Rayleigh waves in a rotating transversely isotropic materials

Electronic Journal «Technical Acoustics»

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.

harmonic waves in x1 -direction in x1x3-plane with displacement components (u1,u2,u3) such that ui = ui(x1,x3,t),i = 1,3, u2 = 0.                                                                           (1)

Generalized Hooke’s law for transversely isotropic body may be written as

_ “1     C 11    C 12    c 13     0    0        0 & 11 _       C 12    C 11    с 13     0    0       0 & 22                    о   n      Л _       C 13    C 13    c 33     0    0       0 & 33 = 0   0   0    c44    0      0 & 23         .      .      . 0    0    0    0    c44       0 & 13 0    0    0    0    0 c 11 - c 12	,Л33    ,                                                 (2) 2 e 23 2 e 13 l 2 ^ 12 j
& 9     0   0   0   0   0 12 L                                    2 J where e ij is the strain tensor such that

-= 2 ( u ij + u ji ) i ^, j = ^1,2,3, (3) a is the stress tensor and c ii >0, i = 1, 3, 4; c 11 c ₃₃ - c ₁₃ 2 >  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

& 11    ^c11^u 1,1 + ^C 13 u 3,3 ,

& 33    C 13 u 1,1 + C 33 u 3,3 ,

^a 13 = ^C 44⁽ u 1,3 + u 3,1 ) •

When a homogeneous body is rotating with a constant angular velocity Q , it is observed that the rate of change of displacement vector u i with respect to time is ( u + Q x u ). In tensor notation this expression may be written as ( u i + £ _ijk Q j u k ) where s _jk is the Levi-Civita tensor. Similarly second derivative with respect to time of u_i becomes (see [14]) u i + Q _ju_j Q i -Q ² u i + 2 £ _jk Q jiu_k . Thus equation of motion & _{ij j} = p u i in the absence of body forces in a rotating medium can be written as follows (see [14]).

& jj = p { u -H u Q Q u + 2 j Q u k } , (5)

where Q = Q (0, 0, 1).

The equations of motion (5) in component form can be written as

& 11,1 ⁺ & 13,3 = ^{p ( u} 1    ^{Q u} 1 ),

& 31,1 ⁺ & 33,3 = ^{p u} 3 •

In view of (4) Eq. (6) can be written as

^C 11 ^u 1,11 ^{+ C} 13 ^u 3,31 ^{+ C} 44 ^{( u} 1,33 ^{+ u} 3,13 ) = ^{p ( u} 1    ^{Q u} 1 ) ,

^C44⁽ u 1,31 ⁺ u 3,11 ) ^{+ C} 13 u 1,13 ^{+ C}33 u 3,33 = P u 3 ^.

The boundary conditions of zero traction are

σ _{3 i} = 0, i = 1,3 on the plane x ₃ = 0.

Usual requirements that the displacements and the stress components decay away from the boundary implies

u_i → 0,     σ _ij → 0( i , j = 1,3) asx ₃ →-∞ .

Considering the harmonic waves propagating in x ₁ -direction, by following Pham & Ogden [13] we write

u_j = φ _j ( y )exp[ ik ( x ₁ - ct )]; j = 1, 3.

In the above equation, k is the wave number, c is the wave speed and y = kx ₃ ; φ _j , j = 1, 3 are the functions to be determined. Substituting (10) into (7) implies

c 44 k Ф + ik ²( c 44 + c 13 ) ф‘ + { k 2 ( pp c ² - c „) + &²ф = 0 , c ₃₃ φ ₃ ′′ + i ( c ₄₄ + c ₁₃ ) φ ₁ ′ + ( ρ c ² - c ₄₄ ) φ ₃ = 0.

The boundary condition (8) by using (4) and (10) can be written as follows:

ic ₁₃ φ ₁ + c ₃₃ φ ₃ ′ = 0,

φ ₁ ′ + i φ ₃ = 0 on the plane y = 0,

while from (9)

φ _j , φ ′ _j → 0 as y → -∞ , j = 1,3 .                                                      (13)

Taking Laplace transform of (11) by using (12)

{k2(c44s2+ρc2-c11)+ρΩ2}φ1(s)+ik2(c44 +c13)sφ3(s)= c44k2{sφ1(0) +φ1′(0)}+ ik2(c44+ c13)φ3(0)                                                       (14)

i ⁽ c 13 ^{+ c} 44 ) ^{s ф} 1 ⁽ s ) ^{+ ( c} 3 3 s ^{2 - c} 44 ⁺ P c ^{2 ) ф} 3 ⁽ s ) = i ^{( c} 44 ⁺ c 13 ^{) ф} 1 ^{(0) +} c 3 3 ^{{ ф} 3 ⁽⁰⁾ s ^{+ ф} 3 ^(0)} •

This implies

c 44 k ²( s φ 1 (0) + φ 1 ′ (0)) + ik ²( c 44 + c 13 ) φ 3 (0)      ik ²( c 44 + c 13 ) s

ϕ ₁( s ) =

i ( c ₄₄ + c ₁₃) φ ₁(0) + c ₃₃( φ ₃(0) s + φ ₃ ′ (0))      ( c ₃₃ s ² - c ₄₄ + ρ c ²)

k ² c ₃₃ c ₄₄ s ⁴ + [ k ²{( c ₄₄ + c ₁₃)² + c ₃₃( ρ c ² - c ₁₁) + c ₄₄( ρ c ² - c ₄₄)} + ρ Ω ² c ₃₃)] s ²

+ ( ρ c ² - c ₄₄){ k ²( ρ c ² - c ₁₁) + ρ Ω ²}

Let s ₁ ², s ₂ ² be the roots (having real parts positive) of denominator of (15). This implies

AAA φ ₁ ( s ) =   ¹ + ² + ³

s - s ₁ s - s ₂ s + s ₁

+ ^{A 4}

s + s ₂

where A1, A2, A3, A4 are the constants to be determined. Taking inverse Laplace transform and applying (13), implies ф1(y) = Ai exP(s 1 y) + A2 exP(s2 y).

In view of (11), (13) and (17)

ф ₃ ( y ) = a 1 A 1 exp( s 1 y ) + a ₂ A 2 exp( s 2 y ),

where a j =

i [ c 44 k ² s² + k ²( p c ² - c 11 ) + p Q ²

k ⁽ c 44 ⁺ c 13 ) s j

, j = 1,2.

Let s ₁² ,

s ₂² be the roots of the denominator of (15), thus we can write

2      2

s 2 + s 2

2 2   ( pc s1 s 2 =-----

[ k {(c 44 + c13) + c 33 (Pc    c 11) + c 44 (Pc k c33c44

~ c 44 )[ k ²( P c 2 - c 11 ) + P Q 2 ]

c ₄₄)} + p Q ² c 33 ]

,

² c ₃₃ c ₄₄

.

Substituting ф ( y ) and ф ₃( y ) from equations (17) and (18) into (12), we get

( ic ₁₃ + c₃₃ a 1 s 1 ) A 1 + ( ic₁₃ + c₃₃ a ₂ s 2 ) A 2 = 0, ( s 1 + i a 1 ) A 1 + ( s 2 + i a ₂ ) A 2 = 0.

For non trivial solution of above linear homogeneous system of equations, the determinant of coefficients must be vanished i.e.

( ic ₁₃ + c ₃₃ a 1 s 1 )( s 2 + i a 2 ) - ( ic₁₃ + c ₃₃ a ₂ s 2 )( s 1 + i a 1 ) = 0.

Simplifying and making the use of (18) and (19)

(pc2 -c44Xk2c132 + cзз{k2(pc2 -cn) + pQ2}]- kpc yjc33c44 {k (pc   c11) + pQ } (pc   c44) = 0, which may be written as pc 2

c33      c11

^c 44 p c 2  1

1 + c11

2 c ₁₃

c ₁₁ c ₃₃

pc²  ,  p Q ²

+        1 +      2

c ₁₁         c ₁₁ k

pc- = 0.

c ₁₁

_T ,      pc ^{2         c} 44   ,     ^c 44

Let u =     , a = 44 , b = 44 , p = c11         c33          c11

² c ₁₃

^c 11 ^c 33

p Q ² k ² c ₁₁

- 1, therefore above equation becomes

( 1 - a ) u ³ +{ 2 p - b + ( 2 - a ) r } u ² + ( p + r )( p + r - 2 b ) u - b ( p + r ) 2 = 0 .

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.

3. RAYLEIGH WAVES SPEED IN SOME TRANSVERSELY ISOTROPIC MATERIALS FOR AN ANGULAR VELOCITY

^ ² Си

If we choose, say I I = —11 , then above equation (21) becomes

к k J P

( 1 - a ) u ³ + ( 2 p - b ) u ² + p ( p - 2 b ) u - bp ² = 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:

c11 = 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, a = -44 = 0.385069,   b = -44- = 0.168966, p =      = 0.284703, r = 0.

c ₃₃                              c ₁₁                             c ₁₁ c ₃₃

Substituting the values of a , b , p and r in equation(22) we get

0.61493 u ³ – 0.40044 u ² – 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 ×1011 N/m2					u	Density kg/m3, P	Rayleigh wave speed, m/s
	c 11	c 12	c 13	c 33	c 44
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 ) u ³ + { a - 2(1 - p ) - b } u ² + {(1 - p )² + 2 b (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.