DOI:.38/NMAT343 Hybrid Elstic olids Yun Li, Ying Wu, Ping heng, Zho-Qing Zhng* Deprtment of Physics, Hong Kong University of cience nd Technology Cler Wter By, Kowloon, Hong Kong, Chin E-mil: phzzhng@ust.hk NATURE MATERIAL www.nture.com/nturemterils Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343. Describing our system using Christoffel s eqution Our system is two-dimensionl elstic solid in squre lttice, nd we consider only elstic wves with displcements in the -y plne. Due to the high symmetry, the stiffness tensor is gretly simplified such tht only c, c nd c eist, nd the constitutive reltion tkes the simple form s T c c, () T = c c T y c y u u i j where T ij is the stress tensor nd ij = is the strin tensor nd u j i is the i displcement in the ith direction. By substituting Eq. into the Newton s second lw, i.e. ρ u t = T where ρ is mss density, nd using plne wve solution of i ij j ( ) the form = ( n ) u uf t v i i, where v is phse velocity, u i denote the wve polriztion vector, nd n i denote the propgtion direction, we find the following equtions of in-plne elstic wves (), i.e., ρv ( ) u c cos ϕ c sin ϕ c c sinϕcosϕ u =. () u y ( c c ) sinϕcosϕ c sin ϕ c cos ϕ u y By solving the seculr eqution of Eq., we obtin two eigenvlues, i.e., phse velocities s functions of the ngle ϕ ( n = cosϕ, n = sinϕ ) ( ) ( ) = ρv c c c c cos ϕ c c sin ϕ, ( ) ( ) = ρv c c c c cos ϕ c c sin ϕ. (3) The corresponding eigenvectors give the polriztions of the plne wves, which cn be ( c c ) obtined from the two orthogonl solutions of the eqution tn β = tn ϕ, c c with β defining the polriztion through y ( ) u tn β = (Ref. ). In generl, the u polriztion is neither longitudinl nor trnsverse. However, for the velocity of pure longitudinl wve nd v ( c ρ ) trnsverse wve. For ϕ π 4 =, the velocity ( ) longitudinl wve nd the velocity ( ) ϕ =, v ( c ρ ) = is = is the velocity of pure v = c c c ρ refers to pure v = c c ρ refers to pure trnsverse wve. NATURE MATERIAL www.nture.com/nturemterils Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343 UPPLEMENTARY INFORMATION If we define the prmeters κ = ( c c ) nd ( c c ) the velocities s v ( κ μ) ρ wves in the cse of ϕ = ; v ( κ c ) ρ μ =, then we cn rewrite = for longitudinl wves nd v = c ρ for trnsverse = for longitudinl wves nd v trnsverse wves in the cse of ϕ = π 4. imple nlysis: = μ ρ for Consider the cse when ρ <, κ, μ > nd c >, which hppens ner the resonnt frequency of κ in bnd gp induced by ρ <, s for the higher negtive bnd in our pper. For ϕ =, we find κ μ <, thus ( ) v = κ μ ρ is rel nd v = c ρ is imginry. Thus only longitudinl wves re llowed. For ϕ = π 4, we find c = ρ is rel nd v = μ ρ is imginry. Thus only longitudinl wves re llowed too. For other directions, if we consider κ is much κ <, thus v ( κ c ) lrger thn other prmeters, then we find v pure longitudinl wves in ny direction. κ ρ nd β ϕ. This indictes lmost Consider the cse when ρ <, κ >, μ nd c >, which hppens ner the resonnt frequency of μ in bnd gp induced by ρ <, s the lower negtive bnd in our pper. For ϕ =, we find κ μ v = κ μ ρ is rel nd v = c ρ is imginry. Thus only longitudinl wves re llowed. For ϕ = π 4, we find v ( κ c ) = ρ is imginry nd v <, thus ( ) = μ ρ is rel. Thus only trnsverse wves re llowed insted. For other directions, the polriztion is neither pure trnsverse or pure longitudinl, but mied type. NATURE MATERIAL www.nture.com/nturemterils 3 Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343. The boundry ective medium theory Fom host teel 4 A unit cell with internl structure 3 Fig.. A squre unit cell with 4 boundries, i.e.,,3,4 We introduce wy to clculte the ective medium prmeters by considering the boundry responses of the metmteril unit cell (boundries should be ll chosen in the host mteril). We consider the unit cell s the bsic element unit tht feels nd responds to the stimultions eerted by the outside wves. Bsiclly, we clculte some certin eigensttes nd clculte the ective force, displcement, stress, nd strin of the unit cell by using the eigenstte fields on the boundries (,,3,4 in the right figure of Fig. ). The ective mss nd moduli cn be further obtined by using the Newton s second lw nd the constitutive reltions. This method normlly gives nonlocl ective medium, but for certin cses such s t low frequency here, we find the ective prmeters do not depend on wve vector nd only depend on frequency. The ective mss density cn be obtined from the Newton s second lw, i.e. m F F ρ = = =, (4) u ω u Here ρ is the ective density. F is the ective net force eerted on the unit cell in the direction. u is the ective displcement of the unit cell in the direction. F nd u cn be obtined s F = T dy T dy T d T d u u dy oft ilicone Rubber = = u dy Hrd ilicone Rubber =, respectively. nd y y = = y= y= The ective moduli cn be obtined from the constitutive reltions, i.e. T = c c, T = c c T = c, y y, (5) 4 NATURE MATERIAL www.nture.com/nturemterils Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343 UPPLEMENTARY INFORMATION which involves only 3 components, i.e. c, c, c s the ective stiffness tensors. Here, T, nd T nd T y re the, nd y components of the ective stress tensor. y re the, nd y components of the ective strin tensor. The Tdy T dy stresses cn be obtined s T = = =, T T dy y= y= T dy =, Tyd Tyd T y= y= ydy Tydy = = Ty = nd Ty =. We note tht Ty = Ty is required for norml liner elstic solids with infinitesiml unit cells. But here the ective medium is bsed on smll but finite unit cell with unit length, thus sometimes we my obtin T T which indictes locl rottions of the unit cell. The udy udy = = strins cn be obtined s y unit cell. y y =, u d u d u dy u dy y y y= y= = = u dy y u dy y y= y= = nd =, which denote the deformtion of the c c c c For simplicity, in the following we use κ = nd μ = insted of c nd c. This is lso becuse tht κ nd μ cn be relted to monopolr nd qudrupolr locl resonnces in physics, respectively. Thus, our prmeter set becomes { ρ, κ, μ, c }. The obtined prmeters for the higher negtive bnd re plotted in Fig.. NATURE MATERIAL www.nture.com/nturemterils 5 Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343 κ (N/m ). -4. 8-8. 8 -. 9 -.6 9 ΓΧ ΓΜ ρ (kg/m 3 ) -4. -6. -8. -. 3 ΓΧ ΓΜ b μ (N/m ) -. 9.8 6.6 6.4 6. 6. 6 4 6 8 4 c ΓΧ f(hz) 4 6 8 4 f(hz) (N/m ) c -. 3 8.8 5 8. 5 7. 5 6.4 5 5.6 5 4.8 5 4 6 8 4 ΓΜ f(hz) d 4 6 8 4 f(hz) Fig.. The ective prmeters for the higher negtive bnd obtined by using the eigensttes long the ΓХ (solid circle) or the ΓΜ (hollow tringle) directions.. κ b. ρ c. μ d c. Note tht μ long the ΓΜ direction nd c long the ΓХ direction cnnot be obtined due to the eigenstte symmetries. From the obtined ective medium prmeters, we cn observe the following results: First, we note tht the ective prmeters re ll lmost rel (i.e. the imginry prts re two orders of mgnitude smller thn the rel prts), indicting tht they re physicl. econd, these ective prmeters cn eplin the polriztions (i.e. trnsverse or longitudinl) of the propgting wves in both negtive bnds, s hve been discussed in the pper. Third, in Fig. b in the pper, which plots the κ for the higher bnd, we note tht the dt points independently obtined in ΓX nd ΓM directions coincide with ech other quite well, s they ll lie on smooth curve. o is μ for the lower bnd plotted in Fig. c in the pper nd ρ in Fig. b here. These indicte tht μ, κ nd ρ re the sme for ll eigensttes with different propgtion directions t single frequency. Fourth, the obtined v l nd v t coincide well with v l nd v t obtined from the bnd structure. We hve reclculted the bnd dispersions by using the obtined ective prmeters, s hve been shown s the crosses in Fig. in the pper. It should be noted 6 NATURE MATERIAL www.nture.com/nturemterils Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343 UPPLEMENTARY INFORMATION tht mss density should be n nisotropic tensor in generl (Refs., 3). We do find some smll nisotropy in the lower bnd. This nisotropy hs been tken into ccount in the clcultion of v nd v. l t 3. The equl-frequency surfce nd slowness curve contour mps of the two negtive bnds Here we plot the equl-frequency surfce s well s the slowness curve, which is the inverse of phse velocity, in contour mps in Fig. 3 s complimentry informtion to the trnsmission spectrum of Figure 4. As cn be seen, in the equl-frequency surfces, both negtive bnds re quite isotropic ner the Brillouin Zone center. But they differ significntly t lrge Bloch wve vectors, e.g. ner the M point. However, the two slowness curve mps re not so different. It should be noted tht in both the lower nd higher negtive bnds, there is only one mode. For the higher negtive bnd, the mode is lmost purely longitudinl in ll directions of the Bloch k vector. For the lower negtive bnd, the mode is purely longitudinl long the ΓΧ direction nd purely trnsverse long the ΓΜ direction. But for other directions between the ΓΧ nd ΓΜ directions, the mode is hybrid mode of longitudinl nd trnsverse wves. Thus it contins some longitudinl component nd some trnsverse component nd there is no wy to seprte one component from the other. NATURE MATERIAL www.nture.com/nturemterils 7 Mcmilln Publishers Limited. All rights reserved.
DOI:.38/NMAT343 k y (π/) k y (π/) c..8.6.4.....4.6.8...8.6.4. k (π/)....4.6.8. k (π/) 56. 58. 6.4 6.6 64.8 67. 69. 7.4 73.6 75.8 78..3.6.9..5.8..4.7.3 k y (π/) b k y (π/) d..8.6.4.....4.6.8...8.6.4. k (π/)....4.6.8. k (π/) 96. 98... 4. 6. 8... 4. 6..4.48.7.96...68.9.6.4 Fig. 3. The equl-frenquency surfce nd slowness curve contour mps.. The equlfrequency surfce of the lower negtive bnd. b. The equl-frequency surfce of the higher negtive bnd. c. The slowness curve, i.e., (phse velocity) -, mp of the lower negtive bnd. d. The slowness curve mp of the higher negtive bnd. References nd Notes. ee e.g., Royer, D. & Dieulesint, E., Elstic wves in solids, pringer, New York (999).. Milton, G. W. & Willis, J. R., On modifictions of Newton's second lw nd liner continuum elstodynmics, Proc. R. oc. A 463, 855-88 (7). 3. Willis, J. R., The nonlocl influence of density vritions in composite, Int. J. olids truct., 85-87 (985). 8 NATURE MATERIAL www.nture.com/nturemterils Mcmilln Publishers Limited. All rights reserved.