Torsional waves in piezoelectric (622) crystal plate

Transcription

1 Proc. Indian Acad. Sci., Vol. 85 A, No. 5, 1977, pp Torsional waves in piezoelectric (622) crystal plate H. S. PAUL AND K. VENKATESWARA SARMA Department of Mathematics, Indian Institute of Technology, Madras 636 MS received 2 August 1976; revised 1 December 1976 ABSTRACT Torsional waves generated in a circular piezoelectric infinite plate of crystal class (622) when excited by an electric potential impulse applied between the centre of one of its flat faces and the opposite face, are investigated. The resulting shearing stress and electric potential are calculated. 1. INTRODUCTION Tim propagation of waves in piezoelectric materials plays an important role in Acousto-electronics particularly in delay lines which are devices for storing information in the form of pulses.' Paul and Rao 2 studied the torsional waves in a circular cylinder of piezoelectric (622) crystal class. Ice and fl-quartz belong to piezoelectric (622) crystal* class. In the present paper we consider the torsional wave propagation in an infinite plate of the some crystal class. An electric potential pulse is applied at a point on the boundary of the plate. Eason et a13 presented a detailed discussion of a somewhat related problem of the determination of the distribution of stress produced by an impulsive point force in an infinite elastic solid. Hankel and Laplace transforms are used in order to derive the solution of the present problem. Expressions for elastic displacement, shear stress and electric potential are obtained for a S-quartz plate of moderate thickness. The shear stress is found to propagate as impulses along any plane parallel to the faces at regular intervals of time. It is shown that the shear waves so generated are reflected at the boundaries repeatedly. This property may be used in the delay lines. 2. BASIC EQUATIONS For an infinite plate (as. shown in figure 1), we use the cylindrical polar coordinates (r,, z). In the case of torsion prollemr, the only 289

2 29 H. S. PAUL AND K. VENKATESWARA SARMA Z= c 6(r)S(t) L I Z=h = Figure 1. non-vanishing component of displacement is the cross-radial component, v (r, z, t) and the electric potential may be taken as, (r, z, t). in this case, the piezoelectric relations for the (622) crystal class 2 are Tg2 ca,v, z + e14 qc, r ; Tre = C66 (v, T Dr = e14v, z E11q, T; DZ = E33 II, Z (1) where c44 and c66 (= (cll c12)/2) are the elastic constants, e14 is the piezoelectric and e, 1 and E33 are the dielectric constants. Comma followed by subscripts denotes the partial derivatives with respect to those subscripts. Tez and TTB denote the shearing stresses while Dr and Dz stand for the components of the electric displacement. The equation of motion and the Gauss's equation div b = are C 44v, xz + C66 [ 2/cbr 2 ) + r-1 (cl/cbr) r- ] v + e14, rz = Pv, tt e14 (y, T + v/r), Z Ell (, rr + r 1, r) E3, ZZ = (2) where p is the density of the material. We assume the solution for the system of eq. (2) in the form = f V (a, z, t) Jj (ra) da ; c = f (a, z, t) Jo (ra) da. (3) where Jl and J are bessel functions of the first kind and of orders I and respectively. Substitution of the results (3) in eq. (2) yields C44V, ZZ c66 a 2 V ae14, Z P V, tt e14a I, z + E11cL 2 E33 #, ZZ =. (4) 3. INITIAL AND BOUNDARY CONDITIONS We consider the infinite piezoelectric plate of thickness h. An electric potential pulse 8 (r) 8 (t) is applied at the origin on the face given by z = where 8 is the dirac delta function. Both the faces (except the origin on the face z = ) are coated with electrodes that are shorted. Hence

3 TORSIONAL WAVES IN PIEZOELECTRIC (622) CRYSTAL PLATE 291 (r,o,t)=8(r)3(t); (r,h,t)=. (5) The dirac delta function can be represented as 4 (r) _ (1/2ir) f a J (ra) da. (6) Hence from eqs (3) and (5) we have (a, o, t) _ (a/27r) 8 (t) ; (a, h, t) =. (7) If the faces z = and z = h are kept free from stress, then Toz (r, o, t) = TBz (r, h, t) =. (8) The initial conditions can be taken as v = v, t = when t =. (9) Let a bar over a quantity denote the Laplace transform of that quantity with respect to time. That is V (a, z, s) L [V (a, z, t)] = f exp ( st) Vdt ^(a,z, ^) L[] and TBz=L[Tez] (1) The electrical boundary conditions are transformed as j (a, o, s) = a/27t ; (a, h, s) =. (11) From eqs (1), (3) and (1), we get Tez = ( 1 /h) f [c44 (dp/dx) ahe 14 j) JI (ra) da (12) where x = z/h, a non-dimensional quantity. Thus the transformed mechanical boundary conditions (8) are TBz = when x = and x = 1. (13) The problem under consideration is specified completely by eqs (1) to (13). 4. SOLUTION OF THE PROBLEM Using Laplace transform to the results (4) we obtain [(d 2/dx 2) (c1h 2a 2 + c-2s 2)] V (ahe14/c44) (d^/dx) = (14) (ahe14/e33) (dv/dx) + [a 2h 2 E (d2/dx2)] = (15)

4 292 H. S. PAUL AND K. VENKATESWARA SARMA where C1 C44/C66, C 2 = C44/(Ph2) and F = E31/ E33 (16) Eqs (14) and (15) may be reduced to where then and [(d4/dx4) (a 2h 2 (c1 + E + k14 2) + (S/C)2) (d2/dx 2) + a 2h 2 E x (c1a 2h 2 + (S/c) 2)] (V, ) = (17) k14 2 = e142/(c44e33) (18) Tf X1 2 and X22 are the roots of the auxiliary equation X4 [h 2 (C1 + e + k14 2) + S 2/(ae) 2)] X2 + Eh 2 (c h 2 +S 2 /(ac) 2) = (19) X12 + X2 2 = h 2 (C1 + E + k 214) + (S/(aC))2 / (2) X32X22 = Eh 2 ( C1h2 + [S/(aC)] 2) (21) Solutions of the differential eqs (17) are 2 V = E [Ai exp (ax4x) + Bi exp ( axix)] 4-1 = 2' e;[a;exp(axix) Bexp( axjx)] (22) {=1 where A{ and B1 are arbitrary constants and the constants ei satisfy the relation (he14l C44) e1 Xi = X8 2 [ c1h 2 + (S/(aC))2] (23) Eqs (21) and (12) enable us to have where TBz = h- 1 j E fi [Ai exp (axgx) B4 exp ( axix)] Jl (ra) da A = a (c44 Xi he 14 eti) ; i = 1, 2. (25) Hence the boundary conditions (11) and (13) along with the results (22) and (24) lead to the system of equations ^.fi (Ai BI) = i=1 4^1 (24) fi [Ai exp (a X1 Bi exp ( a Xi)] = (26)

5 TORSIONAL WAVES IN PIEZOELECTRIC (622) CRYSTAL PLATE 293 z E ei (Ai Bi) = a/2ir {=1 2.' ei [Ai exp (axi) Bi exp ( axi)) =. t=1 Solving the system (26), we have Ai = Bi exp ( 2a Xi) 8t = [( 1)i ahe14xi)j/[2zr C44 (X 2 X2s) (1 exp ( 2a X*))1. (27) Substitution of these results in eqs (22) and (24) determines Y, and 1` x. Inversion of these transformed quantities yields the complete solution of the problem at hand. 5. APPROXIMATE SOLUTION Evaluation of the inverse Laplace transform is complicated for the exact roots of eq (19). Hence we consider the following approximate roots as constructed earlier. 2 X h 2 (C1 + k142) (S/(a C)) = h $E. (28) We note that if the quantity h4 E k142 is added to right-hand side of eq. (21) then the approximate roots in eq. (28) become exact. For practical purposes, the thickness of the plate may not exceed 3 cm. If we consider the infinite plate of moderate thickness then h4 will be very small compared with one meter length. Moreover, since E =.96 and k142 =.29 for f-quartz 2 the term h4 E k14 2 becomes very small in practical situations. Hence these approximate roots are justified. If the approximate roots are first substituted in eqs (23), (25) and (27) and the resulting expressions for Af, Bti, ei and f{ are next substituted in eqs (22) and (24) then we obtain V = e14 Cha 2 (27rC44)-1 X (s2 + (aa)*)i([aeh (E) 112 X cosh (ah (1 x) (E) 112)/ / \ Binh (ah (E)112\) (S2+/ba)2)112 cosh ((1 x) (s2+(ba) 2)112/C)/ sinb ( (S2 + (b a )2)11`2)} (29) = (a/21r) {[sinh (ah (1 x) (E)112) [1 ((a chk14)) 2/(s2 + (aa) 2)]/ sinh (ah (E)112)] + (achk14) 2 sinh [c`-1 (1 x) X (S2 + (ba)2)112]/ [(S 2 + (aa) 2) sinh ((S2 + (ba)2)112 /C)]) (3)

6 294 H. S. PAUL AND K. VENKATESWARA SARMA TO. = ( e14h2/21,) J a2 [(1 ± (ach)2 ( E k14 2)/(s2 + (aa)2)) o X Binh ((1 x) (s 2 + (ba) 2) 1-2/c) sinh (ah (1 x) (E)112) j sinh s 2 + (ba) 2)112/c) sink (ah (E)1'2) J where X Jl (ar) da (31) a2 = (hc) 2 (c1 E + k14 2); b 2 = (hc) 2 (C1 + k14 2) (32) We present the necessary details in order to obtain the inverse Laplace transform of the above results in the appendix. Final expressions for v, and Tz are V = (e14ch/21rc44) f a2 [ (,_I(C^ E + k14 2))112 X Sin (aat) Cosh (ah (1 x) (E) h I 2 ) G (a, x, t) sinh (ah (E)112 ) + a-1 (ch) 2 Ea f sin (aa (t u)) G (a, x, u) du] J1 (ra) da, (33) 21, (' a r sirih ( nh ( ah ) I (;) 112 [8 ( a -1 a (chk14) 2 sin (aat)] t as 1 (chkl 4) 2 f sin (aa Q u)) F (a, x, u) du, Jo (ra) da, (34) 2TrTgx = e14h 2 f a 2 [(F(., x, t) a-' a (ch) 2 (E k14 2 ) t x fsin (aa (t u)) F (a, x, u) du sink (ah (1 x)(e)112 sin (ah (E)112 ) X (8 (t) + a-'1 a (ch)2 (E k142) sin (aat)) ] Jl (ra) da,,j (35)

7 where and TORSIONAL WAVES IN PIEZOELECTRIC (622) CRYSTAL PLATE 295 F(a,x, t) =f(a,x,t) f(a,2 x, t) G(a, x, t) = g(a, x, t) + g(a, 2 x, t) (36) f(a,x,t)_ E [8(t (2n±x)/c) be-'(2n+x)..o X a (t 2 ((2n + x)/c)2)-1' 2 Jl (ba (t 2 ((2n + x)/c) 2) 1(2 ).11(1 (2n + x)/c)] g (a, x, t) = E Jo (ba (t 2 ((2n + x)/c)2)i' 2) H (t (2h + x)/c) 8=o (37) where H is the unit step function. It may be verified that except the initial condition (v, t)t=o = all the boundary and initial conditions are satisfied by the solution derived in eqs (33) to (35). From eq. (33) we find that 27rC44 (v,t)t=o = e14 (E)h12 c2 h2 f as cosh ah (1 x) )E)1I 2 Ji (ra) da. (38) sinh ah (E) 11 The integral on the right-hand side may be expressed in the form f a 3 [exp ( ah ( E)L2 (2n + x)) + exp ( ah ( E) li 2 +^ x (2n + 2 x))) Ji (ra) da. (39) As the improper integral in the expression (39) converges we infer that if h is small compared with 1 meter length then the right hand side of (38) will be also very small. Hence the initial condition (v, t)t=o = is approximately satisfied. 6. PHYSICAL INTERPRETATION From eqs (35) to (37) we find that the expression for the shearing stress contains series of impulses given by 8 (t (2n + x)/c) Z 8 (t (2(n + 1) x)/c) (4) n: *so

8 296 H. S. PAUL AND K. VENKATESWARA SARMA as a factor of the first term while 8 (t) as a factor of the third term. If V S is the velocity of the shear wave given by (C 44/p) 112 then from eq. (16) we have C 2 = Vs 2lh 2. Hence the shear wave travels the distance between the faces x = and x = 1 in time to = 1/c. We consider any plane x = constant, inside the plate, through which initially the impulse 8 (t) propagates. The first term of the expression (4) represents the arrival of the impulse that originated initially from the face x =, at time xto. The first term of the second summation in (4) signifies that this wave is reflected at the face x = 1 and passes through the plane under consideration at time (2 x) to in the backward motion. The second term of the first summation in (4) indicates that the same wave after the reflection at the face x =, propagates along the plane at time (2 + x) t o. The successive terms in (4) imply that the process continues in the foregoing manner and also that the impulsive wave is reflected at both the faces repeatedly. Mason' demonstrates that shear waves with their particle motion parallel to the reflecting surface are reflected into pure shear waves. This property is made use of in delay lines which delay an input wave for finite periods of time. As delay lines are devices for storing electrical. signals for finite lengths of time, they are employed in computers to store information to be extracted at a later stage of calculation. The present problem may be useful in delay lines to store the electrical signals (in the form of impulses) for sufficiently long period of time. In conclusion we make the following observations. The rows of deltas in eq. (4) converge, by Poisson's summation formula, to a generalized function.' In the work of Flinn the solution in the normal mode representation is shown to be the generalized representation of a similar row of deltas, which are the multiple reflections. He states that they satisfy the demands of the uniqueness theorem. Since one of the boundary conditions (5) involve a generalized function,. the corresponding solution in eqs (33) and (34)) may be called a `generalized solution' in the distributional sense, as mentioned by. Stakgold. 9 ACKNOWLEDGEMENT Authors wish to thank the reviewer for his suggestions to improve the presentation of the paper. APPENDIX In order to obtain the inverse Laplace transforms of the results in eqs (29)-(31), we proceed as follows: (s2 + (ba)2)112/(s2 + (aa) 2) _ [ 1 + (b 2 a 2) a 2/ (S 2 + (aa) 2)]/!S2 + (ba)2)112 = ( 1 + E (ahe)2/(s2+(aa)2)]/(s2 + (ba) 2)112 (Al)

9 TORSIONAL WAVES IN PIEZOELECTRIC (622) CRYSTAL PLATE 297 from eq. (32). Also cosh ((1 x) (s 2 + (ba) 2) 1' 2/c) slnh (S 2 {- (ba) 2)1!2/C1 = exp ( (2 x) (S 2 + (ba) 2) 112 /e) + exp ( x (5 (ba)2)1^ 2/C) 2H- / 1 exp (- 2(S 2 + (ba) 2) 112/C) (A2) If we expand the denominator of the right-hand side expression in (A2) into infinite series then the right-hand side of (A2) becomes n_ E exp ( 2n (S 2 + (ba) 2) 112 /c) [exp ( (2 x) (sa'+ (ba)2 )L2/c) For simplicity we put + exp ( x (s 2 + (ba) 2) 112 /c)] (A3) f (a, x, s),' exp ( (2n + x) (s 2 + (ba) 2) 112/c) n=o g (a, x, S) = (S 2 + (ba)2)-112 f (a, x, S). (A4) Therefore from (Al) and (A2), we have (S 2 _r l(ba)2 )1, 2 cosh ((l x) (S 2 + (ba) 2) 112/c) (S 2 {- (aa) 2) sinh ((s2 - f (ba)2)112/ C ) = (1 + E (ach) 2/(S 2 + (aa) 2)) Lb (a. x, s) + b' (a, 2 x, s)]. (A5) Similarly we can show that Binh ((1 x) (s 2 + (ba) 2) 112 /c) sink + (ba)2)1l /c (a, ((s 2 Using Laplace inversion tables5 ' 6 we obtain L-1 [ f (a, x, s)] = f (a, x, t)! x, s) f (a, 2 x, s) (A6) L 1 jg (a, x, s)] = g (a, x, 1) (A7) where f (a, x, t) and g (a, x, t) are defined in eqs (37). REFERENCES 1. Mason, W. P., Physical Acoustics and the Property of Solids, Ch. II, D Von Nostrand Company Inc. (1958). 2. Paul, H. S. and Srinivasa Rao, B., Int. J. Engg. Sci (1969).

10 298 H. S. PAUL AND K. VENKATESWARA SARMA 3. Eason, Fulton and Sneddon, Phil. Trans. R. Soc. London, Series A (1956). 4. Norman Davids, J. App!. Mech (1959). 5. Abramowitz and Stegun, Hand Book of Mathematical Functions, Dover Publications, Inc., New York, p. 12 (1965). 6. Do--tsch, Guide to the Application of the Laplace and Z-transforms, Von Nostrand Reinhold Company, London, II Eng, ed., p. 137 (1971). 7. Lighthill, M. J., Fourier Analysis and Generalized Functions, Cambridge University Press, p. 67 (1958). 8. Flinn, E. A., J. Acoust. Soc. Am (1961). 9. Stakgold, I., Boundary Value Problems of Mathematical Physics Vol. 11, The Macmillan Company, New York p. 42 (1968).