International Journal of Engineering Science xxx (2011) xxx–xxx
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International Journal of Engineering Science
journal homepage: www.elsevier.com/locate/ijengsci
Microstructure continuum modeling of an elastic metamaterial
R. Zhu a, H.H. Huang c, G.L. Huang a,b, C.T. Sun c,⇑
a
b
c
Department of Applied Science, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
Department of Systems Engineering, University of Arkansas at Little Rock, Little Rock, AR 72204, USA
School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA
a r t i c l e
i n f o
Article history:
Available online xxxx
Keywords:
Elastic metamaterial
Microstructure continuum
Wave propagation
Dispersion curve
a b s t r a c t
Elastic metamaterials have unusual microstructures that can make them exhibit unusual
dynamic behavior. For instance, if treated as classical elastic solids, these materials may
have frequency-dependent effective mass densities which may become negative in certain
frequency range. In this study, an approach for developing microstructure continuum models to represent elastic metamaterials was presented. Subsequently, this continuum model
was used to study wave propagation and band gaps in elastic metamaterial with resonators. In contrast to the use of the conventional continuum theory with which the effective
mass density would become frequency-dependent and negative, the main advantage of the
microstructure continuum model is that the local microstructural deformation/motion is
accounted for with the introduction of additional kinematic variables. Moreover, the material constants of the microstructure continuum model are explicitly expressed in term of
the properties of the host medium and the resonator. The accuracy of the microstructure
continuum model was evaluated by comparing dispersion curves of harmonic waves to
those obtained by the finite element analysis based on the exact geometry of the elastic
metamaterial.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
In the recent past, many studies have been devoted to two- and three-dimensional periodic acoustic/elastic metamaterials, based on the similarity between electromagnetic waves and elastic waves (Cummer & Schurig, 2007; Fokin, Ambati,
Sun, & Zhang, 2007; Hou, Fu, & Liu, 2004; Mei, Liu, Shi, & Tian, 2003; Mei, Liu, Wen, & Sheng, 2006). Because of the vector
characteristics of elastic waves and the possible coupling between longitudinal and transverse modes, richer mechanical
phenomena are expected to exist in wave propagation in elastic metamaterials. Generally, elastic metamaterials possess
microscopic to macroscopic heterogeneities and allow comparatively easier fabrications. Practical applications of these systems include mechanical filters, sound and vibration isolators, acoustic waveguides and energy harvesting (Cheng, Xu, & Liu,
2008; Gonella, To, & Liu, 2009; Pai, 2010; Sun, Du, & Pai, 2010).
Elastic/acoustic metamaterials can be modeled as classical homogeneous solids which require the use of negative mass
densities in order to establish the equivalence between the model and the original metamaterial. Liu et al. (2000) fabricated
and investigated an elastic metamaterial based on the idea of localized resonant structures that exhibit low-frequency bandgaps, which was clearly explained by the negative effective mass density (Liu, Chan, & Sheng, 2005) of the metamaterial
within certain frequency range. Li and Chan (2004) first reported theoretically a possibility of the existence of acoustic/elastic metamaterials. They utilized the effective mass density and bulk modulus and showed that both the effective mass
⇑ Corresponding author. Tel.: +1 765 494 5130; fax: +1 765 494 0307.
E-mail address: sun@purdue.edu (C.T. Sun).
0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijengsci.2011.04.005
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
doi:10.1016/j.ijengsci.2011.04.005
2
R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
density and bulk modulus can be simultaneously negative, in the sense of an effective medium. They claimed that the double
negativity is derived from low-frequency resonances. Milton and Willis (2007) presented a rigorous theoretical foundation
for the concept of negative effective mass density of elastic metamaterials. Wave mechanism in the elastic metamaterials
was clearly explained and interpreted in references (Huang & Sun, 2010; Huang, Sun, & Huang, 2009). An effective medium
theory for two-dimensional elastic metamaterials with isotropic scatters embedded in a solid medium was proposed in the
long wavelength limitation (Wu, Lai, & Zhang, 2007). The experimental validation of bandgaps and wave localization in a
one-dimensional elastic metamaterial also has been conducted (Wang, Wen, Wen, & Liu, 2006; Yao, Zhou, & Hu, 2008).
Similar efforts have been taken by working on elastic metamaterials with a macroscopic higher order gradient or non-local elastic response even though they are governed by usual linear elasticity equations (Alibert, Seppecher, & Dell’Isola,
2003). Recently, Milton investigated the macroscopic behavior of an elastic metamaterials integrated with one negative
effective mass (Milton, 2007). Based on his discussion, the microscale effects possibly non-local behavior was explained
by the modification of linear continuum elastodynamic equations. The total stress is not only related with strain but also
with velocity, and the momentum density is not only related with velocity but also with the displacement gradient. A similar
approach proposed by Wang and Sun (2002) is to introduce a micro-inertia continuum model that retains the simplicity of
the classical continuum mechanics while capturing the effects of the microstructure or nanostructure. In this formulation,
the microstructural effects are included in an effective body force term that appears in the equations of motion and the
boundary conditions. The effective body force term is a function of the first and second time derivatives of the macro strain
and is responsible for the dispersive behavior of harmonic waves. By combining this effective body force term with the
macro stress, this continuum model has the form of the classical elasticity theory. One alternative way to solve the aforementioned problem is to employ additional kinematic variables to describe the non-homogeneous local deformation in
the microstructure of the solid. This approach leads to the Cosserat continuum model (Cosserat & Cosserat, 1909) or micropolar model (Toupin, 1962) or the microstructure continuum theory (Eringen & Suhubi, 1964; Huang & Sun, 2007; Mindlin,
1964; Sun & Huang, 2007).
In this paper, we employ a microstructure continuum model to describe the dynamic behavior of an elastic metamaterial
with the microstructure effects accounted for. The major advantage of this technique is that all the material constants in the
model are explicitly expressed in terms of the properties of the host medium and the resonator. For illustrations of this approach, we consider wave propagation in the metamaterial. The accuracy of the present model is verified by comparison
with the results obtained from the finite element method.
2. Microstructure continuum model
We consider a two-dimensional metamaterial consisting of periodic cylindrical cavities of radius a that are carved out
from an elastic matrix, see Fig. 1. This metamaterial was proposed by Milton and Willis (2007). In the center of each cavity
is a mass m which is connected to the cavity with four elastic springs each having spring constant K2 in x2 direction and K3
in x3 direction. For simplicity, we assume that the springs are massless. The distances between two adjacent masses are d2
and d3 in the x2 and x3 directions, respectively. The elastic properties of the matrix are given by the Lame constants km and
lm. It is interesting to mention that Maugin also studied so-called Maxwell-Rayleigh model of a continuum in which there
are small resonators embedded in an elastic matrix (Maugin, 1995; Maugin & Christov, 2001).
The classical continuum theory is not able to account for the local motion of the resonator in the cavity and is not adequate for modeling this type of the elastic metamaterial if local deformation/motion in the microstructure is desired (Huang
& Sun, 2007). On the other hand, microstructure continuum theories which employ additional kinematic variables appear to
be capable of characterizing the dynamic behavior of the elastic metamaterial.
In view of the geometric symmetry of the elastic metamaterial as shown in Fig. 1, a representative volume element (RVE),
for instance, cell (k, l) can be readily identified. In cell (k, l), the position of the internal mass is given by the global coordinates
x2 ¼ xk2 and x3 ¼ xl3 . In addition, we define a local polar coordinate system (r, h) as well as the corresponding local Cartesian
coordinates (X2, X3) with the relationship of X2 = r cos h and X3 = r sin h as shown in Fig. 2.
The local displacements in the RVE are approximated by linear series expansions in terms of quantities which are defined
at the center of the cell. In order to capture the resonant movement of the center mass, the micro-deformations are defined
as follows:
(a) In the center mass of the cell (k, l), (r = 0):
ðk;lÞ
uma
¼ vi
i
ð1Þ
:
(b) In the spring area of the cell (k, l), (0 < r 6 a):
sðk;lÞ
ui
ðk;lÞ k
x2 ; xl3
¼ ui
sðk;lÞ k
x2 ; xl3
þ r cos h/zi
sðk;lÞ k
x2 ; xl3 :
þ r sin h/zi
ð2Þ
(c) In the matrix material of the cell (k, l), (r P a):
mðk;lÞ
ui
ðk;lÞ k
x2 ; xl3
i
¼u
sðk;lÞ k
x2 ; xl3
þ a cos h/2i
mðk;lÞ k
sin h/3i
x2 ; xl3 ;
sðk;lÞ k
x2 ; xl3
þ a sin h/3
mðk;lÞ k
x2 ; xl3
þ ðr aÞ cos h/2i
þ ðr aÞ
ð3Þ
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
doi:10.1016/j.ijengsci.2011.04.005
R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
3
Fig. 1. Microstructures of the elastic metamaterial.
Fig. 2. Representative volume element of the elastic metamaterial.
where i = 2, 3. Thus, the displacement in the matrix is expressed as the displacement at the spring and the matrix interface
plus additional terms which increase linearly with the distance from the interface. By using this expression, the displacement satisfies the condition of continuity at the interface between the spring area and the matrix. The physical interpretation
ðk;lÞ
of the terms in Eqs. (1)–(3) is that the global displacement u
is the displacement at the center of the RVE (the average
i
ðk;lÞ
sðk;lÞ
sðk;lÞ
mðk;lÞ
mðk;lÞ
displacement), v i is the global displacement of the center mass, while /23 ; /32 ; /23 and /32 represent shear misðk;lÞ
sðk;lÞ
mðk;lÞ
mðk;lÞ
cro-motions, /22 ; /33 ; /22 and /33 represent stretch micro-motions.
The displacements are required to be continuous at the interfaces between cell (k, l) and the neighboring cells. It is, however, not possible to require point by point continuity. In this study, the continuity condition on the average displacements at
the interfaces of the cells is required. We have
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
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R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
Z
d2
2
d
22
mðkþ1;lÞ ui
ui
mðk;lþ1Þ ui
ui
d
x3 ¼ 23
mðk;lÞ d
x3 ¼ 23
dx2 ¼ 0;
ð4Þ
dx3 ¼ 0;
ð5Þ
and
Z
d3
2
d
23
d
x2 ¼ 22
mðkþ1;lÞ
mðk;lÞ d
x2 ¼ 22
mðk;lþ1Þ
where ui
and ui
represent the displacement components in the cells (k + 1, l) and (k, l + 1) with i = 2, 3. Substituting
Eq. (3) in Eq. (4) and working out the integrals, we have
ðkþ1;lÞ
ui
ðk;lÞ
ui
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
d ad3
1 þ 1 þ n2 sðkþ1;lÞ
3
mðkþ1;lÞ
sðk;lÞ
mðk;lÞ
mðkþ1;lÞ
mðk;lÞ
ln
/3i
þ /3i /3i
þ /3i
¼ 0;
/3i
/3i
f
d2
2
ð6Þ
where f ¼ dd32 . Similarly, another continuous condition can be obtained from Eq. (5) as
ðk;lþ1Þ
ui
ðk;lÞ
ui
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d ad2
2
sðk;lþ1Þ
mðk;lþ1Þ
sðk;lÞ
mðk;lÞ
mðk;lþ1Þ
mðk;lÞ
/2i
¼ 0;
ln n þ 1 þ n2 /2i
/2i
þ /2i /2i
þ /2i
d3
2
ð7Þ
With the differentiation with respect to the local coordinates x2 and x3, the displacement expansions in Eqs. (1)–(3) can be
used to compute the corresponding strains in the mass-spring system and the matrix, respectively, as
(a) In mass-spring area of the cell (k, l),
sðk;lÞ
sðk;lÞ
sðk;lÞ
sðk;lÞ
sðk;lÞ
sðk;lÞ
sðk;lÞ
esðk;lÞ
¼ /22 ; e33 ¼ /33 ; e23 ¼ e32 ¼ 1=2 /32 þ /23 :
22
ð8Þ
(b) In the matrix material of cell (k, l),
@ cos h sðk;lÞ
@ sin h sðk;lÞ
mðk;lÞ
mðk;lÞ
/22 /22
/32 /32
þa
;
@x2
@x2
@ cos h sðk;lÞ
@ sin h sðk;lÞ
mðk;lÞ
mðk;lÞ
mðk;lÞ
¼ /33 þ a
/23 /23
/33 /33
þa
;
@x3
@x3
a @ cos h 1 mðk;lÞ
a @ cos h sðk;lÞ
a @ sin h sðk;lÞ
mðk;lÞ
mðk;lÞ
mðk;lÞ
mðk;lÞ
sðk;lÞ
mðk;lÞ
/23 þ /32
¼ e32 ¼
/22 /22
/32 /32
/23 /23
þ
þ
þ
2
2 @x3
2 @x3
2 @x2
a @ sin h sðk;lÞ
mðk;lÞ
þ
/33 /33
:
ð9Þ
2 @x2
mðk;lÞ
emðk;lÞ
¼ /22 þ a
22
emðk;lÞ
33
emðk;lÞ
23
Based on the displacement expansions (1)–(3), the kinetic energies stored in the center mass and the matrix are found as
T maðk;lÞ ¼
2 2 1
ðk;lÞ
ðk;lÞ
m v_ 2
þ v_ 3
2
ð10Þ
and
3 2
2
2
1 X
_ mðk;lÞ þ ðJ þ Im 2J Þ /_ mðk;lÞ þ J /_ sðk;lÞ þ J /_ sðk;lÞ
_ Þ2 þ ðJ 1 þ Im
q
Am ðu
2
70
1
2
3 2J 60 Þ /2i
2
3i
2i
3i
2 i¼2
o
mðk;lÞ sðk;lÞ
mðk;lÞ sðk;lÞ
;
þð2J 60 2J 1 Þ/_ 2i /_ 2i þ ð2J 70 2J 2 Þ/_ 3i /_ 3i
T Mðk;lÞ ¼
ð11Þ
respectively, where the coefficients in the expression above are
Im
2 ¼
J 60 ¼
ZZ
ZZ
Am
x23 dAm ;
Im
3 ¼
ar cos2 h dAm ;
Am
ZZ
ZZ
x22 dAm ; J 1 ¼
a2 cos2 h dAm ;
Am
Am
ZZ
2
J 70 ¼
ar sin h dAm :
J2 ¼
ZZ
2
a2 sin h dAm ;
Am
Am
Assume that the unit cell has a unit cross-sectional area, the kinetic energy density in the cell (k, l) is
T ðk;lÞ
av e ¼
1
ðT maðk;lÞ þ T Mðk;lÞ Þ;
d2 d3
ð12Þ
Similarly, based on the defined strain components (8) and (9), the strain deformation energies for the spring-mass system
and the matrix can be written as
W sðk;lÞ ¼ K 2
ðk;lÞ
ðk;lÞ
v2
u
2
2
2
2 2
2
2 sðk;lÞ
sðk;lÞ
ðk;lÞ
sðk;lÞ
sðk;lÞ
2
2
ðk;lÞ
þ K3 u
þ a2 /22
þ a2 /23
v
þ
a
/
þ
a
/
3
3
33
32
ð13Þ
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
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R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
and
W Mðk;lÞ ¼
ZZ
2
2 2
1 mðk;lÞ
mðk;lÞ
mðk;lÞ
mðk;lÞ
mðk;lÞ
km r22 þ r33
þ lm r22
þ r33
þ 2ðr23 Þ2 dAm :
Am 2
ð14Þ
The strain energy density averaged over the volume of cell (k, l) is
W ðk;lÞ
av e ¼
1
ðW sðk;lÞ þ W Mðk;lÞ Þ:
d2 d3
ð15Þ
To obtain a continuum model, we now introduce fields that are continuous in x2 and x3, and whose values at x2 ¼ xk2 and
x3 ¼ xk3 coincide with the values of the actual field variables at the center of the cell. Therefore, we can also consider a strain
energy density W(x2, x3, t) and a kinetic energy density T(x2, X3, t) as continuous functions. Based on Eq. (10) and Eq. (11), the
kinetic energy density in the continuum fields is
3 2
2
1
1 X
_ m þ ðJ þ Im 2J Þ /_ m
_ i Þ2 þ ðJ 1 þ Im
m½ðv_ 2 Þ2 þ ðv_ 2 Þ2 þ q
Am ðu
2
70
3i
3 2J 60 Þ /2i
2
2
2 i¼2
2
2
þ J /_ s
þ ð2J 2J Þ/_ m /_ s þ ð2J 2J Þ/_ m /_ s
þJ /_ s
Ac T ¼
1
2i
2
60
3i
1
2i
2i
70
2
3i
ð16Þ
3i
and the strain energy density in the continuum field is
Ac W ¼ K 2
2
2
2 2
2
2 ðk;lÞ
sðk;lÞ
sðk;lÞ
ðk;lÞ
sðk;lÞ
sðk;lÞ
ðk;lÞ
ðk;lÞ
þ K3 u
u
v2
þ a2 /22
þ a2 /23
v3
þ a2 /33
þ a2 /32
2
3
h 2
2
2
2
1
2
þ Am /m
þ J 4 /s22 /m
þ J 5 /s33 /m
þ J 6 /s32 /m
þ ðkm þ 2lÞ Am /m
22
33
22
33
32
2
2
s
s
i
s
m
m
m
m
þJ 6 /s23 /m
þ 2J 7 /m
þ km Am /m
23
22 /22 /22 þ 2J 8 /33 m /33 /33
22 /33 þ J 7 /22 /22 /33
s
s
s
s
m
m
m
m
þJ 8 /s33 /33 /m
22 þ J 6 /22 /32 /33 /33 þ J 6 /32 /32 /23 /23
2 1 h s
2
2
2
2
AM s
þ 2lm
þ J 6 /22 /m
þ J 6 /s33 /m
þ J 4 /s23 /m
þ J 5 /s32 /m
/23 /m
32
22
33
23
32
4
4
s
s
s
m
s
m
s
m
m
m
m
m
m
þ 2J 6 /s22 /m
22 þ 2J 6 /32 /32 /23 /23 þ 2J 7 /23 /23 /23 þ 2J 8 /23 /23 /32 þ 2J 7 /23 /23 /32
m
ð17Þ
þ2J 8 /s32 /m
/
32
23
where Ac = d2d3, the coefficients in the expression are
J4 ¼
J8 ¼
ZZ
a2
4
sin h dAm ;
2
Am r
ZZ
J5 ¼
ZZ
a2
cos4 h dAM ;
2
Am r
J6 ¼
ZZ
a2
2
sin cos4 h dAM ;
2
Am r
J7 ¼
ZZ
a2
2
sin h dAM ;
2
Am r
a2
cos4 h dAM :
2
Am r
Considering the field variables as continuous functions of x2 and x3, the continuity conditions in Eqs. (6) and (7) can be written as differential relations in the terms of continuous variables as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
i 2a
@u
1 þ 1 þ n2
d3 @/s3i d3 @/m
d3 @/m
3i
3i
/s3i /m
/m
S3i ¼
ln
þ
¼0
m
3i
3i n
@x3 d2
2 @x3
2 @ 3i
2 @/m
3i
ð18Þ
and
S2i ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i 2a
@u
d2 @/s2i d2 @/m
d2 @/m
2i
2i
/m
/s3i /m
ln n þ 1 þ n2
þ
@xm ¼ 0;
2i
2i @x2 d3
2 @x2
2 @x2
2 @x2
ð19Þ
where i = 2, 3. The continuity conditions have thus been turned into constraint conditions between the continuous field
variables.
Considering a fixed region V of the microstructure medium, the displacement equations of motion can then be obtained
by employing Hamilton’s principle for independent variations of the dependent field quantities in V and in a specified time
interval t0 6 t 6 t1 as
Z
t1
Z
d
t0
V
F dV dt þ
Z
t1
dW 1 dt ¼ 0;
ð20Þ
t0
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
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R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
where F = T W, dW1 is the variation of the work done by external forces, and dV is the scalar volume element. To consider
the continuity conditions as subsidiary conditions through the use of Lagrangian multipliers, the variational problem may
then be redefined by using the function
F ¼T W 3
X
ðC2i S2i þ C3i S3i Þ;
ð21Þ
i¼2
where the Lagrangian multipliers C2i and C3i are functions of x2, x3 and t. Since the functional F as given in Eq. (19) depends only on the dependent field variables and their first order derivatives, the system of Euler equations can be written as
2
3
3
X
@ 4 @F 5 @F
¼ 0;
@qr @ @fs
@fs
r¼1
@qr
ð22Þ
m
i ; v i ; /s2i ; /s3i ; /m
where fs represents the 16 dependent variables u
2i ; /3i , C2i and C3i, and qr are the spatial and time variables of x2, x3 and t. A system of 16 governing equations of motion can be obtained from the Euler equation.
3. Wave propagation in the elastic metamaterial
To validate the proposed continuum model, let us first consider one-dimensional longitudinal wave motion along x2
2 ; v 2 ; /s22 ; /22 and C22. Theredirection in the metamaterials. In the case, the motions are described by the field variables u
fore, the energy densities simplify considerably and five equations can be written as
1
€ 2 þ 2K 2 ðU 2 v 2 Þ @ C22 ¼ 0;
½qAm u
Ac
@x2
ð23Þ
1
2 v 2 Þ ¼ 0;
½mv€ 2 þ 2K 2 ðu
Ac
ð24Þ
o
1n
qJ1 /€ s22 þ qðJ60 Ji Þ/€ m22 þ 2a2 K 2 /s22 þ ðkm þ 2lm Þ J4 /s22 /m22 þ J7 /m22 þ lm J6 /s22 /m22
Ac
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2a
ad2
@ C22
ln n þ 1 þ n2 C22 þ
ln n þ 1 þ n2
¼0
d3
d3
@x22
ð25Þ
o
1n
qðJ1 þ Im3 2J60 Þ/€ m22 þ qðJ60 J1 Þ/€ s22 þ ðkm þ 2lm Þ Am /m22 þ J4 /m22 /s22 þ J7 /s22 2J7 /m22 þ lm J6 /m22 /s22 Þ
Ac
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2a
ad2
d2 @ C22
ln n þ 1 þ n2 1 C22 þ
ln n þ 1 þ n2 ¼ 0;
ð26Þ
þ
d3
d3
2 @x22
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2a
@u
d2 @/s22 d2 @/m
d2 @/m
22
22
/m
/s22 /m
ln n þ 1 þ n2
þ
¼ 0;
22
22 @x2 d3
2 @x2
2 @x22
2 @x2
ð27Þ
For longitudinal waves propagating in the X2 direction, the continuum wave fields are given by
2 ¼ B1 exp½iðk2 x2 xt;
u
/m
22
¼ B4 exp½iðk2 x2 xtÞ;
v 2 ¼ B2 exp½iðk2 x2 xtÞ;
/s22 ¼ B3 exp½iðk2 x2 xtÞ;
C22 ¼ B5 exp½iðk2 x2 xtÞ;
ð28Þ
where B1, B2, B3, B4 and B5 are constant amplitudes, k2 is the wave-number along X2 direction, and x is angular frequency.
Substitution of Eq. (26) in the equations of motion refspseqn47(23)–(26) yields five homogeneous equations for B1, B2, B3, B4
and B5. For a non-trivial set of solutions the determinant of the coefficients must vanish to yield the dispersion relation.
The longitudinal wave dispersion curves are also predicted by using the finite element methods. The commercial FE software, ANSYS 11.0, is used. Fig. 3 shows the finite element model of the elastic metamaterial with spring-mass microstructures. Plane-82 element is used to model the host elastic medium, and Mass-21 and Combine-14 elements are used to model
Fig. 3. Finite element model for the elastic metamaterial.
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
doi:10.1016/j.ijengsci.2011.04.005
R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
7
the spring-mass system. In order to obtain the dispersion curves, modal analysis of the metamaterial will be simulated to
obtain the acoustic and optic wave dispersion curves.
Fig. 4 shows the comparison of the normalized dispersion curves for the two modes of the longitudinal wave predicted by
the present microstructure continuum
model and the finite element method. In the figure, the normalized wave frequency is
qffiffiffiffiffiffi
defined as x ¼ xx0 with x0 ¼ 2Km2 and the geometric parameters of the metamaterial are set as dd32 ¼ 1 and da2 ¼ 0:2 The material properties used in the calculation are selected as follows:
(1) The host elastic medium:
Em ¼ 3:50e6
kg
;
mm s2
v m ¼ 0:3; qm ¼ 1:2e 6
kg
:
mm3
ð29Þ
(2) The spring-mass system:
K 2 ¼ 5000
kg
;
s2
K 3 ¼ 20000
kg
;
s2
m ¼ 2:48e 5 kg:
ð30Þ
From Fig. 4, it is seen that the microstructure continuum model is very accurate for the acoustic wave mode prediction.
However, for the optic mode (the higher mode), the micro structure continuum model is adequate for k2d2 < 1.0. It is noted
that the band-gap structure of the metamaterial can be accurately predicted by using the present microstructure theory. It is
also interesting to note that the lower frequency limit of the bandgap can be estimated as the local resonance frequency of
the spring-mass system. The formation of the new band gap can be also explained as repulsion points between two branches
and resonance couplings between the two lowest wave modes (Maugin, 1980).
In the same manner, the transverse wave propagation along x2 direction in the metamaterial can be described by the field
3 ; v 3 ; /s23 ; /m
variables u
23 and C23. Fig. 5 shows the comparison of the normalized dispersion curves predicted by the microstructure continuum model and the finite element method.
Fig. 4. Comparison of dispersive curves of the longitudinal wave propagating in the x2 direction.
Fig. 5. Comparison of dispersive curves of the transverse wave propagating in the x2 direction.
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
doi:10.1016/j.ijengsci.2011.04.005
8
R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
Fig. 6. Comparison of the dispersive curves for the elastic metamaterial
obtained from the microstructure continuum model and the FE method for coupled
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
waves propagating along d = 45°. The length d is defined as d ¼ d2 þ d3 .
Fig. 7. Comparison of the dispersive curves for wave propagation in different directions in the first Brillouin zone for the lowest longitudinal and transverse
modes obtained from the continuum model and the FE method.
It is evident that the metamaterial reduces to a traditional phononic material if the center mass and the springs (the resonator) are removed from the unit cell. For this phononic material the bandgap is located at relatively high wave frequency
range. The absence of the center mass in the unit cell means that no local resonance phenomenon is present. As a result, no
additional degree of freedom (vi) is needed for describing the motion of the internal mass in the microstructure continuum
model. In this case, the longitudinal wave propagation in the phononic structure can be described by four field variables
2 ; /s22 ; /m
u
22 and C22. Similarly, the transverse wave propagating along x2 direction can be described by the field variables
3 ; /s23 ; /m
u
23 and C23. The comparison of the lowest dispersion curves of longitudinal and transverse waves propagating in
the phononic structure predicted by the microstructure continuum model and the FE method also shows very good
agreement.
Consider wave propagation in the two-dimensional elastic metamaterial in which the longitudinal wave and the transverse wave are coupled. Fig. 6 shows the dispersion curves for plane wave propagation in the 2D elastic metamaterial along
d ¼ 45 where d ¼ tan1 kk23 ; k3 is the wave number along x3 direction. The material properties used in the calculation are the
same as those in Fig. 4. In this case, the longitudinal wave and the transverse wave are coupled. Based on the comparisons
with the FEM results, it is found that the microstructure continuum model can still give very good prediction forq
the
lowest
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
2
2
two wave branches and fairly good prediction for the third and fourth wave branches for kd < 1.0. Note that d ¼ d2 þ d3 is
the diagonal dimension of the unit cell. To improve accuracy of the microstructure continuum model for prediction of the
higher wave modes, higher order continuum models are needed (Huang & Sun, 2008).
The comparison of the dispersion curves of the microcontinuum model and the finite element method for the two lowest
wave branches in the first Brillouin is shown in Fig. 7. In Fig. 7, the material properties given by Eqs. (28) and (29) were used.
Excellent agreement between the microstructure continuum model and the FE solutions is observed along all the wave paths.
4. Concluding remarks
In this paper, a quite general approach was introduced to derive field equations for a microstructure continuum model to
represent elastic metamaterials with resonator microstructures. The material constants of the resulting microstructure
Please cite this article in press as: Zhu, R., et al. Microstructure continuum modeling of an elastic metamaterial. Int. J. Eng. Sci. (2011),
doi:10.1016/j.ijengsci.2011.04.005
R. Zhu et al. / International Journal of Engineering Science xxx (2011) xxx–xxx
9
continuum model are explicitly expressed in terms of the properties of host medium and the resonator. It was demonstrated
that the dynamic behavior of the elastic metamaterial could be accurately described by the microstructure continuum model
even for relatively high frequencies and the bandgap frequency range can be accurately predicted. The advantage of the present approach toward treating elastic metamaterials is that there is no need to perform experiments to determine material
constants for the microstructure continuum model.
Acknowledgement
This work was supported by AFOSR Grant #FA9550-10-1-0061. Dr. Les Lee was the program manager. GLH acknowledges
funding from NASA EPSCoR RID.
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