THEORETICAL ANALYSIS AND NUMERICAL MODELING OF ELASTIC WAVE PROPAGATION IN HONEYCOMB-TYPE THIN LAYER

Transcription

1 COMPDYN 2011 III ECCOMAS Thematic Coferece o Computatioal Methods i Structural Dyamics ad Earthquake Egieerig M. Papadrakakis, M. Fragiadakis, V. Plevris (eds.) Corfu, Greece, May 2011 THEORETICAL ANALYSIS AND NUMERICAL MODELING OF ELASTIC WAVE PROPAGATION IN HONEYCOMB-TYPE THIN LAYER B. Y. Tia, B. Tie, D. Aubry LMSSMat (CNRS UMR8579), Ecole Cetrale Paris Grade Voie des Viges, Châteay-Malabry Cedex, Frace (biyu.tia, big.tie, deis.aubry)@ecp.fr Keywords: Hoeycomb sadwich, Elastic wave propagatio, periodic media, Bloch wave trasformatio. Abstract. The preset work is devoted to a theoretical aalysis ad umerical modelig of the pheomea of high-frequecy wave propagatio i hoeycomb-type periodic media. A oedimesioal periodic elastic rod model ad a two-dimesioal periodic elastic beam hexagoal model are cosidered. The Bloch wave direct ad iverse trasformatios are applied to fid geeral or particular solutios. By idetifyig eigefrequecies ad the correspodig eigemodes of the periodic systems, importat iformatio, such as the frequecy badgaps ad the diffracted waves caused by the periodic cells, is obtaied. The aims are to improve our uderstadig of how HF waves are trasmitted ad atteuated i the studied periodic media ad to lik the theoretical aalyses to our umerical assessmets.

2 B. -Y. Tia, B. Tie ad D. Aubry 1 INTRODUCTION Sadwich shells usig hoeycomb-type core are very useful composite materials to desig lightweight but high-stregth structures, especially i aerospace ad aeroautics idustries, for the hoeycomb core provides a efficiet solutio to icrease bedig stiffess without sigificat icrease i structural weight. Whe high-frequecy (HF) flexural waves propagatig i such sadwich shells, the ivolved wavelegths are of the same order of the characteristic legths of the hoeycomb cells, or eve shorter, the hoeycomb cellular microstructure ca iteract with the waves ad highly perturb the propagatio pheomea, as they are govered by the geometry ad the material properties of the cells. Ideed, due to the periodic geometric ad material discotiuities withi the cellular structure, waves ca propagate completely or be atteuated eve stopped, depedig upo the ivolved frequecies ad upo the spatial directio, so we observe the existece of passig ad stop bads i frequecy domai ([1]). As the classical homogeized models caot take ito accout this kid of iteractio, they fail to correctly describe the trasiet flexural behaviors of the sadwich shells i HF rages ([2]). Therefore, relevat aalytical ad umerical models should be established by uderstadig how HF waves propagate i a hoeycomb thi layer ad iteract with its cellular structure. To simplify ad optimize umerical models but still takig ito accout the detail characteristics of the cellular microstructure, the Bloch wave theory is more ad more widely used ow. The basic idea is to trasform a o-periodic fuctio defied i a periodic structure to a set of periodic fuctios havig the same periodicity as the structure. Therefore the study of the origial fuctio i the whole structure ca be replaced by cosiderig the periodic fuctios i a uique cell. The theory was firstly applied i the quatum mechaics to solve the Schrödiger equatio i periodic lattice of particles ([3]). Now it has bee itroduced to the structural mechaics. I the literature, the dispersio equatios relatig the frequecy to the Bloch wave vector have bee give for several types of hoeycomb cells composed by beams ad the effects of the cellular characteristics, for example the slederess ratio ad the iteral agle, have bee discussed i [4] ad [5]. The hoeycomb thi layer that we are iterested is composed by thi-walled hexagoal regular cells, which give rise to a i-plae two-dimesio periodic hexagoal cellular microstructure. Each hexagoal cell has two parallel sides whose thickess is doubled due to the maufacturig process, which forms ribbos with a semi-hexagoal profile ad bods them together i pairs to obtai hexagoal cells. These doubled thickess walls result i plae aisotropic properties of the hoeycomb thi layer. Our first research works preseted herei cosist i the developmet of theoretical ad umerical modelig tools based o the Bloch wave theory ad the i their validatio by applicatio to a oe-dimesioal (1D) periodic elastic rod model ad a two-dimesioal (2D) elastic beam hexagoal model, for which the previously metioed double thickess aspect is cosidered. The Bloch wave modes iside oe cell are cosidered ad the dispersio equatios are obtaied. The paper is orgaized as follows: The sectio 2 itroduces the direct ad iverse Bloch wave trasformatios. The sectio 3 is devoted to the theoretical ad umerical aalyses of the wave propagatio i a 1D periodic structure composed by elastic rods. The modelig of a more complex 2D periodic structure composed by elastic beams is the preseted i the sectio 4, before some cocludig remarks i the sectio 5. 2

3 B. Y. Tia, B. Tie ad D. Aubry 2 BLOCH WAVE THEORY Let us cosider a periodic structure Ω of space dimesio N, a primitive cell Q 0 ad a set of basis vector e i (i=1,...,n), called direct cell basis, are defied so that the etire structure Ω ca be obtaied by repeatig the primitive cell alog the direct cell basis. Dual to the primitive cell Q 0, a reciprocal cell Q 0 * ad a reciprocal cell basis, e j *, are defied, which satisfies the followig relatio: e i. e j * =! ij (1) where! ij is the Kroecker delta. The reciprocal cell is also called the first Brilloui zoe [6]. The areas of Q 0 ad Q 0 * verify the followig equatio: volume(q 0 )volume(q 0 * ) = 1 (2) For ay o periodic fuctio V(x) defied o Ω, the Bloch wave theory states that, for each wave vector k restricted i the first Brilloui zoe Q 0 *, the Bloch trasformatio V B (x,k) of V(x) is a periodic fuctio havig the same periodicity as the periodic structure, which is also called Bloch wave fuctio: B V (x,k) = ( ) ik.(x+ i! i e i ) " V x + i! i e i e (3) The, V(x) ca be recovered by the followig iverse Bloch trasformatio: V( x) = 1 * volume(q 0) B " V (x,k) e!ik.x dk (4) By virtue of the Bloch wave trasformatio, the propagatio pheomea of HF elastic waves, trasmissio, reflectio, coversio ad atteuatio, through the periodic microstructure ca be uderstood by ivestigatig the Bloch wave modes withi the primitive cell, which allows to save lots of the efforts whe doig aalysis ad simulatio. 3 WAVE PROPAGATION IN 1D ELASTIC ROD MODEL The first periodic structure cosidered herei is a 1D elastic rod model, which is a topology of a primitive cell composed by two rigidly joited elastic rods respectively of legths l 1 ad l 2, therefore the period of the 1D medium is λ = l 1 + l 2 (Figure 1). Q 0 * -l 1 0 l 2 =l 1 + l 2 Primitive cell Figure 1: 1D elastic rod model The equilibrium eige equatio of the i-th rod (i=1, 2) reads as: ( ) d! dx E du x # i " dx & % = '( i )2 U x ( ) (5) 3

4 B. -Y. Tia, B. Tie ad D. Aubry where (E i, ρ i ) deote the Youg s Modulus ad the desity, ω the eigefrequecy ad U(x) the correspodig eigemode. By applyig Bloch wave trasformatio to (5), the followig Bloch eige equatio is obtaied:! 2 U B ( x, k) E i ( )!U B x, k!x 2 " 2ikE i!x " E i k 2 U B ( x, k) = "# i 2 U B ( x, k) (6) with k! Q 0 * = [0,2" / #[ the Bloch wave vector. Give k, the goal is to fid the eigevalue! ad the correspodig Bloch eigemode U B (x, k) of the eige problem (6). It is straightforward that the geeral solutio of (6) has the followig aalytical form: " U B k + i (x, k) = a i ei! ' x # c i & + bi e % i " k(! % # c ' x & i (7) where, for the i-th rod, c i = E i! i is the wave velocity ad (a i, b i ) are four costats to be determied. 3.1 Dispersio equatio To calculate the four costats (a i, b i ), (i=1, 2), we cosider the followig the iterface coditios betwee the two rods iside the primitive cell ad the periodic coditios o its extremities: U B (0! ) = U B (0 + ), N B (0! ) = N B (0 + ) U B (!l 1 ) = U B (l 2 ), N B (!l 1 ) = N B (l 2 ) where N B = du B dx! iku B deotes the 1D Bloch axial force vector. Substitutig the geeral solutio form (7) to these coditios (8), a system of four liear equatios is obtaied. To esure that system admits otrivial solutios, its determiat must vaish, which fially gives a relatio betwee k ad ω, called dispersio equatio. For our 1D rod model, the followig aalytical dispersio equatio ca be obtaied: cos(!k) = cos ("T 1 )cos("t 2 ) # ( Z 1 + Z 2 )si("t 2Z 2 2Z 1 )si("t 2 ) (9) 1 where Z i = ρ i c i deotes the characteristic acoustic impedace ad T i = l i / c i the time for wave to propagate through the whole i-th rod. Usig (9), for each give frequecy ω, a Bloch wave vector k = k r + i k im ca be foud ad result i the followig Bloch eigemode i the i-th rod: (8) " U B k r + i (x,k) = a i ei! ' x # c i & e (k x im + b i e % i " k (! % # r c ' x & i e (k im x (10) Therefore, whe k is real (k im = 0), the Bloch wave mode U i B is a propagatig mode, which is trasmitted to the adjacet cells with the same amplitude ad there is o eergy losig whe propagatig through the periodic medium. Otherwise, whe k is complex or pure imagiary (k im 0), U i B is a evaescet Bloch wave mode, it vaishes rapidly whe propagatig to the adjacet cells ad the pure eergy exchagig betwee periodic cells is equal to zero ([1]). The frequecy rages that give real values of k is called passig bad ad the others stop bad. The figure 2 gives the frequecy badgap of the 1D model: The red curves plot the two real solutios of k ad so idicate the passig bads, while the blue curves preset the stop 4

5 B. Y. Tia, B. Tie ad D. Aubry bads, by plottig the imagiary part k im of the two complex solutios of k: j! " + i k im, (j=0,1). We remark that the locatio ad the width of the stop bads maily deped o the ratio betwee the two characteristic acoustic impedaces Z 1 /Z 2 ([7]). 1.0 k (m -1 ) Z 1 /Z 2 = ω (s -1 ) Stop bad Passig bad 3.2 Diffracted wave aalysis Figure 2: Dispersio relatio of eigefrequecy o k, Z 1 / Z 2 = 5 Now, we cosider a icidet plae wave u 0 (x, t) = e i(k 0x!" t) 0 ad ivestigate how its propagatio through the 1D periodic structure that is perturbed by the periodic cells. To do this, the wave solutio u(x, t) is decomposed ito two parts: u(x, t) = [ e ik 0x + u d (x)]e! i" 0t (11) where (k 0, ω 0 ) are the wave vector ad the agular frequecy of the icidet wave. We are iterested i fidig u d (x), the diffracted wave caused by the periodicity of the cells, which idicate us i fact what s the differece betwee the wave motio i a periodic medium ad i a homogeous medium. By substitutig the equatio (11) i the equilibrium eigeequatio (5), we get: d! dx E i "# du d (x) dx %& + ) d! dx E du 0 (x) i "# dx %& + ' i( 2, * 0 u 0 (x)- = /' i ( 2 0 u d (x) (12) where the secod term of the left member is cosidered as a exteral loadig f e due to the icidet wave. By expadig u d (x) as a liear combiatio of eigemode U(x): u d (x) = "! U (x) (13) the Bloch diffracted wave u d B reads as: f e u B d (x, k) = "! (k)u B (x, k) (14) where U B (x, k) is the Bloch eigemode ad! (k) the Bloch coefficiet. 5

6 B. -Y. Tia, B. Tie ad D. Aubry Sice we have already calculated U B (x, k), we look for! (k) ow. Applyig the Bloch wave trasformatio to the equatio (12) to get Bloch equilibrium eigeequatio ad substitutig the equatio (14) ito it, we get: %! i (" 2 # " 2 0 ) (k)u B ( x, k) = f B e (x, k) (15) where f B e is the Bloch exteral loadig that ca be expaded usig the same Bloch modes U B (x, k): f B e (x, k) =! F (k)u B (x, k) (16) Therefore α k ca be obtaied i the followig way:! (k) = F (k) " i (# 2 # 0 2 ) Subsequetly the u d B is got usig (14) ad fially u d is obtaied usig the iverse Bloch trasformatio (4). The figure 3 illustrates the ratio of amplitude betwee u 0 ad u d with two icidet waves with havig respectively two differet frequecies f 0 =2.5 khz ad f 0 =10 khz, whose correspodig wave legths i the first rod, λ 0, are respectively 0.3 ad times of the period. We observe a importat amplificatio of wave propagatio pheomea due to diffracted wave caused by the periodicity of the cells ad the amplificatio level is ot affected a lot by the frequecy of the icidet wave. (17) u d / u 0 Z 1 / Z 2 = 2, T 1 / T 2 = 2 l 1 / l 2 f 0 =2500Hz, λ 0 /λ=0.3 f 0 =10000Hz, λ 0 /λ=0.075 l 1 l 2 x(s) λ=l 1 +l 2 Figure 3: Diffracted wave iside the primitive cell. 4 WAVE PROPAGATION IN 2D ELASTIC BEAM MODEL The secod periodic structure cosidered here is a 2D elastic beam model (Figure 4). It is a topology of a primitive cell composed by three rigidly joited elastic beams ([8]) with the same legth s. The Lamé costats ad the desity of the beams are ( λ, µ, ρ). The thickess of beam (1) is twice of the thickess of the other two beams (2) ad (3) (see Figure 4). Each beam is firstly cosidered i its local basis (s, ), i which the uit vector s is parallel to the beam s axis ad the uit vector is perpedicular to the beam s axis. The the etire model is cosidered i a global basis (x, y). 6

7 B. Y. Tia, B. Tie ad D. Aubry y s (2) (3) (1) s (0, 0) Primitive cell x 4.1 Dispersio relatio Figure 4: 2D elastic periodic structure composed by beams The well-kow Timosheko kietics for thick beams is used, so the displacemet u(s, t) i each beam reads as: u(s,t) = u 01 (s)s + u 02 (s) + t u 13 (s)s (18) where u 01 (s) ad u 02 (s) are the displacemet fields of the middle lie i directio s ad ad u 13 (s) is the rotatio of the fibers. For each beam, the Bloch equilibrium eigeequatios of the 2D model are: d 2 B U 01 ds 2 d 2 B U 02 ds 2 d 2 B U 13 ds 2 ( ) du B 01! 2i k. s! 2i k. s ds ( ) du B 02! 2i k. s ds ( ) du B 13! ( k. s) 2 U B 01 =! "#2 + 2µ U B 01! ( k. s) 2 U B 02 + du B 13 =! "#2 ds µ U B 02, (m = 0, 1) ds! % ' ( k. s)2 + & 12µ ( + 2µ)H m 2 ( * U 13 ) B! 12µ ( + 2µ)H m 2 B du 02 ds =! "#2 + 2µ U B 13 with (U B 01, U B 02, U B 13 ) the Bloch eigemode ad k!q * 0 the Bloch wave vector. I this 2D model, the first Brilloui zoe Q * 0 is also a hexagoal cell. Similar to the aalysis we did for the 1D model, givig k, we look for the eigevalues ad the correspodig Bloch eigemodes of the equatio (19). Accordig to the iterface coditios betwee the beams iside the primitive cell ad the periodic coditios o their extremities as followig: U B(1) 01 + U B(1) 02 = U B(2) 01 + U B(2) 02 = U B(3) B(3) 01 + U 02 U B(1) 13 = U B(2) B(3) 13 = U 13 N B(1) + Q B(1) + N B(2) + Q B(2) + N B(3) + Q B(3) = 0 M B(1) = M B(2) = M B(3) where N B is the Bloch axial force vector, Q B is the Bloch trasverse shear force vector ad M B is the Bloch bedig momet, we get a system of 18 liear equatios. Let the determiat (19) (20) 7

8 B. -Y. Tia, B. Tie ad D. Aubry of the system equal to zero, we ca get fially the dispersio equatios betwee k ad ω. Ufortuately, i the 2D model, we are ot able to explicit the dispersio equatio ad we ca oly plot the dispersio surface umerically (Figure 5). We fid that the ratio of H 0 /H 1 largely affects the width of the stop bads ad the badgap will move to HF rage if we shorter the legths of the beams. (s -1 ) H 0 /H 1 =2 5 CONCLUSIONS k y (m -1 ) k x (m -1 ) Figure 5: Dispersio surface of eigefrequecy upo Bloch wave vector. The classical homogeized models fail to correctly describe the HF trasiet flexural behavior of the hoeycomb-type sadwich shells. It is ecessary to look ito the effects comig from the periodic microstructure of the hoeycomb thi layer. The Bloch wave theory has bee adopted i order to take ito accout the characteristics of the hoeycomb cells while optimizig umerical models ad savig calculatio costs. Theoretical aalyzig ad umerical modelig tools base o the Bloch wave theory ad the FE method has bee developed. They are at first applied to a 1D periodic structure composed by elastic rods ad validated. By cosiderig oly the primitive cell, the dispersio equatio is obtaied, which allows idetifyig the passig ad stoppig bads of frequecy. With a icidet plae wave, the diffracted wave due to the periodicity of the cells is calculated ad amplificatio pheomea are observed idepedetly from the frequecy of the icidet wave. The, a 2D periodic hexagoal regular structure composed by elastic beams is ivestigated. The first result of eigefrequecy badgap is obtaied. Our curret research work is to apply our theoretical ad umerical tools to the hoeycomb-type thi layer composed of periodic thi-walled hexagoal regular cells. REFERENCES [1] D. J. Mead, A geeral theory of harmoic wave propagatio i liear periodic systems with multiply couplig. Joural of Soud ad Vibratio, 27, , [2] A. Grédé, B. Tie, D. Aubry, Elastic wave propagatio i hexagoal hoeycomb sadwich paels: Physical uderstadig ad umerical modellig. Joural de Physique, 134, ,

9 B. Y. Tia, B. Tie ad D. Aubry [3] P. Atkis, R. Friedma, Molecular quatum mechaics. Oxford uiversity press, New York, [4] A. Srikatha, J. Woodhouse, N. A. Fleck, Wave propagatio i two-dimesioal periodic lattice. J. Acoust. Soc. Am, 119(4), , [5] S. Goella, M. Ruzzee, Aalysis of i-plae wave propagatio i hexagoal ad reetrat lattices. Joural of Soud ad Vibratio, 312, , [6] L. Brilloui, Wave propagatio i the periodic structures. Dover Publicatios, New York, [7] B. Y. Tia, B. Tie, D. Aubry, Théorie des Odes de Bloch Appliquée à la Modélisatio de la Propagatio d Odes Elastiques das des Milieux Périodiques. 10 th Colloque Natioal e Calcul des Structures (CSMA 2011), Gies, Frace, May 9-13, [8] C. Davii, F. Ogaro, A Homogeized Model for Hoeycomb Cellular Materials. Joural of Elasticity, Published olie,