Maria Bayard Dühring - Solid Mechanics

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093504-2 <strong>Dühring</strong>, Laude, and Khelif J. Appl. Phys. 105, 093504 2009 FIG. 1. The geometry of the unit cell with periodic boundary conditions and period p. The electrode consists of nickel and has the height h. The substrate is lithium niobate and a PML is placed at the bottom. The electric potential V p is applied at the interface between the electrode and the substrate. potential to the HAR electrodes. This is modeled by applying a positive potential to the electrode in the unit cell and then using periodic boundaries with opposite sign. As the metal is supposed to be perfectly conducting, the electrical potential needs only to be applied at the interface between Ni and LiNbO 3. To prevent reflections of the SAW from the bottom, a PML Ref. 14 is applied as illustrated on Fig. 1. In Ref. 15, PMLs are introduced for time-harmonic elastodynamic problems and are extended to piezoelectric materials in Refs. 16 and 17. The PMLs have the property that the mechanical and electrical disturbances are gradually absorbed in the layers before they reach the outer boundaries. In this way there are no reflections that can disturb the propagation of the SAW. The applied electric potential will introduce mechanical deformations in the solid by the inverse piezoelectric effect and the behavior of the piezoelectric material is described by the following model as found in Ref. 19. A time-harmonic electric potential Vx j,t = Vx je it , 1 with the angular frequency , is applied to the electrode. The mechanical strain S ij and the electric field E j are given by the expressions Sij = 1 2 1 ui + j x j 1 u j and Ej =− i xi 1 V , 2 j x j where ui are the displacements and xi are the coordinates. Note that the Einstein notation is not applied in the expressions in Eq. 2. The parameter j is an artificial damping at position xj in the PML. As the PML is added at the bottom of the structure only 2 is different from 1 and is given by the expression 2x 2 =1−i 2x 2 − x l 2 , 3 where x l is the coordinate at the interface between the regular domain and the PML and 2 is a suitable constant. There is no damping outside the PMLs and here 2=1. A suitable thickness of the PML as well as the value of 2 must be found by calculations such that both the mechanical and electrical disturbances are absorbed before reaching the outer boundaries. However, the absorption must also be sufficiently slow as reflections will occur at the interface between the regular domain and the PML if their material properties are not comparable. The mechanical stresses T jk and the electric displacement D i both depend on the strain and the electric field according to the constitutive relations E T Tjk = c˜ jklmSlm − e˜ ijkEi, 4 S Di = e˜ ijkSjk + ˜ ijEj, 5 E where c˜ jklm are the elastic stiffness constants, e˜ ijk are the S piezoelectric stress constants, and ˜ ij are the permittivity constants. The materials are, in general, anisotropic, and as it is only possible to generate the SAW by the inverse piezoelectric effect in certain directions the material tensors have to be rotated. This is indicated by the tilde above the material tensors. The rotation is done according to Euler’s transformation theory as explained in Ref. 20. Here the relation between the original directional vector rj and the rotated vector rˆ i is given by rˆ i = a ijr j, 6 where a ij is the transformation matrix given by the Euler angles 1, 2, and 3, where the crystal axes are rotated clockwise about the x 3-axis, then the x 2-axis, and finally the x 3-axis again. The material property matrices can then be transformed by the transformation matrix a ij and the Bond stress transformation matrix M ijmn the derivation procedure of M ijmn is defined in Ref. 20 as follows: E E c˜ ijkl= MijmnMklpqcmnpq , e˜ ijk = a ilM jkmne lmn and 7 S S ˜ ij = aikajlkl. The governing equations give the stresses by Newton’s second law and the electric displacement from Gauss law, 1 Tij =− j x j 2 1 Di ui and =0, 8 j x j where is the density of the material. Note again that the Einstein notation is not applied in Eq. 8. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
093504-3 <strong>Dühring</strong>, Laude, and Khelif J. Appl. Phys. 105, 093504 2009 Both mechanical and electrical boundary conditions must be specified to solve the problem. Considering the mechanical conditions, the upper surface is stress-free and the bottom is clamped, streess free surface: Tjkmk =0, 9 clamped surface: u i =0, 10 where mk is the normal unit vector pointing out of the surface. At the upper surface, there are no charges and therefore electric insulation occurs, meaning that the normal component of the electric displacement is zero. At the bottom of the domain it is assumed that the electric potential is zero, whereas at the interface between the electrodes and the substrate the potential is Vp. The electrical boundary conditions are summarized as follows: electrical insulation: Dimi =0, 11 zero potential: V =0, 12 applied positive potential: V = V p. 13 Periodic boundary conditions must be induced for ui and ui,j as well as for V and V ,j at the boundaries l and r, see Fig. 1. When ul represents these quantities on the left boundary l and ur the quantities on the right boundary r, the induced periodic boundary conditions take the form urx j = ulx je−ik2p , 14 where k is the phase propagation constant of the SAW. 21 When looking at the case with an alternating electric potential, k=/2p is used. The piezoelectric problem is solved by a plane formulation obtained by setting Si3 and E3 as well as Ti3 and D3 equal to zero. The governing equations 8 are solved simultaneously to find the dependent variables u1, u2, u3, and V. The model with the PML described above is directly implemented in the partial differential equation application mode of COMSOL MULTIPHYSICS, which is a commercial finite element program. 22 Second order Lagrange elements are used for all dependent variables. III. RESULTS In this section, results are presented for the periodic model of the HAR electrodes. A. Polarization of HAR IDT modes One HAR electrode with periodic boundary conditions is considered. The computational domain is illustrated on Fig. 1 where p=1.4 m is used for definiteness. A harmonic electric potential of V p=1 V is applied to excite the device and the resonance frequencies are found by integrating the deflections along the substrate surface for a frequency range. The first six modes are examined. Three of the modes are found to be mainly polarized in the shear-horizontal SH direction and the other three modes to be mainly vertically polarized VP. In Fig. 2, the half phase velocity fp where f =/2 is the frequency of these six modes is plotted ver- half the phase velocity, fp [ms −1 ] 2500 2000 1500 1000 500 SH2 SH1 VP1 VP2 SH3 VP bulk SH bulk VP3 mode 6 mode 5 mode 4 mode 3 mode 2 mode 1 0 0 0.5 1 1.5 2 2.5 3 aspect ratio, h/2p [−] FIG. 2. Color online Half of the phase velocity fp as function of h/2p for the six first modes for kp=/2. The polarization types are indicated along with the limits for the bulk waves. sus the aspect ratio of the electrode h/2p and the polarization types are indicated. It is seen that the phase velocity is decreasing with increasing aspect ratio for all the modes. The decrease gets bigger for decreasing mode number, and for mode 1 fp is decreased up to 15 times. These modes and their polarization as well as the phase velocity dependence of increasing aspect ratio are in fine agreement with the results presented in Ref. 7, where the modes were found by a combined FEM/boundary element method algorithm. The SH and VP bulk wave limits are marked in Fig. 2 as well and the modes do not appear above these limits as they are dissipated to the bulk. One advantage of using a pure finite element model is that it is easy to plot the displacement fields in the entire structure. Examples are given in Fig. 3 where the deflections u 1, u 2, and u 3 in the x 1-, x 2-, and x 3-directions are plotted for the six modes for h/2p=1. These plots clearly show that all modes of the structure are, in fact, the combination of a vibration in the electrode and of a surface wave in the substrate. The first five modes are fully confined to the surface, whereas mode 6 is leaking into the substrate. The different scaling on the color bars on Fig. 2 indicates the dominant polarization directions for the six modes. In order to clarify these polarizations, a plot for each mode is made in Fig. 4 where the deformations u 1 and u 2 in the x 1- and x 2-directions are plotted with a scaling factor to emphasize in-plane deformations, and the deflection u 3 in the x 3-direction is indicated by the color bar. The plots are focused on the electrode and the regular part of the substrate. The observed modal shapes explain why more and more modes appear in the structure as the aspect ratio increases. When SH-type modes are considered, which are mainly polarized in the x 3-direction, it is observed that the mode shapes in the electrode in this direction are of increasing order for increasing mode number, such that SH1 has a mode shape of order 1, SH2 has one of order 2, and SH3 has one of order 3. The same is found for the modes of VP type where VP1 has a mode of order 1, VP2 has a mode of order 2, and VP3 has a mode of order 3 in both the x 1- and x 2-directions. Thus, for larger aspect ratios, more modes can exist as the electrode is allowed to vibrate with modes of higher order. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
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Optimization of acoustic, optical a
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Title of the thesis: Optimization o
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Resumé (in Danish) Forskningsomr˚
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vi Contents 6 Design of acousto-opt
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4 Chapter 2 Time-harmonic propagati
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62 References [42] A. A. Larsen, B.
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64 References [67] K. Svanberg, “
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Maria Bayard Dühring - Solid Mechanics

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