WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

900 S. Halkjær, O. Sigmund and J. S. Jensen
Table 6. Relative band gap sizes for the three different beam maximization
problems and [13] (non-optimized rod with equal volume fractions
of Aluminium and PMMA). Gaps with numbers in bold have
been optimized.
the beginning and end of the two pronounced band gaps
in the figure. Their values are listed in Table 5. The relative
band gaps are shown in Table 6 (fifth column).
An infinite periodic beam with this base cell was analysed
using (10) and the frequency bands and gaps were
calculated to compare with the experimental results. Very
good agreement was found.
The relative band gaps for the current problem are
higher than those for the coupled problem in Fig. 3. Comparing
with the longitudinal results in Fig. 1, the latter performs
better for the first band gap as expected. Thus, the
optimized design is superior to the non-optimized design
for the first band gap considered.
Design of infinite 2D crystals
In this section we consider the design of infinite two
dimensional crystals with maximum relative band gap
sizes. We optimize the crystals for in-plane polarized
elastic waves assuming very thick plates (i.e. plane strain
assumption) and for bending waves assuming moderately
thick plates (using Mindlin plate theory for the same reason
as considering Timoshenko theory in the previous
section).
In-plane polarized wave propagation in thick elastic
plates is governed by the Navier equations
@
@x
ðl þ 2mÞ @u @v
þ l
@x @y
þ @
@y
m @v @u
þ m
@x @y
þ qw 2 u ¼ 0 ; ð13Þ
@
@x
L B LB L [13]
F L 12 0.9841 0.6956 0.8791
F L 23 0.0011 0.4291 0.4484
F L 34 0.3281 0.4668
F B 12 0.3076 0.3663
F B 23 0.1166 0.0623
F B 34 0.3781 0.3737
F 0.3737
m @u @v
þ m
@y @x
þ @
@y
ðl þ 2mÞ @v @u
þ l
@y @x
þ qw 2 v ¼ 0 ; ð14Þ
where u and v are the two amplitude components and
l; m are the Lamé parameters, defined as
l ¼ En=ðð1 2nÞð1 þ nÞÞ, m ¼ E=ð2ð1 þ nÞÞ. The outof-plane
case is not considered here.
For bending waves in up to moderately thick plates the
governing equations are
Γ
@
@x
@
@x
D @Qx
@x
nD @Qy
@y
@
@y
@
@y
ð1 nÞ D
2
ð1 nÞ D
2
@Qx
@y
@Qy
@x
Gkt @w
@x Qx ¼ qt3 d
12
2 Qx
dt2 ; ð15Þ
@
@y
@
@x
nD @Qx
@x
ð1 nÞ D
2
@
@x
@Qy
@x
ð1 nÞ D
2
@
@y
D @Qy
@y
@Qx
@y
Gkt @w
@y Qy ¼ qt3 d
12
2 Qy
dt2 ; ð16Þ
@
@x
þ @
@x
M
X
Gkt @w
@x
ðGkt QxÞþ
@
@
@y
M K
Fig. 5. Irreducible Brillouin zones for square and rhombic base cells.
Gkt @w
@y
@y Gkt Qy ¼ qt d2w ; ð17Þ
dt2 where t is the plate thickness and other values are defined
previously. Qx denotes the angle between the x-axis and a
cross-section in the yz-plane, and similarly for Qy. Here, inplane
modes are ignored. Note that for n ¼ 0 and no variation
in the y-direction, Eqs. (15) and (17) correspond exactly
to the equations for Timoshenko beam theory (8) and (9).
In both cases, the periodicity conditions for the base
cell are again given by Bloch theory, e.g.
wðr þ RÞ ¼e iR k wðrÞ ð18Þ
and likewise for the other degrees of freedom. Here R is a
lattice vector and k the (two-dimensional) wave vector.
The dynamic response of the 2D crystals can be obtained
by solving Eqs. (13)–(14) or (15)–(17) with boundary
conditions (18) for wave vectors within the irreducible
Brillouin zone. The irreducible Brillouin zones for quadratic
and rhombic base cells are shown in Fig. 5. Assuming
isotropic constituents and imposing 90 degree symmetry
in the square cells and 60 degree symmetry in the rhombic
cells, the irreducible zones can be further reduced to
the triangles indicated in the figure. Finally, it is the experience
that the size of the band gap can be measured by
performing analyses only for wave vectors on the edge of
the triangular zones thus saving significantly in CPU-time.
The analysis is performed using the commercial FE
program FEMLAB which can be called from a MATLAB
script that includes the optimization routine Method of
Moving Asymptotes [17]. Triangular first order elements
have been used in the FE analysis.
Γ