Kundu, Kartik Kumar

Mahalanabis, Ranjit Kumar

Munshi, Tapas Kumar

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Indian J. Phys. 69 A  (6), 639-644 (1995)IJP A- aa international journalDisturbance in a piezo-electric slab sandwiched between an elastic and visco-elastic medium under electrical and mechanical excitationsTapas Kuinar Munshi,Department of Physics, Kharagpur College, Kharagpur-721 305, Midnapore, West Bengal, IndiaKartik Kumar KunduDepartment of Physics. City College. Calcutta-700 009. Indiaand *Ranjit Kumar MahalanabisDepartment of Mathematics. Jadavpur University, Calcutta-700 032. India Received 12 January 1995, accepted 14 July 1995Abstract : An attempt has been made to investigate analytically the mechanical disturbance in a piezoelectric slab (quartz) sandwiched between an elastic U = 0) and a visco­elastic (x -  d) material under electrical and mechanical excitations. The mechanical stress T, = T05u) is introduced at jr = 0 The surface x = 0 is also excited in the direction of y-axis by an electric Held. The problem involves the interaction of two fields viz., electrical and mechanical, and the solution has been obtained with the aid of the operational calculus. The disturbance exhibited by the slab is found to be linear in nature and is of the order of I0~5 m, interestingly with an initial disturbance at t = 0 .Keywords : Piezoelectric slab, elastic and visco-elastic medium, mechanical distuibance PACS No. : 77.65.-jThe mechanical disturbance of piezoelectric materials have been investigated by many workers [1-4]. Such discussions are of enormous importance in view of various practical applications in different fields of science and technology. Researchers [5-7] considered the problems of the piezoelectric slab bounded by an elastic medium on both sides and the bounding surfaces coated by a thin.film of conducting material under different boundary conditions.The present paper deals with a problem of the mechanical displacement in a piezo­electric slab (quartz), sandwiched between an elastic and a visco-elastic material under© 1995 IACS640 Tapas Kumar Munshi, Kartik Kumar Kundu and Ranjit Kumar Mahalanabiselectrical and mechanical excitations. The relevant problems are extensively used in different branches of Acoustical Engineering, particularly in piezo-electric microphone [8,9]. The solution involves the interaction of two fields, i.e., electrical and mechanical, and consequently Maxwell's equations and the equatiqns of elasticity have been used simultaneously to obtain the solution, with the aid of the operational calculus.Let us consider a plane piezo-electric structure consisting of a piezo-electric slab (quartz), bounded by an elastic medium on one of its sides, and visco-elastic medium on the other side. Let x = 0 and x  = d represent the faces of the slab, so that the regions x < 0, 0 < x < d and x > d are occupied by elastic, piezo-electric and visco-elastic material. The mechanical stress 7, = T0d(t) is introduced at x = 0. The surface x = 0 is also excited in the direction of y-axis by an electric field |Ey = Eo sin cot. \ (1)A thin film of perfectly non-conducting material is introduced at the face x = d in order to prevent the electromagnetic wave propagation without disturbing the elastic propagation [10,11]. Now we study the disturbances proceeding in the direction of the jc-axis.Following [1-4], the piezo-electric equations of the medium are given byT , j  = c ijkm  ^Icm + e lij £/• (2)=  ~ e l i j ^ i j  +  ^where T the mechanical stress, D the electric displacement, S the strain, E the electric field intensity, c the elastic compliance at constant electric field, e the piezo-electric coefficient and e the dielectric permittivity. All indices, i, run from 1 to 3 and Einstein's summation convention has been employed.The mechanical displacement u in the direction of the x-axis and the electric field E in the direction of the y-axis satisfy the following equationsd2u / dx2 -  P| / c. d 2u i dt2 = -  ex / c. dE t dxt (4)d 2E / d x 2 -  ped 2E f d t 2 = -  pe^d2 f d t 2. (d u f d x ), (5)where p\ is the density of the piezo-electric material.In the elastic medium, the mechanical displacement u satisfies the equationp d 2u l d t 2 = (A + 2jU) d2u l d x 2, (6)where p  is the density of the elastic material and K pare Lame's constants of the material. In the visco-elastic medium, the displacements satisfies the relationp2 d 2u l d t 2 = {(A0 + 2p0) + (A, + 2 p x) d / dt) d 2u l  dx2 (7)where p2 is the density of the visco-elastic material and Pi are constants.Now we state the boundary conditions as follows :E\ = E0 sin ax at x = 0 (8)T\ « T0S(t) at jc = 0 (9)T\ = T2 at x = d (10)Mi = u2 e X x - d  (11)£ | - 0  at x = d (12)where the suffixes 1 and 2 stand for the entities of the piezo-electric and visco-elastic materials and S(t) is the Dirac delta function.We use the Laplace transform method of parameter p [Re p > 0]. Taking Laplace transform of eqs. (4), (5) and (7), we get(D 2 -  p xp 2 / c ) u x + exDEx!c  = 0, (13)pexp 2Dux + (D2 -  pep2 ) E x = 0, (14)d 2H2 l ~ P2P2^2 f {(^o ) + (A.| + 2p x)p\  = 0, (15)where, D = d I dx and D2 = d 2 / dx2Solving (13) and (14) we getux = C |exp(-m ,x) + C2exp(-m2Jt) + C3exp(m,;t) + C4exp(m2x), (16) where Clf C2, C3 and C4 are constants, and m2, m\  are roots ofm4 -  m2p2 (pxl c  + pe  + M**/c) + p p x ep* I ex = 0 (17)The solution of eq. (15) subjected to the condition u2 -> 0 as x -> *», is given bytt2 = C5 exp(- mfxx), 0 8)where m[ is a root of the equationm2 “ PiP2 1 {(*0 + 2Mo) + (^1 + 2-POp ] "  °*From eq. (16), we get“  Q  + C2 + C3 + C4 .The values of the constants C,, C2, C3 and C4 are obtained by solving the eqs. (16) and (18), with the aid of eqs. (8M 12) and substituting these values in eq. (20) we get ux. The inversion of ux is a complicated task, to get an approximate value, we shall consider small values of time [11,12].173] iB 0|P* *  02P* Wj “ yp%Disturbance in a piezo-electric slab sandwiched etc 641where 0,, 02 and y are positive constants involving material parameters of the problem. Adopting the procedure of approximation [3,6], we getu = -  dtT0A + dexp{-  A ! B . t ) / A  + [ t / 2A -  B t / A 2T0+ e x p ( - A / B . t ) B / A 2 ] / ( 0 l + 02) + e,E0(t -  cosffif)/O)c(0, + 02) (21)where, A = 2 (XQy + 2 p 0y) + p(02 — Q\)IQ\Q2,B = 2(A,y + 2 p ty) + pd(B, + 02) /0 , .Eq. (21) shows the displacement of the piezo-electric slab sandwiched between an elastic and a visco-elastic material under electrical and mechanical excitations. The displacement is found to be partly linear, partly periodic, and partly transient in naLure.For numerical calculations, the standard values of the material constants have been taken [2,13-15], while values like £0, d, 0) and 70 have been chosen suitably to facilitate the numerical calculations as follows : \E0 = 300 V, d = 0.05 m, o>=1.5rad/s, T0 = l N / m 2.The values of the amplitude of free electric field (£0 = 300 v), the mechanical stress (T0 = 1 N/m2) and the thickness of the piezoelectric slab (d= 0.05 m) as considered here are quite in conformity to those of other researchers 12,7]. Again the arbitrary value for frequency of the applied periodic excitations (w = 1.5 rad/s) is conveniently chosen as conceived by 18,15]. The variation of the mechanical disturbances in a piezo-electric slab sandwiched between an elastic and a visco-elastic medium with time is shown in Table 1.T ab le  1. The mechanical disturbance (u) exhibited by the piezo-electric  slab sandwiched betw een an elastic and a visco-elastic medium against time (/)•642 Tapas Kumar Munshi, Kartik Kumar Kundu and Raiyit Kumar Mahalanabist (scc>. m x IO"5 (m)0.0 0.5600 1 0.6150.2 0.6710.3 0.7270.4 0.7830  5 0.8390  6 0.8950.7 0.9510.8 1.0070.9 1.0631.0 1.119From eq. (21), we find that the displacement is partly linear, partly periodic, and partly transient in nature. The contribution of the transient part of eq. (2f) is insignificant compared to the linear and periodic parts and as a result, the disturbance exhibits linearDisturbance in a p ie z o -e le c tr ic  s la b  sandwiched etc643relationship with time (Figure 1). The electric field £  = £„ „„ J« - * * r ; r of ■" ~  -  “ * -  t :moment r -  0 but trom investigauon it is revealed that at t = 0 some disturbance persists.theF igure 1 . The variation of the mechanical disturbance of the piezo­electric slab wiih tinje.This finite response at t = 0 is coming from the time dependent boundary condition T, = Tq&j ) introduced at the face x = 0 of the piezo-elctric slab. Researchers [5-71 obtained similar result in a problem in which piezo-quartz slab is bounded by an elastic medium on both the sides. The order of magnitude of the disturbances obtained therein being 10~7 m and a finite deformation also exists at t = 0. It is very interesting to note that the order of disturbance (10-5 m.) when the piezo-quartz sandwiched between elastic and visco-elastic medium, rises hundred times than the piezo-quartz sandwiched between elastic medium and coaled with a perfectly conducting material.AcknowledgmentThe work was performed under the University Grants Commission, Departmental Special Assistance. Project at the Department of Mathematics, Jadavpur University, Calcutta-700 032, India.References[1] W P Mason P ie z o e le c tr ic  C ry s ta ls  a n d  T h e ir  Application to U ltra so n ic s  (N ew  York : D V Nosirand) p 84 (1950)[2] W G Cady P ie z o e le c tr ic ity  (New York : Me Graw-Hill) (1959)[3] M Redwood J. A c o u s t. Soc . A m . 33 527 (1961)14] R D Mindlin P r o b le m s  o n  C o n tin u u m  M e c h a n ic s  (SIAM . Philadelphia, Pensylvuma) Muskheli&hvih Anniversary Vol. 282 (1961)15] H Banno S o lid  S ta te  P h y s . (Japan) 23 641 (1988)(61 V M Petrov Bull. Acad. Sci. USSR Phys. Ser. (USA) 51 91 (1987)[7] S K Chatteijee Indian J. Mech. Math. 115 (1973)[8] N Imai J. Acoust. Soc. Jpn. (E) 11327 (1990)T9] T K Munshi, K K Kundu and R K Mahalanabis J. Acoust. Soc. Am. 96 (4) 2836 (1994)[10] F Barditi, G Baizilai nnd G Georosa Symposium on Sanies and Ultrasonics (Vancouver, Canada) (1967)[11] K N Baranski, M A Zubareva and IA Yakovlev JETP Lett. (USA) 47291 (1988)[12] T K Munshi, K K Kundu and R K Mahalanabis Acta Phys. Pol. A85 585 (1994)[13] L V Kart ash Sov. Appl. Mech. (USA) 26 440 (1990)[14] C K Hruska J.Appl. Phys. 66 1071 (1989)[15] D F Gibbs Ferroelectrics 32 133 (1988)644 T apas K u m ar Munshi, K artik  K um ar K m d u  a n d  R anjit K um ar M ahalanabis

Disturbance in a Piezo-Electric Slab Sandwiched Between an Elastic and Visco-Elastic Medium Under Electrical and Mechanical Excitations

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Disturbance in a Piezo-Electric Slab Sandwiched Between an Elastic and Visco-Elastic Medium Under Electrical and Mechanical Excitations

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