Bandgaps 171
9.1.2 Square hole arrangement
Zhang et al. [322] proposed a different type of phononic crystal by drilling a periodical array of holes in an aluminum plate, cf. Fig. 9.4. The mechanism that prevents the guided waves from travelling through the structure is the same as described in the previous section (cf. Fig. 9.1).
In general, bandgap structures become more efficient if a strong contrast in the elastic properties of the two materials is given. Typical applications of bandgap structures include sound filters [320, 322] and integrated optical circuits [314].
dl
dh
(a)Square lattice
dl
dh
(b) Triangular lattice
Figure 9.4: Illustration of the square and triangular layout of a lattice structure. The dashed lines indicate the unit cell of the material. The image indicates the alignment (array) of the holes that are crafted into the material.
In contrast to the periodic arrangement of materials, such as described above, also more complex layouts are possible. Due to the introduction of topology optimization to design bandgap structures [314, 318, 319], rather complex arrangements of mostly two materials have been proposed. The approach serves as a means to generate efficient waveguides [314].
Since topology optimization is out of the scope of the current thesis, we refer the interested reader to the thesis by Jensen [314] and the references cited therein.
Bandgaps 172 and validated by means of numerical and experimental data which are available from the present literature.
9.2.1 Fibonacci lattice
Reference [324] lists results for different Fibonacci lattices. They employ the plane wave expansion method to compute the dispersion curves for the structure under investigation.
Gao et al. [325], however, use the FEM to numerically determine bandgaps in periodic Fibonacci structures.
The periodic structure is consists of a combination of tungsten (material B) and silicon (material A) whose material properties are given in Tab. 9.1. The width of the two mate-rials is da =db = 1mm and the plate thickness is L = 1mm. We exemplarily choose the S3-Fibonacci sequence (cf. Eq. (9.5) and Fig. 9.5) to demonstrate the bandgap behavior of such periodic structures.
A B A B A
L
db da
A
B B B A B
F
F
lpi
lf
Lp
lpb
Pi Pb
Figure 9.5: Two-dimensional model of the S3-Fibonacci sequence with loading conditions and dimensioning (three unit cells are exemplarily depicted).
Table 9.1: Material properties of tungsten and silicon. Values of the non-zero components of the elasticity tensor Cand the mass density ρ are taken from [324].
Material Tungsten Silicon resin
C11 502.0GPa 165.7GPa
C12 199.0GPa 63.90GPa
C44 152.0GPa 79.56GPa
ρ 19200kg/m3 2332kg/m3
We record the displacement field one point in front of the Fibonacci sequence (Pi) and at a second point behind the Fibonacci lattice (Pb) to evaluate the structural behavior. The simulations are based on the p-version of the FEM with an isotropic polynomial degree of p= 3and an element size of be = 0.25mm. The evaluation points are located lp = 100mm in front and behind the Fibonacci lattice, respectively. The excitation force acts in a distance of lf = 200mm in front of the periodic array of silicon resin and tungsten. The length of the numerical model is therefore Lp = 520mm. The fundamental layer of the Fibonacci sequence has been repeated N = 40 times. Again, a mono-modal excitation serves to separate the symmetric and anti-symmetric Lamb wave modes.
Bandgaps 173
1E-4 1E-3 1E-2 1E-1 1E+0 1E+1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
r12
f [MHz]
Fibonacci sequence S3
(a)Symmetric Lamb wave modes (excitation by two point forces in x2-direction - out-of-phase)
1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
r12
f [MHz]
Fibonacci sequence S3
(b) Anti-symmetric Lamb wave modes (excitation by two point forces in x1-direction - out-of-phase)
1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
r12
f [MHz]
Fibonacci sequence S3
(c)Symmetric and anti-symmetric Lamb wave modes (excitation by a single point forces in x2-direction)
Figure 9.6: Transmitted spectrum for the Fibonacci sequence S3.
Bandgaps 174 The displacement signal is simulated in a frequency range between 30kHz and 2MHz. To this end, we increase the center frequency fc of a Hann-window modulated sine-burst (cf.
Eq. (7.1)) with n= 5 cycles by 10kHz for every simulation.
In order to evaluate the obtained results, we address the ratio rbi between the Fourier transforms of the displacement signals at points Pb and Pi. The displacement signals ub
andui are therefore subjected to a Fourier transform. The value of the Fourier transforms Ub and Ui at the center frequency fc is then taken and evaluated
rbi(fc) = Ub(fc)
Ui(fc). (9.7)
The results for the transmitted spectra of the symmetric and anti-symmetric guided wave modes are illustrated in Fig. 9.6, showing two distinct stopbands between 570kHz -820kHz and 1.5MHz - 1.75MHz are recognized. These are in good agreement with the results reported in reference [324]. Zhao et al. detected stopbands between 600kHz - 775kHz and 1.45MHz - 1.72MHz. It can be seen, however, that the behavior of the symmetric and anti-symmetric Lamb wave modes differs. This effect can be attributed to the different wavelengths of the individual modes.
Similar results for different Fibonacci lattices are to be found in reference [325] using the FEM to simulate the propagation of guided waves in periodic structures. The main advantage of using higher order FEMs is seen in the fast convergence and therefore a significantly reduced number of degrees-of-freedom.
9.2.2 Square hole arrangement
In order to numerically verify the results presented by Zhang et al. [322], an aluminum plate (material data: Tab. 2.1) with a thickness of tp = 1.27mm is used, cf. Fig. 9.7. The following refers to a square arrangement of holes where the lattice spacing is dl = 1.077mm and the diameter of the holes isdh = 0.762mm. The “volume fraction” of air is accordingly 0.393.
An array of10×20holes is investigated, based on the SCM - due to the symmetry only one half of the plate is modelled. The measurement points Pi and Pb are again located 100mm away from the geometrically perturbed region. To separate the behavior of the symmet-ric and anti-symmetsymmet-ric modes, a mono-modal excitation is reached by applying two point loads similar to the previous example. Consequently, the simulated displacement signal lies in a frequency range between 0.3MHz and 2MHz. To this end, we increase the center frequency fc of a Hann-window modulated sine-burst (cf. Eq. (7.1)) with n = 5 cycles by 10kHz for every simulation.
The cell size for the current example corresponds to the lattice spacing and is therefore be = 1.077mm. The polynomial degree template is again isotopic and chosen as p = 4.
Figs. 9.7c - 9.7e depicts the FC discretization and the corresponding sub-cell grid for the numerical integration (refinement level: k= 5)
As discussed above, phononic materials can be created by choosing a two- or three-dimensional arrangement of two or more materials, exhibiting a strong contrast in their material properties. Reference [322] therefore suggests to use a square or a triangular ar-rangement of holes (filled with air) in an aluminum plate. In their study, Zhang et al. [322]
reported on experimental results showing the existence of bandgaps in such structures.
Bandgaps 175
Pi Pb
F(t)
A A
x1
x2
lPi
l1
l2
lPb
lp
wp
dh
dl
dl
(a)Model of the square hole arrangement
F(t) x1
x3, F(t)
dl
dh tp
Pi Pb
(b) Cross-section A-A
(c) FC discretization and integration sub-cells (refinement level: k= 5)
(d) 10×10sub-cell grid (e) Detail view
Figure 9.7: Three-dimensional model of the square hole arrangement bandgap structure and FC discretization. Geometric dimensions: lPi = 50mm, l1 = 60.1575mm, l2 = 70.6125mm, lPb = 80.77mm, lp = 103.77mm, dh = 0.762mm, dl = 1.077mm, tp = 1.27mm,wp = 21.54mm,tp = 1.27mm.