This article presents the solution of the problem of longitudinal disturbances in a semi-infinite piezoelectric rod being acted by & magnetic field in presence of a body-force

Chaudhury, B

English

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Indian J. Phy •• 43, IO-B, (1969) On longitudinal disturbances in a semi-infinite piezoelectric rod with a body-force in a magnetic field, By B. CHAUDHURI Department oJ 1If athematic. Jadavpur UniverBity, Oalcutta, [ndia, ( Received 19 April 1968) This article presents the solution of the problem of longitudinal disturbances in a semi-infinite piez:oelectric rod being acted by & magnetic field in presence of a body-force. 1. INTRODUCTION The elastic problems on piezoelectric bodies have bee!} thoroughly discussed in the recent papers of Paria (1960), Paul (1961), Sinha (1963, 1965), Giri (1966) and of D.s (1967). A few of these papers have been mentioned here as references. This kind of discussion can be conveniently extended to the corressponding cases in a magnetic field. Banes & Soska (1964) have, perhaps for the tirst time, studied the effect of a magnrtic field on the disturbances in a piezoelectric materi.l in their recent p.per (1964). Recently, Sinh. (1967) has considered the problem of simil.r nature. The object of the presenr paper is to deal with the simple problem of longitudinal disturbances in a semi-infinite piezoelec-tric bar caused by a body-force and a magnetic field. Body-force dependent on the dimension as well as time has been assumed here. The method of Laplace transform has been fOl!.nd effective in making the problem amenable to solution. 2. PROBLEM, EQUATIONS AND BOUNDARY CONDITIONS Let a magnetic tield represented by the magnetic induction B run in a direction perpendicular to the direction of a semi-infinite piezo-eletric bar O":;x":; +oe. Ler a time-dependent displacement be applied at the finite end of the bar ",=0. The problem is to investigate the longitudinal displace-ment n(x,t) of the bar that stems from the interaction of mechanical and electromagnetic fields with a body force F. The basic equations are, therefore, given by Benes & Soska (1964) as follows, Lrmgil'udinu l di8turbance8 in piezo-eleclrie rod A aUII [(8U. a.... az_) L aIG. ] a~, .0l--+<j;1 81" -+- .- + •. IIi;. -- =~az, al aZ. al al a,. 11 (1) ... (2) Where the symbols have their usual meanings as in Benes &. Soska's (1964) For present discussion, the equation of motion is given by .•. (3) -~. where F, being the bodv..force which is assumed to be H(I)e, A , being a costant and H(t) being Heaviside's unit function equal to unity when 1>0 and equal to zero then 1>0 Therefore (3) becomes a... aa,. a~ -.. a" - - 8".B -- -p -at' = -pIl(t)e az' a",at .•• (4) The boundary conditions are ... (5) ... (6) 3. SOLUTlO~ or THB PROBLBM The present problem, being one of the boundary value problems encountered in partial differential equations, the principles of Laplace transform can be used effectiveiV to obtain U (z, II from the equation (4). Defining ij ("',PI =J. t u(z,t)dt, Re(p»O, 1>0 as the Laplace transform of .. (z, I), the equation (4) is transformed to 81 11 a II P 8-V 0 .. az' - 8 I,. B, P -a"""i" - p Jl'D = - -P- .··(7) 12' B. Ohaudhuri Its 901lltion Is given by il = 0,0·,'" + 0,0-,' - p (0",' ...... ,,' Bp ~-p p') where "'" and "'. are the roots of 0..",. - .,,, Bft'" - .pp' = 0 The conditions (5) and (6) give' il ~Oasz~oC at", = 0 Because of (10), (8) is given by - pz[J~Jl D' + 4POIl ii = 0,' using (11), we have p" K 0, = PtH K) Therefore + .~,B1I2C., _ P r.~" p(O .. ~'-e",BpA-pp· ---=~---p~~~--~. p (0" A'--e " ., R P ~ - p p' ) .. -~ [\1"';8' - 4pO,-;- - '", Bl/20 .. X ·e ""'PTa;; 1.'-6", Bpl.-pP') Taking the inverse transform of (13), we obtain ... (8) .\'. (9) .. , (10)' ... (11) (12) (13) Longitudinal disturbances in a pie~o·electric rod 13 and U=P{l..::-:f 1-(v' .... ·B·t4pO .. +., .• B)!20 .. 1]+ + i [ -(~-8){1 .(v'~te. •• B)12C .. I 2P(P'-8') 2P-fP+8)e when where Q ;.. / 'B'+4 '= '[.,BA l"=zpV'r" pC"." 2p Thus we lind that the dispiacement is partly transient in nature for a certain period of time and is partly constant and partly transient after it exceeds a certain period. In conclution, I would like to express my respectful thanks to Dr. R. R. Giri of the department of Mathamatics, Jadavpur University. Calcutta, for his kind help and active guidance. RBFERBNCES Ben ••• F &. Soak •• P. 1964 O,eek. Jour. Phy •• 14, 186. D ••• N. C. 1967 Indian J. PIIy •. 41. 611. OItl, R. R. 1966 Indian J. Theor. Pky •• 14. 6S.n. Olrl. R. R. 1966 B ••. Bourn. Sol. T"hn.-M". Ap1'l. 11, 253 -260 Pari •• O. 1960 Jour. Soi.lIJngg. B ... 4. 361. P.ul~. S.I961 Jour. Soi.lIJngg. Rea. 5. 233. Sioba, D. K. 19671ndian J. Phy •• 42, 784.766 Sluho, 1963, ItWii<>n J. Th .... Phy., 11, 93·99 sum.. D. K. 1965 .. Ze;". Aug. MaJ1I: PAy •• 1, 411. Slqh •• D. K, 1965 II, Prooc.. NoI, lm4 &I, 31., 395. 

On Longitudinal Disturbances in a Semi-Infinite Piezoelectric Rod with a Body-Force in a Magnetic Field.

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On Longitudinal Disturbances in a Semi-Infinite Piezoelectric Rod with a Body-Force in a Magnetic Field.

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