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Proc. R. Soc. A (2012) 468, 1196–1216
doi:10.1098/rspa.10.1098/rspa.2011.0609
Published online 4 January 2012
Analytic theory of defects in periodically
structured elastic plates
BY C. G. POULTON1, *, A. B. MOVCHAN2 , N. V. MOVCHAN2
AND R OSS C. MCPHEDRAN3
1 CUDOS,
Department of Mathematical Sciences, University of Technology,
Sydney, New South Wales 2007 Australia
2 Department of Mathematical Sciences, University of Liverpool, M & O
Building, Peach Street, Liverpool L69 3BX, UK
3 CUDOS, School of Physics, University of Sydney, New South Wales 2006,
Australia
We consider the problem of localized flexural waves in thin plates that have periodic
structure, consisting of a two-dimensional array of pins or point masses. Changing the
properties of the structure at a single point results in a localized mode within the bandgap that is confined to the vicinity of the defect, while changing the properties along an
entire line of points results in a waveguide mode. We develop here an analytic theory of
these modes and provide semi-analytic expressions for the eigenfrequencies and fields of
the point defect states, as well as the dispersion curves of the defect waveguide modes.
The theory is based on a derivation of Green’s function for the structure, which we
present here for the first time. We also consider defects in finite arrays of point masses,
and demonstrate the connection between the finite and infinite systems.
Keywords: flexural waves; biharmonic operator; photonic and phononic band gaps
1. Introduction
The last 20 years have seen remarkable developments in the study and control of
waves in periodically structured media. Much of this research has concentrated on
electromagnetic waves, in which photonic crystals have been shown to exhibit a
number of important applications in both the applied and fundamental sciences;
one specific area in which progress has been particularly impressive is the creation
of high quality-factor (high-Q) resonant cavities (Akahane et al. 2003) and
waveguides in photonic crystals (Gersen et al. 2005). Using an older terminology,
these would be referred to as point and line defects in crystals—in the photonic
case, these defects can be exploited to trap light for optical switching, and to
control the speed at which pulses propagate, leading to dramatically enhanced
nonlinear effects (Monat et al. 2010).
The analysis of elastic solids with periodic systems of defects (inclusions or
voids) brings new challenges that do not appear in the problems of optics and
*Author for correspondence ([email protected]).
Received 6 October 2011
Accepted 2 December 2011
1196
This journal is © 2012 The Royal Society
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Defects in structured plates
1197
electromagnetism governed by Maxwell’s equations. In particular, isotropic elastic
media are characterized by two types of waves, dilatational and shear, which
are coupled via the boundary conditions on the surfaces of embedded elastic
inclusions. This brings new dispersion patterns for elastic Bloch waves, unseen in
problems of electromagnetism (Sigalas & Economou 1993). Equally challenging
is the study of flexural waves in elastic plates, which are governed by a fourthorder partial differential equation. The analysis of dispersion properties of Bloch
waves and simulation of localization around bounded defects were performed
by Movchan et al. (2007, 2009), McPhedran et al. (2009) and Poulton et al.
(2010) for Kirchhoff plates with periodic arrays of defects, and by Movchan et al.
(2011) for structured Mindlin plates. Analytical and numerical scalar models of
localized waveforms in two-dimensional lattice systems have been developed by
Ayzenberg-Stepanenko & Slepyan (2008). The analysis of the dynamic anisotropy
in vector problems of elasticity, including special classes of standing waves in
lattice systems with rotations, was presented by Colquitt et al. (2011); the latter
also discussed the effects of focusing and negative refraction in lattice systems
with rotations.
The design of point and line defects has generally been accomplished using
sophisticated numerical simulations on powerful computers. However, a small
number of groups have developed analytic or semi-analytic methods that
can deliver more physical insight into why particular structures deliver good
performance while others do not (Platts et al. 2002, 2003; Botten et al. 2005;
Sauvan et al. 2005; Wilcox et al. 2005; Thompson & Linton 2007; Mahmoodian
et al. 2009a,b; Busch et al. 2011). One difficulty in developing such methods
is that ideally one would like to model the influence of changing a localized
area in a periodic structure on fields. The fields then cease to be characterized
by their behaviour in a period cell, but are spread out over an infinite area in
two dimensions, or over an infinite volume in three. This causes computational
problems, usually dealt with by computing properties of large but finite systems.
However, this is not a completely satisfactory solution, because finite systems can
show strong surface wave effects, and internal interference effects of the Fabry–
Pérot type not existing in truly infinite systems. One technique to model truly
infinite systems has been termed the Fictitious Source Superposition Method by
Wilcox et al. (2005) and the array scanning method by Thompson & Linton
(2007). These methods consist of solving the quasi-periodic array problem for
a net of values of the Bloch vector sampling the Brillouin zone (BZ), and then
superimposing the solutions to arrive at a defect Green’s function with its source
located at a single point in the crystal lattice. This Green’s function can then
be related to the required point defect solution. For a line defect the method is
similar, except that a propagation vector component, say kx , is chosen, and the
required superposition is only over a set of values sampling the component ky .
The earliest references we know of relating to this method are Zolla et al. (1994)
and Figotin & Goren (2001).
Here, we develop a new analytic theory for the study of localized elastic
vibrations in thin plates. The theory is based on the formulation of the defect
Green’s function for a time-harmonic Kirchhoff plate with a periodic array of
scattering centres, which can take the form of rigid pins or point masses. The
governing equation for this system is the biharmonic equation rather than the
Helmholtz equation. The study of such periodically structured thin plates, known
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C. G. Poulton et al.
as platonic crystals, has been taken up in a number of recent papers (Evans &
Porter 2007; Movchan et al. 2007; McPhedran et al. 2009; Meylan & McPhedran
2011; Movchan et al. 2011). In one of these (McPhedran et al. 2009), a result was
given for the frequency of a defect mode in a finite cluster of pinned points, and
Evans & Porter (2007) have discussed waves guided between two gratings, but the
literature has otherwise been confined to periodic structures without defects. The
generalization we make here opens the study of platonic structures to potentially
useful designs of the type that have been intensively investigated in the field of
photonic crystals, such as high-Q cavities, filters based on point defects coupled
to waveguides, slow wave designs and waveguides with tight bends.
The structure of the paper is as follows. In §2, we develop the theory for
a defect in an array of rigid pins. Our approach is to first construct a quasiperiodic Green’s function and then to apply BZ averaging to derive the defect
Green’s function for the structure. We then continue to find the equations giving
the frequency of both point and line defects, where the defects have been created
by changing the mass of a single point or a line of points. Throughout this
section, we concentrate on the first bandgap for an array of pinned points in
a thin elastic plate. In §3, we analyse Bloch waves and defect modes in an elastic
plate containing an array of finite masses, and establish a connection with the
rigid pins problem of §2. Section 4 presents the discussion of localized vibration
modes in an elastic plate containing a finite cluster of masses, accompanied by
numerical simulations. Note that the outline of the theory given is valid for any
two-dimensional array; however, for brevity, we will usually refer to rectangular
arrays, and give numerical results exclusively for the square array.
2. Point and line defects in platonic crystals
(a) The homogeneous array of pins
We consider a Kirchhoff plate containing a doubly periodic rectangular array of
pins. The flexural displacement is assumed to be time harmonic with the radian
frequency u and amplitude u. The rigid pins are considered as the limit of small
(h)
circular holes Fa = {x : |x − x(h) | < a}, of radius a with clamped edges, as a → 0.
Here, x(h) are the positions of rigid pins within the doubly periodic array on the
plane. The function u satisfies the equation of motion
ru2
u(x) = 0, x ∈ R2 \ ∪h Fa(h) ,
(2.1)
D
together with appropriate boundary conditions—namely, that both the value of
u and its normal derivative vanish at the edge of each small hole (at the location
of the pins):
(2.2)
u(x) = 0, when |x − x(h) | = a
and
vu = 0,
(2.3)
vr |x−x(h) |=a
D2 u(x) −
where r = |x − x(h) |, and h is the bi-index specifying the position x(h) in the array.
Here r is the mass density, u is the radian frequency and D = Es 3 /(12(1 − n2 )),
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Defects in structured plates
1199
with s being the plate thickness, E and n being Young’s modulus and Poisson’s
ratio of the elastic material. The limit, as a → 0, is taken in (2.1)–(2.3) for the
rigid pins problem. In addition, waves in a periodic lattice must obey the Bloch–
Floquet quasi-periodicity condition, which is
(h) ·K
u(x(h) + x) = u(x) eix
,
(2.4)
where the phase change from one cell in the lattice to the next is specified by the
Bloch vector K. For the sake of simplicity, we assume a rectangular lattice, in
which the lattice points are x(h) = (mdx , ndy ), where both m and n are integers.
The BZ is then given by the set of points {K = (kx , ky ) : |kx | ≤ p/dx , |ky | ≤ p/dy }.
To derive the dispersion equation for the Bloch waves, we need the quasiperiodic Green’s function Gp (x − x ; b, K) for the Kirchhoff plate, which—from
Movchan et al. (2007)—is given by
Gp (x − x ; b, K) =
(h) ·K
eix
g(x − x − x(h) , b),
(2.5)
h
with g being the fundamental solution (Evans & Porter 2007) for the
homogeneous Kirchhoff plate, so that
D2 g(x, b) −
ru2
g(x, b) + d(x) = 0,
D
(2.6)
with the explicit representation of the form
1
2
(1)
g(x, b) = − 2 iH0 (b|x|) − K0 (b|x|) .
8b
p
(2.7)
Here
b =u
2
rs
.
D
(2.8)
Then the function (2.5) is re-expanded using the Graf addition theorem as in
Abramowitz & Stegun (1965), and we note that the contribution of the regular
parts of the expansion from pins not located at the origin is to exactly cancel
the regular part of the expansion arising from the central pin (McPhedran et al.
2009). To describe the behaviour near the origin, it is sufficient to retain only the
monopole terms because the limit of pin radius approaches zero; this results in
Gp (x − x ; b, K) =
Proc. R. Soc. A (2012)
1
2
Y0 (br) + K0 (br) + J0 (br)S0Y (b, K)
2
8b
p
2
+ I0 (br)S0K (b, K) ,
p
(2.9)
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1200
C. G. Poulton et al.
where r = |x − x |, and S0Y , S0K are the lattice sums for the array, which are
formally defined by (Chin et al. 1994)
⎫
(h)
S0Y (b, K) =
eix ·K Y0 (b|x(h) |) ⎪
⎪
⎪
⎬
h=(0,0)
(2.10)
(h)
⎪
eix ·K K0 (b|x(h) |).⎪
and
S0K (b, K) =
⎪
⎭
h=(0,0)
The sum S0Y is conditionally convergent when evaluated in direct space, and
S0K is rapidly convergent except when b is small; appropriate computable
representations for these sums are given inter alia by McPhedran et al. (2009).
Taking the limits of equation (2.9) as r → 0, we note that
1
2 K
Y
(2.11)
lim Gp (r; b, K) = 2 S0 (b, K) + S0 (b, K) ,
r→0
8b
p
and in addition, we have
2 K
vGp (r; b, K)
r
Y
= lim
−S0 (b, K) + S0 (b, K) = 0.
lim
r→0
r→0 16b
vr
p
(2.12)
Comparing the limits at the pinned points, we can see that, provided the condition
S0Y (b, K) +
2 K
S (b, K) = 0
p 0
(2.13)
is satisfied, Green’s function Gp obeys the limit conditions (2.2) and (2.3) at the
pinned points x(h), as well as the quasi-periodicity conditions required for the
solution u(x). We can then identify
u(x) = Gp (x; b, K)
(2.14)
with the dispersion equation (2.13) for Bloch waves in an array of rigid pins.
(b) The point defect Green’s function
We now construct the point defect Green’s function—that is, Green’s function
corresponding to the situation when the rigid pin at the origin is released
and replaced by a unit force located at the origin. Up to now, we have
regarded Gp as obeying an inhomogeneous differential equation implied by (2.5)
and (2.6), the sources at the lattice points arising from the discontinuity in
Green’s function’s second-order derivative. We could equivalently specify Gp by
excluding the lattice points from the domain on which the function is defined and
enforcing the appropriate behaviour as x → x(h) : in this way the inhomogeneous
differential equation may be replaced with a homogeneous differential equation
with inhomogeneous boundary conditions (Morse & Feshbach 1953). The quasiperiodic Green’s function then satisfies the homogeneous differential equation
(D2 − b4 )Gp = 0,
Proc. R. Soc. A (2012)
for
x = x(h) .
(2.15)
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Defects in structured plates
1201
From the expansion of Gp given in (2.9), together with the quasi-periodicity
condition, we have the following limiting behaviour at the lattice points x(h) :
2 K
1
(h)
Y
lim Gp (x; b, K) = 2 S0 (b, K) + S0 (b, K) eix ·K .
8b
p
x→x(h)
(2.16)
We can regard equation (2.16) as an inhomogeneous boundary condition that
replaces the driving term in the differential equation. The function Gp is then
determined to within an additive homogeneous part that must vanish at the
lattice points: any such homogeneous solution is of course a Bloch mode of the
system, occuring within the band at values (b, K) for which equation (2.13) is
satisfied. For defect modes existing within the band, this homogeneous solution
must be chosen so that the mode satisfies an outgoing wave condition; however,
within the band gap no such solutions exist and uniqueness, to within a scale
factor, of the defect mode is ensured by the condition that the mode vanishes at
distances far removed from the defect centre.
We now consider an intermediate function g(x,
˜ b), defined by
8b2 Gp (x; b, K)
.
(2.17)
g(x,
˜ b) = Y
S0 (b, K) + (2/p)S0K (b, K) BZ
Here the notation . . .BZ denotes the operation of averaging the vector K over
the BZ, so that, if we assume a rectangular array with x(h) = (mdx , ndy ) where m
and n are both integers, we have
dx dy p/dx p/dy
. . .BZ =
· · · dkx dky .
4p2 −p/dx −p/dy
We note that g˜ satisfies the differential equation
(D2 − b4 )g˜ = 0,
for
x = x(h) ,
(2.18)
BZ = dh,0 .
(2.19)
together with the boundary conditions
(h) ·K
lim g(x,
˜ b) = eix
x→x(h)
The integration over the unit cell renders all the boundary conditions at lattice
points other than the origin homogeneous. Using the expression (2.9) to expand
in the central unit cell, we find that
2
1
K0 (br)
Y0 (br) +
g(r)
˜ = Y
p
S0 (b, K) + (2/p)S0K (b, K) BZ
S0Y (b, K)
J0 (br)
+ Y
S0 (b, K) + (2/p)S0K (b, K) BZ
S0K (b, K)
2
I0 (br).
(2.20)
+
p S0Y (b, K) + (2/p)S0K (b, K) BZ
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C. G. Poulton et al.
We can recover the inhomogeneous differential equation satisfied by g˜ by
re-including the origin and noting that
2
4
2
(D − b ) Y0 (br) + K0 (br) + 8b2 d(x) = 0.
p
We find then that
8b2
(D − b )g(x)
˜ +
(S0Y + (2/p)S0K )
2
d(x) = 0.
4
(2.21)
BZ
The function g˜ therefore represents an inhomogeneous source with weight
8b2 /(S0Y + (2/p)S0K )BZ placed at the origin surrounded by a lattice of rigid
pins.
By re-scaling, one can deduce the defect Green’s function for the array:
−1 1
Gp (x; b, K)
G(x; b) = Y
.
S0 (b, K) + (2/p)S0K (b, K) BZ S0Y (b, K) + (2/p)S0K (b, K) BZ
(2.22)
Using the expansion about the origin in (2.9), one immediately finds that,
provided the weight factor 1/(S0Y + (2/p)S0K )BZ = 0, the function G satisfies
the differential equation
(D2 − b4 )G(x; b) + d(x) = 0
(2.23)
together with pinned boundary conditions at the lattice points x = x(h) , for
h = 0. The special case where 1/(S0Y + (2/p)S0K )BZ = 0 corresponds to a
resonance of the structure; this is discussed in the following section.
(c) Point defects
We consider the equation for a localized state in an array of pins, created by
releasing the pin located at x(h) = 0. The localized state thus created obeys the
differential equation at all but pinned points:
ru2
U (x) = 0 for x = x(h) , with h = 0.
(2.24)
D
In addition, we specify appropriate boundary conditions for an array of pins
at lattice locations x(h) —namely, that both the value of U and its derivatives
vanish at x(h) when h = 0. We note here that by releasing the pin at the origin,
the earlier mentioned differential equation must be satisfied at x = 0.
We now observe that the function g̃(x) is very close to the solution that we
want for U , because g̃(x) obeys the differential equation (2.24) at all points
except for x = 0 and in addition obeys the correct limiting conditions at the lattice
points. All that is required is to make the weight of the source term vanish. To
see this explicitly, we set
2
8b Gp (x; b, K)
,
(2.25)
U (x) = p0 g̃(x, b) = p0 Y
S0 + (2/p)S0K BZ
D2 U (x) −
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Defects in structured plates
where p0 is some arbitrary constant. Around the origin, the expansion for U is
1
2
K
Y
(br)
+
(br)
U (x) = p0 Y
0
0
p
S0 (b, K) + (2/p)S0K (b, K) BZ
S0Y (b, K)
J0 (br)
+ Y
S0 (b, K) + (2/p)S0K (b, K) BZ
S0K (b, K)
2
I0 (br).
(2.26)
+
p S0Y (b, K) + (2/p)S0K (b, K) BZ
In order to satisfy the correct boundary condition for a ‘removed’ pin, the first
term, corresponding to the irregular part, must vanish identically. We therefore
have the condition for the defect state:
1
= 0.
(2.27)
S0Y + (2/p)S0K BZ
The expansion for the defect state in the vicinity of the origin is
S0Y
S0K
2
U (x) = p0 Y
J0 (br) + p0 Y
I0 (br).
p
S0 + (2/p)S0K BZ
S0 + (2/p)S0K BZ
The equation (2.27) can easily be implemented numerically to determine the
frequency of the defect state in a total band gap. The lattice sum S0Y has a firstorder singularity at what is called in the photonic crystal literature the light line
or plane wave propagation condition, which occurs when b = |K|. Consequently,
there is a change of sign at the light line, so that the condition (2.27) requires
that the contributions from inside the light line and outside the light line balance
each other. Numerically, this gives the value b = 2.5372, which agrees well with
the estimate b = 2.538 for the defect mode in a finite set (17 by 17) of pins
from McPhedran et al. (2009). We note that the condition for a resonance (2.27)
corresponds to a singularity of Green’s function for the array (2.22), as is known
from the general treatment in Economou (2006).
We now consider the case where the central pin is released and a finite point
mass perturbation M (which can be positive or negative) is introduced instead.
Assuming that u lies within the stop band, we deduce that the localized state
now obeys the differential equation
M u2
ru2
U (x) −
U (0)d(x) = 0.
(2.28)
D
D
Green’s function G for the localized state obeys equation (2.23), with identical
conditions at the lattice points x(h) as apply for U . Comparing these two
equations, it is apparent that we can write
D2 U (x) −
M u2
G(x)U (0).
D
We then require that the consistency condition
U (x) = −
U (0) = −
Proc. R. Soc. A (2012)
M u2
G(0)U (0)
D
(2.29)
(2.30)
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C. G. Poulton et al.
be satisfied. Rearranging, we obtain
1+
M u2
G(0) = 0.
D
(2.31)
From equations (2.22) and (2.16), we know that
G(0) =
−1
1
1
8b2 S0Y + (2/p)S0K BZ
(2.32)
and so we find that the existence condition for a defect state arising from a single
point mass is
M u2
1
=
−
.
(2.33)
8b2 D
S0Y + (2/p)S0K BZ
The point mass perturbation can be removed entirely by taking the limit as
M → 0. This recovers the equation (2.27).
(d) Line defects
We now consider the case in which we modify an entire line of pins oriented
along the x-axis, either replacing them with point masses or removing them
entirely. The structure is periodic in the x-direction, and so all solutions must
obey the Bloch condition
U (x + mdx x̂) = U (x) eimkx dx ,
(2.34)
where dx is the lattice pitch in the x-direction, x̂ is the unit basis vector along
the x-axis and kx is the Bloch wavenumber. Following the reasoning given in the
previous section, we define a new function g̃ L as
8b2 Gp (x; b, K)
g̃ L (x; b, kx ) = Y
.
(2.35)
S0 (b, K) + (2/p)S0K (b, K) ky
The notation .ky represents an integral over a line in the BZ with respect to ky ,
such that
2p dy /2
.ky =
.dky
dy −dy /2
The function g̃ L then obeys the boundary conditions at lattice points x(h) :
(h) ·K
lim g̃ L (x; b, kx ) = eix
x→x(h)
ky = eikx mdx dn,0 ,
(2.36)
where we have written the bi-index h = (m, n). As in the case for the point
defect, the integral over the BZ has the effect of turning inhomogeneous boundary
conditions into homogeneous ones, this time however leaving the phased array of
Proc. R. Soc. A (2012)
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1205
Defects in structured plates
sources at lattice points on the x-axis. Following identical reasoning to that of
the previous section, Green’s function for a line defect is
−1 1
Gp (x; b, K)
(2.37)
GL (x; b, kx ) = Y
S0 + (2/p)S0K ky S0Y + (2/p)S0K ky
and it satisfies the differential equation
∞
ru2
GL (x − x ; b, kx ) +
d(x − x + mdx x̂) eimkx dx = 0.
D GL (x − x ; b, kx ) −
D
m=−∞
(2.38)
The solution that corresponds to removing an entire row of pins is then
2
8b Gp (x; b, K)
,
(2.39)
U (x, kx ) = p0 g̃ L (x; b, kx ) = p0 Y
S0 + (2/p)S0K ky
2
with the resonance condition
1
Y
S0 + (2/p)S0K
=0
(2.40)
ky
required to remove the array of sources, and thus obtain a solution that satisfies
the differential equation everywhere. The left-hand side of this relation is a
function of kx ; therefore, we may expect a curve of solutions (kx , u). This is the
dispersion curve for the waveguide created by removing the pins.
The dispersion diagram for the wave guided by a square array of pinned points
with a line defect created by releasing an entire row of pins is shown in figure 1.
The guided wave b value runs over roughly the top third of the band gap range,
and exceeds p before decreasing towards it as b tends to the edge of the BZ. It is
notable that the wave guide range of b includes the point defect value b = 2.5372,
as this means it will be possible to construct systems coupling point defects to
wave guides.
If we now replace the row of released pins with point masses M , the solution
will satisfy the differential equation
∞
M u2
ru2
U (x) −
U (0)
d(x + mdx x̂) eimkx dx = 0.
D U (x) −
D
D
m=−∞
2
(2.41)
Comparing (2.38) with (2.41), we can write the solution to the line defect
problem as
U (x) = −
M ru2
GL (x; b, kx )U (0).
D
(2.42)
The consistency condition that must be satisfied by a valid solution is then
1=−
Proc. R. Soc. A (2012)
M ru2
GL (0; b, kx ),
D
(2.43)
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1206
C. G. Poulton et al.
3.0
2.5
2.0
b
1.5
1.0
0.5
0.5
1.0
1.5
kx
2.0
2.5
3.0
Figure 1. Dispersion diagram, showing b versus kx , for the guided waves in a plate having a square
array of pinned points with a line defect forming the waveguide. The dispersion curve for the elastic
plate with no pinned points is the lowest line, while that for the guided wave is the middle, curved
line. The dashed line corresponds to b = p.
or, using the formulae (2.37) and (2.16), we deduce
1
Y
S0 (b, kx , ·) + (2/p)S0K (b, kx , ·)
=−
ky
M b2
.
8rh
(2.44)
Removing the row of additional masses entirely by taking the limit as M → 0
recovers the resonance condition (2.40).
3. Band structure for Bloch waves in a plate with a periodic array of masses
We consider a Kirchhoff plate containing a doubly periodic rectangular array
of additional point masses m. The flexural displacement is assumed to be time
harmonic with the radian frequency u and the amplitude u. The function u
satisfies the following equation of motion:
D2 u(x) −
mu2 ru2
u(x) −
u(x(h) )d(x − x(h) ) = 0.
D
D
(3.1)
h
As in §2, the solution must have quasi-periodicity specified by the Bloch vector
K, thus
(h) ·K
u(x(h) ) = u(0) eix
Proc. R. Soc. A (2012)
.
(3.2)
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1207
Defects in structured plates
7
6
X
5
4
b 3
G
M
2
1
0
–6
–4
X
–2
0
G
2
4
M
6
8
X
Figure 2. Dispersion diagram, showing b versus K , for the Bloch waves in the plate, containing
a doubly periodic array of point masses, with m/r = 5. The dispersion curves correspond to the
three segments bounding the irreducible BZ, as shown in the inset.
By comparing (3.1) and (3.2) with the equation for the quasi-periodic Green’s
function (2.9), we deduce
u(x) = −
mu2
u(0)Gp (x; b, K),
D
(3.3)
mu2
u(0)Gp (0; b, K).
D
(3.4)
with the compatibility condition
u(0) = −
By cancelling u(0) and using the representation (2.9), we derive the dispersion
equation relating b to K as follows:
2 K
mb2
Y
(3.5)
S0 (b, K) + S0 (b, K) = 0.
1+
8rh
p
It is important to note that in deriving this result, we used
2
lim Y0 (br) + K0 (br) = 0.
r→0
p
The solutions of the dispersion equation (3.5) representing the first three bands
are shown in figure 2, for a square lattice.
The most striking feature in figure 2 is the presence of an acoustic band, absent
in the corresponding diagram for an array of rigid pins (McPhedran et al. 2009).
We note that, as the mass ratio m/r increases towards infinity this acoustic band
becomes flatter and flatter, finally collapsing into the axis b = 0 in the limit.
Another interesting feature of figure 2 is that the second band for the segment
X − M coincides exactly with the dispersion curve for the unstructured plate,
given by
(3.6)
b = p2 + ky2 .
Proc. R. Soc. A (2012)
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C. G. Poulton et al.
7
6
X
5
4
b 3
G
M
2
1
0
–6
–4
X
–2
0
G
2
4
M
6
8
X
Figure 3. Dispersion diagram, which shows b versus K , for the case when m/r = 20. The dispersion
curves correspond to the three segments bounding the irreducible BZ as shown in the inset. It is
noted that the second and third bands are practically unaffected by the change in the mass ratio
m/r, and they look the same as in figure 2. The acoustic band lowers down with the increase of
m/r, which leads to an increase in the width of the first stop band.
This is true for all values of m/r, and it shows that the propagation of the elastic
waves in the mass-loaded plate is totally unaffected by the loading, for this specific
direction of propagation. As noted in McPhedran et al. (2009), this occurs because
the second band is sandwiched between two planes giving dispersion surfaces for
the unstructured plate. At each of these planes, the lattice sum S0Y diverges,
so that the second band cannot cross either plane, and so coincides with them
at their intersection. Note that such intersections occur for every higher pair of
bands along symmetry lines, so that the higher bands do not have total band
gaps; the segment G − M is the only region where there exists a partial band-gap
between bands 2 and 3, which touch at the X point. Band 4 touches band 3 at
the G point, and so on for higher bands. Thus, the gap between the acoustic band
and the second band is the only total gap, for any value of the mass loading. As
illustrated in figure 3, the width of the band gap increases with the increase of
m/r. The second and third bands in figure 3 are practically unaffected by the
change in m/r, whereas the acoustic band moves downward compared with that
in figure 2.
The asymptotic analysis of the dispersion equation (3.5) for small b and |K|
leads to the simple relationship
m 1/4
.
(3.7)
|K| = b 1 +
rs
In particular, when m/rs → ∞ the slope of the corresponding dispersion
curve reduces to zero. For finite m/rs, it lies below the first band for the
unstructured plate.
(a) A point perturbation within the system of masses
For a frequency u from the stop band separating the first and the second
dispersion bands in figure 2 there are no propagating Bloch modes. For such a
Proc. R. Soc. A (2012)
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1209
Defects in structured plates
frequency, we consider the exponentially localized Green’s function G(x; b), which
satisfies the equation
D2 G(x; b) −
ru2
mu2 G(x(h) ; b)d(x − x(h) ) = 0.
G(x; b) + d(x) −
D
D
(3.8)
h=(0,0)
Physically, this function represents the flexural displacement in the periodically
m-mass-loaded plate, with the central mass being replaced by a time harmonic
point force of unit amplitude. The quantity b is related to the radian frequency
u by the formula (2.8).
We wish to relate this Green’s function to the localized defect mode created
by replacing the mass m in the central cell by a different value m + M ≥ 0. This
localized mode U (x; b) satisfies the equation
(m + M )u2
ru2
U (x; b) −
U (0; b)d(x)
D
D
U (x(h) ; b)d(x − xh ) = 0.
D2 U (x; b) −
−
mu2
D
(3.9)
h=(0,0)
Next, we introduce the quasi-periodic Green’s function Gm for the mass-loaded
plate, with the full periodic array of masses (m) being present. This obeys the
governing equation
D2 Gm (x, x ; b, K) −
ru2
(h)
eix ·K d(x − x(h) − x )
Gm (x, x ; b, K) + d(x − x ) +
D
h=(0,0)
−
mu2 Gm (x(h) , x ; b, K)d(x − x(h) ) = 0.
D
(3.10)
h
As in §2, the notation ·BZ is used for the average, with respect to K, over the
BZ. Taking the average of equation (3.10) and noting that
(h) ·K
eix
BZ = dh,0 ,
we deduce
D2 Gm BZ (x, x ; b) −
−
ru2
Gm BZ (x, x ; b) + d(x − x )
D
mu2 Gm BZ (x(h) , x ; b)d(x − x(h) ) = 0.
D
(3.11)
h
By comparing equation (3.8) with (3.11) and using the uniqueness of the localized
Green’s function G(x; b), we deduce
G(x; b) =
Proc. R. Soc. A (2012)
Gm BZ (x, 0; b)
.
1 − (mu2 /D)Gm BZ (0, 0; b)
(3.12)
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1210
C. G. Poulton et al.
A similar rescaling relates the defect mode U to the localized Green’s function G
(m + M )u2
U (0; b)G(x; b).
D
Putting x = 0 in (3.13) leads to the compatibility condition
U (x; b) = −
1+
(3.13)
(m + M )u2
G(0; b) = 0.
D
(3.14)
2
D/(rs) be the radian frequency of the localized defect mode
Let uM = bM
corresponding to the perturbation M of the mass in the central cell. Then,
combining (3.14) and (3.12), we derive
2
M uM
(3.15)
Gm BZ (0, 0; bM ) = 0.
D
An equivalent version of this equation gives the perturbation mass M as a function
of a specified value of b chosen within the total band gap
1+
M
= −(b4 Gm BZ (0, 0; b))−1 .
rs
(3.16)
The final step in this argument is to relate Green’s function Gm to
the representation (2.9) of the quasi-periodic Green’s function Gp for the
homogeneous Kirchhoff plate. By comparing (3.10), with x = 0, and (2.5) we
deduce
−1
mu2
Gp (x, 0; b, K) = 1 −
Gm (x, 0; b, K).
(3.17)
Gm (0, 0; b, K)
D
Hence, Gm (0, 0; b, K) can be represented via Gp (0, 0; b, K) as follows:
Gm (0, 0; b, K) =
Gp (0, 0; b, K)
1+
mb4
Gp (0, 0; b, K)
rs
.
(3.18)
Furthermore, after the averaging over the BZ, we obtain the relation
−1 mb4
.
+ Gp−1 (0, 0; b, ·)
Gm (0, 0; b, ·)BZ =
rs
(3.19)
BZ
This has been written in a form suitable for computations because Gp−1 goes to
zero at ‘light lines’ rather than diverging. Using (3.19) and (3.16), we derive the
mass perturbation formula
−1 rs
m
=− 1+
.
(3.20)
M
mb4 Gp (0, 0; b, ·)
BZ
For the case of m/(rs) = 5, the mass perturbation M /m is plotted as a
function of b in figure 4, which indicates that to increase the frequency of
the defect mode the mass of the central cell should be reduced. In particular,
in the limit M = −m, the corresponding frequency represents the defect mode for
the extreme case when the point mass has been entirely removed from the central
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1211
Defects in structured plates
−0.3
−0.4
m/ρ = 20
m /ρ = 5
M/m
−0.5
−0.6
−0.7
−0.8
−0.9
−1.0
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
b
Figure 4. Defect modes: m/r = 5 and m/r = 20. Mass perturbation ratio M /m versus b.
cell. We also note that the small reduction in mass generates a defect mode near
the lower edge of the band gap, and the curve M /m versus b is steepest near
that band edge. It appears that the lower edge of the stop band is reached for the
perturbation mass ratio M /m close to −0.32, which suggests that a finite mass
perturbation is required to create a defect mode even in the close proximity of
the boundary of the stop band.
As an interesting special case, we consider a system approaching the defect in
an array of pinned points created by removing the constraint on the central point.
Formally, as b > 0, the array of pinned points is obtained by letting m tend to
infinity. To obtain the equation for the defect frequency and the corresponding
b, we replace M by −m + e and take the limit as e/m → 0. Referring to (3.16)
and (3.19), we derive the first-order expansion
rs rs 2 1
1
rs
,
=− +
−m + e
m
m b4 Gp (0, 0; b, ·) BZ
and hence
1
rs
.
e 4
b Gp (0, 0; b, ·) BZ
In the limit as e → 0, we obtain the formula that defines the frequency of the
defect mode for the plate containing an array of pinned points:
1
= 0.
(3.21)
Gp (0, 0; b, ·) BZ
(b) A line defect within an array of point masses
Consider a waveguide created by altering the masses located on the x-axis,
so that each mass m is replaced by m + M . The displacement amplitude U is
assumed to satisfy the one-dimensional quasi-periodicity condition
U (x + ndx e1 ) = U (x) eindx kx .
Proc. R. Soc. A (2012)
(3.22)
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1212
C. G. Poulton et al.
It also satisfies the differential equation
D2 U (x) −
−
∞
(m + M )u2
ru2
U (x) −
U (0)
d(x − ndx e1 ) einkx dx
D
D
n=−∞
(3.23)
∞
mu2 U (0, ldy )
d(x − ndx e1 − ldy e2 ) einkx dx = 0,
D
n=−∞
l=0
where the delta function terms represent the inertial contribution from point
(m)
masses. In turn, the ‘phased line defect’ Green’s function GL for the mass-loaded
plate satisfies the equation
D
2
∞
ru2 (m)
GL (x, x ; b, kx ) +
−
d(x − x − ndx e1 ) einkx dx
D
n=−∞
(m)
GL (x, x ; b, kx )
−
∞
mu2 (m)
GL ((0, ldy ), x ; b, kx )
d(x − ndx e1 − ldy e2 ) einkx dx = 0. (3.24)
D
n=−∞
l=0
The comparison of (3.23) and (3.24) leads to the relation
−
m+M 2
(m)
u U (0)GL (x, 0; b, kx ) = U (x).
D
(3.25)
(m)
In a way similar to (3.12), we obtain the connection between GL and the
quasi-periodic Green’s function Gm for the doubly periodic array of point masses:
(m)
GL (x, 0; b, kx ) =
Gm ky (x, 0; b, ky )
,
1 − (mu2 /D)Gm ky (0, 0; b, kx )
(3.26)
where ·ky stands for the average with respect to ky within the BZ. Furthermore,
similar to (3.15) we derive the dispersion equation for flexural waves localized
within the waveguide along the x-axis:
1+
M u2
Gm ky (0, 0; b, kx ) = 0.
D
(3.27)
One can also take into account the connection (3.19) between the quasi-periodic
Green’s functions Gp and Gm . In the limit, as m → ∞ and M → 0, we obtain the
dispersion equation for waves propagating through a waveguide, along the x-axis,
surrounded by the arrays of rigid pins. Similar to (3.21), we deduce
1
= 0,
(3.28)
Gp (0, 0; b, kx , ·) ky
which is equivalent to
Proc. R. Soc. A (2012)
1
S0Y (b, kx , ·) + (2/p)S0K (b, kx , ·)
= 0.
ky
(3.29)
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1213
Defects in structured plates
2
1
10
0
5
5
10
Figure 5. Flexural displacement for the localized defect mode in an 11 × 11 cluster, with
m/(sr) = 5. The spectral parameter value is b = 2.5, and the corresponding defect mass ratio
is M /m = −0.948.
4. Discussion: localized vibrational modes in a finite cluster
The band diagram in figures 2 and 3 contains a band gap, and the possible
location of the frequency of a defect mode is in the total band gap of an infinite
periodic system. The presence of the full band gap in figures 2 and 3 suggests that
a strong localization of waves will occur in this frequency region. This means that
the study of wave properties and defect modes in relatively small finite systems
could be expected to yield accurate predictions for the periodic case.
Consider a finite array of point masses m arranged at points O(q) (q1 , q2 ), where
q1 and q2 are integers running from −N to N , omitting the point (0, 0) at the
origin, and the notation q = (q1 , q2 ) is used for a multi-index. A mass m + M
is placed at O(0) , at the centre of the cluster. The notations P, P0 are used for
the cluster excluding the origin, and for the complete cluster with the origin.
The equation of motion for the amplitude W of the time-harmonic transverse
displacement is
ru2
(m + M )u2
W (x; b) −
W (0; b)d(x)
D
D
mu2 W (O(h) ; b)d(x − Oh ) = 0.
−
D
D2 W (x; b) −
h∈P
Proc. R. Soc. A (2012)
(4.1)
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1214
C. G. Poulton et al.
(a)
(b)
1.0
0.5
10
0
5
5
10
1.0
0.5
0
–0.5
10
5
5
10
Figure 6. Flexural displacement for the defect mode: m/(sr) = 5. The spectral parameter values
and the corresponding defect mass ratios are: (a) b = 2.079, M /m = −0.599351 and (b) b = 2.02,
M /m = −0.33916.
The notation W (h) is also used for the nodal displacements W (O(h) ). By
representing W via Green’s function g(x, b) from (2.7)
M + m 2 (0)
mu2 (q)
W (x; b) = −
u W g(x, b) −
W g(x − O(q) , b),
(4.2)
D
D q∈P
we write the consistency equations for the nodal amplitudes W (q) in the form
M + m 2 (0)
mu2 (q)
W (p) +
W g(O(p) − O(q) , b) = 0, p ∈ P0 .
u W g(O(p) , b) +
D
D q∈P
(4.3)
The equation (4.3) can be programmed in matrix form, where it is required that
the determinant of the matrix is zero. A better conditioned form of the equations
for this purpose is
M
rs
(p)
(p)
(0)
W
+
g(O
,
b)
+
W (q) g(O(p) − O(q) , b) = 0, p ∈ P0 ,
+
1
W
b4 m
m
q∈P
(4.4)
where we also note that g(0, b) = −i/(8b2 ).
In the numerical implementation of (4.4), the matrix is of dimension (2N +
1)2 × (2N + 1)2 . The magnitude of the determinant of the matrix has a sharp
minimum as soon as N is sufficiently large (N ≥ 3). Subject to this condition, the
mass ratio M /m for a given b is in excellent agreement with the value coming
from the treatment for the defect in the infinite array as described in §3. Given the
value of M /m for which the determinant is minimized, the vector of displacement
amplitudes W (q) at the nodal points of the cluster can be found by solving a
system of linear algebraic equations (after putting W (0) = 1).
We give in figures 5 and 6 plots of the real part of the displacement amplitude
versus position within the cluster. Note that the displacement amplitude has only
been computed at integer mesh points, but an interpolation procedure has been
applied to generate a smooth surface passing through the calculated points, to
Proc. R. Soc. A (2012)
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Defects in structured plates
1215
enable better visualization of the displacement pattern. The imaginary part of
the displacement amplitude takes values that are negligible compared with those
of the real part.
The first value of b chosen (figure 5) lies in the middle of the band gap, which
runs between 2.018 and p. The mode is accordingly tightly confined at the centre
of the cluster. As we move down in b, we approach the edge of the stop band, and
the mode becomes less localized, and it develops an oscillating part surrounding
the central peak. This is particularly evident for figure 6b, which has b very close
to the band edge.
C.G.P. and R.C.M. acknowledge support from the Australian Research Council through its
Discovery Grants Scheme, and, in the latter case, from the Liverpool Research Centre in
Mathematics and Modelling.
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