114
Finite Strain Theory of Love Waves By KRISHNA DE 1)
Summary - The propagation of Love waves is discussed on the basis of finite strain theory. The primary Love wave is found to be associated with a secondary Rayleigh wave and a tertiary Love wave. Numerical calculations are presented for two values of the wave-velocity; the results show that the theory of Love waves based on infinitesimal strain, is not applicable to short period waves. 1. Introduction The propagation of surface waves in an elastic half-space has been based on the classical theory of Elasticity which takes account only of infinitesimal strains. An exact theory of the propagation of such waves should be based on the theory of finite strain. TRtJ~SDELL [3] Z) has discussed the general theory of body waves following finite strain theory. For an isotropic body he has calculated the wave speeds for first order and second order principal waves. HAYES and R~vLIN [4] have examined the propagation of surface waves in an initially homogeneously strained body assuming an expression for the energy density function W in the isotropic incompressible material. We confine ourselves in this paper to an approximate treatment of Loves waves based on the assumption that the non-linearity is a weak one allowing the expansion of the stresses and displacements in powers of a small real parameter e. The solution is taken to the third order as the exact solution is too complicated. It is expected that the smallness of the higher order terms will give an indication of the range of validity of the usual theory of Love waves. The primary motion being assumed to be a steady periodic wave train due to Love wave whose fundamental simple harmonic component has argument K ( x - e t), the second order motion can be expected to consist of the first harmonic with argument 2 K ( x - c t). But it is found to be Rayleigh wave motion. The third order motion, however, is a Love wave consisting of a second harmonic with argument 3 K ( x - c t). The cumbrous nature of the solution precludes the possibility of evaluation of the nth order motion and the subsequent consideration of convergence of the solution. Numerical calculations have therefore been carried out with values of the parameters, which agree with values obtained for the earth. x) Department of Mathematics, Bengal Engineering College, Howrah-3, West Bengal, India. 3) Numbers in brackets refer to References, page 123.
Finite Strain Theory of Love Waves
115
It is shown from the results that the theory of Love waves based on the consideration of infinitesimal strain is not valid for short period waves though they may furnish a good approximation to the exact solution for long period waves. It is worthmentioning that if we apply the results obtained by TRUESDELL [3] to the particular form of the strain energy and the displacement assumed by us then there is only one shear wave-velocity and one compressional wave-velocity as compared with nine obtained by him. Our choice of the strain energy function is dictated by the need to consider numerical values relevant to the earth; we have been unable to obtain such values for real materials with a more complicated strain energy function.
2. Statement of the problem We consider an isotropic, elastic layer in welded contact along the plane y = 0 with a semi-infinite isotropic medium M. The homogeneous upper layer M' of thickness H is bounded above by the free horizontal surface y = - H . Following GREEN and SPRATT [1], we expand the stresses and displacements in powers of a small real parameter e, the displacement is expressed in the form:
1),(x, y, t) + r 2 W,(x, y, t) + ~3 el(x, y, t)
i = (1, 2, 3)
(2.1)
where 1)3 = V ( y ) 1)1 ~
e 'k(:'-c'~
1)2 ~" 0 .
This results from the assumption that the primary wave is a Love wave and ~, ~V and 0 are displacement vectors corresponding to first, second and third order. The usual expression for the stress in terms of the strain-energy density function W for isotropic elastic material is z ~ J = ~ g ~ J + T B ~ j + p G ~j
(2.2)
where ~, T, p are proportional to the derivatives of W with respect to the strain invariants I~, 12, 13 (defined in GREEN and ZERNA [2]). Our procedure is valid for a general strain-energy function W; but since we are interested only in applying the results to Love waves, we confine ourselves to a particular form of W containing only those two constants whose values are known for materials composing the earth. We therefore assume 2+2# # W8 J~ - } J2 (2.3) where 2, # are Lam6's constants and J~ =/1 - 3,
J2 = I2 - 2 11 + 3,
J3 --/3 - / 2 + I, - 1.
(2.4)
116
K. De
(Pageoph,
This form of W reduces to the usual form for infinitesimal elasticity theory if we retain only the quadratic terms in J1 and J2. Expansion in powers of e, substitution in the equations of motion "ciJ[li =- O f i (2.5) leads to the equations of motion 8203 82V3 0 8203 8X 2 "I--
8y 2
8 T 11 8X
8 T 12 -? ~
8 T 12
(2.6)
# 8t 2
t~2W1 = Q 8t 2
OT 22
(2.7)
82W2
8 x - + 7}-y = Q 8t 82W3 8 2 W 3 0~2W3 8x 2 + 6322 ~ 8t 2
(2.8)
where the stress tensor z 'J is expressed in the form Z ij = e S ij + e 2 T ij -Jr...
(2.9)
giving Sl3
803
# ~xx'
$23 =/~ f l y '
(2.1o)
S~J = 0
for all other i, j. and
T**=
+#
2~x
T22=
+~
2~-y \ a y / j + ~ (
T12 r2,
\ O x / j +~ 2 - - + vy
kay/J
Txx+\&]j
/eWl aW2 a 3aoq
(2.11)
aw3
T 13 = T 31 = # -
0x T 23 = T 32 =
# aw3 Oy
On substituting in (2.6) from (2.1) we get,
V(y) = P' e -u~s'y + Q' eirs'y i n =Pe-KSY
in
M
M'
(2.12)
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Finite Strain Theory of Love Waves
117
where
V
S = ~l ~o/e21
i
Since the displacements and the stress components are continuous at the interface y = 0 and the upper boundary y = - H is stress-free, we obtain from the 1st order boundary conditions the following equations connecting P', Q' and P, p'+Q'=p #S
P' - Q' = - P i #' S'
(2.14)
p ' = Q' e -21Ks'it
so that t a n K S' H -
#S
(2.15)
ffS'
which is the frequency equation for Love waves. 3. Second order calculations
The equations (2.6) and (2.8) are of the same form while the stress components corresponding to v3 and 1413are also of the same form as is clear from (2.10) and (2.11). Thus the second order Love-type wave motion satisfies the same frequency equation as the first order Love-type wave motion though the two are not related in any way. Hence the second order motion of Love-type can be determined only if we stipulate that the Love-wave motion in the medium is known, For instance, if we assume that such motion is entirely of the first order the amplitude of the second order motion o f this type vanishes. We now consider the remaining part of the second order wave motion which is of Rayleigh-type. The substitutions
w; =
l(y) e 21K x-c
W; = g2(y) e iK x_.
t
(3.1)
made in (2.7) and (2.11) lead to a pair of ordinary differential equations whose solution is gl(Y) = a e ~'y + b e -~'r + c' e p'y + d e -a'y
+ K(2' - - + 2 #') E (p,2 e-2irS'y + Q,2 e2,Ks'y) 4 i D' +
P' Q' i(2' + 2 #') (1 + S '2) 2(~' c 2 _ 2' - 2 p')
(3.2)
K.De
118
(Pageoph,
g2(Y) = a e~'y K~ - b e -~'y K~ + c' e e'' K~ - d e -a'y K~ + (p,2 e-2U,;S,y __ Q , 2 e2mS'y) x
(0' c2 - 2' - 2/.t') (2' + 2 f ) i K E (2' + #') 4 S' D'
_KS'#'L'(2'+2#')
(3.3)
K(2'+2#')(1+S,2)}
4 D'(2' + #')
4 S(2' + #') i2
+Ka(2 '+2#')(1+S 2' + #'
)e,Q,y
where a, b, c ', d are constants, ~,' 8' are roots of the equation, f ( Z + 2 #') m 4 + 4 KZ{(2 ' + 2/z') (0' c2 - 2' - 2 #') + (2' + #,)2 + #'(0' e 2 - #')} m2 + 16 K4(O ' c z - 2' - 2 # ' ) ( ~ ' c 2 - #') = O.
"~
(3.4)
and
K~-2,_~tt,(2K
(0'c 2-
- 2' +
+ /~-~(e' ~
-2#')
J
~') E = (2' + f ) S'2(1 + S 'z) + S'2(1 + S 'z) (2' + 2/z') - (r ~ - ~ ' ) (1 + s '~)
G
~' ( ~
+-:t'
-
-
2
(3.5)
J
D' = f ( 2 ' + 2 #') S '4 - S'2{(2 ' + 2 bt') (0' c2 - 2' - 2 p') + (2' + #,)2 + #'(O' cZ - #')} + (0' c2 - f ) (0' c 2 - 2' - 2/~'). The solutions (3.1) and (3.2) are displacements in the layered medium M ' . For the medium M, we substitute W1 = f l ( Y ) e2iK(x-ct) W2 = f2(Y) e2iKO,_ct ) ~ (3.6) in (2.7) with (2.11) and get the displacement components in the medium M in the form KL p2 e-2~sy f l ( Y ) = m e -~y + n e -~y + __ 4 iD _
fdY)=-me
-~'yK~-ne
-prKz
K L +~(0c2_2-2~)~-~(2
(3.7)
L
( 2D
K
+ 2~)(1-S
} 2)
(3.8) p2 e-gKsY
Vol, 80, 1970/III)
Finite Strain Theory of Love Waves
119
where m, n are constants, c~,fi are the roots of the equation (3.4) with 2, kt, (p replacing 2', #', q~' and g l --/]. "-k
-~- .... {g (~o c 2 -- ,~ -- 2 #)
~
i
{l~fi 2K 2K +T(0c2-2-2/0
K2-2+~
}
(3.9)
L = - 4 i K(2 +/~)(1 - S 2) {$2(2 + 2 # ) - (0 c2 - / 0 - $2} D = #(2 + 2#) S 4 + 2 $2{(2 + 2#) (0 c2 - 2 - 2 #) + (2 + #)2 + #(0 c2 - P)} + 2(0 c2 - #) (0 c2 - 2 - 2 #). The same boundary conditions produce the following six equations a, b, c, d, m and n a+b+c'+d-m-n+U=O. a K~ - b K~ + c'K~ - d K ~ + mK~ + nK2 + V = 0 . a #'(o~ + 2 i K K ~ ) - b #'(o~ + 2 i K K~) + c' If(fi + 2 i K K ~ ) - dlz'(fi + 2 i K K~)
+ m #(~ + 2 i K KI) + n/~(fi + 2 i K K2) + W= 0. a{(2' + 2 #') 7K~ + 2 i K 2 ' } + b{(2' + 2 #') c~K~ + 2 i K 2 ' } + c'{(2' + 2/f) fl K 2 + 2 i K 2'} + d{(2' + 2/~') fl K~ + 2 i K 2'} + m{(2 + 2 #) ~ K, + 2 i K 2 } + n{(2 + 2 # ) f i K z + 2 i K 2 } + x = 0 a(cr + 2 i K Kf) e -~m - b(~ + 2 i K K;) e~m + c'(fi + 2 i K K2) e -pu - d ( ~ + 2 i K 1,:;) d " + y = o .
a{~ K;(2' + 2 #') + 2 i 2' K} e -~m + b{o~ K~(2' + 2 #') + 2 i 2' K } e ~'n
+ c'{fi K2(2' + 2 i f ) + 2 i 2' K} e -e~ + d{fi Kd(2' + 2/f) + 2 i 2' K} eaR + Z = 0
where, U
Q,2)
(2'+2p')KL' =
4 i D'
(p,2 +
i(2'+2#') +
(
0 C~p,Q,
2-
/, ]
KLp 2
2(0' c 2 - 2' - 2 #')
- 4 iD
V = (p,_2 Q,2) {i K(2' + 24(2,/~')+(0'#')c2S'- D'2'- 2 #') E _ K S'4 #'D'(2,E(2,++#,)2/f) K(2' + 2 /~') (
K2(2' + 2 i f ) ( 2 0 ' C 2 ~ p ' Q 2 -
4 S'(2' + / d )
pQiK~SL + ( 2D
'
_
/f ] J
+
iK(~ +
L
/*' ]
iK(2+2/1)(2_0;_2)}
4SD
4S
W= #'(2' + 2 #') K z S' E (Q,2 _ p,2) + 2 i K #,(p,2 _ Q,2) 2 D' 3iK(o'cZ-2'-21~')E KS'(2'+2#')L' 3(1 + S'Z)~ x
{
q- K 2 S '
4S'D'
-
, ( p , 2 _ Q,2) q_ i t / K 2 S p 2 .
8D'
2S'
J
"
120
K. D e
(Pageoph,
X - - ( 2 ' +2#')I-2iKS'(P'Z+Q'2){3iK(o'~
SD,(2'+2#')E-ss,(I+S':) + (2' + 2 #') i K
S ' ( Q '2 -
+~
p,2) q_
Q'
5#'KZE 2 D' (p,2 + Q,2)
_ P' Q' K f ( 2 ' + 2#')(1 + S'z)_ p, K 2 p 2 _ 6#' S' (0' c ~ -
3
~')
x{~+4s(OC2-31OD-4~KSL2D K
L
K (2+2#)(l_S2)}_(2+2#)KZSZp2
I1 K2 Lp2 - # K2 p2
2D
K 2 S'
Y - 2 D' (2' + 2/~') E(Q '2 e - 2iKs'H- p,2 eZ,KS'n) + 2 i K P
f3iKL' x [ 8 S ' D ' (0, c 2 _ 2 , _ 2 # ,
)
'2
(e 2'Ks'n- Q,2 e-2,Ks,n)
3K KS'EsD,( 2 ' + 2 # ' ) - 8 S , ( 1 + S ' 2 )
}
3 i K3(1 + S '2) n' Q' H - K 2 S'(Q '2 e -2'Ks'n - n,2 e2,~s'n) Z = - 2 2 ' ( + 2 #') i K S'(P '2 e2'~:s'n - Q,a e-ZiKS'n) -
{3(0' c 2 - 2 ' - 2 # ' ) i K E K S ' I l, K } 8S'D' 8D' (2' + 2/f) - 8 S~ (2' + 2 #') (1 + S'2) 3K2 K 2 S ,2 + 2 - (1 + S '2) P' Q' 2 (P' ei~S'n - Q' e-i~s'n)z
x
(p,2 e2iKS'It e-2itcs'n) +Q,2
# ' K 2 L' + ~ (2D 2'+2/~')
K [,1'(,~,t -Jr-2 #t)
(1 q- S ,2)
P' Q'
-
(0' c 2 -
;r -
2
~')
- #' KZ(P ' eIKs'n - Q' e-;~s'n)2.
4. Third order calculations Here we consider the displacement corresponding to the coefficients of e 3 in (2.1). The stress components for these displacements are R 11 = R 22 = R 33 = 0 ,
R12
['c3U~ OU2
~v3 ~W3
3W3 ~v3'] + Ox a y /
R, 3
faU3(aW,
~W2
1(c3v3"]2
= #l~-y - \ Ox + 3 y + 2 \ 3 x ]
l[c~v3"N2~~v3}
+ 2\ ~y ] I Oy J
(4.1)
Vol. 80, 1970/III)
12l
Finite Strain Theory of Love Waves
As from w 3 we have seen W3 = 0, the equations of motion corresponding to the cornponents U~ and U2 are
02UI
02U2
0y 2 + OXOy
02U1
0 02U1
/
/
/z 0t 2 = 0
~2U2 0 02U2 -- O.
O\ 0~ + O\ 2
[
# Ot2
(4.2)
/
Hence these displacements are free from the effect of primary and secondary wave motion. But the equation of motion for the displacement U3 is, 0R 13 0R 23 (OT ~1 OTa>] Ov3 (OT 12 0T22~ 0v3
..... 0x +-Uy-y + \ ~
@ Z 11 02V3 ~~.
+ 2
-+
0y/~+\-LZ
zl 2 02U3
o--~ + o~
+ 0y/~
T22 02u3 (6813
0S23~
~y~ + \ ~ - x + oy /
(4.3)
•
+
+
+s
)~-bU +
+
+
= 0 ~ g ~ - + et-~ \ ex + ey / + ~ V \ ox + - # - / J " Putting the values of R ~j' s, T ij' s and s ~ s from (4.1), (2.11) and (2.10) respectively in (4.3), we get a differential equation of second degree the solutions of which in the two media are of the forms
U~ = {P3 e-3iKS'y + Qt3 e3iKS'y _[_M} e 3iK(x-ct) in and,
U3 = {P3 e-KSY + N} e 3iK(x-C~ in
M
M' (4.4)
where M and N are given by extremely cumbrous expressions involving the constants occurring in the first and second order calculations. The constants P~, Q; and P3 are found from the boundary conditions as stated before. The expressions for those have not been given here as they are extremely cumbrous. However, the values of the constants are obtained numerically and given below.
5. Discussions Since (see (2.14)) P ' and Q' are linearly proportional to P, the constants a, b, c, d are proportional to p2 and P~, Q; and P3 are proportional to p3. It is clear from (3.1), (3.6) and (4.4) that associated with a first order Love-wave, there is a second order Rayleigh wave and a third order Love-wave. Numerical calculations have been carried out under the assumption that Poisson's condition (2 = #) holds for the media, the values of the rigidity p and density P being taken as follows y = 60.30 /~' -- 33.57 ~ = 2.85 6 ' = 2.59.
122
K. De
(Pageoph,
The velocity of Love-waves in such a system varies between 3.60 km/sec to 4.56 km/sec (page 214 of [5]). Calculations have been carried for c=4.56 and c=3.80 as those represent conditions near two extreme limits of long periods and short period Love waves respectively, y=0
.2761
iv 2
y=-
H
2
1.030
y=-H 1.112
c = 4.56 km/sec .3781
.2675
.8028
7390 c = 3.80 km/sec 4567
V2
P;
Q;
P3
or3
V3
V3
V3
V3
c = 4.56 km/sec
.0314
.0478
.0405
.0822
c = 3.80 km/sec
2305
3720
3698
37200
F r o m the above table it is clear that for long period waves, the amplitudes of the second and third order motions are of the order of V2(y) and V3(y) where V(y) is the amplitude of the primary Love-wave. Since V(y) is small, it seems reasonable to suppose that our expansion procedure in terms of a parameter ~ gives the solution approximately with the primary Love-wave predominating. In the case of short period waves, the amplitude of the second and third order motions are of the order of 103 V2(y) and 10 4 V3(y). We conclude from this result that the usual theory of Love waves based on infinitesimal strain is not applicable to waves of short period. Some support for our conclusion is furnished by a comparison of observed and theoretical dispersion curves for Love waves given by ([3], p. 212). The comparison shows awide discrepancy between the two for short period Love waves but fairly good argument for long period Love waves. For a complete theoretical treatment of Love waves it seems therefore necessary to apply the exact non-linear theory of elasticity.
Acknowledgement I take this opportunity of thanking Dr. M. Mitra for his kind help throughout the work.
Vol. 80, 1970/III)
Finite Strain Theory of Love Waves
123
REFERENCES [1] A. E. GREEN and E. B. SPRATT, Proc. Roy. Soc. [A] 224 (1954), 347-361. [2] A. E. GREEN and W. ZERNA, Theoretical Elasticity (Oxford Press, 1954). [3] C. TRUESDELL,Arch. Rat. Mech. and Anal. 8 (1961), 263-296. [4] M. HAYESand R. S. RIVLIN, Arch. Rat. Mech. and Anal. 8 (1961), 358-380. [5] W. M. EWING,W. S. JARDETZKYand F. PRESS, Elastic Waves in Layered Media (McGraw Hill Book Company, 1957). (Received 14th July 1969)
