Wave characterisation in a dynamic elastic lattice: lattice flux and circulation Текст научной статьи по специальности «Физика»

In memory of Professor G.I. Barenblatt

УДК 539.4

Wave characterisation in a dynamic elastic lattice: lattice flux and circulation

G. Carta1, I.S. Jones1, N.V. Movchan2, and A.B. Movchan2

1 Mechanical Engineering and Materials Research Centre, Liverpool John Moores University, Liverpool, L3 3AF, United Kingdom 2 Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, United Kingdom

A novel characterisation of dispersive waves in a vector elastic lattice is presented in the context of wave polarisation. This proves to be especially important in analysis of dynamic anisotropy and standing waves trapped within the lattice. The operators of lattice flux and lattice circulation provide the required quantitative description, especially in cases of intermediate and high frequency dynamic regimes. Dispersion diagrams are conventionally considered as the ultimate characteristics of dynamic properties of waves in periodic systems. Generally, a waveform in a lattice can be thought of as a combination of pressure-like and shear-like waves. However, a direct analogy with waves in the continuum is not always obvious. We show a coherent way to characterise lattice waveforms in terms of so-called lattice flux and lattice circulation. In the long wavelength limit, this leads to well-known interpretations of pressure and shear waves. For the cases when the wavelength is comparable with the size of the lattice cell, new features are revealed which involve special directions along which either lattice flux or lattice circulation is zero. The cases of high frequency and wavelength comparable to the size of the elementary cell are considered, including dynamic anisotropy and dynamic neutrality in structured solids.

Keywords: elastic waves, lattice structures, dynamic anisotropy, lattice flux and circulation

DOI 10.24411/1683-805X-2018-16016

Характеристики волн в динамической упругой решетке: поток и циркуляция в решетке

G. Carta1, I.S. Jones1, N.V. Movchan2, A.B. Movchan2

1 Ливерпульский университет имени Джона Мурса, Ливерпуль, L3 3AF, Великобритания 2 Ливерпульский университет, Ливерпуль, L69 7ZL, Великобритания

В работе дана новая характеристика дисперсии в упругой векторной решетке с точки зрения поляризации волн. Рассматриваемая проблема имеет особое значение при анализе динамической анизотропии и стоячих волн в решетке. Операторы потока и циркуляции в решетке позволяют получить количественное описание, особенно в случаях средне- и высокочастотных динамических режимов. Дисперсионные диаграммы традиционно рассматриваются в качестве важных характеристик динамических свойств волн в периодических системах. Обычно форму волны в решетке можно представить как комбинацию волн, подобных P- и S-волнам. Однако прямая аналогия с волнами в континууме не всегда очевидна. Предложен последовательный алгоритм описания формы волны в решетке с помощью понятий потока и циркуляции в решетке, который в длинноволновом пределе приводит к известным интерпретациям P- и S-волн. Для случаев когда длина волны сопоставима с размером ячейки решетки, выявлены новые характеристики, а именно особые направления, вдоль которых поток или циркуляция в решетке равны нулю. Рассмотрены случаи с большими значениями частоты, где длина волны сравнима с размером элементарной ячейки, включая динамическую анизотропию и динамическую нейтральность в структурированных твердых телах.

Ключевые слова: упругие волны, решетчатые структуры, динамическая анизотропия, поток и циркуляция в решетке

1. Introduction

Scaling laws and asymptotic analysis are of great importance in mathematical modelling of physical phenomena, as described in fundamental monographs [1-4]. Important examples include boundary layer fluid flow, reaction-diffusion problems and fracture, especially fracture in

structured media. In particular, an elastic lattice represents a simple example of a structured solid, where propagation of cracks brings qualitatively new features compared to the continuum models, as discussed in [5-8]. Asymptotic analysis of structured solids has led to the well-developed approach of homogenisation, which is highly efficient in static

© Carta G., Jones I.S., Movchan N.V., Movchan A.B., 2018

problems [9-12]. Dynamic homogenisation, and in particular dynamic anisotropy of certain standing waveforms, was systematically studied in [13-15]. In dynamics a lattice, which would be considered as isotropic in a static homogenised state, displays new features associated with wave dispersion and dynamic anisotropy (see, for example, [16, 17]). Green's functions in dynamic lattices versus homogenisation approximations were considered in [18]. In recent years, the questions of polarisation of elastic waves in lattice systems and dynamic anisotropy were also raised in [19-23].

For a general periodic elastic lattice, the dispersion diagram is conventionally considered as the ultimate characteristic of the dynamic properties of waves in such a system. However, if one is interested in the combination of pressure and shear in a particular lattice waveform, the answer is not always obvious. The present paper shows a coherent way to characterise lattice waveforms in terms of so-called lattice flux and lattice circulation. Of course, in the long wavelength limit, this leads to well-known interpretations of pressure and shear waves. However, for the cases when the wavelength is comparable with the size of the lattice cell, new features are revealed which involve special directions along which either lattice flux or lattice circulation is zero.

2. Governing equations and dispersion. A periodic elastic lattice

The system under consideration is an infinite, periodic triangular array of point masses m, connected by massless elastic links of stiffness c and length l. We assume that the masses are constrained to move in the (xl, x2)-plane and that their displacements are small. A repetitive cell of the system is shown in Fig.1, a.

In the time-harmonic regime, the governing equations of the periodic triangular lattice can be written in the following matrix form [24]:

(C-ro2M) U = 0, (1)

where ro is the radian frequency, U = (Uj, U2)T is the displacement vector and M = ml 2 is the mass matrix, with I2 being the 2x2 identity matrix. Moreover,

U(x + ^t(1) + n2t^ ) = U(x>;

(2h = i

¿k-Tn

(3)

where x = (x1, x2)T is the position vector, n = (nl, n2)T is the multi-index and T = (t(1), t(2)) is the periodicity matrix. The vectors t(1) = (l, 0)T and t(2) = (//2, V3//2)T are shown in Fig. 1, a.

In what follows, we use the normalised quantities x = x/1, U = U/1, T = T/1, k = kl,

Z = Zl, I = I M = M/m, C = C/c, (4)

cb = m/yl Cm.

The dispersion relation is obtained by looking for nontrivial solutions of Eq. (1), and it is given by

bb4 - tr(C)bb2 + det(C) = 0. (5)

The equation above admits two positive solutions:

œ (1)(k ) = <

œ (2)(k ) = .

tr(C ) -V tr2(C ) - 4det(C )

2 '

tr(C ) + <J tr2(C )- - 4det(C )

(6)

The two dispersion surfaces of the periodic system are presented in Fig. 1, b. We note that the dispersion surfaces meet at six points, called the Dirac points. The positions of the six Dirac points in the k - plane are indicated in Fig. 6 by points D.

After determining the eigenfrequencies cb(1) = bb (1)(k) and m(2) =m(2) (k), the corresponding eigenvectors (or eigenmodes) U(1) = U(1)(k) and U(2) = U(2)(k) can be obtained from the vector equations of motion (1), which are rewritten below in the form

cos Z + cos I

3 - 2 cos (Z + ! ) —

V3(cos |-cosZ)u ( j) =

(œ( j))2

u(j +

u 2j) = o,

V3(COS I -COS Z ) u j) +

3 - 3(cOS C + COS !) - (¿&( j) )2

(7)

u/ 2j) = o.

with j = 1, 2. We note that the eigenvectors corresponding

(

C = c

3 - 2cos (Zl +1) - 1/2[cos (ZD + cos (|)] >/3/2[cos (|l) - cos (Z/)]

V^/2[cos(|/) - cos(Zl )]

3 - 3/2[cos(Zl ) + cos (|l )]

(2)

is the stiffness matrix, where k = (kj, k2)T is the wave vector, Z = k[/2 + V3k2/2 and | = k^/2-V3k2/2.

In the derivation of Eq. (1), Bloch-Floquet quasi-peri-odicity conditions have been used, which allow a single repetitive cell to fully characterise the dispersive properties of the infinite system. The Bloch-Floquet conditions account for a phase shift between neighbouring cells and can be expressed by

to the lower and upper dispersion surfaces are real and or-

thogonal.

3. Lattice flux and circulation

In this section, we introduce the discrete definitions of flux and circulation for the lattice described in Sect. 2. These definitions will be used to characterise the nature of the waves that can propagate in the discrete medium.

148

Carta G., Jones I.S., Movchan N.V., Movchan A.B. / Ou3UHecKan Me30MexaHum 21 6 (2018) 146-156

Fig. 1. Periodic cell of the triangular elastic lattice (a), lower and upper dispersion surfaces of the discrete system (b)

3.1. Displacements of the lattice particles

The time-harmonic displacement of a lattice particle for a given eigenmode is obtained by multiplying the eigenvector by the phase shift in time, while the displacements of all the other particles are shifted according to the Bloch-Floquet conditions. If the coordinates of the central node of the lattice unit cell shown in Fig. 1, a are identified as x0 = (0,0), then the coordinates of the central node of the cell n (n e Z2) are given by x(n'0) = x0 + Tn, where T is the periodicity matrix defined in Sect. 2. Accordingly, the time-harmonic displacement of a lattice particle for a given eigenmode j = 1, 2 is given by

là(j}(x, f) = Re(U(j)ei(ô('*-kTn)) =

U( j)cos(co( j) t - k • Tn),

:,( j)

(8)

where time t has been normalised as t = t^c/m and U (j )is a unit vector.

From Eq. (8) it is apparent that the trajectory of a generic particle in the lattice is a straight line.

3.2. Definitions of lattice flux and circulation of the discrete displacement vector field

For a triangular lattice considered here, the lattice flux and lattice circulation of the displacement vector û are defined as

~ V3 -

Ofl = û • f, (9)

- V3 -

ra =-i2y(û x f ) •e 3, (10)

respectively, where the vector f is given by

f = (2 sin (Z +1) + sin Z + sin |, V3[sin Z - sin |])T.(11)

Since u is a real time-harmonic quantity (see Eq. (8)), the flux O% and the circulation rfi are pure imaginary time-harmonic quantities. We will derive formulae (9)-(11) in Sect. 3.2.1.

Since the eigenvectors calculated for the lower and upper dispersion surfaces are real and orthogonal, f % (j) =-i x xV3/2(e3 x u(j)) • f = miV3/2u(0 • f = mO%(i) with i ± j. Therefore, the amplitude of the flux of the displacement field determined for one dispersion surface is equal to the amplitude of the circulation of the displacement field calculated for the other surface.

3.2.1. Interpretation of lattice flux and circulation

In a two-dimensional continuum, the flux and circulation of a vector field F are defined by

OF = I F •n ^ (12)

3Q

Tf = I F • t ds, (13)

a^

respectively. In the equations above, is the boundary of the area ^ where the flux and circulation are calculated, n is the outward unit normal vector, t is the unit tangential vector, s is the coordinate that traces the boundary and e3 is the unit vector perpendicular to the (x1, x2)-plane and parallel to the x3- axis.

In order to apply the continuum concepts of flux and circulation to the lattice, which is a discrete system, we consider a generic nodal point of the lattice and its six nearest neighbours. We can denote by ^ the domain inside the hexagon having the six neighbouring points as vertices and by the boundary of the hexagon (Fig. 2). We consider the flux and circulation of the normalised displacement vector u defined at the six nodal points. The vectors n and t in (12) and (13) are identified, respectively, as the unit normal vectors and unit tangential vectors to the sides of the hexagon, as illustrated in Fig. 2, a and 2, b, respectively. We observe that the vector sum of the two unit vectors n (or t) at each nodal point is a radial (or tangential) vector with respect to the circle circumscribing the hexagon, shown as a dashed line in Fig. 2, a (or Fig. 2, b).

Using the Bloch-Floquet conditions (3), the (normalised) flux and circulation of the displacement field are given by

Ofi = I u.n dS = I{ueikl • (a/3, 0)t + ue~ikl • (-V3 0)T +

+ û e

+ û e

- + ûe

.- iZ

— + û e

V3 2

2 '

= i—{¿1[2sin (Z +1) + sin Z + sin | + + U2V3[sin Z - sin |]}

and

К. 0) - -

(-1/2, Y3/2) 0

(■<3/2, 1/2)

(-1/2, -f3/2)

(1/2, <3/2)

Fig. 2. Hexagonal cell, occupying the domain Q with the boundary 9Q, relative to a generic point in the lattice. In the unit normal vectors (a) and in the unit tangential vectors are shown (b)

ffi = J u•tds =-

I ue!tl • (0, V3)T + Ue~ik! • (0, - V3)T +

ЭП

+ ue'

( 3 Тз

2 2

+ ue

V3

+ ue'

-iÇ

V3

+ ueK

r 3 A v 2' 2

J3 ~ ~

= i-{-Mj>/3[sm Z - sin |] +

+ w2[2sin(Z + |) + sin Z + sin |]}, (15)

respectively. In the definitions above, s = s/l is the normalised coordinate. This interpretation is consistent with Eqs. (9)-(11).

3.2.2. The long wavelength limit

In the long wavelength limit as | k | — 0. Equations (9) and (10) have the following asymptotic approximations: 6U ~ iAhexU • k for |k |— 0, (16)

ffi ~ -iAhex (U x k) • e3 for | k | —^ 0, (17)

where Ahex = 3V3/2 is the (normalised) area of the hexagon.

It is apparent from Eqs. (16) and (17) that in the long wavelength limit, where the lattice approximates a continuum, shear and pressure waves are characterised by OU = 0 and fu = 0, respectively. For large values of the wave vector, we will distinguish between flux-free waves and circulation-free waves referring to Eqs. (9) and (10), as the continuum concepts of pressure and shear waves are inappropriate for large | k |.

3.2.3. Properties of the vector f

As evident from Eqs. (9) and (10), the vector f plays a key role in the characterisation of waves in a triangular lattice, analogous to the wave vector k in a continuum with respect to flux and circulation.

The components of the vector f are plotted in Fig. 3. For k2 = 0 (Fig. 3, a) f2 = 0, while for k1 = 0 (Fig. 3, b) f = 0. Moreover, we note that for k2 = kj/>/3 (Fig. 3, c) f = fJJÏ, while for k%2 = k>/3 (Fig. 3, d) f, = fV3. The vectors f and k are thus parallel in the directions given by arctan k2/kj = (n -1)n/6 with n = 1, ..., 12, and in the neighbourhood of the origin of the k - plane where f ~ 3k. The relation between vectors f and If is also illustrated in Fig. 3, e, where both vectors f and If are plotted in the same diagram.

3.3. Evaluation of lattice flux and circulation using lattice derivatives

The expressions (14) and (15) for lattice flux and lattice circulation, respectively, can also be derived using the notions of divergence and curl of a discrete field. In a continuum, the definitions (12) and (13) for flux and circulation can be written as

Of =/V-F dQ, (18)

rF = J (Vx F) • e3dQ,

(19)

respectively, using the divergence theorem and Stokes' theorem. In this section, we show that the definitions of lattice flux and lattice circulation are fully consistent with the notion of lattice derivatives through the discrete forms of divergence theorem and Stokes' theorem.

In order to define the divergence and curl of a vector field in a discrete system, we use the definition of lattice derivatives given in [25]. Referring to Fig. 4, we denote by bj ( j = 1, ..., 6) the oriented link vectors of the hexagon sides, which are given by

( i /TV

b1 = -b4 =

1 2'

~2'

£ 2

b2 =-b5 =(-1, 0)T

л 2

f k

2n/f3

n/T(3

k7 0

-n/Kl

y\ v * t X n v f ;

.V \

-2n/(3

-4n/3 -2n/3

/ / i \ \N>

« ' / / / 7 V v A' ✓ / 7 J j

x1 / / / /

A/f 1 1

\\ \ » > * V \ \

\ \ A

v s /

0

ki

2 n/3

4n/3

Fig. 3. Components of the vector f calculated for k2 = 0 (a), % = 0 (b), k2 = kjjl (c), k2 = (rf). Vector plots in the k - plane of f and k (e)

The (normalised) lattice derivatives associated with the links are then defined as follows:

V 1g = g(xB ) - g(XA )' V2g = g(XC ) - g(XB X V3g = g(Xd ) - g(Xc X V4g = g(X£ ) - g(XD X (21)

V5g = g(XF ) - g(XE )' V6g = g(X A ) - g(XF )'

where g is a generic lattice quantity, that is assumed to satisfy Bloch-Floquet conditions. We note that

ivjg = o.

j=1

The Cartesian discrete forms V^ and V~2 of the derivatives df and df are defined as

à * ^v x - 2 i(b1)j V j, - j1 (22)

where (b^) j and (b2) j are the components of b j in the x1-and x2 - direction, respectively.

Substituting Eqs. (20) and (21) into (22) and using the Bloch-Floquet conditions (3) for the lattice quantity g, the

Fig. 4. Oriented link vectors for the hexagonal unit cell of the discrete system

Cartesian lattice derivatives take the following forms:

v^ g=3[2 sin (Z+I)+sin Z+sin i]>

- tJi . - (23)

Vx2 g = -^[sin Z- sin I].

Accordingly, we define the (normalised) lattice divergence and lattice curl of the displacement field u as

VU = Vx Uj +VXj U2 = = 3[2sin (Z +1) + sin Z + sin |]Uj +

and

+ [sin Z - sin |]m2

(VXu) • e3 = ^¿2 -Vx2¿1 =

= 3 [2sin(Z + |) + sin Z + sin |]w2 + + —33[ - sin Z + sin |]mj ,

(24)

(25)

respectively.

Comparing Eqs. (24) and (25) with Eqs. (14) and (15), we notice that the lattice flux Ofi and the lattice circulation rfi have the form

^fi = ^hexV • u, ru = ihex (VX Û) • e^, (26)

which is consistent with the continuum definitions of flux and circulation given by Eqs. (18) and (19), respectively.

4. Wave characterisation in the k - plane

As shown in Sect. 3.1, the trajectories of the lattice par-

ticles are straight lines. Generally, these lines are neither

parallel nor perpendicular to the vector f, defined in (11).

This implies that flux and circulation are generally non-

zero, according to (9) and (10), for a general mode. We

denote by P(j) (j = 1, 2) the acute angle between the vec-

tor f and the displacement u(j), corresponding to the lower

( j = 1) or upper ( j = 2) dispersion surface.

In a continuum, the angle between the displacements of

the lattice particles and the wave vector k is used to iden-

tify the type of waves travelling in the medium. In particu-

lar, if this angle is 0 (or n/2), we refer to pressure (or

shear) waves. The discrete medium considered here represents a good approximation of a continuum only when | k | ^ 0. In this limit, vectors f and k are parallel, as dis-

cussed in Sect. 3.2.2. This means that P = 0 (or P = n/2)

indicates pure pressure (or pure shear) waves. Accordingly,

in the long wavelength limit where the discrete medium

approximates a continuum, flux-free and circulation-free waves represent shear and pressure waves, respectively.

The displacement vector can be decomposed into a component parallel to the vector f, characterised by zero cir-

culation, and a component perpendicular to the vector f, for which the flux is zero. When P = 0 (or P = n/2) only the former (or latter) component is different from zero. In

this case, waves are circulation-free (or flux-free). We shall see that there are also some special cases where both the flux and circulation are zero. In general, waves in the lattice may be thought as a "mixture" of circulation-free and flux-free waves. This mixture can be characterised by the value of the angle p. Polarisation of elastic waves in continuous media and in crystals is discussed, for example, in [26] and [27], respectively. The notion of polarisation of elastic waves in a chiral lattice is presented in [21].

We indicate by U. (or UN), where j = 1, 2, the component of the displacement vector that is parallel (or perpendicular) to the vector f. We emphasise that r Uj) = 0 and = 0. Accordingly, the total circulation f Uj ^ = f(^,

UN _ _ ( ... U UN

while the total flux =^U .

As discussed in Sect. 3.2, flux and circulation vary harmonically in time with the same frequency as the displacement U. Flux and circulation differ in phase; however, the phase difference is not relevant for our purposes. Hence it will not be examined in this paper. In what follows, we denote by |g| the amplitude of a generic time-harmonic quantity g.

From Eqs. (9)-(11) we obtain the following expressions for the amplitudes of lattice flux and lattice circulation:

|0fi|=^rlf|cosp, f. If| sinP,

(27)

(28)

where | f |= f2 + f22 and P is taken between 0 and n/2: P = arccos(U-f/| f |) if Uf >0, P = n-arccos(U-f/|f|) if U-f<0. The diagrams of the amplitudes of flux and circulation in the k- plane are shown in Figs. 5, a and 5, b. We note that only two diagrams are presented in Fig. 5 because 16U1) | = |f U2) | and 16U2) | = |fU1) |, as discussed in Sect. 3.2.

When k — 0, the flux and circulation for both surfaces tend to zero, while the ratios 1| / |f U^ | — 0 and | fU2)|/ |0U2)| — 0. This means that waves associated with the lower dispersion surface are flux-free (or of shear-type in the continuum definition) and those corresponding to the upper dispersion surface are circulation-free (or of pressure-type in the continuum definition). For large values of the wave vector, the flux and the circulation strongly depend on k.

Figure 5, c shows how the angles P(1) and P(2) change with the wave vector k. When | k | — 0, P(1) — n/ 2 while P(2) — 0. Interestingly, there are 12 directions, given by arctan (k2/k) = (n -1)n/6 with n = 1, ..., 12, where P(1) = = n/ 2 and P(2) = 0. Accordingly, waves characterised by a wave vector falling on one of these lines are either flux-free or circulation-free. This is extremely important, since we maintain "pure" flux-free or circulation-free waves even though the magnitude of the wave vector is not small, as in the long wavelength limit. We also note that P(1) = 0 and P(2) = n 2 on the sides of the hexagon having its vertices at the six Dirac points of the lattice (see points D in Fig. 6).

Fig. 5. Three-dimensional representations in the k- plane of the amplitudes | ®-1' | = |r%2) | (a) and | ®(2) | = |r(1' | (b), as well as the angles P(1) (c) and P(2) (d), calculated in the triangular lattice

This hexagon has considerable significance in characterising the nature of wave propagation, as will be seen in the following sections.

We now classify the nature of a propagating wave as a function of its wave vector. There are regions of "pure" flux-free wave propagation, regions of "pure" circulation-free wave propagation and regions where the waves have non-zero flux and circulation. The stationary points of the dispersion surfaces and the hexagon through the Dirac points will be seen to have special significance.

2n/V3 n/Tl3

k 0

-n/ Tf3 -2n/t|3

\ \ \! a f \ f / a / / /

\ a f \ \ \ / / o f f

fi' / / a / / / \ \ \ a f \ \

/ / v / f a m f \ \ \

Fig. 6. Stationary and Dirac points of the dispersion surfaces, whose properties are given in Tables 1 and 2. The crosses represent stationary points (points A) and Dirac points (ponts D) for both surfaces, while the dots indicate stationary points (points F) only for the upper surface

4.1. Stationary points and Dirac points

The stationary and Dirac points of the lower and upper dispersion surfaces are indicated in Fig. 6. Points D denote the Dirac points, which are at the vertices of a hexagon. Points A coincide with the middle points of the hexagon sides. Points A and D are characteristic of both dispersion surfaces, while points F are stationary points only for the upper dispersion surface. The properties of the stationary and Dirac points of the lower (upper) dispersion surface for the triangular lattice are detailed in Table 1 (Table 2).

At the points A and D, both the flux and the circulation are zero for both dispersion surfaces. This is due to the fact that the vector f = 0 at those points (see also Fig. 3). Furthermore, at the Dirac points D, the coefficients of ¿/1 and

Table 1

Stationary points of the lower dispersion surface of the triangular lattice

Point ki k2 (1 Type

±2n 0

A 0 ± 2tc/V3 V2 Saddle point

±n ±rc/V3

D ± 4n/ 3 0 3/V2 Maximum

± 2n/ 3 ± 2rc/V3

Table 2

Stationary points of the upper dispersion surface of the triangular lattice

Point k k1 &2 Type

±2n 0

A 0 ± 2л/л/3 V6 Maximum

±n ±лД/3

D ± 4п/3 0 3/V2 Minimum

± 2п/3 ± 2n/V3

±4arccos (V7/4) 0

F ±4arccos (3/4) ± 2n/V3 9/4 Saddle point

±[2n- arccos (-1/8)] ±[2VV3 -V3arccos(-1/8)]

± arccos (-1/8) ^>/3arccos(-V8)

U2 in (7) are equal to zero, implying that the eigenvectors can be chosen arbitrarily. This is consistent with the Dirac points being associated with isotropy.

At the points F, which are stationary points only for the upper dispersion surface, | Og1' | = |f U2) | = 0 while 1O~2)| = = if® | * 0. This occurs because the points F lie on the radial lines of zero flux for the lower surface and on the radial lines of zero circulation for the upper surface.

4.2. Wave characterisation in the k -plane

In Fig. 7 we detail the wave type for any value of the wave vector inside the hexagon passing through the Dirac points for the lower and upper dispersion surface. The diagram is repeated in a doubly periodic array outside the hexagon. The values of k where waves are flux-free or circulation-free are shown as dashed and dot-dashed lines, respectively, while the points where waves are characterised by both zero flux and zero circulation are indicated by crosses. In the grey regions, waves have both nonzero flux-free and nonzero circulation-free components.

We observe that the crosses, where both the flux and the circulation are zero, are at the intersections between the dashed and dot-dashed lines, where either the flux or the circulation is zero.

4.2.1. Lines of zero flux or zero circulation

In what follows, we give the analytical description of the results presented in Fig. 7.

Equation (1) can be written in normalised form as

(

BU =

B,, В

B11

В21 в

12

V ü 1

22

ü 2

= 0 with B = C -rô2M, (29)

that leads to U1 =-B12 U2 /B11 if B11 * 0 or U2 = = -B21UJB22 if B22 * 0. According to (9) and (29), flux is zero if one of the following conditions is met: Af = A2/1, if Bn * 0,

B21f2 = B22f1, if B22 * 0 We note that B11 = B11(ro) and B22 = B22(ro), while B12, B21, f, f2 are independent of at.

We start by proving that O(2) = 0 in the hexagon passing through the Dirac points. The left-hand side and right-hand side terms of (301) are given by

B11(&(2)) f2 =V3cos 'k

(k 1 (V3£ 1

2

V У

sin

2cos

( k 1 1

2

V У

cos

- 2cos k1 - V2>

xp - cosk1 + cos(2k1) + [2cosk1 - 1]x

x cos(V3k2) - 2

cos

( i 1 2

V У

+ cos

( 3k 1 2

V /-I

+ cos

(V3k2 H1/2

and

(31)

В12Л = 2л/з

2cos

x sin2

( К1 2

( k 1 (V3k 1

+ cos

(Jbh 1

ki 2

V У

sin

(32)

respectively. We will investigate the horizontal and inclined lines of the hexagon separately.

1. k2 = (2n + 1)2n/V3(rc e Z).

The equation in (301) is clearly satisfied since here sin (V3£2/2) = 0. However,

-2[cos (k1 /2) + cos k ]

if - 2n/3 + 4nn< k1 < 2n/3 + 4nn, (33) 0 otherwise.

B„(&(2)) =

Hence, (30j) is valid only on the horizontal sides of the hexagons connecting the Dirac points of the lattice. For 2n/3 + 4nn< k1 < 10n/3 + 4nn, B22(rô(2)) * 0, but (302) is not satisfied.

---Flux-free -----Circulation-free X

2n/f3 n/TÍ3

-n/ TÍ3 - 2n/ íÍ3

Flux-free and circulation-free

Nonzero flux and nonzero circulation

D

D

-) / A J <-------> \ r / \ 1 / k- A A 1 a

¡-K A/' i--- \ A V 1 / --¥- - - ' / \ - \ / A P

J\ \ 5 / 1 \ > 1 A ^ (D. ^ r / i

2n/-Í3 n/ll3

t2 0 -n/i/3 ■ 2n/V3

/ A .f f f. \ ! / A A

—D> r /F E-»— \ A v i / ' / \ - \ F \ / A D

V

\ \ ) F \F (D iA D /

2n/3 4n/3 2n -2n -4n/3 -2n/3 0 2n/3 4n/3 2n

h

Fig. 7. Characterisation of waves in the triangular lattice for the lower (a) and upper dispersion surface (b)

-2n -4n/3 -2n/3 0 ki

2. = ±V3£j + (2« +1) (n eZ). In this case, Eqs. (31) and (32) become

Ai(®(2)) f2 = W^cos x [1 + 2 cos k + 12 + 4cos k1 |]

( k ' 2

V y

sin

( % 1 2

V y

sin

( 3k1 1

(34)

and

B12 f = ±V3csc ksin2

( k ' 2

v y

[sin k + sin(2k1)]2, (35)

respectively. Formulae (34) and (35) are identical for 2tc/3 + Inn < k1 < 4n/3 + 2nn, namely at the inclined sides of the hexagons passing through the Dirac points. We also point out that

J%\

B11(rô(2)) = - sin

kL 2

v y

[1+ 2cosk1+ | 2 + 4cosk1 |], (36)

which is zero only in a countable set of points of the k - plane.

Next, we focus on the 12 directions passing through the origin of the k - plane, given by

arctan (k2/k1) = (n -1)n/6 (n = 1,..., 12). We want to prove that = 0 along these lines. For ffl = ffl(1), the right-hand side of (301) is the same as in (32), while the left-hand side of (301) is given by

B11(rô(1)) f2 =V3cos

( ^ ( 2

v y

sin

J3L

2cos

( k 1 № 1

2

V y

cos

- 2cosk +

+ V2|3 - cos k1 + cos(2k1) + [2cos k1 - 1]x x cos(V3k2) - 2 cos

( k1 2

V y

+ cos

( 3k11

x cos

(sk 11V2'

(37)

1. k2 = n 4n/S (n e Z).

The equation in (301) is clearly satisfied, because both (31) and (32) are zero considering that on these lines sin (V3k2/2) = 0. Since

-2[cos (kj/2) - cosk1]

if - 4n/3 + 4nn < k1 < 4n/ 3 + 4nn, (38)

0 otherwise,

B„(®(1)) =

the relation (301) is valid only inside the hexagons, as shown in Fig. 7. In the ranges of k1 where Bn(rô(1)) = 0, the relation (302) is not satisfied.

2. k1 = n4n (n e Z).

Bn(rô(1)) = 0, hence relation (301) cannot be used. However, relation (302) is satisfied, because B21 = 0, / = = 0 and

B22 (rô(1) ) = 2 - 2cos

1

(39)

which is zero only in a countable set of points. 3. k2 =±kJS +n4n/V3(n eZ). In this case, Eq. (301) is satisfied everywhere apart from a countable set of points. In particular,

Bn(ffl(1))f 2 = Bnf1 = +3V3 sin2

and

Bn(rô(1)) f2 = 3sin

(% 1

( k11 2

V y

sin k1

(40)

(41)

4. k2 = ±Sk + n4n/V3 (n e Z). Along these lines, the left-hand side and right-hand side terms of (301) are given by

Ai(®(1)) f2 = W3[1 + 2cos k1-12 + 4cos k1 |]x

x sin

( k11 2

V y

[sin k1 + sin(2k1)],

(42)

B12 f = ±V3csc k1sin"

( k.1 2

v y

[sin k1 +sin(2k1)]2, (43)

respectively. The last two formulae are equivalent if -2n/ 3 + +2nn < k1 < 2n/3 + 2nn. In this case,

B11 (bb(1)) = ^{2 |cosk - cos (2k%1) | +

+ [ - cos k%1 + cos(2k%1)]}, (44)

which is not zero except in a countable set of points.

5. Illustrative examples

Here we show how waves with specific values of the wave vector k propagate in the triangular lattice.

We start by considering a large value of | k in particular k%1 = 2.15 and k2 = 3.19. This wave vector lies in one of the grey regions of Fig. 7, characterised by "mixed" nonzero flux-free and nonzero circulation-free components. The corresponding frequencies in the lower and upper dispersion surface are bb(1) = 1.96 and bb(2) = 2.24, respectively. The components of the vector f are given by f = 0.04 and f = 0.61. Using (27), we derive the amplitudes of flux and circulation, namely | oU1)| = |fU2)| = 0.386 and | oU2)| = |f U1)| = 0.366. The angles between the vector f and the straight-line trajectories of the particles for the two dispersion surfaces are P(1) = 0.76 rad and P(2) = 0.81 rad. The trajectory of a lattice particle calculated for the lower (upper) dispersion surface is shown in Fig. 8 by a dashed (dot-dashed) line. We point out that the two trajectories are perpendicular to each other, because the corresponding eigenvectors are orthogonal.

The motions of the particles in the lattice for the two dispersion surfaces, both calculated for kf = 2.15 and k2 = = 3.19, are shown in the videos included in the supplementary material in Ref. [28]. The total displacements and the components of the displacements tangential to f and normal to f are displayed in the videos. The magenta and green arrows represent the wave vector k and the vector f, respectively. Videola-Videolf illustrate that the total motion of each lattice particle is the superposition of a flux-free and a circulation-free component.

For a smaller value of the wave vector, say k1 = 0.20 and k2 = 0.30, the lattice approximates a continuum. In this case, the vectors k and f are parallel to each other. The motion is of shear (or pressure) type in the lower (or upper) dispersion surface. Accordingly, the tangential (or normal) component of the displacement vector is very small (see Video2a-Video2f in the supplementary material in Ref. [28]).

~ ~ t

In Video3a-Video3f, the wave vector (k1, k) =

T

= (2.15,1.24)T, which is on the line inclined by 30° with respect to the k1 -axis. Although the modulus of the wave vector is large, the lattice motions corresponding to the lower and upper surfaces are similar to those observed in Video2a-Video2f for a small wave vector. The reason is that on the line kxj k2 = V3 the flux (or circulation) is zero in the lower (or upper) surface, as shown in Fig. 7, a (or Fig. 7, b), hence the tangential (or normal) component to f is null.

1

We note that flux-free waves in Video2a represent the case of small | k | and hence are similar to shear waves in a continuum. On the other hand, in Video3a, which also illustrates flux-free waves, the wavelength is comparable to the size of the elementary cell and | k | is finite. Similar observations apply to circulationfree waves, shown in Video2d and Video3d for small and finite values of | k respectively.

Video4a-Video4f show the lattice motion at the stationary point A with coordinates k = n and k2 =n/V3 (see Fig. 6 and Table 1). Standing waves are clearly visible from the videos. In addition, we note that the tangential (normal) component of the displacement is zero in the lower (upper) dispersion surface, because P( ) = n/ 2 (P(2) = 0). Finally, Video5a-Video5f present the motion of the lattice particles when the wave vector is on the hexagon connecting the Dirac points, shown in Fig. 6. For the calculations, we have taken k1 = 3.67 and k2 = 0.91. In this case, the normal (tangential) component of the displacement is equal to zero in the lower (upper) dispersion surface, since p(1) = 0

(P(2) =V 2).

6. Conclusions

The paper has addressed a fundamental issue of quantitative and graphical characterisation of the dynamic anisot-ropy of waves in a vector elastic lattice. Compared to the earlier work [21], which discussed wave polarisation in elastic structured media, an additional viewpoint has been proposed and fully implemented to take into account effects of pressure and shear in the dynamic elastic lattice. Although in the long-wave regime corresponding to a continuum approximation the matter appears to be straightforward, additional lattice constraints make coupling complex in the intermediate and high frequency regimes.

Fig. 8. Trajectory of a generic particle in the lattice corresponding to the lower (dashed line) and upper (dot-dashed line) dispersion surface, determined for k =2.15 and k2 = 3.19. The wave vector k, the vector f and the angles P(j) are also plotted in the figures

In particular, we have identified quantitative characteristics of waveforms in an elastic lattice to generalise the notion of pressure and shear waves used in the continuum. The cases of high frequency and wavelength comparable to the size of the elementary cell are considered in detail. These include dynamic anisotropy and dynamic neutrality in the context of waveforms classified by the lattice flux and lattice circulation attributed to them.

Acknowledgments

The authors would like to thank the EPSRC (UK) for its support through Programme Grant No. EP/L024926/1.

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Received November 15, 2018, revised November 15, 2018, accepted November 22, 2018

Сведения об авторах

Giorgio Carta, Liverpool John Moores University, UK, [email protected], Ian S. Jones, Liverpool John Moores University, UK, [email protected] Natalia V. Movchan, University of Liverpool, UK, [email protected] Alexander B. Movchan, Prof., University of Liverpool, UK, [email protected]