non-linear strain waves in an elastic plate is studied. The governing non-linear equations for longitudinal and shear waves are obtained. A threshold value of the external medium parameter is found that separates the existence of either one-dimensional (or plane) localized strain wave or two-dimensional localized strain wave. A considerable increase in the amplitude of the wave is found during the formation of the two-dimensional localized strain wave from an arbitrary initial pulse.
The second order elastic moduli λ1 ≡ λξηξη and λ2 ≡ λξξηη of a perfect two-dimensional (2D) hard disk crystal are determined by the constant thermodynamic tension Monte Carlo method. The elastic moduli show the free-volume-like density dependence, , where ρ0 is the density at close packing, and prove to be close to those for hexagonal planes of the three-dimensional (3D) fcc and hcp crystals of hard spheres. The Kosterlitz-Thouless, Halperin-Nelson, and Young elastic constant is estimated to approach the universal value 16π in the phase coexistence region of the system.
The paper describes a nonlinear phenomenon of interaction of waves that may be a reason for the existence of high-amplitude wave humps on the sea surface. The properties of extreme elevation area in the vicinity of the intersection point of two long-crested shallow water waves travelling in slightly different directions are evaluated in the framework of two-soliton solution of the Kadomtsev–Petviashvili equation. The extent of the area where surface elevation exceeds a certain level is estimated analytically based on the study of corresponding surface isolines. Shown is that this area is very narrow in the direction of its propagation. The extreme slopes at the front of the interaction soliton may eight times exceed the maximum slope of the interacting solitons.
In this paper, the derivation of higher-order continuum models from a discrete medium is addressed, with the following aims: (i) for a given discrete model and a given coupling of discrete and continuum degrees of freedom, the continuum should be defined uniquely, (ii) the continuum is isotropic, and (iii) boundary conditions are derived consistently with the energy functional and the equations of motion of the continuum. Firstly, a comparison is made between two continualisation methods, namely based on the equations of motion and on the energy functional. They are shown to give identical results. Secondly, the issue of isotropy is addressed. A new approach is developed in which two, rather than one, layers of neighbouring particles are considered. Finally, the formulation and interpretation of boundary conditions is treated. By means of the Hamilton–Ostrogradsky principle, boundary conditions are derived that are consistent with the energy functional and the equations of motion. A relation between standard stresses and higher-order stresses is derived and used to make an interpretation of the higher-order stresses. An additional result of this study is the non-uniqueness of the higher-order contributions to the energy.
Poisson's ratio in materials is governed by the following aspects of the microstructure: the presence of rotational degrees of freedom, non-affine deformation kinematics, or anisotropic structure. Several structural models are examined. The non-affine kinematics are seen to be essential for the production of negative Poisson's ratios for isotropic materials containing central force linkages of positive stiffness. Non-central forces combined with pre-load can also give rise to a negative Poisson's ratio in isotropic materials. A chiral microstructure with non-central force interaction or non-affine deformation can also exhibit a negative Poisson's ratio. Toughness and damage resistance in these materials may be affected by the Poisson's ratio itself, as well as by generalized continuum aspects associated with the microstructure.
Anisotropies of Young's modulus E, the shear modulus G, and Poisson's ratio of all 2D symmetry systems are studied. Simple necessary and sufficient conditions on their elastic compliances are derived to identify if any of these crystals are completely auxetic, non-auxetic and auxetic. Particular attention is paid to 2D crystals of quadratic symmetry. All mechanically stable quadratic crystals are characterized by three parameters belonging to a prism with the stability triangle in the base. Regions in the stability triangle in which quadratic materials are completely auxetic, non-auxetic, and auxetic are established. Examples of all types of auxetic properties of crystals of oblique and rectangular symmetry are presented.
Lattice Dynamical Foundations of Continuum Theories
- A Askar
A. Askar, Lattice Dynamical Foundations of Continuum Theories, World Scientific,
Singapore, 1985.