Dispersion Properties of a Closed-Packed Lattice Consisting of Round Particles

  • Vladimir I. ErofeevEmail author
  • Igor S. Pavlov
  • Alexey V. Porubov
  • Alexey A. Vasiliev
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 90)


A two-dimensional discrete model for a hexagonal (closed-packed) lattice with elastically interacting round particles possessing two translational and one rotational degrees of freedom is considered. The linear differential-difference equations are obtained by the method of structural modeling to describe propagation of longitudinal, transverse and rotational waves in the medium. The dispersion properties of the model are analyzed. Existence of a backward wave is revealed. The numerical estimations of threshold frequencies of acoustic and rotational waves are given for some values of microstructure parameters.


Structural modeling Hexagonal lattice Round particles Microstructure parameters Dispersion properties 

5.1 Introduction

Prediction of physical and mechanical properties of media with microstructure and adequate description of dynamic (wave) processes [1] require mathematical models taking into account the presence of several scales (structural levels) in a medium, their self-consistent interaction and the possibility of energy transfer from one level to another [2]. It should be emphasized that the actual values of the “microstructure” of the medium in a specific problem can lie both in the range of nanometers or angstroms, and in the field of microns and even on larger scales. From the viewpoint of the methodology of theoretical research, the absolute values of the “microstructure” are not so important, as the smallness of some scales with respect to others.

Investigation of wave processes in crystal lattices can be carried out by the method of structural modeling [3, 4, 5, 6, 7]. Modeling by this method starts with a selection of a certain minimum volume (a structural cell that is analog of the periodicity cell in the crystalline material) in the bulk of a material represented by a regular or a quasiregular lattice consisting of particles of finite sizes. Such a cell is capable of reflecting the main features of the macroscopic behavior of this material [8]. First, a discrete model is elaborated within the scope of this method. Only at the next stage, one can pass to the continuum approximation. Structural models in explicit form contain the geometric parameters of the structure—the size and shape of the particles, on which, ultimately, the effective moduli of elasticity depend [5]. By changing these parameters, we can control the physical and mechanical properties of a medium. Such investigations are very important, for instance, for the photonic and phononic crystals [9, 10, 11].

The term “photonic crystals” appeared in the early 1990s for media having a periodic system of dielectric inhomogeneities giving rise to emergence of zones opaque both for light and electromagnetic waves [12]. From a general viewpoint, a photonic crystal is a superlattice or a medium, in which an additional field has been artificially created, and its period is of some orders greater than the basic lattice period. The behavior of photons is radically different from their behavior in the ordinary crystal lattice if the optical superlattice period is comparable with the length of the electromagnetic wave. They do not transmit the light with a wavelength comparable with the lattice period of the photonic crystal and determine the effect of the light localization. Photonic lattices are in the gap between the atomic crystal lattices and the macroscopic artificial periodic structures.

Subsequently, natural or artificial periodic structures became known as “phononic” crystals (acoustic superlattices) by analogy if they consist of non-pointwise particles, in which the length of the acoustic waves is comparable with the lattice period [9, 13, 14, 15]. The velocity of propagation of elastic waves in solids is about \({10}^{5}\) times less than the light wave velocity. Therefore, all effects inherent to photonic crystals should take place in acoustics, but for significantly lower frequencies. High interest in materials of this type is caused by the unique properties of the materials that enables one to apply them in many fields, primarily, in nanoelectronics. The ordering of the geometric structure is typical for the periodic (crystalline) media. It is a decisive factor leading to anisotropy of the properties of crystals and to the predominance of the collective motions of the wave type in the crystal lattice [16]. The dispersion properties of the phononic crystal representing a rectangular lattice consisting of ellipse-shaped particles were analyzed in [17].

It is interesting to note that examples of materials, in which the presence of various structural levels is very clearly manifested, can be also found in geophysics. For instance, the internal structure of rocks, in particular, hydrocarbon reservoirs, is different on various scales and determines their specific physical properties. First of all, it concerns such physical properties as thermal and electrical conductivity, hydraulic and dielectric permittivity. Methods of the theory of effective media are employed in geophysics for elaboration of different-scale mathematical models of such media. The construction of models is performed according to the principle “from small inhomogeneities to large ones”. For each scale, a model medium is constructed with the given parameters. Its equations establish relationships between the parameters of the model and the measured physical properties of the rock. The role of the model parameters can be played by the characteristics of the shape and orientation, degree of ordering of the inhomogeneities, and the degree of their connectivity [18, 19]. In this case, inhomogeneities mean the grains of minerals, particles of organic matter, cracks and pores filled with various fluids. In addition, such models can be used, in particular, for solving problems of geomechanical modeling [20]. Obviously, such approach to construction of models for physical properties of media resembles with the structural modelling method in mechanics of microstructured solids.

In exploration geophysics, interest has recently increased to unconventional reservoirs of hydrocarbons and to reservoirs with complicated production conditions [20]. Such objects include gas-hydrate formations and rocks of “shale oil/gas”. In particular, gas hydrates, as distinct from traditional hydrocarbons, have a crystalline structure. “Shale oil/gas” rocks are characterized by a rather large (more than 30%) content of clay minerals, the crystal lattice of which contains intracrystalline water. Due to that, the elastic properties of clay minerals are drastically changed. Therefore, studies of processes occurring at the level of the crystal lattices of such media, which influence and give rise to the interrelationships of the physical properties mentioned above, are of great importance. When different-scale mathematical models of physical properties of such rocks are constructed, these studies should precede the study of properties on nano-and micro-scales.

Using the structural modeling method, a discrete model of a two-dimensional close-packed lattice consisting of rigid non-deformable round particles is elaborated in this paper. Between the particles there is the so-called porous space—a medium, through which force and moment interactions between the particles are transmitted. If the appropriate model is used to solve a geophysical problem, it is possible to suppose that this porous space is filled with a fluid, for example, an intracrystalline water. Next, the dispersion properties [21] of such a lattice are analyzed. An influence of the microstructure of the crystal on its dispersion properties is also shown, and theoretical estimates of the threshold frequencies of the acoustic and optical phonons are obtained for some values of the microstructure parameters.

5.2 Discrete Model for a Hexagonal Lattice Consisting of Round Particles

We consider a two-dimensional hexagonal closed-packed lattice (or triangle, as it is mentioned in [22]) consisting of homogeneous round particles (grains or granules) with masses M and diameter d. In the initial state, they are located in the lattice sites and the distance between the mass centers of the neighboring granules are equal to a, see Fig. 5.1. Each particle has three degrees of freedom: translational degrees of freedom \(u_{i,j} \) and \(w_{i,j}\) for the displacement of the mass center of the particle with the number \(N = N(i, j)\) along the axes xand y, and the rotational degree of freedom \(\varphi _{i, j} \) for the rotation with respect to the mass center (Fig. 5.2). The kinetic energy of the particle N(i, j) is
$$\begin{aligned} T_{i,j} =\frac{M}{2} \left( \dot{u}_{i,j}^{2} +\dot{w}_{i,j}^{2} \right) +\frac{J}{2} \dot{\varphi }_{i,j}^{2}, \end{aligned}$$
where \(J=Md^{2} /8\) is the moment of inertia of the particle about the axis passing through its mass centre. The upper dot denotes derivatives with respect to time.
Fig. 5.1

Hexagonal lattice with round particles

Fig. 5.2

Kinematical scheme

Fig. 5.3

Scheme of the force interactions and introduced notations

It is assumed that each particle interacts only with six nearest neighbors in the lattice. Simulation of the interactions between the particles is performed by means of the so-called “spring” model. Such a model is used in many works, see, e.g., [3, 23, 24, 25, 26, 27, 28, 29]. In this paper, the central and non-central interactions of the neighboring granules are simulated by elastic springs of three types [30]: central (the corresponding spring is designated by number 1 and has rigidity \({K}_{0}\)), non-central (2 and 3 with rigidity \({K}_{1}\)), and “diagonal” (4 and 5 with rigidity \({K}_{2}\)). The interactions of tension-compression type are modeled by the central and non-central springs. The torques of the particles are provided by the springs of the \({K}_{1}\) type. Springs with the rigidity \({K}_{2}\) characterize the force interactions of the particles at the shear deformations. The points of junctions of the springs \({K}_{1}\) and \({K}_{2}\) coincide with the apexes of the regular hexagon inscribed in the round particle (Fig. 5.3).

It should be noted that six pairs of diagonal springs connecting the central particle with the six nearest neighbors in the lattice have the same rigidity \({K}_{2}\). But if the rigidities of the diagonal springs in pairs are different, then there is a lattice with a chiral microstructure. Dynamical properties of such lattices were discussed, particularly, in Refs. [31, 32].

The displacements of the granules are supposed to be small in comparison with the sizes of the elementary cell of the lattice. The energy of each particle provided by deviation of the particle from the equilibrium state is determined by the strain energy of the springs connecting this particle with the six nearest neighbors in the lattice. These six particles can be numbered by two ways: either by the number of the row, where the particle is located (Fig. 5.3), or by the coordinates of the mass centers of these particles on the circle of unit radius. In order to construct a discrete model, it is more convenient to use the first method. In this case, 1 is added to the first index of the particles, if they are located to the right of the particle N(i, j) (in Fig. 5.3, these particles have the numbers \(n = 0, 1, 5\)), and \(-1\) is added, if the particles are to the left of it (these are particles \(n=2, 3, 4\)). Similarly, 1 is added to the second index of the particles located above the particle N(i, j) and \(-1\) is added, if the particles are below it (respectively, for particles with numbers n = 0 and n = 5, the second index remains equal to j). Thus, the potential energy due to the interaction of the particle N(ij) with six nearest neighbors in the lattice \((i+m_{1} ,\, j+m_{2} )\), where \(m_{1} =\pm 1\) is the shift of the number along the horizontal axis and \(m_{2} =0,\; \pm 1\) is the shift of the number along the vertical axis, is described by the formula
$$\begin{aligned} \begin{array}{c} U_{i,j} =\frac{1}{2} \sum \limits _{(m_{1} ,{} m_{2} )}\left( \frac{K_{0} }{2} D_{1(m_{1} ,\, m_{2} )}^{2} + +\frac{K_{1} }{2} (D_{2(m_{1} ,\, m_{2} )}^{2} +D_{3(m_{1} ,\, m_{2} )}^{2} )+\right. \\ +\left. \frac{K_{2} }{2} (D_{4(m_{1} ,\, m_{2} )}^{2} +D_{5(m_{1} ,\, m_{2} )}^{2} \right) . \end{array} \end{aligned}$$
Here \(D_{l(m_{1} ,\; m_{2} )} \) are the elongations of the springs connecting the central particle N with its six neighbors, \(l=1,2,3,4,5\) is the spring number in Fig. 5.3. Equation (5.2) contains an additional factor 1/2, since the potential energy of each spring is equally divided between two particles connected by this spring. Expressions for the elongations of the springs \(D_{l(m_{1} ,\; m_{2} )} \) calculated in the approximation of smallness of the quantities \(\varDelta u_{m_{1} ,m_{2} } =\,=\) \((u_{i+m_{1} ,j+m_{2} } -u_{i,j} )/a \sim \varDelta w_{m_{1} ,m_{2} } =(w_{i+m_{1} ,j+m_{2} } -w_{i,j} )/a\sim \) \(\sim \varphi _{i,j} \sim \varepsilon \) (here \(\varepsilon<<1\) is a measure of the cell deformation, \(m_{1} =\pm 1\), \(m_{2} =0,\; \pm 1\)) and \(\varPhi _{m_{1} ,m_{2} } ={\left( \varphi _{i,j} +\varphi _{i+m_{1} ,j+m_{2} } \right) /2} =\) \(=\varphi _{i,j} -0,5a\varDelta \varphi _{m_{1} ,m_{2} }<<{\pi / 2} \) have the form:
$$ D_{1(m_{1} ,m_{2} )} =\frac{a}{2} \left( m_{1} \varDelta u_{m_{1} ,\; m_{2} } +m_{2} \sqrt{3} \varDelta w_{m_{1} ,\; m_{2} } \right) , $$
$$ D_{1(m_{1} ,\, 0)} =m_{1} a\varDelta u_{m_{1} ,\; 0} , $$
$$ D_{2,3m_{1} ,\, m_{2} )} =\frac{a}{4} \left( 2m_{1} \varDelta u_{m_{1} ,\; m_{2} } +2m_{2} \sqrt{3} \varDelta w_{m_{1} ,\; m_{2} } \mp m_{2} d\sqrt{3} \varDelta \varphi _{m_{1} ,\; m_{2} } \right) , $$
$$\begin{aligned} D_{2,3(p,\, 0)} = m_{1} a\left( \varDelta u_{m_{1} ,\; 0} \pm \frac{d\sqrt{3} }{4} \varDelta \varphi _{m_{1} ,\,0} \right) , \end{aligned}$$
$$ D_{4(m_{1} ,\, m_{1} )} =\frac{m_{1} a}{2r_{0} } \left( (a-2d)\varDelta u_{m_{1} ,\; m_{1} } +a\sqrt{3} \varDelta w_{m_{1} ,\; m_{1} } +d\sqrt{3} \varPhi _{m_{1} ,\; m_{1} } \right) , $$
$$ D_{5(m_{1} ,\; m_{1} )} =\frac{m_{1} a}{2r_{0} } \left( (a+d)\varDelta u_{m_{1} ,\; m_{1} } +(a-d)\sqrt{3} \varDelta w_{m_{1} ,\; m_{1} } -d\sqrt{3} \varPhi _{m_{1} ,\; m_{1} } \right) , $$
$$ D_{4,5(m_{1} ,\, 0)} =\frac{a}{2r_{0} } \left( m_{1} (2a-d)\varDelta u_{m_{1} ,\; 0} \pm d\sqrt{3} \varDelta w_{m_{1} ,0} \pm m_{1} d\sqrt{3} \varPhi _{m_{1},\,0} \right) , $$
$$ D_{4,5(\mp 1,\, \pm 1)} =\frac{a}{2r_{0} } \left( \mp (a+d)\varDelta u_{\mp 1,\; \pm 1} \pm (a-d)\sqrt{3} \varDelta w_{\mp 1,\pm 1} +d\sqrt{3} \varPhi _{\mp 1,\pm 1} \right) , $$
$$ D_{4,5(\pm 1,\, \mp 1)} =\frac{a}{2r_{0} } \left( \pm (a-2d)\varDelta u_{\pm 1,\; \mp 1} \mp (a-d)\sqrt{3} \varDelta w_{\pm 1,\mp 1} -d\sqrt{3} \varPhi _{\pm 1,\mp 1} \right) , $$
where \(r_{0} =\sqrt{a^{2} -ad+d^{2} } \) is the length of the undisturbed spring \({K}_{2}\). In expressions for \(D_{2,3} \) and \(D_{4,5} \), the upper symbols in ± and \(\mp \) are taken for the springs of types 2 and 4, whereas the lower ones are necessary for the springs 3 and 5.
It should be noted that Eq. (5.3) have been obtained with the accuracy up to the linear terms having the order \(\varepsilon ^{1}\). Substitution of these expressions in Eq. (5.2) leads to the following expression for the potential energy per cell with the number N = N(i, j) with accuracy up to quadratic terms:
$$ \begin{array}{l} U_{i,j} =\gamma _{1} (\varDelta u_{1,0}^{2} +\varDelta u_{-1,0}^{2} )+\gamma _{2} (\varDelta u_{1,1}^{2} +\varDelta u_{-1,-1}^{2} +\varDelta u_{1,-1}^{2} +\varDelta u_{-1,1}^{2} )\,+ \\ \ \\ +\,\gamma _{3} (\varDelta w_{1,0}^{2} +\varDelta w_{-1,0}^{2} +\varPhi _{1,0}^{2} +\varPhi _{-1,0}^{2} +\varPhi _{1,1}^{2} +\varPhi _{-1,-1}^{2} +\varPhi _{1,-1}^{2} +\varPhi _{-1,1}^{2}\,+ \\ \ \\ \quad +\,\varDelta w_{1,1} \varPhi _{1,1} -\varDelta w_{-1,-1} \varPhi _{-1,-1} +\varDelta w_{1,-1} \varPhi _{1,-1} -\varDelta w_{-1,1} \varPhi _{-1,1} )\,+ \end{array} $$
$$ +\,\sqrt{3} \gamma _{3} (-\varDelta u_{1,1} \varPhi _{1,1} +\varDelta u_{-1,-1} \varPhi _{-1,-1} +\varDelta u_{1,-1} \varPhi _{1,-1} -\varDelta u_{-1,1} \varPhi _{-1,1} )\,+$$
$$\begin{aligned} +2\gamma _{3} (\varDelta w_{1,0} \varPhi _{1,0} -\varDelta w_{-1,0} \varPhi _{-1,0} )\,+ \end{aligned}$$
$$ \begin{array}{l} +\,\gamma _{4} (\varDelta w_{1,1}^{2} +\varDelta w_{-1,-1}^{2} +\varDelta w_{1,-1}^{2} +\varDelta w_{-1,1}^{2} )\,+ \\ \ \\ +\,\gamma _{5} (\varDelta \varphi _{1,0}^{2} +\varDelta \varphi _{-1,0}^{2} +\varDelta \varphi _{1,1}^{2} +\varDelta \varphi _{-1,-1}^{2} +\varDelta \varphi _{1,-1}^{2} +\varDelta \varphi _{-1,1}^{2} )\,+ \\ \ \\ +\,\gamma _{6} (\varDelta u_{1,1} \varDelta w_{1,1} +\varDelta u_{-1,-1} \varDelta w_{-1,-1} -\varDelta u_{1,-1} \varDelta w_{1,-1} -\varDelta u_{-1,1} \varDelta w_{-1,1} ). \end{array}$$
Here the coefficients \(\gamma _{1} \),..., \(\gamma _{6} \) are
$$\begin{aligned} \gamma _{1} =\frac{a^{2} }{2} (K_{0} +2K_{1} +\frac{(2a-d)^{2} }{2r_{0}^{2} } K_{2} ), \end{aligned}$$
$$\begin{aligned} \gamma _{2} =\frac{a^{2} }{8} (K_{0} +2K_{1} +\frac{2a^{2} -2ad+5d^{2} }{r_{0}^{2} } K_{2} ), \gamma _{3} =\frac{3a^{2} d^{2} }{4r_{0}^{2} } K_{2} , \end{aligned}$$
$$\begin{aligned} \gamma _{4} =\frac{3}{8} a^{2} (K_{0} +2K_{1} +\frac{2a^{2} -2ad+d^{2} }{r_{0}^{2} } K_{2} ), \end{aligned}$$
$$\begin{aligned} \gamma _{5} =\frac{3a^{2} d^{2} }{16} K_{1} , \gamma _{6} =\frac{\sqrt{3} }{4} a^{2} (2K_{1} -\frac{2a^{2} -2ad-d^{2} }{r_{0}^{2} } K_{2} ). \end{aligned}$$
The linear equations of motion for our lattice are obtained using the variational principle, where parts of the Lagrangian are defined by Eqs. (5.1) and (5.4),
$$ M\ddot{u}_{i,j} -\frac{2\gamma _{1} }{a^{2} } (u_{i+1,j} -2u_{i,j} +u_{i-1,j} )\,-$$
$$\begin{aligned} -\,\frac{2\gamma _{2} }{a^{2} } (u_{i+1,j+1} +u_{i-1,j-1} +u_{i+1,j-1} +u_{i-1,j+1} -4u_{i,j} )\,- \end{aligned}$$
$$\begin{aligned} -\frac{\gamma _{6} }{a^{2} } (w_{i+1,j+1} +w_{i-1,j-1} -w_{i+1,j-1} -w_{i-1,j+1} )\,- \end{aligned}$$
$$-\,\frac{\sqrt{3} \gamma _{3} }{2a} (-\varphi _{i+1,j+1} +\varphi _{i-1,j-1} +\varphi _{i+1,j-1} -\varphi _{i-1,j+1} )=0, $$
$$ M\ddot{w}_{i,j} -\frac{2}{a^{2} } \gamma _{1} (w_{i+1,j} -2w_{i,j} +w_{i-1,j} )\, - $$
$$ -\,\frac{2}{a^{2} } \gamma _{2} (u_{i+1,j+1} +u_{i-1,j-1} +u_{i+1,j-1} +u_{i-1,j+1} -4u_{ij} )\,- $$
$$\begin{aligned} -\frac{1}{a^{2} } \gamma _{6} (w_{i+1,j+1} +w_{i-1,j-1} -w_{i+1,j-1} -w_{i-1,j+1} )\,- \end{aligned}$$
$$\begin{aligned} -\,\frac{1}{a} \gamma _{3} (\varphi _{i+1,j} -\varphi _{i-1,j} )\,- \end{aligned}$$
$$-\,\frac{1}{2a} \gamma _{3} (\varphi _{i+1,j+1} -\varphi _{i-1,j-1} +\varphi _{i+1,j-1}\, -\varphi _{i-1,j+1} )=0, $$
$$\begin{aligned} \begin{array}{l} M\ddot{\varphi }_{i,j} -(\frac{16}{a^{2} } \gamma _{5} -4\gamma _{3} )(\varphi _{i+1,j} +\varphi _{i-1,j} +\varphi _{i+1,j+1} +\varphi _{i-1,j-1} \,+ \\ +\,\varphi _{i+1,j-1} +\varphi _{i-1,j+1} -6\varphi _{i,j} )\,+ \frac{8}{a} \gamma _{3} (w_{i+1,j} -w_{i-1,j} )\,+\\ +\,48\gamma _{3} \varphi _{i,j} -\frac{4\sqrt{3} }{a} \gamma _{3} (u_{i+1,j+1} -u_{i-1,j-1} -u_{i+1,j-1} +u_{i-1,j+1} )\,- \\ -\,\frac{4}{a} \gamma _{3} (-w_{i+1,j+1} +w_{i-1,j-1} -w_{i+1,j-1} +w_{i-1,j+1} )=0. \end{array} \end{aligned}$$
Here \(R=\sqrt{{J / M}} =d/8\) is the radius of inertia of the microparticles of the medium with respect to the mass center.

Equations (5.6)–(5.8) obtained in this section can be used for numerical simulation of the response of the system to the external dynamic effects in a wide range of frequencies up to the threshold values [30]. It should be noted that non-neighboring interactions are frequently introduced, if the particles are the material points [24]. It leads to the differences in the Lagrangian, which gives the higher-order derivatives in the continuum limit. In this paper the particles of the lattice are finite size bodies. They possess both translational and rotational degrees of freedom. In the low-frequency long-wavelength approximation, when the rotational mode does not propagate, the three-mode system is reduced to the two-mode system containing the fourth-order derivatives in equations for the longitudinal and transverse modes [30]. These equations are called equations of the second-order gradient elasticity. Thus, we achieve appearance of the higher-order derivatives in the governing equations by another way. Further, the dispersion properties of Eqs. (5.6)–(5.8) will be analyzed.

5.3 Derivation of the Dispersion Equation

The lattice with round particles considered in Sect. 5.2 represents a system with N degrees of freedom, which is described by coupled equations (5.6)–(5.8), see [16]. Introduction of normal mode variables, makes equations of motion independent [33], and the arbitrary motion of the system can be represented as a superposition of normal vibrations. This approach is very convenient both for the theoretical analysis of the problem and for the physical interpretation of the obtained results. Similar concepts can be also introduced for distributed systems, where interacting waves of various types can propagate. A generalization of the concept of the normal vibrations of concentrated systems to “not closed” wave systems (boundless media, waveguides, tubes, rods, strings, and etc.) gives rise to the normal wave, i.e. traveling harmonic waves in the linear systems with constant parameters, in which an absorption and scattering of energy are negligible [34, 35].

In order to study the collective motions arising in an arranged crystalline structure, we will pass to the normal oscillations. Let us consider solutions of the equations of motion representing plane monochromatic waves, for which the displacements are
$$\begin{aligned} \begin{array}{c} {u\left( \mathbf {N},t\right) =u_{0} \exp \left[ i\left( \omega \left( \mathbf {q}\right) \, t-\mathbf {q}\mathbf {N}\right) \right] } \\ \ \\ {w\left( \mathbf {N},t\right) =w_{0} \exp \left[ i\left( \omega \left( \mathbf {q}\right) \, t-\mathbf {q}\mathbf {N}\right) \right] } \\ \ \\ {\varphi \left( \mathbf {N},t\right) =\varphi _{0} \exp \left[ i\left( \omega \left( \mathbf {q}\right) \, t-\mathbf {q}\mathbf {N}\right) \right] } \end{array} \end{aligned}$$
Here \(\omega =\omega \left( \mathbf {q}\right) \) is a wave frequency regarded as a continuous function of the wave vector \(\mathbf {q}=\left( q_{1} ,q_{2} \right) \) that defines both the direction of the wave propagation in the Cartesian coordinate system \(\left( x,y\right) \) and the wave length \(\lambda ={2\pi / q} \) \(\left( q=\left| \mathbf {q}\right| \right) \). The vector \(\mathbf {N}=\left( i,j\right) \) fixes the lattice sites. Arbitrary collective motions can be represented as a superposition of monochromatic waves. Substitution of Eq. (5.9) into Eq. (5.6)–(5.8) results in a set of equations in the matrix form for determination of the amplitudes of displacements,
$$\begin{aligned} \left( \begin{array}{ccc} {M\omega ^{2} -d_{11} } &{} {d_{12} } &{} {d_{13} } \\ {d_{21} } &{} {M\omega ^{2} -d_{22} } &{} {d_{23} } \\ {d_{31} } &{} {d_{32} } &{} {M\omega ^{2} -d_{33} } \end{array}\right) \cdot \left( \begin{array}{c} {u_{0} } \\ {w_{0} } \\ {\varphi _{0} } \end{array}\right) =\left( \begin{array}{c} {0} \\ {0} \\ {0} \end{array}\right) , \end{aligned}$$
where the matrix elements are
$$\begin{aligned} d_{11} =\frac{8\gamma _{1} }{a^{2} } \sin ^{2} (\frac{q_{1} a}{2} )+\frac{8\gamma _{2} }{a^{2} } (1-\cos (\frac{q_{1} a}{2} )\cos (\frac{q_{2} a\sqrt{3} }{2} )), \end{aligned}$$
$$\begin{aligned} d_{22} =\frac{16\gamma _{3} }{a^{2} } \sin ^{2} (\frac{q_{1} a}{2} )+\frac{8\gamma _{4} }{a^{2} } (1-\cos (\frac{q_{1} a}{2} )\cos (\frac{q_{2} a\sqrt{3} }{2} )), \end{aligned}$$
$$\begin{aligned} \begin{array}{c} d_{33} =\frac{8}{a^{2} } \left[ \frac{8\gamma _{5} +4a^{2} \gamma _{3} }{d^{2} } +\left( \gamma _{6} \sqrt{3} -\gamma _{3} (1+\frac{2a}{d} )\right) \right. \times \\ \times \left. \left( \sin ^{2} (\frac{q_{1} a}{2} )-\cos (\frac{q_{1} a}{2} )\cos (\frac{q_{2} a\sqrt{3} }{2} )\right) \right] , \end{array} \end{aligned}$$
$$ d_{12} =d_{21} =\frac{8\sqrt{3} }{3a^{2} } (\gamma _{3} -\gamma _{4} )\sin (\frac{q_{1} a}{2} )\sin (\frac{q_{2} a\sqrt{3} }{2} ), $$
$$\begin{aligned} d_{13} = -\frac{d^{2} }{8} d_{31} =i\frac{2\sqrt{3} \gamma _{3} }{a} \cos (\frac{q_{1} a}{2} )\sin (\frac{q_{2} a\sqrt{3} }{2} ), \end{aligned}$$
$$d_{23} =-\frac{d^{2} }{8} d_{32} =-i\frac{2\gamma _{3} }{a} \sin (\frac{q_{1} a}{2} )\left( \cos (\frac{q_{2} a\sqrt{3} }{2} )+2\cos (\frac{q_{1} a}{2} )\right) . $$
The solvability condition for Eq. (5.10) with coefficients defined by Eq. (5.11) leads to a bi-cubic dispersion equation for \(\omega \),
$$\begin{aligned} M^{3} \omega ^{6} +F_{1} \omega ^{4} +F_{2} \omega ^{2} +F_{3} =0, \end{aligned}$$
where \(F_{1,2,3} \) are the wave vector functions:
$$\begin{aligned} F_{1} =d_{11} +d_{22} +d_{33} , \end{aligned}$$
$$\begin{aligned} F_{2} =d_{11} d_{22} +d_{11} d_{33} +d_{22} d_{33} -d_{12} d_{21} -d_{13} d_{31} -d_{23} d_{32} , \end{aligned}$$
$$\begin{aligned} F_{3} =-d_{11} d_{22} d_{33} +d_{11} d_{23} d_{32} +d_{22} d_{13} d_{31} +d_{33} d_{12} d_{21} +d_{12} d_{23} d_{31} +d_{13} d_{32} d_{21} . \end{aligned}$$
Dividing Eq. (5.12) by \(K_{0}^{3} \) and substituting the relationship, \(\varpi =\omega \sqrt{M/K_{0} } \), one can obtain the dispersion equation in the dimensionless form:
$$\begin{aligned} \varpi ^{6} -f_{1} \varpi ^{4} +f_{2} \varpi ^{2} +f_{3} =0, \end{aligned}$$
where \(f_{1} =F_{1} /K_{0} ,\) \(f_{2} =F_{2} /K_{0}^{2} ,\) and \(f_{3} =F_{3} /K_{0}^{3} \). Thus, the left-hand side of Eq. (5.14) contains three variables: frequency \(\omega \) and the components of the wave vector, \(q_{1} \) and \(q_{2} \). Moreover, the coefficients of Eq. (5.14) depend on the relative particle size d / a and on two parameters of the force and couple interactions: \(K_{1} /K_{0} \) and \(K_{2} /K_{0} \).

Two lattices correspond to each of the crystal structure: a direct lattice and a reciprocal one. A direct lattice is a lattice in ordinary space and a reciprocal one is a lattice in abstract reciprocal space, where distances have a dimension of the reciprocal length, in fact, it is the Fourier transform of the direct lattice [36]. The diffraction pattern represents a reciprocal crystal lattice map, just as the microscopic image is a map of the real crystal structure.

The primitive unit cells which constitute the periodic reciprocal lattice in the Bloch wave vector space are referred to as Brillouin zones [36]. The first Brillouin zone can be regarded as a primitive cell of the reciprocal lattice that possesses point symmetry of this lattice. Indeed, if we construct the first Brillouin zone around each node of the reciprocal lattice (the origin should be located in the node), then such zone would entirely fill the entire space without overlapping with each other. From this fact it follows, in particular, that the volume of the first Brillouin zone is equal to the volume of the primitive cell of the reciprocal lattice.

The structure of the Brillouin zones is defined only by crystal structure and depends neither on the type of particles forming the crystal, nor on their interaction. The physical meaning of the Brillouin zone boundaries consists in that they show the following values of the wave vectors or the electron quasi-pulses, in which the electron wave cannot propagate in a solid [36].

Next, we will analyze the dispersion properties of the medium in the first Brillouin zone and on its boundary depending on the values of the microstructure parameters.

5.4 Dispersion Properties of Normal Waves

Like in the solid-state physics, each normal lattice vibration can be associated with a certain type of quasiparticle—phonon [37]. The considered system has a longitudinal acoustic (LA) phonon, a transverse acoustic (TA) one, and an optical rotational (OR) phonon [38]. We pass to the polar coordinate system \(q_{1} =q\cos \theta \), \(q_{2} =q\sin \theta \), in Eq. (5.12), where q is the wave vector module and the angle \(\theta \) indicates the direction of the plane wave propagation with respect to x-axis in the direct lattice. In particular, in the case of propagation of the plane waves, when \(q_{2} \equiv 0\) and, hence, \(d_{12} \equiv d_{13} \equiv d_{21} \equiv d_{31} \equiv 0\), Eq. (5.14) is substantially simplified since the longitudinal phonons become independent in it:
$$\begin{aligned} (\varpi ^{2} -\frac{d_{11} }{K_{0} } )\left( (\varpi ^{2} -\frac{d_{22} }{K_{0} } )(\varpi ^{2} -\frac{d_{33} }{K_{0} } )-\frac{d_{23} }{K_{0} } \frac{d_{32} }{K_{0} } \right) =0, \end{aligned}$$
where \(\varpi ={\omega / \omega _{0}} \), \(\omega _{0} =\sqrt{M/K_{0} } \) and the coefficients of Eq. (5.14a) have the form:
$$\begin{aligned} d_{11} =\frac{8}{a^{2} } \left[ \gamma _{1} \sin ^{2} (\frac{qa}{2} )+\gamma _{2} (1-\cos (\frac{qa}{2} ))\right] , \end{aligned}$$
$$\begin{aligned} d_{22} =\frac{8}{a^{2} } \left[ 2\gamma _{3} \sin ^{2} (\frac{qa}{2} )+\gamma _{4} (1-\cos (\frac{qa}{2} ))\right] , \end{aligned}$$
$$d_{33} =\frac{8}{a^{2} } \left[ \frac{8\gamma _{5} +4a^{2} \gamma _{3} }{d^{2} } +\left( \gamma _{6} \sqrt{3} -\gamma _{3} (1+\frac{2a}{d} )\right) \left( \sin ^{2} (\frac{qa}{2} )-\cos (\frac{qa}{2} )\right) \right] , $$
$$ d_{23} =-\frac{d^{2} }{8} d_{32} =-i\frac{2\gamma _{3} }{a} \left( \sin (\frac{qa}{2} )+\sin (qa)\right) .$$
From Eqs. (5.14a) and (5.15) it follows that each wave mode has both minimum and maximum, which values depend on microstructure parameters. Thus, for example, along the \(\varGamma -K\)-axis the frequency of the longitudinal phonons has a local maximum \(\omega _{LA}^{\max } =\sqrt{2\left( 4(\gamma _{1} +\gamma _{2} )+{\gamma _{2}^{2}/ \gamma _{1} } \right) /Ma^{2} } \) at the point \(q={2\left( \pi -\arccos \left( \gamma _{2} /2\gamma _{1} \right) \right) / a} \). Consequently, by varying the microstructure parameters, it is possible to specify certain dispersion properties of the crystal [17, 39].
Analysis of solutions of the dispersion Eq. (5.14) is performed for the following values of the microstructure parameters: \(d/a=0.1\), \(K_{1} /K_{0} =0.5\), \(K_{2} /K_{0} =0.3\). The dispersion curves calculated along directions \(\theta =0^{\circ } \) (\(\varGamma \)–K), \(\theta =30^{\circ } \) (\(\varGamma \)–M) and along the boundary of the Brillouin zone (\(K-M\)) are shown in Fig. 5.4.
Fig. 5.4

Dispersion curves of the hexagonal lattice

From Fig. 5.4 it is visible that in the \(\varGamma -M\)-direction the frequency increases monotonically, when the wave number grows, up to the boundary of the Brillouin zone, and in the \(\varGamma -K\)-direction the frequency of the longitudinal phonons has a local maximum \(\varpi \approx 2.96\) located at the point \(q={2\left( \pi -arctg\left( 3\sqrt{7} \right) \right) / a} \). In the interval \({2\left( \pi -arctg\left( 3\sqrt{7} \right) \right) /a}<q<{4\pi / 3} a\) the group velocity of rotational phonons is negative: \(v_{gr} = {d\omega / dq} <0\). This area is called a backward-wave region [36]. Usually, a field of the negative group velocity exists for optical phonons in lattices with a complex structure, when more than one particle is present in the Bravais lattice [36]. Here, a similar situation takes place for acoustic phonons in a simple lattice. The presence of a backward wave in a medium is associated with the phenomenon of negative refraction provided that the surface of equal frequencies is convex. The longitudinal mode has the maximum frequency \(\varpi \approx 3.65\) that is achieved on the boundary of the first Brillouin zone at point (\(q = {2\pi / \sqrt{3} a} \)). At this point, the group velocity is equal to zero and therefore a signal with such a frequency cannot propagate in a crystal lattice. This restriction can be dropped only for nonlinear perturbations [35], when anharmonic terms are taken into account in equations of motion. The frequency of the transverse phonons has the maximal value \(\varpi \approx 3.38\) at the point K. The rotational (optical) mode has two threshold frequencies: the minimum \(\varpi (0)\approx 3.44\) and maximum \(\varpi \approx 4.15\) ones. In the frequency range \(0\le \varpi \le 3.38\) the system has LA-and TA-modes. In the interval \(3.28<\varpi <3.44\) there is only a longitudinal mode and for frequencies \(3.44\le \varpi \le 3.65\) there are longitudinal and rotational modes. And, finally, in the high-frequency range \(3.65<\varpi \le 4.15\), only the rotational mode is present in the system (Fig. 5.4).

It should be noted that in the continuum approximation the rotational mode has only one threshold frequency—minimal, whereas the longitudinal and transverse modes have no threshold frequencies [39]. If a medium consists of material points (d = 0), then there are no rotational phonons in the medium.

Figure 5.5 shows maps of equal frequencies for longitudinal, transverse, and rotational phonons (for LA-mode \(\varpi \) = 0.7, 1.0, 1.5, 2.1, 2.7, 3.0, 3.2, 3.3, for TA-mode \(\varpi \) = 0.7, 1.0, 1.5, 2.1, 2.7, 3.0 and for OR mode \(\varpi \) = 3.2, 3.3, 3.5, 3.6, 3.7, 3.8). The horizontal axis represents the projection \(q_{x} \) of the wave vector, and along the vertical axis—\(q_{y} \). The boundaries of the first Brillouin zone are indicated by a dashed line.
Fig. 5.5

Maps of equal frequencies for the hexagonal lattice

Figure 5.5 shows that lines of equal frequencies are circles for small values of the wave number. Hence, the crystal structure behaves like an isotropic medium in the long-wavelength range. However, when the wavelength decreases (the magnitude of the wave vector increases), the properties of acoustic anisotropy begin to appear. In this case, the transverse waves become anisotropic ones faster than the longitudinal waves do. For \(\varpi = 3.22\) the map of equal frequencies reproduces completely the structure of the hexagonal lattice at issue.

5.5 Conclusions

A two-dimensional discrete model of a close-packed (hexagonal) lattice consisting of rigid non-deformable round particles of finite sizes is elaborated in this paper using the structural modeling method. This model can be used, for example, for description of wave processes in phononic crystals and in geophysics—for studying physical properties of rocks.

Dispersion properties of such a medium have been analyzed for some values of the microstructure parameters. The analysis showed the existence of a backward wave, i.e. the wave whose phase and group velocities are oppositely directed. Moreover, if in the long-wavelength (continuum) approximation (when the characteristic length of an acoustic wave is much larger than the lattice period) the hexagonal lattice with round particles is isotropic in terms of acoustic properties, then in the short-wavelength (discrete) approximation it is anisotropic, and the transverse waves become anisotropic ones faster than the longitudinal waves do. The rotational mode has two threshold frequencies: maximum and minimum. In the ranges of low (\(0\le \varpi <2.33\)) and high (\(2.98<\varpi \le 3.28\)) frequencies, there are two wave modes in the system, whereas for \(2.33<\varpi \le 2.98\) all three wave modes (longitudinal, transverse and rotational) are present in the system. The greatest value of the frequency of longitudinal phonons is reached at the boundary of the Brillouin zone at the point M (\(q={2\pi / \sqrt{3} a}\)).

The similar results were obtained in [17], where the dispersion properties of the discrete model of the rectangular lattice consisting of ellipse-shaped particles were analyzed. But, in contrast to the hexagonal lattice considered here, the rectangular and square lattices are anisotropic in terms of acoustic properties even in the long-wavelength approximation. Moreover, the continuum approximations of the hexagonal and square lattice were considered in [30], where the analytical relationships between the macroelasticity constants of the medium and microstructure parameters were found. These relationships appeared to be different for the hexagonal and square lattices.

Dispersion properties of elastic waves in 3D phononic crystals, where rotational degrees of freedom had been taken into account, were considered in [40].

Transformation from a discrete model to a continuum one is suitable when the long-wavelength processes are studied [30]. In this case, a comparison of the elaborated model with the well-known continuum theories becomes possible. For adequate description of the dispersion properties in the short-wavelength range (for instance, frequency band gaps), it is necessary either to remain within a discrete model, or to pass to a generalized continuum model, for example, in the framework of the multi-field approach [25, 26] or on the base of Pade approximations [41].

Nonlinear plane waves in media with a hexagonal lattice consisting of material points were studied in the scope of multi-field models in Refs. [6, 7]. The lattice models with finite-size particles generalize such models and enable one to take into account rotational wave effects. Earlier, multi-field models were elaborated only for square lattices [25, 26]. Nowadays, construction of multi-field models for hexagonal lattices with finite-size particles is, in our opinion, of prime interest.

The results obtained in this work can be suggested for modeling of artificial periodic structures consisting of particles of non-zero sizes comparable with the wavelengths of the considered phenomena, and possessing predetermined dispersion properties.



The research was carried out within the framework of the Russian State assignment to IAP RAS (topic No 0035-2014-0402, State Registration No 01201458047, V.I.E. and I.S.P.), as well as under the financial support of the Russian Foundation for Basic Research (projects No 18-08-00715-a, 16-08-00776-(V.I.E. and I.S.P.), 16-08-00971-a (I.S.P. and A.A.V.), and 16-01-00068-a (A.V.P.)) and the Ministry of Education and Science of the Russian Federation within the framework of the basic part of State Work for scientific activity (project 9.7446.2017/8.9, A.A.V.).


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Vladimir I. Erofeev
    • 1
    • 2
    Email author
  • Igor S. Pavlov
    • 1
    • 2
  • Alexey V. Porubov
    • 3
    • 4
    • 5
  • Alexey A. Vasiliev
    • 6
  1. 1.Mechanical Engineering Research Institute of Russian Academy of SciencesNizhny NovgorodRussia
  2. 2.Nizhny Novgorod Lobachevsky State UniversityNizhny NovgorodRussia
  3. 3.Institute of Problems in Mechanical EngineeringSaint-PetersburgRussia
  4. 4.St. Petersburg State UniversitySaint-PetersburgRussia
  5. 5.St. Petersburg State Polytechnical UniversitySaint-PetersburgRussia
  6. 6.Department of Mathematical ModellingTver State UniversityTverRussia

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