WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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Phononic band-gap optimization<br />
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/Ii~L~ I , /~l band 5, (o5<br />
.t.1 Ir B tA: U............... I' ' band 4, 04<br />
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A~-_t~ ~~. ' 'L<br />
X -"J / band<br />
k<br />
3, (3<br />
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|r J<br />
band 2, (02<br />
!<br />
I<br />
&"tg,,^f -,;!|i :1g is<br />
/ " 'band 1, (o<br />
_~B Fr x M r k<br />
Figure 2. (a) The irreducible Brillouin zone indicating the wave vectors to be searched for the<br />
general two-dimensional case (grey area). For square symmetry, the wave equation only has<br />
to be calculated for k-vector values along the curve F-X-M-F. (b) Sketch of band structure<br />
indicating lowest five eigenvalues for wave vectors along the line F-X-M-F in the irreducible<br />
Brillouin zone.<br />
Table 1. Material parameters for the examples<br />
pl P2 E1 E2<br />
contrast (103 kg m-3) (103 kg m-3) (GPa) (GPa) v<br />
high 1 2 4 20 0.34<br />
low 1 2 4 9 0.34<br />
which is recognized as a complex eigenvalue problem with K(k) the stiffness matrix<br />
and M the mass matrix. Equation (2.9) should be solved for any wave vector k,<br />
but due to the periodicity we may restrict the wave vector to the first Brillouin zone<br />
k E [-Tr, 7r]d, where d is the dimension (Brillouin 1953). Due to the square symmetry<br />
of the base cell, the area can be restricted further to the triangle defined by the lines<br />
F -+ X, X -+ M and M -+ F (see figure 2a). In principle, the whole triangle should<br />
be searched, but, although unproven, many researchers claim that the information<br />
required can be obtained by searching points only on the boundary lines. Figure 2b<br />
shows a sketch of how the results of the FE analysis are presented in a band diagram<br />
for the coupled problem (2.4), (2.5) solved for the five lowest eigenvalues.<br />
Throughout this paper we will use two sets of material parameters for the exam-<br />
ples. They are denoted the 'high-contrast' and the 'low-contrast' cases (see table 1).<br />
Band diagrams calculated for base cells of 2 cm x 2 cm composed of two material<br />
phases (high-contrast) are shown in figure 3. First we show the band diagrams for<br />
the in-plane coupled modes (2.4) and (2.5) for pure material phases 1 and 2 in fig-<br />
ure 3a and figure 3b, respectively. It is seen that, for these homogeneous materials,<br />
eigenmodes (i.e. propagation modes) exist for all frequencies. Figure 3c shows the<br />
band structure for square phase 2 cylinders in a matrix of phase 1. It is seen that in<br />
this case there is a range of frequencies with no corresponding eigenmodes, i.e. there<br />
is a band gap between the third and fourth bands (from 56 to 64 kHz, corresponding<br />
to a relative band-gap size of Af/fo = 0.14). This means that no elastic waves with<br />
frequencies within the band gap may propagate through the structure. The band-gap<br />
zone is indicated with hatched regions in the diagram. Figure 3d shows the band-<br />
gap structure for the scalar case (2.6). Here there is a band gap between the first<br />
Phil. Trans. R. Soc. Lond. A (2003)<br />
I<br />
1005