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Surface waves in a nonlinear magnetothermoelastic perfect conductor
Pergamon
Int. J. Engng Sci. Vol. 33, No. 10, pp. 1435-1448, 1995
0020--7225(95)00009-7
SURFACE WAVES MAGNETOTHERMOELASTIC
Copyright t~) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00
IN A NONLINEAR PERFECT CONDUCTOR
I. A. Z. HEFNI Department of Mathematics, Faculty of Education, Cairo University Branch at Fayum, Egypt
A. F. G H A L E B l)epartment of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
G. A. M A U G I N t Laboratoi:re de Mod61isation en M6canique, associ6 au CNRS, Universit6 Pierre-et-Marie Curie, Tour 66, 4 Place Jussieu, 75252 Paris Cedex 05, France Abstract--We use the basic equations given in a previous paper by Hefni et al. [1] to investigate the nonlinear surface wave propagation (SWP) in an isotropic magnetothermoelastic (MTE) half-space of perfect electric conductivity subjected to an initial constant magnetic field normal to the sagittal plane. The first and second order approximations are derived respectively in terms of one and two unknowns. 1]ae governing equations for the first-order approximation reduce to those of classical MTE but include perturbing parameters. Neglecting these parameters as compared to unity reduces the problem to the corresponding one in thermoelasticity for which the phase velocity and attenuation constant are presented both numerically and graphically for aluminium, copper and lead. The results show that the variation of the phase velocity and the attenuation constant in the thermoelastic case have similar behaviour but higher values as compared to those of magnetoelasticity.
1. I N T R O D U C T I O N
The object of the present paper is to obtain a solution for the nonlinear problem of surface wave propagation (SWP) in a magnetothermoelastic (MTE) half-space of perfect electric conductivity subjected to an initial magnetic field normal to the sagittal plane, within the frame of a model including the first level of nonlinearity. The first and second asymptotic solutions are derived respectively in terms of one and two arbitrary constants. The phase velocity and attenuation of the waves are obtained and discussed both numerically and graphically. The governing equations for the first order approximation reduce to those of a classical MTE medium but they include perturbing parameters. In addition, the case eo = 0 reduces the equation of dispersion to the well-known Rayleigh equation for surface waves in a purely elastic half-space. On the other hand, there are several studies dealing with the linear SWP in MTE half-space with infinite or finite electric conductivity and subjected to an initial magnetic field. For example, Kaliski and Nowacki [2, 3] studied SWP in a MTE half-space having infinite or finite electric .conductivity, and subjected to thermal shock on its boundary, while the coupling between deformation and temperature was neglected. The first of these problems was extended by Massalas and Dalamangas [4] by taking into account the thermoelastic coupling, while Sharma anti Chand [5] extended the second one by considering the thermoelastic coupling and a finite thermal relaxation time. For more details, the reader may refer to the references cited in the above mentioned articles. To the authors knowledge, there has been no attempt yet to examine the nonlinear SWP in a MTE half-space of a perfect electric conductor subjected to an initial magnetic field normal to the sagittal plane. This problem will be the main purpose of investigation of the present paper. tTo whom all correspondence should be addressed. 1435
I. A. Z. HEFNI et al.
1436
2. THE BASIC EQUATIONS We consider an anisotropic, homogeneous and centrosymmetric MTE half-space with infinite electric conductivity and placed in contact with a vacuum. The boundary of the half-space is assumed to be free of tractions and thermally isolated. According to Maugin [6, p. 274] and Hefni et al. [1]. Equations (2.1)-(2.15), in the reference configuration, have the following form: 2.1 Bulk equations
O2UL Po ~ = T~L,r + FL
(2.1)
OffJ Ot
= O,
•
~r,r
0o
= 0,
~
~ = 0
(2.2)
0o
"
poCO - OO'CKLUK, L -- QK, K + " ~ LKL(~)K~L) -- "-~ "CKLM,v(UK, L UM, N) -- p~.
(2.3)
2.2 Boundary conditions
NK[T~L] = 0 N ^ I~1 = K ,
N . [~1 = 0,
(2.4)
N ^ [~1 = 0
N[QK]=O. kL 0oJ
(2.5) (2.6)
In the above equations T~L is the total first Piola-Kirchhoff stress, ~ is the magnetic induction and Q is the heat flux. These functions obey the following constitutive equations It U TKL = CKLMNUM,N + "I~KLO + "BKLMN~M~N+ TKLMNPQ M,NU&o
(2.7)
+ ~:kLI~NOUM,N+ B'KLMNeeUM,N~e~e 0
~K = IZKL~L -- BKLMN~LUM,N+ LKL~LO
(2.8)
0
QK = (KKL+ KrLMNUM,N E + K'xLO)O, L.
(2.9)
The material tensors used in these equations are constant and defined by Abd-Alla and Maugin [71. We shall study the bulk equations (2.1)-(2.3) with the boundary conditions (2.4)-(2.6) under the following assumptions: (i) All field functions U=(U1, U2,0), 0 and Ij depend on space-time variables X = (X1, X2, 0) and t, and have power series expansions in terms of a small parameter ~, such that (U, 0) = ~ ¢"(U ("), 0(m), n=l
I~= [I(°) + ~ e"ll("),
(2.10)
n=l
where II(°) is a constant initial magnetic field. Following Tiersten and Baumhauer [8] and Kalayanasundaram [91, we truncate the infinite series solution (2.10) retaining only terms corresponding to n = 1, 2. (ii) The material is assumed to be isotropic. Isotropic representations for the material tensors may be found in Abd-Alla and Maugin [71. (iii) The MTE medium is free of body forces and heat sources, and occupies the half space {X2 >0}; on the other hand the half space {X2 <0} is a vacuum of magnetic
1437
Surface waves in a nonlinear magnetothermoelastic perfect conductor
×2
Fig. 1. Setting of the problem.
permeability/z~ and electric permittivity c~'. Hence, in vacuum, Maxwell's equations reduce to
0E* 17^11"=~* 0t ' 17^E* = - g o
17.E*=0
(2.11)
17"1]* =0.
--, Ot
(2.12)
In vacuum all field functions and constants will be noted by an asterisk, Let N = (0, - 1 , 0) be the unit outward normal to the limiting plane X2 = 0 (see Fig. 1). Taking these considerations into account, equations (2.1)-(2.6), (2.11) and (2.12) reduce to
(A +/z)17(~ r" U in)) +/.*V2U (") + 2"B3V(II (°)" II(n)) + "B2[(II(°)" 17)[j(n) + 1~(0)(17. [j(n))] + B_2 2 (l](°) . O(o))v2U<.) + a2([~(°) "17)U(") - ~(0)17z(~(0). U ( . ) ) _ z0~.) _ ao
0 2 U (n)
ot 2
_ V (n)
= (2.13)
i~(n) =__{[B311 ( 1 o o)(v. U(n)) +/~2((It(°)V)U(~) + V(I~(°) • U ( m ) ) - Lll(°)0(n)] + B(m},, /Zo
(2.14)
K°V20 (") - p o C O (m - Ooz(V- 0 (~)) -OoLIt (°)" ft (n) = V(3n)
(2.15)
~(") = 0
or
E ~") =/~oerL~tU(L")l~) + l:(~),
(2.16)
where E ~ ) is the n t h order electric field evaluated in the fixed Galilean frame R c at time t. The quantities V ("), B ('') and F(m vanish for n = 1, while for n = 2 their expressions are given by V(~) = -r(2) I K L , ~~, T~)L -- - "B K L M N 4 q,,(:t)~(1) At ~IN
+ .o,t
YKLMNPQt-/
irl(1)
i?(1) _t_ , t a t
M,N~P,Q
~
gKLMN
L = 1, 2
(2.17)
r~(l)IT(l)
~
~-J M , N
+ ]t~,t
IT(l) [~(0)~(1) ..[. (1) (0)
°KLMNPQt'JM,
V(2)
[/gE
I/(1)
..l_
K7 t
lO(1)~ ,a(1)
00
= ~"~rLMU~M,U -- "~KL" ,,r",t- - -~" L K L ( ~ )
N~,JqP ~ l O
..~ 00 a-'
~P
t i t ( l ) II(1)
~e )
(2.18)
~
(2.19)
2 "KLMU~'~K,L'-'M,N,
B ~ ) _- - Lo' a K L M N ~ 'a(1)17(1) - LKa[}(~)O °) IL 'J M,N
(2.20)
~:~) = ~XLM0 O)sL~(°)00)L t ,JM -- [Ba[)~)(V • U 0)) +/~2(~ (°)" 17)U O) + (IJ(°)" U °)) - ,--N~'(l)''U.M. (')'-' (2.21) 2.3 In the half-space {9(2 < 0 }
v ^ [,., =
° bOt° ) '
17. ~(m = 0
(2.22)
O~(n) 17^E(m=-g°
Ot '
17"~(m=0"
(2.23)
1438
I.A.Z.
et al.
HEFNI
2.4 O n the limiting p l a n e {X2 = O}
+ 1 {9~){9~))U~,~ + (tz + B3 {9~){9(M0))U(~)K + 2B~SzL{9~){9~) + B~({9~){9(c~) + {9~){9(L °)) + B2{9~)(11(°). V)Ok'0 + {9~){9~)v • tJ ~) - {9~)(~(o). v ) u ~ > + 8,,L(~ ~°). v)(tj ~°~. tJ ~))
- t~'(°)~(°)U(O)~.L ,U M,r -- {9~){9~)U(~?L) -- *SZL0(") + T~L)] = 0 (2.24)
{9(3~)- ~(r> = ~r>,
~ > - ~ > = o,
{gr ) - ~r)= K(3o>
(2.25)
~(r ) - ~(~)= o
(2.26)
0
[/Zo{9(#) - B,(V" U(")){9(2°)- B2(({9(°)" V)U (") + ({9(o). U(~)),2) + L~(O)o(.) _ B(#)] = 0 (2.27) [0!~') + Q(2~)] = 0,
(2.28)
where Q ( K ) = O,
/'1(2) - - 1"~-- / I?'E r/(1) ~K -- K o \ ' % K L M N ~ M , N
(2.29)
-I- /'¢Tt LI(1)~/~(1) ) "~'KL v }V,L"
In what follows, we shall study the solution of the first-order and second-order approximations for the bulk equations (2.13)-(2.15) and the boundary conditions (2.25)-(2.29) for an initial magnetic field parallel to the Xa-axis such that 11(°) = (0, O, {9(3o) = H).
(2.30)
3. O R D E R ONE IN For the first-order approximation, we have the following system of equations. 3.1 I n the h a l f space {X2 > O } ~2.(I) +e2TU(II~2_[_(~'2__7.2~IT(I ) .4- 2"B3H r,(1) ~ ~(1) ~'LL" 1,11 , t'Tl~J2,12 -*'/3,1 - - - - v , 1 - - I J l , t t -
Po ~.2 I R 1 )
..k 7,2 It(1)
T*-"2,11 - - t" L t d 2 , 2 2 + .-2(CL.
2"B3H
__ ,~2 "~I r(1) v T ] v 1,12 +
I7(1)--0
(3.1)
g l ( l ) - - td2,tt l l ( 1 ) --- 0 t",2
(3.2)
L H 0(1)
(3.3)
Po
Po
'~(1) - - - *'13,2
Po
0 {91 = 0 ,
{9(1) = B 3 H ( V *
{92 = 0 ,
U (1)) -
/Zo
/-*o
K°V20 (1) - poCO} 1) - 0o'r(V • V°)),, - OoLH{9~I) = 0 E~,) -- - F~0" .. u l, rLd( ,2.t) )
E(2t) = la°O • H U O) l,t)
~2c = c~(1 + 81),
g~ = c2r(1 + 82)
BaH 2 81
20oC 2 ,
E(3t) = 0
B3H 2
82
2poCk-
(3.4) (3.5) (3.5')
(3.5")
Equations (3.1)-(3.5) are none other than equations (2.1)-(2.5) obtained earlier by Nowacki
Surface waves in a nonlinear magnetothermoelastic perfect conductor
1439
[10] except for additional terms which we have underlined. Carrying equation (3.3) into (3.1), (3.2) and (3.4) yields 7/(1)2 + • L'-'1,11
,~2 lr?'(1) .a_ ( ~ 2 _ 7.2'~1r(1) t" T v 1,22 ~ t" TJL" 2,12 -- A a(1) v,1 -- It(l) 'Jl,tt --- 0
(3.6)
e2~,(1) T~2,11 "F e2LU~I~2 , -{- ( e 2 -- 7,2Xrr(1) t.T]tJl,12 _ A nv,( l2) _ U~l)t = 0 (V2-- 1£~0
(1)- n/it'(l)
K Ot/
(3.7)
--F Uil~t) = 0
~,,a. it
(3.8)
where we have set 0
e 2 "~"C2(1 -Jr"EH) ,
EH =
2"B3B3 H2 "F ~1 potxoC 2
(3.9) g o
OoZ tl B =-U-,
A = __r(1 + ~ o ) , Po ~°1
2"BaLH2
R =
~~12 B s L H 2
lZo'C
poC (1 - e ~ 3 )
(3.10)
~ 3 = O°LEH2 poCtxo
]Zo,:
(3.11)
Equations (3.6)-(3.8) indicate that the MTE problem for a perfect conductor is reduced to one of pure thermoelasticity in which the usual quantities gz., gr, A, B a n d / ( now depend on the magnetic characteristics of the problem through the parameters ~n, E~I, ~ 2 and E°3. The above equations are satisfied in the region {X2 > 0}. At the limiting plane {X2 = 0} we have the following boundary conditions (1 - 2.'~)U(11 + /'/(1) __ 0 ,
-/~ -- "B3H2
"J 1,2 --
2poC~
(3.12)
[(~_2,~2(1_/,5.~ur(1) +e2U~1~_]]tJX,l , _v O(1)+(2"B3o(31)2'B3~(¢)) H = 0 Po \ Po Po / 8 ( 1 ) - ~ t l ) = ~..~1) (1) = 0, ,2
~,o) = ].1.,0~(1) _ */.~0~12
~ 1 ) -- ~(1) = ~..~(1),
E~I) _ ]~1) = ~1)=
H(~o- '
~11) =. j~(1) = 0
hIl(1) H'0./tJ2,t for
(3.13)
0
(3.14)
E(31) _ ~(1) = 0
(3.15)
X 2 <0.
(3.16)
Now, let us consider the bulk equations (3.6)-(3.9). For plane wave harmonic solutions propagating along the Xa-direction (Fig. 1), and decaying in time and depth (from the plane X 2 = 0), we assume that ( U (1), 0 (1)) _- (17j(1),
O(1))eI(~t-K,X,-K2X2).
(3.17)
As a result of a previous paper by Hefni et al. [11], the global solution is 3 ( U(K1)' 0(1)) = E I,'K(j),/f'r(i)Oll~)ei(t~t-KiX,-Kzo,X2) j=l
(3.18)
where ~](1) K2(1) ^ (1) 1(1) = -- K1 U2(1)' ~T(1)
=
1(2) ~/0) =
g l /.T/O) K2(2 ) 2(2), K1
frO)
1(3) K2(3), ~ 2 ( 3 ) , ES$3-10-E
(1) = 0 (1)
(3.19)
0(1) =
in2 ^(~) A 'K2(2) U2(2)
(3.20)
0(1) :
in3 fro) A'K2(3)'~2(3)
(3.21)
I . A . Z . HEFNI et al.
1440 with 2
K20)
2 =KI_K K2iD
nj-
0)2
_ K 2 _ K 2,
K2= --
A' =--A
Cr-2, 0) 2
2 - n j,
(3.22)
e2
K2 = e-~L'
j = 2, 3
(3.23)
[K 2 + q(1 + co)] • X/[K 2 - q(1 + ea)] 2 + 4qK2L 2
(3.24)
(tlpo)(Oov/Ko)(K°/poC)(1 + col)(1 + e~2) ~o =
C'~.(1 - ¢~3)
'
i0)(1 - ~ 3 )
q = (KO/poC) .
(3.25)
Substituting from equations (3.19)-(3.21) into (3.3)3 and using (3.23) one obtains 3 ~i 1) = ~ I~(3}})ei(~'-K'x'-K2°~v:), 1~1
(3.26)
where
iH
~,1) •13(1)--Ft -U,
ff[o
L
)
0 l~2]i,T(1)
~(1) [ k B 3 - ~ T n j - - U 3 - - L j V 2 ( i ) for ] = 2,3. "13(j)_ /Zo~22(1)
(3.27)
It remains to determine expressions for the constant amplitudes in terms of one of them and the dispersion relation through the boundary conditions (3.12)-(3.15). Hence, let us start by the field equations in vacuum (2.23) and (2.24) which can be written as V2
1 0 2 \ * ..
----|[~(1), C2 Ot2] ~
~(1)) _~ 0,
c2 =
1
(3.28)
/Zo~o
where Co is the velocity of light in vacuum. For a solution of the type (3.17), we have
(~(o,~(1)=(~(1),~(l))ei(,ot-r,x,-r~x2)
(3.29)
with *2
K 2 = K 2 - K 2,
K2
_~0)2
c2.
(3.30)
Carrying equation (3.29) into (2.23) and (2.24) and taking into account (3.16) and (3.14)2 implies that •
•
(El 1), E(2')) = ~ ( - K * , K 0, ~'0)
~1) _.~ ~(21) = 2(31) ~___~(31) __ 0.
(3.31)
Equations (3.5)1 and (3.31)1 give the following expression for ~(3~):
~¢31)= iHK2 (-1(2'). K~
(3.32)
Surface waves in a nonlinear magnetothermoelasticperfect conductor
1441
Now, let us consider the boundary conditions (3.12)-(3.15). These equations, by substituting from equation (3.7), and using (3.14)1, (3.19)2 and (3.3)3 reduce to /d" /'~/(1) iV" 7"/(1) (1) I/" [ f r O ) frO) ~ (1) 1 *~'2(1) ~'- 1(11 q" ,v~2(2)~J 1(2) + K2(3)|~'T1(3)) + ~tXlKt-t 2(1) + tJ 2(2) -1- U 2 ( 3 ) ) ( --
2A) = 0
(3.33)
Kt[(E2L -- 2g'~(1 - '~)](0t~]) + f/(1) 11 + K~2)6"~2) 1) + K2(3)G(3)) ,~ 1~) + 0~p~)) + c=2L(K~.)6"~(1) ,'~(0/~ll + 01~l) - D(0(3~])
-
+ t" 2(2)fz(1) .-- L" f / ( 2(3)J 1 ) % ---- 0
K2(2)011/+ K2(3)0/~ / = 0
(3.34)
(3.35) 0
fl = Z
1 + 2"B3HZL.
po
~
~zL = ~
1+
po#oe~
/"
*
)"
2iH2K~"B3
B 3 = "B3 - "B3,
D -
(3.36)
,
poKE
Carrying equations (3.20)2 and (3.21)2 into (3.35) yields ~](1) n2/'1(1) 2(3) ~-- - - - t.J 2(2) •
(3.37)
n3
Hence equation (3.21) reduces to 7(1 _ 1(3)
Kin2 fro) K 2 ( 3 ) n 3 v 2(2)~
in2
0(31 ) - -
A'K2(3)
fro) tJ 2(2).
(3.38)
Substituting from (3.19)1, (3.20)1, (3.37) and (3.38)1 into (3.33), gives a relation between the constant amplitudes t-~ t'm) 2(1) and vfr(1) 2(2) [2(1 -- fi)K~ -- K~-]U2(1) 2 ^ ( 1 1 + 2(1 -/~)K2(1 - nn3],.., 2 ~ f m2(2) ) =0.
(3.39)
Similarly, equation (3.34) reduces to 2~1-/~)K2(2)K2(1)0(}{)
+
{~2A2(1
K2(2)n2A2'~ K2(3)n3A2/ K2(3)na/ +D(l_n2/[
K2(3)]
A' 1-
2 ( 1 - / ~ ) K2
- - - "w-~--v'~2% - 2 / / U2(~) = 0, 2(1 -- A)/~. 1 -- .r~TJ)
n3/L
(3.40)
where A~ = K~ - n2,
(3.41)
A~ = K 2 - n3.
On making zero the determinant of the system (3.39) and (3.40) and setting K2(1) = i(1 - ¢)1r2, Ki
¢~,,2
K2(s) = iss( 1 K] \ -~]
,
j=2,3
(3.42)
where ~02 ¢ = - cgKl 2 2,
e2
~ = ~e~y '
N~ ~1,2 Sj = 1 + (1 - ~/~)) '
~
(3.43)
~ = - -~
~ Nj = K?'
]=2,3
(3.44)
1442
I.A.Z.
HEFNI et al.
a~er some calculations, we arrive at the dispersion relation Eo)=O
D(~, ~ o ) = F ( ~ ) - G ( ~ ,
(3.45)
with
(
(2(1 - ,~)~¢)\2
-
F(s¢) = 4(1 -/~)2(2 - so)m(1 - ~1~)~,2 szs3( N3 - Nz) (N3s3 - NzsE) + w
G(~, ~o) =
(3.46) (3.47)
K2(3) ,/
(2i,,~3H2~ (
~2
In studying these equations, we assume for simplicity that the parameter en is very small as compared to unity and related terms may be neglected [12, 13]. Similarly, the remaining parameters E°;, j = 1, 2, 3, may be neglected as compared to unity. Hence, we have "r
__. C2L
A=.a,=--, Po
or eL=e,.=cL,
/
(3.49)
Eo = \po] \ KO ] ~oC '
where eo is none other than the usual parameter in pure thermoelastic media which is small for most materials at normal temperatures [14, equation (3.8)]. For eo = 0 one can show that equation (3.24) and (3.44)2 imply that N2 I
= 0,
s2 = 1.
(3.50)
I EO~0 Consequently, the dispersion relation (3.45) reduces to 4(1 - ~)1/2(1 - ~/~.)1/2 = (2 - ~)2,
(3.51)
which is the Rayleigh equation for surface waves in a purely elastic half-space. This equation is known to reduce to a cubic one in ~ [zero is an obvious root of equation (3.51)], the nonzero positive root which is less than unity represents the only admissible solution [15]. For later use we shall denote this solution by ~ such that F(~) = 1 and ~ < 1. To deduce the change of the phase velocity c and attenuation constant S corresponding to the case Eo # 0 but small, we assume that 3(3 = 6r + i6i) is the increment of ~ due to the coupling between the strain and temperature fields. Then following Massalas [16] and assuming that W is small and may be discarded (since W is proportional to co), we obtain c = Re g l S = - I r a K1 --- S*
6---
c (1+ fL
- G(~:, ~-o)1 OG OF l j•
[1
(3.52)
8i > 0
(3.53) (3.54)
1443
Surface w a v e s in a n o n l i n e a r m a g n e t o t h e r m o e l a s t i c p e r f e c t c o n d u c t o r Table 1
cL ( k m . s - l ) ( k m • s -1) cR ( k m " s -1) to* (s -1) S* ( m - ' )
cr
Aluminium
Copper
Lead
6.42 3.04 2.85 4.76 x 1011 7.28 x 107
5.01 2.27 2.13 2.25 × 1011 4.49 x 107
1.96 0.69 0.65 1.35 × 101° 6.89 x 106
and OF(f) = -1 1 1 Os~ e=~ (1 - 0.5~) -~ 2st(1 - ~/sr) ~-2(1 - ~--'-~
(3.55)
ac
(3.56)
- ~) (1 - ~/~.)2 where
(3.57)
S* = p°CcL
CR : C T ~ 1/2,
Ko
to* = p°CcZ Ko ,
f~
= to to,
v2,3 : ~ { [ 1 + i(1 -~e°)] q: ~ f [ 1
(3.58)
i ( 1 ~ %)]2 +-~}.
(3.59)
Values of the fundamental constants for aluminium, copper and lead at room temperature are given in Table 1 [17, 18]. Following Chadwick and Sneddon [14], the calculation will be obtained for f~ lying in the range 10 -5 <- f~ -< 1. The variation of Ac and S/S* for aluminium, copper and lead with f~ are cited in Tables 2 and 3 [(a) and (b) denote the cases for Eh = 0.01 and eh = 0.002 respectively]. The graphs of Ac and S in the neighbourhood of to* for aluminium are shown in Figs 2 and 3. Tables and Figs 2, 3 obtained here show that the variations of the phase velocity and attenuation constant with f~ and e0 have a similar behaviour but weaker values in the magnetoelastic problem as compared to those of the thermoelastic one.
Table 2
Ac=(C-C~n~ \ Aluminium
1 10 - I 10 -2 10 -3 10 -4 10 - s
CR
/
Copper
(a)
(b)
(a)
7.27 X 10 -6 1 . ! ) 3 × 1 0 -4 2.86 × 10 -4 2.89 × 10 -4 2.95 x 10 -4 7.51 x 10 -4
1.47 x 10 -5 3 . 8 4 × 1 0 -4 5.66 x 10 -4 5.73 x 10 -4 5.78 × 10 -4 7.51 X 10 -4
5.27 × 10 -6 1.56×10 4 2.44 x 10 -4 2.48 × 10 -4 2.53 × 10 -4 6.42 X 10 -4
Lead
(b) 1.66 3.11 4.83 4.90 4.94 6.42
× x × × x X
10 -5 11-4 10 -4 10 -4 10 -4 10 -4
9.09 4.71 1.09 1.14 1.16 2.95
(a)
(b)
x × x x X ×
1.84 X 10 -6 9.42×10-5 2.18 X 10 -4 2.25 X 10 -4 2.27 X 10 -4 2.95 X 10 4
10 -7 10-s 10 -4 10 -4 10 -4 10 -4
I. A . Z. H E F N I
1444
et al.
Table 3
s s* t2
Aluminium
(a) 1 10 t 10 -2 10 3 10 -4 10 -5
9.78 2.82 5.12 6.28 1.74 1.35
x x X X X X
Copper
(b) 10 -5 10 -5 10 -7 10 -9 10-to 10 n
X x x X X X
1.95 5.58 1.01 9.96 2.25 1.35
Lead
(a) 10 -4 10 -5 10 6 10 9 10 -1° 10-1t
x x X X X X
8.03 2.58 4.98 6.19 1.72 1.33
(b) 10 -5 10 -5 10 -7 10 -9 10 -1° 10-n
1.61 5.10 9.79 1.10 2.22 1.33
X x X X X X
(a) 10 -4 10 -5 10 -7 10 -8 10 -1° 10-11
2.89 1.57 4.58 5.08 1.78 1.39
x x X X X X
(b) 10 -5 10 -5 10 -7 10 -9 10 - t ° 10 -12
x x X X X X
5.79 3.11 9.04 1.12 2.29 1.39
10 - s 10 -5 10 -7 10 - s 10 - I ° 10 -12
4. O R D E R TWO IN In the preceding section, we obtained the solution of the first order approximation, in the ^ ^ 1) general form (3.18), (3.26) for X2 > 0. The amplitudes U#/j ), 0/} / and t~u) are related by equations (3.19)-(3.21), (3.27), (3.37), (3.38) and (3.39), and can be written as functions of one of them only, e.g. fr(1) v 2 ( 2 ) = Uo. The general first-order approximation thus reads 3
(U(K1), 0(1), ~(1))
= Uo E (UK(j), O(j), ~3(j))e i(°Jt-K'x'-r2q)X2),
K = 2, 3
(4.1)
]=1
where
2K1K2(1)(1--~-~) UI(,)-
(K 2 - K20)) -
2K~(1-n~) 02(1) = ( K 2 - K2(1))
'
gl
-
U1(2) = K2(2) '
U2(2) = 1
-Kin2
-
/-]2(3) =
--
U](3) = g 2 ( 3 ) n 3 '
0.)
= O,
0(2) -
in2
,
A K2(2)
~3(1) =0 ,
~3(j)
iH /z0K2(j)
(4.2) (4.3)
n2
(4.4)
n3
.
03 -
in3
(4.5)
A'K2(3)
[(/~3 --~)nj-B3KL]U2(j) 02-
and
j=2,3.
(4.6)
For simplicity we shall neglect the dependence of the longitudinal and transverse wave
S,,1133
----C 8
--~:8
20
= 0.01 =0.02
40111"I05
/-
lO
/
/
r~
4
Fig. 2. A t t e n u a t i o n versus frequency.
0.1
N.
1
Fig. 3. Change in velocity versus frequency.
Surface wavesin a nonlinearmagnetothermoelasticperfect conductor
1445
velocities on the magnetic field. For the second-order approximation, the system of equations (2.13)-(2.22) gives ,- U L L'-' ,t(2) "2
+ c2U~2~2 +,
(c~-cZr)U~2.~2, - AO!~ ) - '-'L,'r(Z)-_ U2o{j~. Plje 2i°, + ~,j,~p,j~e2'°J.}
(4.7)
}
(4.8)
p 2j,e2~°J+ E. P2Pe2'°"
c2U~2.I, + c2U~2)22 + (c 2 - vT,~2'U(2)a,*2- ,t,,~aa(2)_u~2~,= U2o .
,
(V 2 - -='--~0 ` ) -
K Ot/
I, s
(4.9)
B ( U ~ 2 } t q- U(22~t)~-
'
]~(2) ~___].~oHU~2~+
j
- J
),"
(V" U(1))U~I,? -
LH
--~
O(1)u~l)t,
BcH(V.U(1)~U(1)+L--~-~D(1)II(1) 0
E(z2)
. ]~rTT(2) ~-~ - - / ~ 0 1 1 t . J l , t - -
]
1,t
tl
t.Jl, t
~)(2)= ~)(22)=E(32)= 0
~,:i~,= htb~v, u,~)- Lo(~'l + [ ~ n v hi,0
L #0
"
(4.11) (4.12)
u `'-L-
~0
o"] ~
=-H (B3v. v(~'- Lo `2,) + v~[E e,,e 2°' + X W,,,e~°~] /.%
(4.10)
(4.13)
~•
with Oj = 19 -- K 2 u ) X 2 ,
0 = tot - K I X ~ ,
3
E--E, j ]=1
O~s = 0 - K2u,)X2
(4.14)
K2(js) = K2(j) + K2(O
(4.15)
2
3
3
E--EE, j,s 1=1 s=l
(4.16)
j#s
where Pq and Wqs are, respectively, the algebraic sums of the quantities which propagate with the phases 2Oj and 2Ojs of the inhomogeneous terms of the equation which is denoted by subscript i. According to Hefni et al. [11], the system of equations (4.7)-(4.9) may have a solution
U~2)=U~{~Alje2iOj+~,Qlpe2iO,,
- j
(4.17)
1,s
(419,
I.A.Z. HEFNI et al.
1446 where
[w:+ ,,,, "
A3i =
Dj
ni J
A~-
_K1 _ Aq + aEj,
O~-
K1 K2js QIjs + qzys,
,
a3j = 1
a2j -
K2U )
( Vj
+
i A ' ( K 2 - n i ) A) 3 j
njKE(j)
(4.21)
2
1 ( iA'(K~-n/) Q3j,) q2ys= njK2# Rj~ + 2
Vj = (KaP~j+ KE(~)P2i) 4c2 , Dj=(K2L-nj)+ q[1
(4.20)
Dj
R# =
(Kx W~j~+ K2#W2ix) 4c2
(K~-nJ),o] , nj
(4.22)
(4.23)
n l = K 2 - K z.
(4.24)
Equations (4.20) and (4.19) indicate that the second-order solution of approximation for 0 `2) can be obtained directly from the first-order solution. On the other hand, equations (4.21) and (4.22) represent six equations for 12 unknowns (Ai#, Aq for i2, 3 and j, s = 1, 2, 3). In order to determine the relations between these unknowns, we consider the boundary conditions (2.25)-(2.28) taking into account the equations in vacuum (2.23) and (2.24). Hence, we have the following equations:
4.1 On the limiting plane {Xe =0} U(2) 2,1 + it(2)+ t-~1,2 e2iO[E G1, + ~ Glj,] = 0 j
(c 2-'~'2wr'2) z't'T]tFl'l
(4.25)
j,s
~2 tr(z) ----T *B(32)~(32))+e2i°[~, PO 0 `2) + 2"("B3[?(32)PO ~ j Gz, + ~" G2,,] = 0
-['- L'Lt"J2'2
O'2)+e2i°[2G3,+~)'~G3#]=O ,2 " 1
~(2)_ 6(2)= ~ 2 ) ,
(4.27)
j,s
~ 2 ) _ ~ 2 ) = ~(2),
~ ( 2 ) - ~ 2 ) = O,
(4.26)
]&O~2) --/XS~(22)=
0
~ 2 ) - ~(32)= O.
(4.28) (4.28')
Equations (4.12)3 and (4.28')imply that ~2), ~ 2 ) a n d ~2)vanish identically.
4.2 In vacuum {X2 <0} (~2), ~(z2),~(32))= E(12),E(22), ~(3=))e2i(,,,-r~x,-x'~x2)
(4.29)
where (~2), j~(22))=
(-K~', K,). fi0(.O
(4.30)
Surface waves in a nonlinear magnetothermoelastic perfect conductor
1447
The constant amplitude ~(32)may be obtained directly from equations (4.28)2 and (4.30). Now, let us return to the boundary conditions (4.25)-(4.28)1, which may be reduced to
~ (K, A2j-F K2(j)A1y+~-J)-I- ~ (K,A21s + K2(DA1js-t-~) =0
(4.31)
~', (K,Alj + g/K2(j)A2j+ 02j) + '~ (KIAljs + ~K2jsA2j~+ j j,s
(4.32)
02j'~) = 0
~J (K2(j)A3j +i-~) + ~ (KzjsA3js +---~'-) " iG3j~ = 0
(4.33)
[ 2HB___~a U2 p 02j = ict L po 4j - ~ A3j]
(4.34)
"a[2Hff-3U2p4j,-~A3js]
(4.35)
where
(~2j, =
PO
t L
~/=~-2'
a
c2_c2.
(4.36)
Equation (4.33) gives a relation between the variables ~, K1 and ~o, similar to the dispersion relation (3.45). While equations (4.21), (4.22)(, (4.31) and (4.32) represent a system of eight linear inhomogeneous algebraic equations in 12 unknowns, one can obtain a particular solution for this system by substituting from equations (4.21) and (4.22) into (4.31) and (4.32) and assuming that Alj and A2j for j = 1, 2, 3 are independent of Q~js and Q2j, for js = 12, 13, 23. In this way, we obtain two similar independent groups of inhomogeneous linear algebraic equations Alibi + A1 = 0, J
j,s
~ A u + A2 = 0
(4.37)
J
Qljsbjs + Q~ = 0,
~ Q~j~ + Q2 = 0,
j,s
(4.38)
where
bj =
(K~(i) - K~) K2U)
'
al = ~. Kla2j + T } '
b# =
(K~/~ - K~)
Kzjs
Q1 = X. Klq2js+
l
(4.39)
(4.40)
I,s
1
A2
Q2
(1 - ~/)K~ ~ (3'K2(j)a2j + G2j), J
1
(1
i
~'~ (3'K2j~+ 02j~).
"y)K 1 j,s
(4.41)
The system of equations (4.37), (4.21), (4.38) and (4.22) may be solved to obtain the first group (i.e. Alj and A2j) and the second (i.e. Q~js and Qajs) in terms of one of Alj for the first group
1448
I.A.Z.
HEFNI et al.
and one of Ql# for the second. For example, these two groups may be expressed respectively in terms of AI3 and 0123 as follows: Kl All + a21,
A21 -
A22 = -
K2( 1)
gl A23 --- - - - A 1 3 K2(3) Q212 --
K1 --
K~12
K--L1A12 + a22, K2(2)
(4.42)
+ a23
Ql12 + q212,
K1 Ql13 K21~
Q213 = - - -
q'- q213,
gl 0123 ÷ q223 Q223 = -- - K223
(4.43)
where All =(bl -
0112
(b2A2- A,)
A13 + (b, - b2) '
(b13 -- b23) (b13Q2 - Qa) (bl2- b13) Q123 + (b12_ b13) ,
(bl - ~iI
A12 = (b:
(b~A2- A1)
(4.44)
A~3 -~ ( b z - bl)
(b12 --/923) Q123 + (b12Q2 - Q1) 0113 - (b13 b12) (b13- b12)
(4.45)
5. C O N C L U S I O N S
We have studied the nonlinear SWP in a magnetothermoelastic half-space of infinite electric conductivity subjected to an initial constant magnetic field normal to the sagittal plane. As in the magnetoelastic case with finite electric conductivity, the system of equations splits into two surface problems, one concerning a Rayleigh mode of pure thermoelastic form and the other a Bleustein-Gulyaev mode (B-G) with thermoelastic coefficients modified by the magnetic field. As in Hefni et al. [11] the B - G mode cannot exist. A dispersion relation is obtained and discussed for a numerical example. It is shown, in particular, that the variations of the phase velocity and attenuation constant with the angular frequency of the wave and the "modified" thermoelastic coefficient have similar behaviour, but with higher values, as compared to the corresponding magnetoelastic case considered in Hefni et al. [11].
REFERENCES [1] I. A. Z. HEFNI, A. F. GHALEB and G. A. MAUGIN, One-dimensional bulk waves in a nonlinear magnetoelastic conductor of finite electric conductivity. To be published. [2] S. KALISKI and W. NOWACKI, Bull. Acad. Polon. Sci. Ser. Sci. Technol. 10, 1 (1962). [3] S. KALISKI and W. NOWACKI, Bull Acad. Polon. Ser. Sci. 10, 159 (1962). [4] C. MASSALAS and A. DALAMANGAS, Lett. Appl. Engng Sci. 21, 991 (1983). [5] J. N. SHARMA and D. CHAND, Int. J. Engng Sci. 26, 951 (1988). [6] G. A. MAUGIN, Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam (1988). [7] A. N. ABD-ALLA and G. A. MAUGIN, Int. J. Engng Sci. 28, 589 (1990). [8] H. F. TIERSTEN and J. C. B A U M H A U E R , J. Appl. Phys. 45, 4272 (1974). [9] N. KALAYANASUNDARAM, Int. J. Engng Sci. 19, 435 (1981). [10] W. NOWACKI, Bull. Acad. Polon. Sci. Ser. Sci. Technol. 12, 485 (1962). [11] 1. A. Z. HEFNI, A. F. GHALEB and G. A. MAUGIN, Surface waves in a nonlinear magnetoelastic conductor of finite electric conductivity. To be published. [12] J. W. DUNKIN and A. C. ERINGEN, Int. J. Engng Sci. 1, 461 (1963). [13] G. A. MAUGIN and A. HAKMI, J. Acoust. Soc. Am. 77, 1010 (1985). [14] P. CHADWICK and I. N. SNEDDON, J. Mech. Phys. Solids, 6, 223 (1958). [15] K. F. GRAFF, Wave Motion in Elastic Solids. Ohio State U.P., Columbus (1975). [16] C. MASSALAS, Acta Mechanica. 65, 51 (1986). [17] F. C. MOON, Magnetosolid Mechanics. Wiley, New York (1984). [18] B. R. ROBERT, Metallic Materials Specification Handbook, Third Edition. Spon, New York (1980).
(Received 1 November 1994; accepted 7 January 1995)
Int. J. Engng Sci. Vol. 33, No. 10, pp. 1435-1448, 1995
0020--7225(95)00009-7
SURFACE WAVES MAGNETOTHERMOELASTIC
Copyright t~) 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9.50+ 0.00
IN A NONLINEAR PERFECT CONDUCTOR
I. A. Z. HEFNI Department of Mathematics, Faculty of Education, Cairo University Branch at Fayum, Egypt
A. F. G H A L E B l)epartment of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
G. A. M A U G I N t Laboratoi:re de Mod61isation en M6canique, associ6 au CNRS, Universit6 Pierre-et-Marie Curie, Tour 66, 4 Place Jussieu, 75252 Paris Cedex 05, France Abstract--We use the basic equations given in a previous paper by Hefni et al. [1] to investigate the nonlinear surface wave propagation (SWP) in an isotropic magnetothermoelastic (MTE) half-space of perfect electric conductivity subjected to an initial constant magnetic field normal to the sagittal plane. The first and second order approximations are derived respectively in terms of one and two unknowns. 1]ae governing equations for the first-order approximation reduce to those of classical MTE but include perturbing parameters. Neglecting these parameters as compared to unity reduces the problem to the corresponding one in thermoelasticity for which the phase velocity and attenuation constant are presented both numerically and graphically for aluminium, copper and lead. The results show that the variation of the phase velocity and the attenuation constant in the thermoelastic case have similar behaviour but higher values as compared to those of magnetoelasticity.
1. I N T R O D U C T I O N
The object of the present paper is to obtain a solution for the nonlinear problem of surface wave propagation (SWP) in a magnetothermoelastic (MTE) half-space of perfect electric conductivity subjected to an initial magnetic field normal to the sagittal plane, within the frame of a model including the first level of nonlinearity. The first and second asymptotic solutions are derived respectively in terms of one and two arbitrary constants. The phase velocity and attenuation of the waves are obtained and discussed both numerically and graphically. The governing equations for the first order approximation reduce to those of a classical MTE medium but they include perturbing parameters. In addition, the case eo = 0 reduces the equation of dispersion to the well-known Rayleigh equation for surface waves in a purely elastic half-space. On the other hand, there are several studies dealing with the linear SWP in MTE half-space with infinite or finite electric conductivity and subjected to an initial magnetic field. For example, Kaliski and Nowacki [2, 3] studied SWP in a MTE half-space having infinite or finite electric .conductivity, and subjected to thermal shock on its boundary, while the coupling between deformation and temperature was neglected. The first of these problems was extended by Massalas and Dalamangas [4] by taking into account the thermoelastic coupling, while Sharma anti Chand [5] extended the second one by considering the thermoelastic coupling and a finite thermal relaxation time. For more details, the reader may refer to the references cited in the above mentioned articles. To the authors knowledge, there has been no attempt yet to examine the nonlinear SWP in a MTE half-space of a perfect electric conductor subjected to an initial magnetic field normal to the sagittal plane. This problem will be the main purpose of investigation of the present paper. tTo whom all correspondence should be addressed. 1435
I. A. Z. HEFNI et al.
1436
2. THE BASIC EQUATIONS We consider an anisotropic, homogeneous and centrosymmetric MTE half-space with infinite electric conductivity and placed in contact with a vacuum. The boundary of the half-space is assumed to be free of tractions and thermally isolated. According to Maugin [6, p. 274] and Hefni et al. [1]. Equations (2.1)-(2.15), in the reference configuration, have the following form: 2.1 Bulk equations
O2UL Po ~ = T~L,r + FL
(2.1)
OffJ Ot
= O,
•
~r,r
0o
= 0,
~
~ = 0
(2.2)
0o
"
poCO - OO'CKLUK, L -- QK, K + " ~ LKL(~)K~L) -- "-~ "CKLM,v(UK, L UM, N) -- p~.
(2.3)
2.2 Boundary conditions
NK[T~L] = 0 N ^ I~1 = K ,
N . [~1 = 0,
(2.4)
N ^ [~1 = 0
N[QK]=O. kL 0oJ
(2.5) (2.6)
In the above equations T~L is the total first Piola-Kirchhoff stress, ~ is the magnetic induction and Q is the heat flux. These functions obey the following constitutive equations It U TKL = CKLMNUM,N + "I~KLO + "BKLMN~M~N+ TKLMNPQ M,NU&o
(2.7)
+ ~:kLI~NOUM,N+ B'KLMNeeUM,N~e~e 0
~K = IZKL~L -- BKLMN~LUM,N+ LKL~LO
(2.8)
0
QK = (KKL+ KrLMNUM,N E + K'xLO)O, L.
(2.9)
The material tensors used in these equations are constant and defined by Abd-Alla and Maugin [71. We shall study the bulk equations (2.1)-(2.3) with the boundary conditions (2.4)-(2.6) under the following assumptions: (i) All field functions U=(U1, U2,0), 0 and Ij depend on space-time variables X = (X1, X2, 0) and t, and have power series expansions in terms of a small parameter ~, such that (U, 0) = ~ ¢"(U ("), 0(m), n=l
I~= [I(°) + ~ e"ll("),
(2.10)
n=l
where II(°) is a constant initial magnetic field. Following Tiersten and Baumhauer [8] and Kalayanasundaram [91, we truncate the infinite series solution (2.10) retaining only terms corresponding to n = 1, 2. (ii) The material is assumed to be isotropic. Isotropic representations for the material tensors may be found in Abd-Alla and Maugin [71. (iii) The MTE medium is free of body forces and heat sources, and occupies the half space {X2 >0}; on the other hand the half space {X2 <0} is a vacuum of magnetic
1437
Surface waves in a nonlinear magnetothermoelastic perfect conductor
×2
Fig. 1. Setting of the problem.
permeability/z~ and electric permittivity c~'. Hence, in vacuum, Maxwell's equations reduce to
0E* 17^11"=~* 0t ' 17^E* = - g o
17.E*=0
(2.11)
17"1]* =0.
--, Ot
(2.12)
In vacuum all field functions and constants will be noted by an asterisk, Let N = (0, - 1 , 0) be the unit outward normal to the limiting plane X2 = 0 (see Fig. 1). Taking these considerations into account, equations (2.1)-(2.6), (2.11) and (2.12) reduce to
(A +/z)17(~ r" U in)) +/.*V2U (") + 2"B3V(II (°)" II(n)) + "B2[(II(°)" 17)[j(n) + 1~(0)(17. [j(n))] + B_2 2 (l](°) . O(o))v2U<.) + a2([~(°) "17)U(") - ~(0)17z(~(0). U ( . ) ) _ z0~.) _ ao
0 2 U (n)
ot 2
_ V (n)
= (2.13)
i~(n) =__{[B311 ( 1 o o)(v. U(n)) +/~2((It(°)V)U(~) + V(I~(°) • U ( m ) ) - Lll(°)0(n)] + B(m},, /Zo
(2.14)
K°V20 (") - p o C O (m - Ooz(V- 0 (~)) -OoLIt (°)" ft (n) = V(3n)
(2.15)
~(") = 0
or
E ~") =/~oerL~tU(L")l~) + l:(~),
(2.16)
where E ~ ) is the n t h order electric field evaluated in the fixed Galilean frame R c at time t. The quantities V ("), B ('') and F(m vanish for n = 1, while for n = 2 their expressions are given by V(~) = -r(2) I K L , ~~, T~)L -- - "B K L M N 4 q,,(:t)~(1) At ~IN
+ .o,t
YKLMNPQt-/
irl(1)
i?(1) _t_ , t a t
M,N~P,Q
~
gKLMN
L = 1, 2
(2.17)
r~(l)IT(l)
~
~-J M , N
+ ]t~,t
IT(l) [~(0)~(1) ..[. (1) (0)
°KLMNPQt'JM,
V(2)
[/gE
I/(1)
..l_
K7 t
lO(1)~ ,a(1)
00
= ~"~rLMU~M,U -- "~KL" ,,r",t- - -~" L K L ( ~ )
N~,JqP ~ l O
..~ 00 a-'
~P
t i t ( l ) II(1)
~e )
(2.18)
~
(2.19)
2 "KLMU~'~K,L'-'M,N,
B ~ ) _- - Lo' a K L M N ~ 'a(1)17(1) - LKa[}(~)O °) IL 'J M,N
(2.20)
~:~) = ~XLM0 O)sL~(°)00)L t ,JM -- [Ba[)~)(V • U 0)) +/~2(~ (°)" 17)U O) + (IJ(°)" U °)) - ,--N~'(l)''U.M. (')'-' (2.21) 2.3 In the half-space {9(2 < 0 }
v ^ [,., =
° bOt° ) '
17. ~(m = 0
(2.22)
O~(n) 17^E(m=-g°
Ot '
17"~(m=0"
(2.23)
1438
I.A.Z.
et al.
HEFNI
2.4 O n the limiting p l a n e {X2 = O}
+ 1 {9~){9~))U~,~ + (tz + B3 {9~){9(M0))U(~)K + 2B~SzL{9~){9~) + B~({9~){9(c~) + {9~){9(L °)) + B2{9~)(11(°). V)Ok'0 + {9~){9~)v • tJ ~) - {9~)(~(o). v ) u ~ > + 8,,L(~ ~°). v)(tj ~°~. tJ ~))
- t~'(°)~(°)U(O)~.L ,U M,r -- {9~){9~)U(~?L) -- *SZL0(") + T~L)] = 0 (2.24)
{9(3~)- ~(r> = ~r>,
~ > - ~ > = o,
{gr ) - ~r)= K(3o>
(2.25)
~(r ) - ~(~)= o
(2.26)
0
[/Zo{9(#) - B,(V" U(")){9(2°)- B2(({9(°)" V)U (") + ({9(o). U(~)),2) + L~(O)o(.) _ B(#)] = 0 (2.27) [0!~') + Q(2~)] = 0,
(2.28)
where Q ( K ) = O,
/'1(2) - - 1"~-- / I?'E r/(1) ~K -- K o \ ' % K L M N ~ M , N
(2.29)
-I- /'¢Tt LI(1)~/~(1) ) "~'KL v }V,L"
In what follows, we shall study the solution of the first-order and second-order approximations for the bulk equations (2.13)-(2.15) and the boundary conditions (2.25)-(2.29) for an initial magnetic field parallel to the Xa-axis such that 11(°) = (0, O, {9(3o) = H).
(2.30)
3. O R D E R ONE IN For the first-order approximation, we have the following system of equations. 3.1 I n the h a l f space {X2 > O } ~2.(I) +e2TU(II~2_[_(~'2__7.2~IT(I ) .4- 2"B3H r,(1) ~ ~(1) ~'LL" 1,11 , t'Tl~J2,12 -*'/3,1 - - - - v , 1 - - I J l , t t -
Po ~.2 I R 1 )
..k 7,2 It(1)
T*-"2,11 - - t" L t d 2 , 2 2 + .-2(CL.
2"B3H
__ ,~2 "~I r(1) v T ] v 1,12 +
I7(1)--0
(3.1)
g l ( l ) - - td2,tt l l ( 1 ) --- 0 t",2
(3.2)
L H 0(1)
(3.3)
Po
Po
'~(1) - - - *'13,2
Po
0 {91 = 0 ,
{9(1) = B 3 H ( V *
{92 = 0 ,
U (1)) -
/Zo
/-*o
K°V20 (1) - poCO} 1) - 0o'r(V • V°)),, - OoLH{9~I) = 0 E~,) -- - F~0" .. u l, rLd( ,2.t) )
E(2t) = la°O • H U O) l,t)
~2c = c~(1 + 81),
g~ = c2r(1 + 82)
BaH 2 81
20oC 2 ,
E(3t) = 0
B3H 2
82
2poCk-
(3.4) (3.5) (3.5')
(3.5")
Equations (3.1)-(3.5) are none other than equations (2.1)-(2.5) obtained earlier by Nowacki
Surface waves in a nonlinear magnetothermoelastic perfect conductor
1439
[10] except for additional terms which we have underlined. Carrying equation (3.3) into (3.1), (3.2) and (3.4) yields 7/(1)2 + • L'-'1,11
,~2 lr?'(1) .a_ ( ~ 2 _ 7.2'~1r(1) t" T v 1,22 ~ t" TJL" 2,12 -- A a(1) v,1 -- It(l) 'Jl,tt --- 0
(3.6)
e2~,(1) T~2,11 "F e2LU~I~2 , -{- ( e 2 -- 7,2Xrr(1) t.T]tJl,12 _ A nv,( l2) _ U~l)t = 0 (V2-- 1£~0
(1)- n/it'(l)
K Ot/
(3.7)
--F Uil~t) = 0
~,,a. it
(3.8)
where we have set 0
e 2 "~"C2(1 -Jr"EH) ,
EH =
2"B3B3 H2 "F ~1 potxoC 2
(3.9) g o
OoZ tl B =-U-,
A = __r(1 + ~ o ) , Po ~°1
2"BaLH2
R =
~~12 B s L H 2
lZo'C
poC (1 - e ~ 3 )
(3.10)
~ 3 = O°LEH2 poCtxo
]Zo,:
(3.11)
Equations (3.6)-(3.8) indicate that the MTE problem for a perfect conductor is reduced to one of pure thermoelasticity in which the usual quantities gz., gr, A, B a n d / ( now depend on the magnetic characteristics of the problem through the parameters ~n, E~I, ~ 2 and E°3. The above equations are satisfied in the region {X2 > 0}. At the limiting plane {X2 = 0} we have the following boundary conditions (1 - 2.'~)U(11 + /'/(1) __ 0 ,
-/~ -- "B3H2
"J 1,2 --
2poC~
(3.12)
[(~_2,~2(1_/,5.~ur(1) +e2U~1~_]]tJX,l , _v O(1)+(2"B3o(31)2'B3~(¢)) H = 0 Po \ Po Po / 8 ( 1 ) - ~ t l ) = ~..~1) (1) = 0, ,2
~,o) = ].1.,0~(1) _ */.~0~12
~ 1 ) -- ~(1) = ~..~(1),
E~I) _ ]~1) = ~1)=
H(~o- '
~11) =. j~(1) = 0
hIl(1) H'0./tJ2,t for
(3.13)
0
(3.14)
E(31) _ ~(1) = 0
(3.15)
X 2 <0.
(3.16)
Now, let us consider the bulk equations (3.6)-(3.9). For plane wave harmonic solutions propagating along the Xa-direction (Fig. 1), and decaying in time and depth (from the plane X 2 = 0), we assume that ( U (1), 0 (1)) _- (17j(1),
O(1))eI(~t-K,X,-K2X2).
(3.17)
As a result of a previous paper by Hefni et al. [11], the global solution is 3 ( U(K1)' 0(1)) = E I,'K(j),/f'r(i)Oll~)ei(t~t-KiX,-Kzo,X2) j=l
(3.18)
where ~](1) K2(1) ^ (1) 1(1) = -- K1 U2(1)' ~T(1)
=
1(2) ~/0) =
g l /.T/O) K2(2 ) 2(2), K1
frO)
1(3) K2(3), ~ 2 ( 3 ) , ES$3-10-E
(1) = 0 (1)
(3.19)
0(1) =
in2 ^(~) A 'K2(2) U2(2)
(3.20)
0(1) :
in3 fro) A'K2(3)'~2(3)
(3.21)
I . A . Z . HEFNI et al.
1440 with 2
K20)
2 =KI_K K2iD
nj-
0)2
_ K 2 _ K 2,
K2= --
A' =--A
Cr-2, 0) 2
2 - n j,
(3.22)
e2
K2 = e-~L'
j = 2, 3
(3.23)
[K 2 + q(1 + co)] • X/[K 2 - q(1 + ea)] 2 + 4qK2L 2
(3.24)
(tlpo)(Oov/Ko)(K°/poC)(1 + col)(1 + e~2) ~o =
C'~.(1 - ¢~3)
'
i0)(1 - ~ 3 )
q = (KO/poC) .
(3.25)
Substituting from equations (3.19)-(3.21) into (3.3)3 and using (3.23) one obtains 3 ~i 1) = ~ I~(3}})ei(~'-K'x'-K2°~v:), 1~1
(3.26)
where
iH
~,1) •13(1)--Ft -U,
ff[o
L
)
0 l~2]i,T(1)
~(1) [ k B 3 - ~ T n j - - U 3 - - L j V 2 ( i ) for ] = 2,3. "13(j)_ /Zo~22(1)
(3.27)
It remains to determine expressions for the constant amplitudes in terms of one of them and the dispersion relation through the boundary conditions (3.12)-(3.15). Hence, let us start by the field equations in vacuum (2.23) and (2.24) which can be written as V2
1 0 2 \ * ..
----|[~(1), C2 Ot2] ~
~(1)) _~ 0,
c2 =
1
(3.28)
/Zo~o
where Co is the velocity of light in vacuum. For a solution of the type (3.17), we have
(~(o,~(1)=(~(1),~(l))ei(,ot-r,x,-r~x2)
(3.29)
with *2
K 2 = K 2 - K 2,
K2
_~0)2
c2.
(3.30)
Carrying equation (3.29) into (2.23) and (2.24) and taking into account (3.16) and (3.14)2 implies that •
•
(El 1), E(2')) = ~ ( - K * , K 0, ~'0)
~1) _.~ ~(21) = 2(31) ~___~(31) __ 0.
(3.31)
Equations (3.5)1 and (3.31)1 give the following expression for ~(3~):
~¢31)= iHK2 (-1(2'). K~
(3.32)
Surface waves in a nonlinear magnetothermoelasticperfect conductor
1441
Now, let us consider the boundary conditions (3.12)-(3.15). These equations, by substituting from equation (3.7), and using (3.14)1, (3.19)2 and (3.3)3 reduce to /d" /'~/(1) iV" 7"/(1) (1) I/" [ f r O ) frO) ~ (1) 1 *~'2(1) ~'- 1(11 q" ,v~2(2)~J 1(2) + K2(3)|~'T1(3)) + ~tXlKt-t 2(1) + tJ 2(2) -1- U 2 ( 3 ) ) ( --
2A) = 0
(3.33)
Kt[(E2L -- 2g'~(1 - '~)](0t~]) + f/(1) 11 + K~2)6"~2) 1) + K2(3)G(3)) ,~ 1~) + 0~p~)) + c=2L(K~.)6"~(1) ,'~(0/~ll + 01~l) - D(0(3~])
-
+ t" 2(2)fz(1) .-- L" f / ( 2(3)J 1 ) % ---- 0
K2(2)011/+ K2(3)0/~ / = 0
(3.34)
(3.35) 0
fl = Z
1 + 2"B3HZL.
po
~
~zL = ~
1+
po#oe~
/"
*
)"
2iH2K~"B3
B 3 = "B3 - "B3,
D -
(3.36)
,
poKE
Carrying equations (3.20)2 and (3.21)2 into (3.35) yields ~](1) n2/'1(1) 2(3) ~-- - - - t.J 2(2) •
(3.37)
n3
Hence equation (3.21) reduces to 7(1 _ 1(3)
Kin2 fro) K 2 ( 3 ) n 3 v 2(2)~
in2
0(31 ) - -
A'K2(3)
fro) tJ 2(2).
(3.38)
Substituting from (3.19)1, (3.20)1, (3.37) and (3.38)1 into (3.33), gives a relation between the constant amplitudes t-~ t'm) 2(1) and vfr(1) 2(2) [2(1 -- fi)K~ -- K~-]U2(1) 2 ^ ( 1 1 + 2(1 -/~)K2(1 - nn3],.., 2 ~ f m2(2) ) =0.
(3.39)
Similarly, equation (3.34) reduces to 2~1-/~)K2(2)K2(1)0(}{)
+
{~2A2(1
K2(2)n2A2'~ K2(3)n3A2/ K2(3)na/ +D(l_n2/[
K2(3)]
A' 1-
2 ( 1 - / ~ ) K2
- - - "w-~--v'~2% - 2 / / U2(~) = 0, 2(1 -- A)/~. 1 -- .r~TJ)
n3/L
(3.40)
where A~ = K~ - n2,
(3.41)
A~ = K 2 - n3.
On making zero the determinant of the system (3.39) and (3.40) and setting K2(1) = i(1 - ¢)1r2, Ki
¢~,,2
K2(s) = iss( 1 K] \ -~]
,
j=2,3
(3.42)
where ~02 ¢ = - cgKl 2 2,
e2
~ = ~e~y '
N~ ~1,2 Sj = 1 + (1 - ~/~)) '
~
(3.43)
~ = - -~
~ Nj = K?'
]=2,3
(3.44)
1442
I.A.Z.
HEFNI et al.
a~er some calculations, we arrive at the dispersion relation Eo)=O
D(~, ~ o ) = F ( ~ ) - G ( ~ ,
(3.45)
with
(
(2(1 - ,~)~¢)\2
-
F(s¢) = 4(1 -/~)2(2 - so)m(1 - ~1~)~,2 szs3( N3 - Nz) (N3s3 - NzsE) + w
G(~, ~o) =
(3.46) (3.47)
K2(3) ,/
(2i,,~3H2~ (
~2
In studying these equations, we assume for simplicity that the parameter en is very small as compared to unity and related terms may be neglected [12, 13]. Similarly, the remaining parameters E°;, j = 1, 2, 3, may be neglected as compared to unity. Hence, we have "r
__. C2L
A=.a,=--, Po
or eL=e,.=cL,
/
(3.49)
Eo = \po] \ KO ] ~oC '
where eo is none other than the usual parameter in pure thermoelastic media which is small for most materials at normal temperatures [14, equation (3.8)]. For eo = 0 one can show that equation (3.24) and (3.44)2 imply that N2 I
= 0,
s2 = 1.
(3.50)
I EO~0 Consequently, the dispersion relation (3.45) reduces to 4(1 - ~)1/2(1 - ~/~.)1/2 = (2 - ~)2,
(3.51)
which is the Rayleigh equation for surface waves in a purely elastic half-space. This equation is known to reduce to a cubic one in ~ [zero is an obvious root of equation (3.51)], the nonzero positive root which is less than unity represents the only admissible solution [15]. For later use we shall denote this solution by ~ such that F(~) = 1 and ~ < 1. To deduce the change of the phase velocity c and attenuation constant S corresponding to the case Eo # 0 but small, we assume that 3(3 = 6r + i6i) is the increment of ~ due to the coupling between the strain and temperature fields. Then following Massalas [16] and assuming that W is small and may be discarded (since W is proportional to co), we obtain c = Re g l S = - I r a K1 --- S*
6---
c (1+ fL
- G(~:, ~-o)1 OG OF l j•
[1
(3.52)
8i > 0
(3.53) (3.54)
1443
Surface w a v e s in a n o n l i n e a r m a g n e t o t h e r m o e l a s t i c p e r f e c t c o n d u c t o r Table 1
cL ( k m . s - l ) ( k m • s -1) cR ( k m " s -1) to* (s -1) S* ( m - ' )
cr
Aluminium
Copper
Lead
6.42 3.04 2.85 4.76 x 1011 7.28 x 107
5.01 2.27 2.13 2.25 × 1011 4.49 x 107
1.96 0.69 0.65 1.35 × 101° 6.89 x 106
and OF(f) = -1 1 1 Os~ e=~ (1 - 0.5~) -~ 2st(1 - ~/sr) ~-2(1 - ~--'-~
(3.55)
ac
(3.56)
- ~) (1 - ~/~.)2 where
(3.57)
S* = p°CcL
CR : C T ~ 1/2,
Ko
to* = p°CcZ Ko ,
f~
= to to,
v2,3 : ~ { [ 1 + i(1 -~e°)] q: ~ f [ 1
(3.58)
i ( 1 ~ %)]2 +-~}.
(3.59)
Values of the fundamental constants for aluminium, copper and lead at room temperature are given in Table 1 [17, 18]. Following Chadwick and Sneddon [14], the calculation will be obtained for f~ lying in the range 10 -5 <- f~ -< 1. The variation of Ac and S/S* for aluminium, copper and lead with f~ are cited in Tables 2 and 3 [(a) and (b) denote the cases for Eh = 0.01 and eh = 0.002 respectively]. The graphs of Ac and S in the neighbourhood of to* for aluminium are shown in Figs 2 and 3. Tables and Figs 2, 3 obtained here show that the variations of the phase velocity and attenuation constant with f~ and e0 have a similar behaviour but weaker values in the magnetoelastic problem as compared to those of the thermoelastic one.
Table 2
Ac=(C-C~n~ \ Aluminium
1 10 - I 10 -2 10 -3 10 -4 10 - s
CR
/
Copper
(a)
(b)
(a)
7.27 X 10 -6 1 . ! ) 3 × 1 0 -4 2.86 × 10 -4 2.89 × 10 -4 2.95 x 10 -4 7.51 x 10 -4
1.47 x 10 -5 3 . 8 4 × 1 0 -4 5.66 x 10 -4 5.73 x 10 -4 5.78 × 10 -4 7.51 X 10 -4
5.27 × 10 -6 1.56×10 4 2.44 x 10 -4 2.48 × 10 -4 2.53 × 10 -4 6.42 X 10 -4
Lead
(b) 1.66 3.11 4.83 4.90 4.94 6.42
× x × × x X
10 -5 11-4 10 -4 10 -4 10 -4 10 -4
9.09 4.71 1.09 1.14 1.16 2.95
(a)
(b)
x × x x X ×
1.84 X 10 -6 9.42×10-5 2.18 X 10 -4 2.25 X 10 -4 2.27 X 10 -4 2.95 X 10 4
10 -7 10-s 10 -4 10 -4 10 -4 10 -4
I. A . Z. H E F N I
1444
et al.
Table 3
s s* t2
Aluminium
(a) 1 10 t 10 -2 10 3 10 -4 10 -5
9.78 2.82 5.12 6.28 1.74 1.35
x x X X X X
Copper
(b) 10 -5 10 -5 10 -7 10 -9 10-to 10 n
X x x X X X
1.95 5.58 1.01 9.96 2.25 1.35
Lead
(a) 10 -4 10 -5 10 6 10 9 10 -1° 10-1t
x x X X X X
8.03 2.58 4.98 6.19 1.72 1.33
(b) 10 -5 10 -5 10 -7 10 -9 10 -1° 10-n
1.61 5.10 9.79 1.10 2.22 1.33
X x X X X X
(a) 10 -4 10 -5 10 -7 10 -8 10 -1° 10-11
2.89 1.57 4.58 5.08 1.78 1.39
x x X X X X
(b) 10 -5 10 -5 10 -7 10 -9 10 - t ° 10 -12
x x X X X X
5.79 3.11 9.04 1.12 2.29 1.39
10 - s 10 -5 10 -7 10 - s 10 - I ° 10 -12
4. O R D E R TWO IN In the preceding section, we obtained the solution of the first order approximation, in the ^ ^ 1) general form (3.18), (3.26) for X2 > 0. The amplitudes U#/j ), 0/} / and t~u) are related by equations (3.19)-(3.21), (3.27), (3.37), (3.38) and (3.39), and can be written as functions of one of them only, e.g. fr(1) v 2 ( 2 ) = Uo. The general first-order approximation thus reads 3
(U(K1), 0(1), ~(1))
= Uo E (UK(j), O(j), ~3(j))e i(°Jt-K'x'-r2q)X2),
K = 2, 3
(4.1)
]=1
where
2K1K2(1)(1--~-~) UI(,)-
(K 2 - K20)) -
2K~(1-n~) 02(1) = ( K 2 - K2(1))
'
gl
-
U1(2) = K2(2) '
U2(2) = 1
-Kin2
-
/-]2(3) =
--
U](3) = g 2 ( 3 ) n 3 '
0.)
= O,
0(2) -
in2
,
A K2(2)
~3(1) =0 ,
~3(j)
iH /z0K2(j)
(4.2) (4.3)
n2
(4.4)
n3
.
03 -
in3
(4.5)
A'K2(3)
[(/~3 --~)nj-B3KL]U2(j) 02-
and
j=2,3.
(4.6)
For simplicity we shall neglect the dependence of the longitudinal and transverse wave
S,,1133
----C 8
--~:8
20
= 0.01 =0.02
40111"I05
/-
lO
/
/
r~
4
Fig. 2. A t t e n u a t i o n versus frequency.
0.1
N.
1
Fig. 3. Change in velocity versus frequency.
Surface wavesin a nonlinearmagnetothermoelasticperfect conductor
1445
velocities on the magnetic field. For the second-order approximation, the system of equations (2.13)-(2.22) gives ,- U L L'-' ,t(2) "2
+ c2U~2~2 +,
(c~-cZr)U~2.~2, - AO!~ ) - '-'L,'r(Z)-_ U2o{j~. Plje 2i°, + ~,j,~p,j~e2'°J.}
(4.7)
}
(4.8)
p 2j,e2~°J+ E. P2Pe2'°"
c2U~2.I, + c2U~2)22 + (c 2 - vT,~2'U(2)a,*2- ,t,,~aa(2)_u~2~,= U2o .
,
(V 2 - -='--~0 ` ) -
K Ot/
I, s
(4.9)
B ( U ~ 2 } t q- U(22~t)~-
'
]~(2) ~___].~oHU~2~+
j
- J
),"
(V" U(1))U~I,? -
LH
--~
O(1)u~l)t,
BcH(V.U(1)~U(1)+L--~-~D(1)II(1) 0
E(z2)
. ]~rTT(2) ~-~ - - / ~ 0 1 1 t . J l , t - -
]
1,t
tl
t.Jl, t
~)(2)= ~)(22)=E(32)= 0
~,:i~,= htb~v, u,~)- Lo(~'l + [ ~ n v hi,0
L #0
"
(4.11) (4.12)
u `'-L-
~0
o"] ~
=-H (B3v. v(~'- Lo `2,) + v~[E e,,e 2°' + X W,,,e~°~] /.%
(4.10)
(4.13)
~•
with Oj = 19 -- K 2 u ) X 2 ,
0 = tot - K I X ~ ,
3
E--E, j ]=1
O~s = 0 - K2u,)X2
(4.14)
K2(js) = K2(j) + K2(O
(4.15)
2
3
3
E--EE, j,s 1=1 s=l
(4.16)
j#s
where Pq and Wqs are, respectively, the algebraic sums of the quantities which propagate with the phases 2Oj and 2Ojs of the inhomogeneous terms of the equation which is denoted by subscript i. According to Hefni et al. [11], the system of equations (4.7)-(4.9) may have a solution
U~2)=U~{~Alje2iOj+~,Qlpe2iO,,
- j
(4.17)
1,s
(419,
I.A.Z. HEFNI et al.
1446 where
[w:+ ,,,, "
A3i =
Dj
ni J
A~-
_K1 _ Aq + aEj,
O~-
K1 K2js QIjs + qzys,
,
a3j = 1
a2j -
K2U )
( Vj
+
i A ' ( K 2 - n i ) A) 3 j
njKE(j)
(4.21)
2
1 ( iA'(K~-n/) Q3j,) q2ys= njK2# Rj~ + 2
Vj = (KaP~j+ KE(~)P2i) 4c2 , Dj=(K2L-nj)+ q[1
(4.20)
Dj
R# =
(Kx W~j~+ K2#W2ix) 4c2
(K~-nJ),o] , nj
(4.22)
(4.23)
n l = K 2 - K z.
(4.24)
Equations (4.20) and (4.19) indicate that the second-order solution of approximation for 0 `2) can be obtained directly from the first-order solution. On the other hand, equations (4.21) and (4.22) represent six equations for 12 unknowns (Ai#, Aq for i2, 3 and j, s = 1, 2, 3). In order to determine the relations between these unknowns, we consider the boundary conditions (2.25)-(2.28) taking into account the equations in vacuum (2.23) and (2.24). Hence, we have the following equations:
4.1 On the limiting plane {Xe =0} U(2) 2,1 + it(2)+ t-~1,2 e2iO[E G1, + ~ Glj,] = 0 j
(c 2-'~'2wr'2) z't'T]tFl'l
(4.25)
j,s
~2 tr(z) ----T *B(32)~(32))+e2i°[~, PO 0 `2) + 2"("B3[?(32)PO ~ j Gz, + ~" G2,,] = 0
-['- L'Lt"J2'2
O'2)+e2i°[2G3,+~)'~G3#]=O ,2 " 1
~(2)_ 6(2)= ~ 2 ) ,
(4.27)
j,s
~ 2 ) _ ~ 2 ) = ~(2),
~ ( 2 ) - ~ 2 ) = O,
(4.26)
]&O~2) --/XS~(22)=
0
~ 2 ) - ~(32)= O.
(4.28) (4.28')
Equations (4.12)3 and (4.28')imply that ~2), ~ 2 ) a n d ~2)vanish identically.
4.2 In vacuum {X2 <0} (~2), ~(z2),~(32))= E(12),E(22), ~(3=))e2i(,,,-r~x,-x'~x2)
(4.29)
where (~2), j~(22))=
(-K~', K,). fi0(.O
(4.30)
Surface waves in a nonlinear magnetothermoelastic perfect conductor
1447
The constant amplitude ~(32)may be obtained directly from equations (4.28)2 and (4.30). Now, let us return to the boundary conditions (4.25)-(4.28)1, which may be reduced to
~ (K, A2j-F K2(j)A1y+~-J)-I- ~ (K,A21s + K2(DA1js-t-~) =0
(4.31)
~', (K,Alj + g/K2(j)A2j+ 02j) + '~ (KIAljs + ~K2jsA2j~+ j j,s
(4.32)
02j'~) = 0
~J (K2(j)A3j +i-~) + ~ (KzjsA3js +---~'-) " iG3j~ = 0
(4.33)
[ 2HB___~a U2 p 02j = ict L po 4j - ~ A3j]
(4.34)
"a[2Hff-3U2p4j,-~A3js]
(4.35)
where
(~2j, =
PO
t L
~/=~-2'
a
c2_c2.
(4.36)
Equation (4.33) gives a relation between the variables ~, K1 and ~o, similar to the dispersion relation (3.45). While equations (4.21), (4.22)(, (4.31) and (4.32) represent a system of eight linear inhomogeneous algebraic equations in 12 unknowns, one can obtain a particular solution for this system by substituting from equations (4.21) and (4.22) into (4.31) and (4.32) and assuming that Alj and A2j for j = 1, 2, 3 are independent of Q~js and Q2j, for js = 12, 13, 23. In this way, we obtain two similar independent groups of inhomogeneous linear algebraic equations Alibi + A1 = 0, J
j,s
~ A u + A2 = 0
(4.37)
J
Qljsbjs + Q~ = 0,
~ Q~j~ + Q2 = 0,
j,s
(4.38)
where
bj =
(K~(i) - K~) K2U)
'
al = ~. Kla2j + T } '
b# =
(K~/~ - K~)
Kzjs
Q1 = X. Klq2js+
l
(4.39)
(4.40)
I,s
1
A2
Q2
(1 - ~/)K~ ~ (3'K2(j)a2j + G2j), J
1
(1
i
~'~ (3'K2j~+ 02j~).
"y)K 1 j,s
(4.41)
The system of equations (4.37), (4.21), (4.38) and (4.22) may be solved to obtain the first group (i.e. Alj and A2j) and the second (i.e. Q~js and Qajs) in terms of one of Alj for the first group
1448
I.A.Z.
HEFNI et al.
and one of Ql# for the second. For example, these two groups may be expressed respectively in terms of AI3 and 0123 as follows: Kl All + a21,
A21 -
A22 = -
K2( 1)
gl A23 --- - - - A 1 3 K2(3) Q212 --
K1 --
K~12
K--L1A12 + a22, K2(2)
(4.42)
+ a23
Ql12 + q212,
K1 Ql13 K21~
Q213 = - - -
q'- q213,
gl 0123 ÷ q223 Q223 = -- - K223
(4.43)
where All =(bl -
0112
(b2A2- A,)
A13 + (b, - b2) '
(b13 -- b23) (b13Q2 - Qa) (bl2- b13) Q123 + (b12_ b13) ,
(bl - ~iI
A12 = (b:
(b~A2- A1)
(4.44)
A~3 -~ ( b z - bl)
(b12 --/923) Q123 + (b12Q2 - Q1) 0113 - (b13 b12) (b13- b12)
(4.45)
5. C O N C L U S I O N S
We have studied the nonlinear SWP in a magnetothermoelastic half-space of infinite electric conductivity subjected to an initial constant magnetic field normal to the sagittal plane. As in the magnetoelastic case with finite electric conductivity, the system of equations splits into two surface problems, one concerning a Rayleigh mode of pure thermoelastic form and the other a Bleustein-Gulyaev mode (B-G) with thermoelastic coefficients modified by the magnetic field. As in Hefni et al. [11] the B - G mode cannot exist. A dispersion relation is obtained and discussed for a numerical example. It is shown, in particular, that the variations of the phase velocity and attenuation constant with the angular frequency of the wave and the "modified" thermoelastic coefficient have similar behaviour, but with higher values, as compared to the corresponding magnetoelastic case considered in Hefni et al. [11].
REFERENCES [1] I. A. Z. HEFNI, A. F. GHALEB and G. A. MAUGIN, One-dimensional bulk waves in a nonlinear magnetoelastic conductor of finite electric conductivity. To be published. [2] S. KALISKI and W. NOWACKI, Bull. Acad. Polon. Sci. Ser. Sci. Technol. 10, 1 (1962). [3] S. KALISKI and W. NOWACKI, Bull Acad. Polon. Ser. Sci. 10, 159 (1962). [4] C. MASSALAS and A. DALAMANGAS, Lett. Appl. Engng Sci. 21, 991 (1983). [5] J. N. SHARMA and D. CHAND, Int. J. Engng Sci. 26, 951 (1988). [6] G. A. MAUGIN, Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam (1988). [7] A. N. ABD-ALLA and G. A. MAUGIN, Int. J. Engng Sci. 28, 589 (1990). [8] H. F. TIERSTEN and J. C. B A U M H A U E R , J. Appl. Phys. 45, 4272 (1974). [9] N. KALAYANASUNDARAM, Int. J. Engng Sci. 19, 435 (1981). [10] W. NOWACKI, Bull. Acad. Polon. Sci. Ser. Sci. Technol. 12, 485 (1962). [11] 1. A. Z. HEFNI, A. F. GHALEB and G. A. MAUGIN, Surface waves in a nonlinear magnetoelastic conductor of finite electric conductivity. To be published. [12] J. W. DUNKIN and A. C. ERINGEN, Int. J. Engng Sci. 1, 461 (1963). [13] G. A. MAUGIN and A. HAKMI, J. Acoust. Soc. Am. 77, 1010 (1985). [14] P. CHADWICK and I. N. SNEDDON, J. Mech. Phys. Solids, 6, 223 (1958). [15] K. F. GRAFF, Wave Motion in Elastic Solids. Ohio State U.P., Columbus (1975). [16] C. MASSALAS, Acta Mechanica. 65, 51 (1986). [17] F. C. MOON, Magnetosolid Mechanics. Wiley, New York (1984). [18] B. R. ROBERT, Metallic Materials Specification Handbook, Third Edition. Spon, New York (1980).
(Received 1 November 1994; accepted 7 January 1995)