The results presented here are part of a study of the interaction between an electro-mechanical transducer plate element and wavelengths in the test structure that are comparable to the plate dimensions. The steady time-harmonic problem is reduced to a system of singular integral equations for determining the tractions generated at the interface between the plate and half-space where a thin viscous couplant is applied. This system is solved numerically for various values of couplant viscosity and incident wavelength. Graphical results for interface tractions and reflected power are presented

Angel, Y.

Bogy, D. B.

English

The results presented here are part of a study of the interaction between an electro-mechanical transducer plate element and wavelengths in the test structure that are comparable to the plate dimensions. The steady time-harmonic problem is reduced to a system of singular integral equations for determining the tractions generated at the interface between the plate and half-space where a thin viscous couplant is applied. This system is solved numerically for various values of couplant viscosity and incident wavelength. Graphical results for interface tractions and reflected power are presented.</p

Bogy, D.

Digital Repository @ Iowa State University (ISU)

SCATTERING OF SURFACE ACOUSTIC WAVES FROM AN ELASTIC PLATE ATTACHED TO A HALF-SPACE WITH A VISCOUS COUPLANT D. B. Bogy andY. Angel University of California, Berkeley Berkeley, California 94720 ABSTRACT The results presented here are part of a study of the interaction between an electro-mechanical transducer plate element and wavelengths in the test structure that are comparable to the plate dimen-sions. The steady time-harmonic problem is reduced to a system of singular integral equations for de-termining the tractions generated at the interface between the plate and half-space where a thin viscous couplant is applied. This system is solved numerically for various values of couplant viscosity and incident wavelength. Graphical results for interface tractions and reflected power are presented. INTRODUCTION We are interested in the interaction between electro-mechanical transducers and the elastic waves in a test medium in cases when the wave-lengths are comparable to the dimensions of the transducer and the propagation vector is not normal to the interface. The ultimate goal of this NSF funded project at Berkeley is to predict what mechanical disturbance is input to the transducer from the electrical output signal. As a first step we will try to predict the electrical output given the mechanical input. Under the conditions of in-terest the mechanical loading is not spatially uniform over the contact face of the transducer. We will therefore need a plate theory for the piezoelectric element with electroded surfaces which can accommodate the kinds of mechanical and electrical boundary conditions that are appropriate to the transducer design and mounting and the electrical circuitry to which it is connected. Such a plate theory has been derived in Bugdayci and Bogy [l]. The work presented here is also only a pre-liminary part of this project, but here we focus on the purely mechanical problem of determining the interaction of an elastic plate with a half-space. Specifically, we seek to understand the stresses generated between the half-space and plate when a surface wave impinges on the plate that is attached through a viscous couplant. The dependence of these interface stresses on couplant viscosity and the incident wavelength are the primary concern. We also calculate the power of the reflected and transmitted surface waves as a function of these parameters since these quantities can be compared with experiments. The results summarized here are derived in complete detail in Angel [2] and Angel and Bogy [3]. PROBLEM FORMULATION AND REDUCTION TO INTEGRAL EQUATIONS Figure l depicts the plane strain problem of a surface wave impinging on an elastic plate that is in contact through a viscous couplant with the half-space. All the physical parameters of the problem are shown. 248 Surface Wove AR,w, u0 X!~ ~~.a,' p' f- [I =:::!~:::31-· x' _,-I I I r--- 2 b ____, 1 ~ Viscous Couplanl K ~~~~------.-~~--------~-----x~ !J.,CT, p Fig. l Geometry, material properties and co-ordinates for half-space, plate and couplant. Plate and half-space are in contact over lx1 I < b, i.e., couplant thickness is neglected. The problem is easier to formulate when con-sidering the plate as a 3-dimensional continuum. Then, in the reduction to a 2-dimensional plate theory all the required boundary conditions and interior loading will be incorporated. The con-ditions on the plate are a2 2(xi, ~· t) = a21 (xi, ~· t) = 0, I xi I < b, (l) in which ti(xi,t) and t2(xi,t) represent the com-ponents of the traction vector applied to the bottom surface of the plate. The half-space is assumed to be traction free on the boundary outside the contact region. So the half-space boundary conditions are (2) The adhesive bond conditions between the plate and half-space are expressed as u2(x1,o,t) = -u2(x1, - ~,t), cr21 (x1 ,o, t) = K[u 1 (x1 ,o, t)-uj (x1, - ~· t)], t1 (x1, t) = -t] (x1, t), t 2(x1, t) = t2(x 1, t), I x1 1 < b . (3) We assume an incident Rayleigh wave with e-iwt time dependence traveling in the half-space in the x1-direction is scattered by the plate. Therefore the displacement and stress fields in the half-space are decomposed into their incident and scattered parts as u = u(I) + u(S) a a a' (4) in which u(I) and cr(I) are prescribed by the in-a aS cident Rayleigh wave. The scattered fields in the half-space can be obtained from Lamb's [4] solution. A complete and thorough re-derivation of this solution is carried out in Angel [2] in order to arrive at a form that is suitable for our computations. It is convenient to decompose the problem into its physically sym-metric and antisymmetric parts with respect to x1 and to denote them respectively by superscripts "S" and "A". In time-reduced form the scattered dis-placements on the boundary of the half-space can be expressed as b u~s)s(x 1 ,o) = I Ga6 (x 1 -s)t~(s) ds -b u(S)A(x 0) a 1 ' -b in which Ga6(x1-s) are the surface displacement Green's functions of Lamb's solution. (5) After reducing the problem in (1) to its corresponding plate theory formulation and decom-posing into plate extension and bending, we obtain the solutions in the form b uj5(xj) I Gj 5(xj .sltj 5(s) ds 0 u25(xj) I: G2~(xj.slt65 (s) ds ujA(xj) I: GjA(xj.sltjA(s) ds u2A(xj) =I: G2~(xj.slt6A(s) ds (6) in which Gj 5 , G]A are the symmetric and anti sym-metric Green's functions for the plate extension and G2~· G2~ arP. those for plate bending. The integral equations determining the inter-face tractions result from use of (5), (6) together with the appropriate form for the incident Rayleigh wave in (1)-(4) and they appear in dimensionless form, after appropriate differentiations and in-tegrations,as follows 249 IX _L Il V _L ia t 1 (s)ds + G11 (x,slt1 (slds -1 -1 l Il V L _L + k- Gj (x,slt1 (s)ds 0 2 -L E 2 tl (x) 2(1-E ) 0 + _ _,_l~,--1 t (sl -L L 11 i i-, d( + <,1 (,-(Jt1 (()d( 21T(l-E2) -1 -1 Il v _s Il v s G21 (1 .slt1 (slds + G22 (1 .slt2(s)ds -1 -1 1 Il v 5 _s -+ k- G2 2(l,slt2(slds = f 3 0 in which superscript "V" denotes integration from -1 to x, "A" denotes differentiation, K21 and 1<22 are defined by l -----=-....,2,----- + s g n ( x ) 1<2 2 ( x ) 2n(l-E )x A 2 Gl2(x) = - E 2 8(x) + Kl2(x) 2(1-E ) and where (8) a= ~/wKb, E2 (l-2o)/2(l-o), k = ~·~~ • (9) The symbol 8(x) in (8) is used to formally denote :__L the Dirac delta function. The functions f (x) and f 3 stem from the incident Rayleigh wave and are defined by -S f 1(x) (i~0/sR)[cos(sRx)-cos(sR)J, (l 0) in which sR and u0 are the dimensionless Rayleigh slowness and amplitude and where n~ is the real root in 0 < n2 < l of the Rayleigh equation (ll) (13) ANALYSIS AND NUMERICAL TREATMENT OF INTEGRAL EQUATIONS The system of integral equations is singular and of the first or second kind depending on a and E. The methods of Muskhelishvili [5] are used to determine the types of singularities in-herent in the solution. This also depends on the two parameters a and E, which are defined in (9). Four possibilities arise for the analytical form of the shear and normal interface tractions _L _L t 1(x) and t 2(x). These are summarized below: 250 Case 1: 0 <a< oo, 0 < E < l//:2 (viscous coupling) _L t 1(x) is bounded L t2(x) has a square root singularity at lxl = l Case 2: a+ oo, 0 < E < l/1:2 (smooth contact) _L t 1 (x) = 0 _L t 2(x) has a square root singularity at lxl = l Case 3: a= 0, E = 0 (welded contact, in-compressible half-space) L L both t 1(x) and t2(x) have square root singularities at lxl = l. Case 4: a = 0, 0 < E < l//:2 (welded contact, compressible half-space) _L _L both t 1(x) and t 2(x) have oscillatory square root singularities at lxl = l . We are concerned here with Case l for which _L the analysis reveals that the functions t 1(x) and _L t 2(x) have the forms \>{here _L L 2 lz L t 1 (l)e (x) + (1-x ) <Pl (x), [x.ifL=S \ l , if L = A L L S,A (14) in which <P1(x) and <P2(x) are bounded functions. The numerical methods in Erdogan and Gupta [6] are used to replace the system of integral _L _L equations for t 1 (x), t 2(x) by an algebraic system L L for the bounded functions <P 1 (~k)' <P 2(nk) and the _L value t 1(l) in which ~k and nk' k = l,2, ... ,n are interpolation points defined as roots of Chebychev polynomials and are given by the expressions ( 1Tk) [(2k-l)1TJ 1:;k cos n+T , nk cos 2n (16) l ,2, ... , n In order to solve this complex linear system, we decompose all complex functions into real and imaginary parts denoted by superscripts "~" an~ "I" and obtain a real linear system that 1s tw1ce as large. The system was solved numerically for the real quantities LR LI LR LI ¢1 (~:;k)' ¢1 (~:;k)' ¢2 (nk)' ¢2 (nk)' for L =Sand A, i.e., for the symmetric and anti-I. 5r-----------------,------------------, . 5 * ~ or=======--------~~--------~====~ '?e.-symmetric parts of the problem. -. 5 NUMERICAL RESULTS AND DISCUSSION The parameter values for the numerical results are 0 = 0.25, 0' = 0.25, k = 2, p'/p = 0.5, * K K/1\lP= (O.Ol,O.l,l,lO), h = h/b 0.2 ' -u0 l, 2/AR = (0. l through 1.4), (17) SR * SR Figures 2 and 3 show ¢1 (x)/K and ¢2 (x) for three * values of the dimensionless viscosity parameter K. Similar results were obtained for the other functions. Since these functions contribute directly to the interface shear and normal stress, we can con-clude that the magnitude and functional form of these quantities depend strongly on the viscosity. As would be expected, it also is strongly dependent on the incident wavelength. We have also calculated the reflected and transmitted power as a function of the incident wavelength for different values of the couplant viscosity. Figure 4 shows the reflected power curves. The resonance and anti-resonance pattern de-pends strongly on K*. We note that for the low viscosity cases, K* = 0.01, 0.1, resonances occur near 2b/AR = 0.25, 0.75*and l .25 whereas for the high viscosity cases, K = l, 10, they occur closer to 2b/AR = 0.5 and l .0. In an attempt to compare our results with those of Simons [7], who studied the same problem for an extensible membrane welded to a half-space, we took h/b = 0.01 and K* = 100. The reflected power curve showed similar resonances and anti-resonances and the actual values were as close as could be expected given the difference in the two models and in the parameters. 251 - I . 0 - I . 5 .__ ______ __._ ________ ......._ ______ __..__ ______ _J - I . 0 - . 5 0 X . 5 I . 0 Fig. 2 Symmetric real part of interface shear stress function defined in Eq. (14) for parameter values 0 = 0' = 0.25, ~·;~ = 2, p'/p = 0.5, h/b = 0.2, u0/b = l, * 2b/AR = 0.7 and K = (0.01, l, 100), SR * [t1 (1)/K = -0.02354, -0.3703, -1.537]. SUMMARY We have considered the problem of interaction of elastic surface waves in a half-space with an elastic plate attached through a viscous couplant. The interface tractions transmitted between the plate and half-space are strongly dependent on the couplant viscosity and incident wavelength. The results obtained are preliminary to the pre-diction of piezoelectric transducer output and its dependence on wavelength (or frequency) and couplant viscosity. ACKNOWLEDGEMENT This work was performed under the sponsorship of NSF Grant Eng 77-17150. [l] REFERENCES N. Bugdayci and D.B. Bogy, "A two-dimensional theory for piezoelectric layers used in electromechanical transducers," Report No. UCB/AM-80-2, Department of Mechanical En-gineering, University of California, Berkeley, March 1980. [2] Y. Angel, "Scattering of Rayleigh waves by a plate attached to a half-space through a viscous couplant," Ph.D. dissertation, University of California, Berkeley, 1980. [3] [4] [5] [6] [7] 0: Y. Angel and D.B. Bogy, "Scattering of Ray-leigh waves by a plate attached to a half-space through a viscous couplant," in preparation. H. Lamb, "On the propagation of tremors over the surface of an elastic solid," Philos. Trans. R. Soc. London, Ser. A 203, l-42, 1904. N.I. Muskhelishvili, Singular integral equations, Noordhoff Int. Pub., 1977. F. Erdogan and G.D. Gupta, "On the numerical solution of singular integral equations," Quart. Appl. Math. 30, 525-534, 1972. D.A. Simons, "Scattering of normally incident Rayleigh waves by thin strips," J. Acoust. Soc. Am. 60, 5, 1100-1107, 1976. .2r---------------.---------------~ }("*= I . I CJ)~ 0 17-t--t-------+------~-t---:~ - I -. ZL----~----~----~---~ - I . 0 - 0 5 0 X 0 5 I • 0 Fig. 3 Symmetric real part of interface normal stress function defined in Eq. (14) for parameter values cr = cr' = 0.25, w'/w = 2, p'/p = 0.5, h/b = 0.2, u0/b = l, 2b/AR = 0.7 and K* = (0.01, l, 100) 0 252 ·2 2•10 ·-~---.---r------.---,----.-----. u .s Q) ;< 0 Q. '-.. d "0 -4 ~ 10 Q) ;;::: Q) 0: . -........ 1< = 10 .-·r·, , // 0 I; \ 0 \ I /1.o · l! I I I I / I i I o.o1 i ,,, ....... \, i 1/ \ it'' 0.1 \ . I I \ i \ I :1 I lf I 1l i 106 0~-~~~~-~0L.6--0L.8--~10--~I.L2 _ __JI4 2b/XR Fig. 4 Power of reflected surface wave versus in-cident wave number for parameter values cr = cr' = 0.25, w'/w = 2, p'/p = 0.5, h/b = 0.2, u /b = l and K* = (0.01, 0.1, l' l 0) 0 ° SUMMARY DISCUSSION Mr. Kinsley (Cincinnati): I wonder if you have looked at the dispersive effect of the coupling between the half space and plate. What happens as your B gets very large? What happens to the singularity at the ends of the plate? David Bogy (University of California, Berkeley): Let me consider the second question first. You can solve the problem. Nothing happens to the singularity. One could surmise the results by looking at this last curve here which is radiated power as a function of 2B over AR. If B gets very large, you approach the results for an infinite plate, which is what this result must be asymtotic to. So I think the width of B really does not matter in the singularity calcula-tion. If B gets large, that means the wavelength is getting relatively small. That means I will have a lot of oscillation under mY plate. And that means I have to take a lot more interpolation points in the numerical scheme, and that costs more money. With regards to your first question, I have not considered the dispersion effects of viscosity, I have not looked into that, and I'm not quite sure I understand what dispersion effect you're talking about. William Pardee, Chairman (Science Center): I think further questions should be reserved for private discussion. Thank you, David. 253 

Scattering of Surface Acoustic Waves from an Elastic Plate Attached to a Half-Space with a Viscous Couplant

Bogy, D

Angel, Y

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Scattering of Surface Acoustic Waves from an Elastic Plate Attached to a Half-Space with a Viscous Couplant

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