The Rayleigh-Lamb dispersion equation for a viscoelastic plate

MECHANICS RESEARCH COMMUNICATIONS Vol. 20(3), 215-222, 1993. 0093-6413/93 $6.00 + .00 Copyright (c) 1993

Printed in the U.S.A. Pergamon Press Ltd.

T H E RAYLEIGH-LAMB DISPERSION E Q U A T I O N F O R A VISCOELASTIC PLATE

Daniel Nkemzi Department of Theoretical Mechanics, University of Nottingham, Nottingham, NG7 2RD

(Received 19 August 1992; acceptedfor print 24 August 1992)

Introduction For a semi-infinite isotropic linearly elastic or viscoelastic plate in plane strain, the propagation condition for time harmonic waves propagating in the plane of the plate is known as the Rayleigh-Lamb dispersion equation. Because of its important role in the calculation of the in-plane Green's function for transient elastodynamic fields, the Rayleigh-Lamb equation has been studied extensively. In the case of an elastic plate, and mainly due to Mindlin [1], knowledge of the nature and behaviour of the solutions of the dispersion equation is now more or less complete. For a viscoelastic plate the study is less complete. The problem has previously been addressed by Chervinko and Senchenkov [2] who assumed that the viscoelastic material was what they called "rubber - IRP-1347", and by Tanaka and Kon-no [3]. The latter modelled the plate as a Standard Linear Solid and solved for the fundamental mode (and two higher harmonics) numerically. In this paper we extend the results of Tanaka and Kon-no in three specific ways: (i) we suggest a systematic procedure of identifying and calculating all the higher harmonics, Oi) we highlight the relationship between the elastic and viscoelastic solutions, and (iii) we investigate the parametric dependence of the solutions on the viscoelastic t i m e constants. In Section 2 the equations governing the motion of a linearly viscoelastic solid are stated. These are then used in Section 3 to derive the Rayleigh-Lamb dispersion equation for a viscoelastic plate with traction-free faces. The dispersion equation is a complex-valued transcendental equation. Though in principle there exist methods for solving certain classes of transcendental equations in closed form, such as the method of Burniston and Siewert [4], in practice all but the simplest equations of this type must be solved numerically. Thus in Section 3 we describe a systematic numerical procedure for solving the dispersion equation. In the final section, Section 4, numerical solutions for various values of material constants are presented and discussed. It is found that the nature of the solutions (variation of frequency with wavenumber) depends crucially on the values of the viscoelastic time constants. Thus we find that for certain values of the viscoelastic time constants, the viscoelastic solution is, apart from the anticipated attenuation in the direction of propagation, very similar to the elastic solution; while for other values the two solutions are qualitatively very different. 215

216

D. NKEMZI

Basic equations As usual a dot and a subscript comma will denote differentiation with respect to time and a spatial coordinate respectively. For any continuous medium subjected to external forces, the following two facts are well known. From the analysis of strain (kinematics) the relationship between strain eij and displacement u i is given by the six independent equations eij

(l)

- -~(uj, i + u i , j )

provided the displacement gradients are sufficiently small. From the analysis of stress (dynamics) which uses Newton's Laws of Motion, the stress tensor ~ij is symmetric and the three equations of momentum become, in the absence of a body forde 7ij,j

(2)

- pti i

where p is the density. For a non-heat-conducting, isotropic linearly viscoelastic solid, the relationship between stress and strain at time t and position x = (x 1,x~,x3), can be written as t

zij(t ) - 2 I

t

/~(t-T)

¢ij(T)

dr + 6 i j

-co

I

k(t-7)

ekk(r)

dr

.

(3)

-co

In terms of the displacements, equation (3) can also be written as t

PUi = I

t

g(t-z)ui,jj

dT + I

-co

[#(t-r)

+ k(t-z)]

uj,ji

dr

(4)

-oo

In (3) and (4), /a(t) and X(t) are independent response functions. For two dimensional motion in which all the stress components are independent of one Cartesian coordinate (x 3 say), the displacement equations of motion, equation (4), uncouple into the two sets t Ptii=

I

t /~(t-7)

ui,jj

dr + I

-oo

It'(t-z)

+ X(t-r)]

uj,ji

dr

(5)

-oo

i,j

= 1,2

and t

PiJ3 = I

/a(t-z)

u3,jj

j = 1,2

dr

(6)

-oo

Equation (5) represents coupled dilatation and shear waves (associated with motion in plane strain) known as P-SV waves. Equation (6) represents shear waves (associated with anti-plane strain motion) known as SH waves. SH waves will not be discussed further. For harmonic P-SV waves of frequency co, and wavenumber K, propagating in the x,-direction with particle displacement polarised in the (x~,x2) plane, the stress and displacement can be written in the form ri J - Tij(x2 ) e i(cot-Kxl)

i,j

- 1,2

(7)

WAVES IN VISCOELASTIC PLATES

u i - Ui(x2)e t(c°t-I
i -

217

1,2

.

(8)

Equations (3) and (5), with the help of (7)-(8), can be combined and rearranged in the form dS dx 2

-

^

s

,

(9)

where

0

-iK

tKX*

0

_p~2

0

0

4K2F*(X*+F*) (X*+2#*)

(x*+2~*) A -

1

0

iKk*

0

,(I0)

(X*+2F*)

(X*+2#*) 1 -F

0

~.o 2

O

-iK

u,]t

(11)

All the relevant components of stress are completely determined by the vector S. In (10), F* and X* are functions of co and are given by oo

oo

/~*(c~) - I 0 / ~ ( t ) e - i c o t

X*(co) - IoX(t)e-iC°t dt .

dt

(12)

For viscoelasticity of the Standard Linear Solid type, the functions /~* and >,* are of the form

F* ( ~ )

" #o

Ll+i~z r s

),* (¢0)

2e#*(ta) 1-21,

'

(13)

where /x 0, ¢ c and ¢ r are positive constants with ¢ c > Or, and ~, is the (assumed) constant Poisson's ratio. The numerical results in Section 3 are based on the Standard Linear Solid viscoelastic model. While the Standard Linear Solid may not represent the behaviour of any real viscoelastic material, it does however exhibit the four most c o m m o n features of a viscoelastic solid, namely, instantaneous elasticity, creep, stress relaxation and creep strain. In particular it creeps to a finite limit under a fixed stress and relaxes to a positive non--zero stress under a constant strain. It will return to its original state if a stress is applied and then removed.

The dispersion equation Consider a semi-infinite plate of thickness 2h and whose top and bottom faces are free of traction. Choose a Cartesian coordinate system OxTx2x3 such that the ( x l , x 2 ) plane coincides with the midplane of the plate and the faces of the plate are at x 2 = -*h. Then in terms of the value of S at an arbitrary point x 0 (x 0 • x2), equation (9) has the unique solution - exp[(x:×0)^]s(×0)

.

(14>

In particular, the solution at the top face of the plate in terms of that at the bottom face

218

D. NKEMZI

can be written as S(h) - exp(2hA)S(-h)

[P,,

P,21

P21 where P l l ,

S(-h)

(15)

,

P22

Pa2, P2~, P22 are 2 x 2 submatrices

For a plate with traction-free faces, equation (15) becomes

(16) U(h)

where

1)21

P22

U(-h)

0 = [ 0 , 0 ] T , U(h) = [ U 2 ( h ) , U l ( h ) ] T, U ( - h ) = [ U 2 ( - h ) , U I ( - h ) ] T

.

Equation (16) has (non-trivial) solutions if and only if det(P12)

(17)

= 0 .

Equation (17) is the propagation condition for time harmonic waves. the matrix P1 2 has the explicit representation

#~K P12 - ~

-(q~2-1) 2 s i n q l Kh q1

2

-4q2sinq2Kh

I

It can be shown that

2i(q2-1)(cosqlKh-cosq2Kh) _(q~_l)2

[ ~ ' 2 i ( q ~ - l ) ( c ° s q l K h - c ° s q 2 Kh)

sinqzKh + 4qlsinqlKh q2 (18)

where ql =

[

p~2

= [p~2

1] 5 q2

(X*+2~*)K 2

1]:~

(19)

[#.K 2

On substituting (18) into (17), the Rayleigh-Lamb equations tanqzKh + ~ - 0 tanqlKH ({:12-1)2 ,

(20)

tanqlgh ~ 0 tanq2K h + ( q ~ - l ) 2 ,

(21)

are obtained.

Equation (20) corresponds to symmetric modes of vibration in which

U 1 (X 2 )

=

U 1 (-X 2 )

U2(X 2 ) =

-LI(-X2)

,

while equation (21) represents anti-symmetric modes, for which U 1 (X 2 )

--

-U 1 (-X2)

,

U2(X 2 )

-- U 2 ( - X 2 )

We seek solutions K = K(oJ) to (20)-(21) of the form K - k-i~

,

k ;~ 0 ,

c~ ;~ 0 ,

co ;~ 0 .

(22)

The restriction c~ ;~ 0 ensures that the solution remains bounded as x -~ o% while k and co

WAVES IN VISCOELASTIC PLATES

219

which represent the physical wavenumber and frequency are necessarily non-negative. For general values of c0, the complex-valued dispersion equations (20)-(21) can only be solved by numerical iteration. To do that we write the wavenumber in terms of its real and imaginary parts, K = k-ic~, and the consider each dispersion equation as a nonlinear system in k and u, with u as parameter. For a given c0, each system has infinitely many solutions (k,t~). However for a given set of material constants we are only interested in the solutions k(oJ), ~c0), as continuous functions of u. These are the modes of propagation and constitute a discrete set of curves in (k,c~,c0) space. Hence to identify the different modes, it suffices to determine the points of intersection of these modes with a given surface in (k,cx,o~) space. One such surface is given by c0 = 0. It can be shown that on this surface equations (20)-(21) reduce, respectively, to s i n 2 K h + 2Kh - 0 ,

(23)

.

(24)

sfn2Kh

-

2Kh

-

0

Now, all non-trivial roots of the equations sin 2Kh ± 2Kh = 0 are complex. The usual method of locating these roots is by trial and error; see e.g. Vasudevan and Mal [5]. However, it can be shown, Nkemzi [6] that the large roots and of (23) and (24), respectively, are given by the asymptotic formulae Kns Kna

Kns

1 (ln(4n~) ~r~

-

1 - ~-~ +

1 {ln(4nr) Kna " 2-h

i

[/$n-ln(2/3n)'~)' /3n

JJ

+ 1 ln(2c~n)]~ ~n + i [ an an j j

,

~n

-

'

an -

(4n-1)

(4n+1)~

1~ ,

(25)

,

(26)

where n is a positive integer. Other sequences of roots are -K n , - K n a , . ± K n , ._*Kna . The formulae (25)-(26) were used to calculate starting values for ~he iteraUve soSiuuon of the dispersion equations. Though in principle the determination of the complex roots of the dispersion equation appears to be straightforward, in practice the solution of this equation is far from routine. When the material is viscoelastic, the real part of the complex wavenumber is no longer a monotonic increasing function of frequency. This makes it difficult to track the various branches by numerical iteration with frequency as parameter. Further, the fact that c~ < < k at high frequencies can lead to numerical precision problems. The exact nature of these difficulties and their resolution can be found in [6].

Dispersion curves For the purpose of numerical calculations it is convenient to work in terms of the dimensionless variables I(

-

k-i~

-

KH

,

L ~ j

and the dimensionless time constants

% " '~ / # * ( ° ) / ~ L

p

j

~" H

-

" -/#*
H

Numerical solutions to dispersion equations are usually presented in the form of plots of frequency against wavenumber (other choices such as phase velocity or wavelength are possible). These plots are known as the frequency spectrum and each branch or dispersion curve represents a m o d e of propagation. Here we have plotted the (dimensionless) frequency on the vertical axis. The complex wavenumber is the independent variable with its real part

220

D. NKEMZI

represented by the positive horizontal axis and its imaginary part by the negative horizontal axis. Figure 1 depicts the frequency spectrum of an elastic plate (drawn in three-dimensional perspective) while Figures 2 - 4 show that of a viscoelastic plate for different values of the time constants ~c and 7r- For each figure the Poisson's ratio is 0.35. The elastic plate solution corresponds to ~c = ~r = 0. For the viscoelastic plate spectrum, all solutions K(~0) are complex. There are no real or imaginary branches. This contrasts with the elastic plate spectrum which admits real, imaginary, and complex branches. Another difference between the elastic and viscoelastic spectra concerns s~mmetry about the planes k = 0 and & = 0. Figures 1-4 show only the fourth octant of (k,&,&) space. However, it is well known that for all frequencies^&, the elastic spectrum is symmetric about the planes k = 0 and & = 0. That is to say if K 0 satisfies the dispersion equation then so do - K and ± K0" For the viscoelastic spectrum, this is only true for low frequencies. At high frequencies K 0 and - K 0 both satisfy the dispersion equation but not their complex conjugates. Specifically, it is found that all the viscoelastic branches which emanate from the plane ~ = 0 in the fourth and third quadrants of the K-plane terminate in the fourth octant. Those originating from the first and second quadrants finish up in the second octant. Thus, since we are interested in all solutions for which k > 0, we have also included in Figures 2-4 those branches of the solution which originate from the third quadrant of the complex wavenumber plane (drawn in dotted lines). However, in this case only the high frequency portion of these branches, on which k ~ 0, is shown. T h e solution for ?c = 0.5, ~r = 0.1 (Figure 2) is rather peculiar. For these values of the time constants the imaginary part of the wavenumber ~ is very much larger than one, even at high frequencies. This means that for these values of the time constants, the waves decay rapidly with distance along the plate and are in effect non-propagating. When the time constants are increased to ~c = 5.0, ~r = 1.0, we get the spectrum shown in Figure 3. Here we find that the attenuation factor & is much smaller than the physical wavenumber. Furthermore, at high frequences & is virtually constant. For very large values of the viscoelastic time constants (Figure 4) the viscoelastic spectrum becomes more elastic-like; with the waves being virtually undamped and the "terrace-like look", characteristic of the elastic spectrum, becoming more pronounced. This is because as the time constants tend to infinity the complex modulus /~ asymptotes the instantaneous modulus of the viscoelastic material.

Acknowlegements I am grateful to Dr. W. A. Green for many helpful discussions. This work was done while on a Cameroonian Government studentship. The financial support is gratefully acknowledged. References [I] [2] [3] [4] [5] [6]

Mindlin, R.D.: Mathematical Theory of Vibration of Elastic Plates. In Proc. l l t h Am. Symp. on Frequency Control, p.l (US Army Corps. Eng. Labs., New Jersey)(1957). Chervinko, O.P. and Senchenkov, I.K.: Viscoelastic Waves in a Layer and in an Infinite Cylinder. Soviet Appl. Mech. 22, I136 0987). Tanaka, K. and K o n - n o , A.: Harmonic Waves in a Linear Viscoelastic Plate. Bull. of the JSME, 23, 185 (1980). Burniston, E.E. and Siewert, C.E.: The Use of Riemann Problems in Solving a Class of Transcendental Equations. Proc. Camb. Phil. Soc. 73, 111 (1973). Vasudevan, N. and Mal, A.K.: Response of an Elastic Plate to Localized Transient Sources. ASME J. App]. Mech. 52, 356 (1985). Nkemzi, D.: Dynamics of Elastic Viscoelastic Sandwich Plates, p.61. Ph.D. thesis, University of Nottingham (1991).

WAVES IN VISCOELASTIC PLATES

II

221

. i~I

::

~ ~ -

,,._~

•%..>%.~( ~(" ~ ~" _@-<-~:<>...-'~ X Y" J-o + ' , X X . , ~ Figure 1

Frequency as a function of complex wavenumber for an elastic plate (the dotted lines correspond to complex branches).

....

!zli i

, ,I,,;7,'I

"

,,, ,,," ,,, ,,,',;,'

,

;::::;iil

, / ", / ",,,,," , / ,,/ s,,,

| /

'

.;;iii 7

,,,

"~

•

/

"

/

" t #s • ,,t o

/

"" ,, ,, ,, ,;:',

:t,~, ' ,;,;,,,..,,,, ';",",,",,",,¢ /

•

,,' / (

"

lo '

"

IS '

Figure 2

"

la "

! ¢Olllil

" "

IlillUllll

- 25 "

lo

is

lo

f l'ill -S" S' ¢ONPLEX ¥AYEHUMOEE

INTI-$YIHETRIC 140DES Io" IS" zo"

Frequency as a function of complex wavenumber for a viscoelastic plate with ~c = 0.5, ¢c = 0.1.

222

D. NKEMZI

/ /////

i

/ ~o

Figure

ii I Is

COXPLEX UAVE~UXtSE~

1o

S

s

so

15

~o

25

Frequency as a function of complex wavenumber for a viscoelastic plate with ~c = 5.0, ~r = 1.0.

3

.J-"Jiiiiii iiiiiiiiiiiii li ii: _

25

2O

s5

to

5

S

10

S5

2O

~S

zo

'

_

15

COflPLEX ¥&¥ENUNEER

Figure 4

Frequency as a function of complex wavenumber with ~c = 50.0, ~c = 10.0.

-

t0

_'

S 5 COKPLEX ~AVENUgBEg

IO

for a viscoelastic plate

IS'

20"

25