CC BY
53Electronic Journal «Technical Acoustics» http://www .ejta.org
2007, 5
A. Rehman1, A. Khan2, A. Ali3
1 Department of Mathematics, Quaid-I-Azam, University, Islamabad, Pakistan, e-mail: [email protected]
2Dean Faculty of Sciences, Karakurum International University, Gilgit, Northern Areas, Pakistan, e-mail: [email protected]
3Department of Mathematics, Quaid-I-Azam, University, Islamabad, Pakistan, e-mail: [email protected]
Rayleigh waves in a rotating transversely isotropic materials
Received 22.10.2006, published 07.02.2007
Rayleigh wave speed in a rotating transversely isotropic material is studied. Speed in some transversely isotropic materials is calculated by choosing an angular velocity. Rayleigh wave speed is also calculated in non-rotating medium. It is observed that rotational effect plays a significant role and increases the speed of Rayleigh waves.
1. INTRODUCTION
Rayleigh wave speed in isotropic materials has been discussed by a number of researchers but in this article we consider transversely isotropic materials. Elastic waves in a transversely isotropic material were studied by P. Chadwick [1] and the same was studied in the rotating materials by Ahmad and Khan [2]. Wave propagation simulation in an elastic anisotropic (transversely isotropic) solid was also discussed by Carcione and Kosloff [3]. P. Chadwick proved that three waves can propagate in the medium.
In 1885, first time Rayleigh [4] studied the waves propagated along the plane surface of elastic solid, therefore, surface waves are known by his name. After that a number of researches [5-12] studied the Rayleigh wave speed by using different techniques in different kind of materials. Recently Pham & Ogden [13] discussed the Rayleigh waves speed in orthotropic elastic solids.
In this article rotational effects on Rayleigh waves speed on a transversely isotropic medium are studied.
2. BOUNDARY VALUE PROBLEM & SECULAR EQUATION
Consider the semi-infinite stress-free surface of transversely isotropic material. We choose the rectangular co-ordinate system in such a way that x3-axis is normal to the boundary and
the body occupies region x3 < 0. Also it is supposed that the body is rotating about x3-axis
which is axis of symmetry of the material. Like Pham & Ogden [13], we consider the plane
harmonic waves in x -direction in x1x3-plane with displacement components (u1,u2,u3) such that
U = u(x1,x3,t),i = 1,3, u2 = 0. (1)
Generalized Hooke’s law for transversely isotropic body may be written as
G11 " " C11 C12 C13 0 0 0 " G11
G22 C12 c11 C13 0 0 0 G22
G33 C13 c13 c33 0 0 0 G33
G23 0 0 0 C44 0 0 2 2
G13 0 0 0 0 C44 0 2 G13
G12 0 0 0 0 0 1c 1 _ 2 G12 _
2 _
(2)
where g.. is the strain tensor such that
i, j = 1,2,3,
(3)
CTÿ is the stress tensor and cii >0, i = 1, 3, 4; c11c33 - c13 > 0 which are the necessary and
sufficient conditions for the strain energy of the material to be positive definite. By using the above equations one can write
Ö"l1 C11U1,1 + C13U3,3,
G33 C13U1,1 + C33U3,3 ■
G13 c44(ui,3 + U3,1) •
(4)
When a homogeneous body is rotating with a constant angular velocity Q, it is observed that the rate of change of displacement vector ui with respect to time is (u + Q x u ). In tensor
notation this expression may be written as (u + sijkQjuk) where sijk is the Levi-Civita tensor.
Similarly second derivative with respect to time of ui becomes (see [14])
ui +Q ujQi -Q2ui + 2sijkQjUk. Thus equation of motion aij j = pui in the absence of body
forces in a rotating medium can be written as follows (see [14]).
aV,j = p{Ui +QjUjQ - Q ui +2SvkQjUk },
(5)
where Q = Q(0, 0, 1).
The equations of motion (5) in component form can be written as
G1U + ^ 13,3 = p(U1 — Q U1 ) ,
G31,1 + G33,3 = pU3 .
In view of (4) Eq. (6) can be written as
c\\U\,w + c13u3 31 + c44 (u133 + u3,13) = p(U1 - Q2u1),
C44 (U1,31 + U3,11) + C13U1,13 + C33U3,33 = pU3 .
(6)
(7)
The boundary conditions of zero traction are
C3i = 0, i = 1,3 on the plane x3 = 0. (8)
Usual requirements that the displacements and the stress components decay away from the
boundary implies
u. ^ 0 , Cj ^ 0 (i, j = 1,3) as x3 ^ -œ . (9)
Considering the harmonic waves propagating in x1 -direction, by following Pham & Ogden
[13] we write
u} =$j(y)exp[ik(x -ct)] ; j = 1,3. (10)
In the above equation, k is the wave number, c is the wave speed and y = kx3; (f>j, j = 1, 3 are
the functions to be determined. Substituting (10) into (7) implies
c44 k 2^1+ ik 2(c44 + c13)^3 + {k 2(p2 — cn) + pQ 2}^x = 0,
C33$3 +i(c44 + c13)^1 + (pc — c44)^3 = 0 .
The boundary condition (8) by using (4) and (10) can be written as follows:
+ c33fi3 = 0,
+ i$3 = 0 on the plane y = 0,
while from (9)
(11)
(12)
^ 0 as y ^-œ, j = 1,3. (13)
Taking Laplace transform of (11) by using (12)
{k2 (c44s2 + pc2 - Cjj) + pQ2 }$ (s) + ik2 (c44 + c13 )sÿ3 (s) =
C44k 2{s^1(0) + ^(0)} + ik 2(C44 + C13)^3(0) (14)
i(c13 + C44)s^1(s) + (c33s - C44 + p )^3(s) = i(c44 + C13)^1(0) + C33{^3(0)s + ^3(0)} •
This implies
C44k2 (sA (0) + ^'(0)) + ik2 (C44 + C13 ¥3 (0) ik2 (C44 + C13)s
i(c44 + CM (0) + c33(^3(0)s + ^(0)) (c33s2 - C44 + Pc2)
(Pl( S) =
(15)
k c33c445 + [k {(c44 + c13) + c33(pc — c„) + c44(pc — c44)} + pQ c33)]s +(pc2 — c44){k 2(pc2 — c„) + pQ2}
Let s2, s2 be the roots (having real parts positive) of denominator of (15). This implies
11 \— A A A3 a4
i^(s) 1-1-1-, (16)
s — s1 s — s2 s + s1 s + s2
where A1, A2, A3, A4 are the constants to be determined. Taking inverse Laplace transform and applying (13), implies
fa (y) = 4 exP(S y) + A2 exp(s2 y). (17)
In view of (11), (13) and (17)
(y) = a1 A1 exP(s1 y) + a2 A2 exP(s2y), (18)
¡[c^k sj + k (pc — cn) + pQ Let s2, s2 be the roots of the denominator of (15), thus we can write
where a =-------------—-------------------------, j = 1,2.
j k 2(c44 + cn) 'J ’
2 , 2 = [k {(C44 + C13) + c33(pc - c11) + c44(pc - C44)} + C33]
S1 + S2 =------------------------------------------~2----------------------------------------:
k c33c44
„ 2 s 2 = (pc 2 - c44)[k 2(pc 2 - c„) + pQ2]
(19)
k c c
33 44
Substituting </\(y) and ^3(y)from equations (17) and (18) into (12), we get
(c13 + c33alsl ) ) + (ic13 + c33a2 s2 )2 = 0,
( + iax ) A + (s2 + ia2 ) )2 = 0.
(20)
For non trivial solution of above linear homogeneous system of equations, the determinant of coefficients must be vanished i.e.
(ic13 + c33aj sj)(s2 + ia2) - (ic13 + c33a2 s2)(sj + ia1) = 0.
Simplifying and making the use of (18) and (19)
(pc2 -c44)[k2cn2 + C33{k2(pc2 -cn) + pQ2}]-kpc VC33C44 ^{k2(pc2 - cn) + pQ2}(pc2 - c44) = 0, which may be written as
2
P c44
c33 c11 c11
44
c!2? , pc2 , pO2
-+------1+-
1 + 2
11 k c11
p2 c44 -, a = —
c11 c33
c11k
P- = 0.
w w c c2 pQ2
Let u = , a = , b = -44, p = ——, r = —--------1, therefore above equation becomes
cn c33 c11 c11c33 k cu
(1 — a)u3 +{2p — b + (2 — a)r} u2 + (p + r)(p + r — 2b)u — b (p + r)2 = 0 . (21)
This can be solved for u for different materials and for different values of angular velocities by Cardado’s rule (see Cowles and Thompson [15]). Also one can solve it by numerical methods or simply by using computer software MATHEMATICA or MATLAB.
2
2
c11 c33
c11
c11
c
3. RAYLEIGH WAVES SPEED IN SOME TRANSVERSELY ISOTROPIC MATERIALS FOR AN ANGULAR VELOCITY
If we choose, say | — | = —, then above equation (21) becomes
^ k J p
(1 — a )u3 +(2 p — b )u2 + p (p — 2b )u — bp2 = 0 (22)
and can be solved for u for different materials as follows.
For an example we choose Cadmium. Stiffness elastic constants for Cadmium are (see Rahman and Ahmad [16]) as follows:
cu = 1.16 x 1011 N/m2, c13 = 0.41 x 1011 N/m2,
c33 = 0.509 x 1011 N/m2, c44 = 0.196 x 1011 N/m2, c c c2
a = = 0.385069, b = = 0.168966, p = ^^ = 0.284703, r = 0.
c33 cn cnc33
Substituting the values of a , b, p and r in equation(22) we get
0.61493 u3 - 0.40044u2 - 0.0151545 u - 0.136957 = 0. (23)
This implies u = 0.459112, therefore c = 2482.45 m/s.
Similarly we can determine speed c for other transversely isotropic materials as is evident from the following table.
Table 1. For rotating materials
Material Stiffness X1011 N/m2 u Density kg/m3, P Rayleigh wave speed, m/s
cii C12 C13 c33 C44
Cadmium 1.16 0.42 0.41 0.509 0.196 0.459112 8642 2482.45
Cobalt 2.59 1.59 1.11 3.35 0.71 0.264378 8900 2773.75
Titanium boride 6.90 4.10 3.20 4.40 2.50 0.443684 4500 8249.12
Zinc 1.628 0.362 0.508 0.627 0.385 0.321827 7140 2708.88
Magnesium 0.5974 0.2624 0.217 0.617 0.1639 0.296935 1740 3299.80
If we choose Q = 0 (non-rotating case), then the above equation (21) becomes (1 — a)u3 + {a — 2(1 — p) — b}u2 + {(1 — p)2 + 2b(1 — p)}u — b(1 — p)2 = 0 .
In the following table Rayleigh wave speed in non-rotating transversely isotropic materials is calculated.
Table 2. For non-rotating materials
Material Cobalt Cadmium Titanium boride Zinc Magnesium
Rayleigh wave speed, m/s 2685.55 1404.77 5983.28 2045.01 2894.65
CONCLUSIONS
Above results showed that rotational effect plays a significant role and increases the speed of the Rayleigh waves for a finite angular velocity of the materials.
REFERENCES
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