Supplementay mateial fo the pape Platonic Scatteing Cancellation fo Bending Waves on a Thin Plate M. Fahat, 1 P.-Y. Chen, 2 H. Bağcı, 1 S. Enoch, 3 S. Guenneau, 3 and A. Alù 2 1 Division of Compute, Electical, and Mathematical Sciences and Engineeing, King Abdullah Univesity of Science and Technology (KAUST, Thuwal, 23955-6900, Saudi Aabia, 2 Depatment of Electical and Compute Engineeing, The Univesity of Texas at Austin, Austin, TX, 78712, USA, 3 Aix-Maseille Univesité, CNRS, Centale Maseille, Institut Fesnel, UMR 7249, 13013 Maseille, Fance. (Dated: Mach 11, 2014 Abstact This is the supplementay mateials section outlining the bihamonic equation deivation, its bounday conditions and the expession of scatteing coefficients fo the coe-shell system. 1
I. SET-UP OF THE BIHARMONIC EQUATION IN ANISOTROPIC THIN- PLATES A. Subdomain govening equation To pefom ou simulations, we use the FEM commecial softawae COMSOL Multiphysics 1. The bihamonic equation 2 is obtained by a set of two coupled PDE involving two independent functions V and U, whee U is the displacement field of the plate. It is staightfowad to show that V and U satisfy the set of equations 3,4.( ζ 1 U + λ 1 V = 0.( ζ 1 V + λ 1 β0 4 U = 0, whee ζ is an inhomogeneous anisotopic 2D tenso and λ is an inhomogeneous coefficient of the mateial (to be specified. In cylindical coodinates and assuming that ζ, ζ θ (components of ζ and λ depend only in adial coodinate, these equations become 1 ( ζ U + 1 ζ θ 2 2 θ U λ 1 V = 0 1 ( ζ V + 1 ζ θ 2 2 θ V λ 1 β 4 0 U = 0. The equation satisfied by the out-of-plane displacement U is then 3 6.(ζ 1 (λ.(ζ 1 U λ 1 β 4 0 U = 0. (3 In cylindical coodinates and assuming that U = n= U n(e i n θ, the equation satisfied by W n wites ( ζ (λ( 1 ( ζ U n n2 ζ θ 2 U n n2 λ ζ θ (1 ( U n n2 ζ ζ θ U n 2 λ 1 β0 4 U n = 0. (4 The next step is to find the physical signification of the paametes used in Eq. (3 and to link them to the physical paametes of the elastic plate. (1 (2 To do so, we can suppose that the coefficients of Eq. (3 ae constants and ae expessed in tem of the homogeneous paametes ρ 0, E 0, ν 0 and h 0 (density, Young modulus, Poisson atio and hight of the plate espectively, {E 1 0 12 (1 ν 2 0 h 2 0 ρ 0 }{λ 1 ζ 2 } {E 1 12 (1 ν2 0 ρ}. (5 h 2 0 2
One possible physical choice is the following one 3,4 ζ = E 1/2, and λ = ρ 1. (6 B. Bounday conditions in weak fomulation When consideing popagation in finite media, the bihamonic equation [Eq. (3] is geneally supplied with appopiate bounday conditions that ae of thee types: simply suppoted, fixed (clamped o Diichlet-type and feely vibating (Neumann-type. In tems of cylindical coodinates, they can be witten espectively as follows (fo a plate of adius a U =a = 0, M =a = 0, (7 U =a = 0, =a = 0, (8 with M M =a = 0, (V 1 M t θ =a = 0, (9 = D[ 2 U + ν(1/ U + 1/ 2 2 θ U], M t = D(1 ν(1/ 2,θ U 1/2 θ U] and V = D ( 2 U + 1/ U + 1/ 2 θ 2U 1/ θm t. The fist condition given in Eq. (7 means the plate does not expeience any deflection and that bending moments ae zeo. The second condition in Eq. (8 says that the bounday of the plate does not expeience any deflection and that it must be hoizontal (the deivative is zeo. The last one given in Eq. (9 expesses that the plate is feely vibating. The fist two conditions can be easily implemented in the commecial softwae Comsol 1. Howeve, the thid one poses a seious convegence poblems and is had to fomulate in a coect manne. On Geneal fom (PDE module of Comsol, the ode of equations is impotant, since in weak fomulation the fist equation is multiplied by the test function U and the second one by the test function test(v. The natual bounday conditions appea when integating by pat the following system V + β 4 U = 0 U + V = 0 (10 3
They ae given by n V = g U (11 n U = g V whee the U and V indices on the bounday flux tems ae the to emind with which test function they should be multiplied. As most of the cases we have studied wee linked with cylindical geometies, we will conside the special case of constant adius cicles (with nomal vecto n pointing towads the oigin. The system (11 becomes V/ = g U (12 / = g V The fist condition in (9 which means that thee is no bending at the bounday of the plate can be witten fo a constant in cylindical coodinates as 2 U 2 + ν ( 1 + 1 2 2 U θ 2 = 0 (13 Using the development of the Laplacien in cylindical coodinates pemit us to e-wites Equation (13 in this fom ( 1 V + 1 2 U ( 1 + ν 2 θ 2 + 1 2 U = 0 (14 2 θ 2 This can also be witten as follows in ode to compae with (12 ( V = 1 ν 1 2 U 2 θ 2 (15 which clealy means that ( V g V = 1 ν 1 2 U 2 θ 2 The last step is to expess this equation in catesian system because as we know the package Comsol isn t adapted to othe types of coodinates (cylindical o spheical. To do so, we will use the coespondance / ( n = /n / ( t = /t (16 (17 4
whee t is the tangential unitay vecto (pependicula with the nomal vecto n. We emak that this vecto is always twicely applied, so that its diection doesn t matte. This double application is also impotant fo the weak fomulation which consists in integating by pat and by using a function test. fom Unde all these assumption, Equation (16 tuns to be implemented in the following weak ( test(v V test(v g V = 1 ν + T test(v T U We now tun to the the second condition in (9 expessing that the genealized Kichhoff stess is zeo can be witten ( 2 U 2 + 1 + 1 2 U + 2 θ 2 (1 ν ( 1 θ 2 U θ 1 2 θ (18 (19 This condition can be tansfomed, and we finally get the expession of g U g U = (1 ν ( 1 θ θ ( 1 2 θ (20 Finally, we can integate by pats and multiply by a function test test(u to obtain the appopiate fom ( ( test(ug U = (1 ν T test(v T 1 T U (21 II. SCATTERING COEFFICIENTS The incident field is a plane wave e ik 0cos θ and can be developed in this way U inc = n ε n i n J n (k 0 cos nθ (22 The scatteed field must satisfy the adiation condition and can be developed in tem of the cylindical Hankel functions and the modified Bessel ones U scatt = n ε n i n [A n H (1 n (k 0 + B n K n (k 0 ] cos nθ (23 Inside the cloaking shell (a s < < a c, the field must emain finite at = a s, then U cloak = n ε n i n [C n Y n (k c + D n K n (k c + E n J n (k c + F n I n (k c ] cos nθ (24 5
The field inside the obstacle is given by U int = n ε n i n [G n J n (k s + H n I n (k s ] cos nθ (25 The scatted field can be made identically zeo if the scatteing coefficients A n = B n = 0 fo evey n. As pointed out, in the fa-field we have B n = 0. Thus, we have to calculate the invese matix and to find the conditions to impose to have a zeo-scatteed field. The scatteing coefficients A n of the coe-shell systems of Fig. 1 ae given in Eq. (2 of the manuscipt. These ae expessed as atios A n = Ãn/d n. Ã n ae the deteminants given in Eq. (4 of the manuscipt. The emaining tems d n ae also deteminants and they could be expessed as: H (1 n (k 0 a c K n (k 0 a c Y n (k c a c K n (k c a c J n (k c a c I n (k c a c 0 0 0 0 Y n (k c a s K n (k c a c J n (k c a c I n (k c a c J n (k s a s I n (k s a s k 0 H (1 n (k 0 a c k 0 K n(k 0 a c k c Y n(k c a c k c K n(k c a c k c J n(k c a c k c I n(k c a c 0 0 0 0 k c Y n(k c a s k c K n(k c a s k c J n(k c a s k c I n(k c a s kj n(k s a s ki n(k s a s S H (k 0 a c S K (k 0 a c S Y (k c a c S K (k c a c S J (k c a c S I (k c a c 0 0 0 0 S Y (k c a s S K (k c a s S J (k c a s S I (k c a s S J (k s a s S I (k s a s T H (k 0 a c T K (k 0 a c T Y (k c a c T K (k c a c T J (k c a c T I (k c a c 0 0 0 0 T Y (k c a s T K (k c a s T J (k c a s T I (k c a s T J (k s a s T I (k s a s (26 with same notations of the paametes as in the manuscipt. The scatteing coefficients fom clamped obstacles and stess-fee holes of the same adius a s could also be obtained fom the geneal case above by emoving the sixth and eighth lines and last two columns, and second and fouth lines and last two columns fom the 8 8 deteminants of density-dependent objects, espectively. One obtains thus 6 6 deteminants that can be used to descibe scatteing fom these obstacles. 1 http://www.comsol.com. 2 A. N Nois, and C. Vemula, J. Sound Vib. 181, 115-125 (1995. 3 M. Fahat, S.Guenneau, S. Enoch, and A. B. Movchan, Phys. Rev. B 79, 033102 (2009. 6
4 M. Fahat, S.Guenneau, and S. Enoch, Phys. Rev. Lett. 103, 024301 (2009. 5 M. Fahat, S. Guenneau, S. Enoch, A. B. Movchan, and G. Petusson, Appl. Phys. Lett. 96, 081909 (2010. 6 M. Fahat, S. Guenneau, S. Enoch, and A. B. Movchan, EPL-Euophys. Lett. 91, 54003 (2010. 7