Propagation of Stoneley Waves at an Interface Between Two Microstretch Elastic Half-spaces S. K. TOMAR DILBAG SINGH Department of Mathematics, Panjab University, Chandigarh-160014, India. (e-mail:
[email protected]) (Received 8 September 20051 accepted 31 May 2006)
Abstract: Frequency equations for Stoneley waves at unbonded and bonded interfaces between two dissimilar microstretch elastic half-spaces have been derived. Eringen’s theory of microstretch elastic solid material is employed for mathematical analysis. It is found that Stoneley waves in a microstretch elastic solid medium are dispersive. Numerical computations are performed for a particular model to study the variation of phase velocity with respect to wavenumber, and reveal that for small values of non-dimensional wave number, the effect of microstretch property of the medium has a significant effect on dispersion curves, while no effect is observed for higher values of the non-dimensional wavenumber. The results of Tajuddin, Murty and Stoneley have been derived as particular cases of the present problem, and some other interesting particular cases have also been discussed.
Keywords: Stoneley wave, microstretch, micropolar, frequency equation, unbonded interface, bonded interface.
1. INTRODUCTION The study of surface wave propagation along the free boundary of an elastic half-space or along the interface between two dissimilar elastic half-spaces, has been a subject of continued interest for many years. These waves are well known in the study of seismic waves, geophysics and non-destructive evaluation, and there is a rich literature available on surface waves in terms of classical elasticity (see, e.g., Achenbach, 19731 Brekhoviskikh 19601 Bullen and Bolt, 19851 Ewing et al., 19571 Love, 19111 Udias, 1999). Surface waves propagating along the free boundary of an elastic half-space, non-attenuated in their direction of propagation and damped normal to the boundary are known as Rayleigh waves in the literature, after their discoverer (Rayleigh, 1885). The phase velocity of Rayleigh wave is a single valued function of the parameters of an elastic half-space. These waves are nondispersive and their velocity is somewhat less than the velocity of shear waves in unbounded media. The Rayleigh wave is a coupled compressional-shear system propagating with a unique velocity. Later, Stoneley (1924) investigated the possible existence of waves similar to surface waves and propagating along the plane interface between two distinct uniform elastic solid half-spaces in perfect contact, and these are now universally known by his name. Stoneley waves can propagate on interfaces between either two solid media or a solid medium and a
Journal of Vibration and Control, 12(9): 995–1009, 2006 1 2006 SAGE Publications 1
DOI: 10.1177/1077546306068689
996 S. K. TOMAR and D. SINGH
liquid medium. These waves are harmonic waves propagating along the interface between two elastic half-spaces, produce continuous traction and displacement across the interface and attenuate exponentially with distance normal to the interface in both the half-spaces, provided the range of the elastic constants of the two solids lie within some suitable limits (Scholte, 1947). Stoneley (1924) obtained the frequency equation for propagation of Stoneley waves and showed that such interfacial waves can exist only if the velocity of distortional waves in the two half-spaces is approximately same. Since then a number of problems concerning the propagation of Stoneley waves along the solid2solid and fluid2solid boundary have been discussed by several researchers, including Strick and Ginzbarg (1956), Lim and Musgrave (1970), Chadwick and Currie (1974), Barnett et al. (1985), Hsieh et al. (1991), Abbudi and Barnet (1990), Goda (1992), Ashour (1999), Ansell (1972), Murty (1975a, 1976), Edelman and Wilmanski (2002), Tajuddin (1995), Sambaiah et al. (1986), Abd-Alla and Ahmed (2003), Abd-Alla (1999), Ghosh (1991), Ghosh et al. (2001). Murty (1975b) discussed the wave propagation at an unbonded interface between two elastic half-spaces. The idea of an unbonded interface was given by Shin-Jang Chang (1971), who said that an interface could be described as unbonded if it obeyed the following ideal crack boundary conditions: (i) the normal component of the particle displacement vector is continuous, (ii) the normal component of the stress tensor is continuous, and (iii) the shearing stress vanishes along the ideal crack. Tajuddin (1995) discussed the existence of Stoneley waves at such an interface between two micropolar elastic half-spaces, and employed Eringen’s theory of micropolar elasticity to show that Stoneley waves can exist at an ideal unbonded interface between two micropolar elastic half-spaces under suitable conditions. Eringen (1990, 1999) developed a linear theory of microstretch elastic solids which is a generalization of the linear theory of micropolar elasticity and a subclass of the theory of micromorphic materials (Eringen and Suhubi, 19641 Suhubi and Eringen, 1964). In microstretch elastic solids, the material particles can undergo translation, rotation and stretching. The motion in microstretch elastic solids is characterized by seven degrees of freedom (three for translation, three for rotation and one for stretch). The transmission of load across a differential element of the surface of a microstretch elastic solid is described by a force vector, a couple stress vector and a microstress vector. The theory of microstretch elastic solids differs from the theory of micropolar elasticity in that there is an additional degree of freedom called scalar stretch and an additional kind of stress called the stretch vector. Materials such as porous elastic solids filled with gas or inviscid fluid, asphalt, composite fibers, etc., fall into the category of microstretch elastic solids. Many researches have investigated the problems of waves and vibrations in microstretch media. Notable among them are Kumar et al. (2002), Singh (2002), Iesan and Scalia (2001, 2003), Midya (2004), and Tomar and Garg (2005). Recently, Eringen (2004) developed a linear electro-magnetic theory of microstretch elasticity. The book written by Eringen (1999) is very useful on the subject. In the present paper, we have discussed the propagation of Stoneley waves at an unbonded/bonded interface between two microstretch elastic solid half-spaces and the frequency equations for Stoneley waves are derived, analyzed and compared numerically with that of micropolar media. It is found that Stoneley waves are dispersive in microstretch medium and there is a significant effect of microstretch on dispersion curve. The results of some earlier workers have been reduced as a particular case of the present formulation.
INTERFACE BETWEEN TWO MICROSTRETCH ELASTIC HALF-SPACES 997
2. BASIC EQUATIONS Equations of motion in a linear homogeneous and isotropic microstretch elastic solid in the absence of body forces and body couple are given by Eringen (1999, pp. 254–255) 1 2 2 1 2 8 c1 3 c32 414 5 U2 2 c22 3 c32 4 6 14 6 U2 3 c32 4 6 1 3 30 44 7 U5 2 1 2 8 c4 3 c52 414 5 12 2 c42 4 6 14 6 12 3 620 4 6 U 2 2620 1 7 15 8 c62 4 2 4 2 c72 4 2 c82 4 5 U 7 25
(1) (2) (3)
where c12 7 13 3 272895 c22 7 7895 c32 7 K 895 c42 7 89 j5 c52 7 1 3 289 j5 620 7 c32 8j5 c62 7 2 0 89 j5 c72 7 231 839 j5 c82 7 230 839 j5 39 0 7 30 89 3 and 7 are Lame’s parameters, K 5 5 and are micropolar constants, 30 5 31 and 0 are microstretch constants, 9 is the density of the medium, j is the micro-inertia, U and 1 are the displacement and microrotation vectors respectively, and 4 is the scalar microstretch. Dots above U, 1, and 4 on right hand sides of equations (1) to (3) indicate the temporal derivatives and other symbols have their usual meanings. The constitutive relations in a linear isotropic microstretch elastic solid medium are given by Eringen (1999) as
kl
7 3Ur5r kl 3 71Uk5l 3 Ul5k 2 3 K 1Ul5k 2 klr r 2 3 30 4 kl 5
(4)
m kl
7 r5r kl 3 k5l 3 l5k 5
(5)
7 0 4 5k 5
(6)
k
where kl is the force stress tensor, m kl is the couple stress tensor, and k is the microstretch vector. The comma in the subscript denotes the partial derivative with respect to spatial k coordinates viz Uk5l 7 U xl We will discuss two-dimensional problems in the x 2 z plane. Therefore, in this plane, we will take U 7 1u1x5 z2 05 1x5 z225
1 7 105 L1x5 z25 025
4 7 41x5 z25
Introducing the potentials and such that u7
3 5 x z
7
2 z x
we obtain from equations (1) to (3) the following 13 3 27 3 K 24 2 3 30 4
7 9
2 5 t 2
(7)
17 3 K 24 2 2 K L
7 9
2 5 t 2
(8)
998 S. K. TOMAR and D. SINGH
4 2 L 3 K 4 2 2 2K L 6 0 4 2 4 2 230 4 2 2 231 4
7 9j
2 L t 2 5
7 39 j
(9)
24 t 2
(10)
Assuming time harmonic variation of 5 5 L and 4 as follows 9 4 1x5 9 5 9 L5 9 5 5 L5 4 1x5 z5 t2 7 5 z2 exp 216t 5
(11)
where 6 is angular frequency and is related to the wavenumber and phase velocity c through the relation 6 7 c. Substituting (11) into equations (7) to (10), we obtain 9 13 3 27 3 K 24 2 9 3 30 4
9 7 2962 5
(12)
17 3 K 24 2 9 2 K L9
9 7 2962 5
(13)
9 7 29 j62 L5
(14)
9 7 239 j6 4
(15)
4 2 L9 3 K 4 2 9 2 2K L9 29
9 2 230 4 2 231 4 9 6 0 4 4 2
2
9 while equations (13) and (14) are Note that equations (12) and (15) are coupled in 9 and 4, 9 By elimination procedure, it can be seen that coupled in 9 and L. 3
4
39 j62 2 231 3 0 962 3 320 3 6 0 3 0 13 3 27 3 K 2 6 962 139 j62 2 231 2 9 9 3 145 2 7 05 6 0 13 3 27 3 K 2 3 4 5
962 3 K 2 9 j62 2 2K 44 3 42 3
17 3 K 2 4 56 9 K 62 9 j62 9 2 9 7 0 3 2 2 1 L5
17 3 K 2 K 44 3
5 42 (16)
(17)
The solutions of equations (16) and (17) can be obtained easily1 finally, the time harmonic solutions of equations (7)–(10) can be written as 2 Ae Rz 3 Be2Rz 3 Ce Sz 3 De2Sz e11 x26t2 5 2 1 28 7 1 Rz 7 a Ae 3 Be2Rz 3 b Ce Sz 3 De2Sz e11 x26t2 5 2 1 7 Ee Pz 3 Fe2Pz 3 Ge Qz 3 H e2Qz e11 x26t2 5 2 1 28 7 1 Pz 7 c Ee 3 Fe2Pz 3 d Ge Qz 3 H e2Qz e11 x26t2 5
7 4 L
1
(18) (19) (20) (21)
where A5 B5 C5 D5 E5 F5 G and H are unknown and a5 b5 c and d are coupling parameters given by
INTERFACE BETWEEN TWO MICROSTRETCH ELASTIC HALF-SPACES 999 8 7 a 7 2 13 3 27 3 K 212 2 3 R 2 2 3 962 830 5 7 1 2 8 b 7 2 13 3 27 3 K 2 2 2 3 S 2 3 962 830 5 7 7 8 8 c 7 17 3 K 212 2 3 P 2 2 3 962 8K 5 d 7 17 3 K 212 2 3 Q 2 2 3 962 8K 5 while the expressions of R5 S5 P and Q are given by R 5S 2
2
34 5 1 3 0 962 3 320 39 j62 2 231 7 2 3 2 6 0 3 0 13 3 27 3 K 2
9 4 52 3 0 962 3 320 962 139 j62 2 231 2 39 j62 2 231 3 246 5 6 0 3 0 13 3 27 3 K 2 6 0 13 3 27 3 K 2 2
P 25 Q2 7 2 2 94
1 2
34
9 j62 2 2K
962 3 K 2 3
17 3 K 2
9 j62 2 2K
962 3 K 2 3
17 3 K 2
52
5
962 19 j62 2 2K 2 246
17 3 K 2
The ‘3’ sign corresponds to the quantities R 2 and P 2 , while the ‘2’ sign corresponds to the quantities S 2 and Q 2 . The coupling parameters a and b are obtained by substituting equations (18) and (19) into equation (7) and parameters c and d are obtained using equations (20) and (21) in equation (8).
3. PROBLEM FORMULATION We consider two linear isotropic homogeneous microstretch elastic solid half-spaces H1 and H2 with different elastic properties, using Cartesian axes such that the upper half-space H2 occupies the region 2 z 0 and the lower half-space H1 occupies the region 0 z Quantities with subscript 1 correspond to the half-space H1 , and quantities with subscript 2 correspond to the half-space H2 . The x-axis is taken along the plane of separation of H1 and H2 . In order to discuss Stoneley waves at the interface z 7 0, we take the following solution of equations of motion (7) to (10). The expression of potentials in medium H1 are
41 1 L1 and in medium H2 are
1
2 Be2R1 z 3 De2S1 z e11 x26t2 5 7 8 7 a1 Be2R1 z 3 b1 De2S1 z e11 x26t2 5 1 2 7 Fe2P1 z 3 H e2Q 1 z e11 x26t2 5 7 2P1 z 8 7 c1 Fe 3 d1 H e2Q 1 z e11 x26t2 5
1 7
(22) (23) (24) (25)
1000 S. K. TOMAR and D. SINGH
42 2 L2
1
2 Ae R2 z 3 Ce S2 z e11 x26t2 5 7 8 a2 Ae R2 z 3 b2 Ce S2 z e11 x26t2 5 7 1 2 7 Ee P2 z 3 Ge Q 2 z e11 x26t2 5 7 P2 z 8 7 c2 Ee 3 d2 Ge Q 2 z e11 x26t2 5
2 7
(26) (27) (28) (29)
where Re1Ri 5 Si 5 Pi 5 Q i 2 0. The quantities ai 5 bi 5 ci 5 di , Pi 5 Q i 5 Ri and Si are defined in the same way as their counterparts without subscripts.
4. BOUNDARY CONDITIONS At an unbonded interface, we assume that the interface between the half-spaces is frictionless, so that shear traction is absent and shear displacement is discontinuous at the interface. Thus, for an unbonded interface, there is continuity of the normal component of the displacement vector, stress tensor, couple stress tensor, microrotation, scalar microstretch and microstretch tensor, while the shear components of the stress tensor vanishes across the interface. At a bonded interface, by contrast, we assume that the half-spaces are in perfect contact. Thus, for a bonded interface, there is continuity of the components of the displacement vector, microrotation vector, scalar microstretch, stress tensor, couple stress tensor and microstretch tensor at the interface. Mathematically, these boundary conditions can be expressed as At z 7 0, [ zz ]1 7 [ zz ]2 5 [m zy ]1 7 [m zy ]2 5 [ z ]1 7 [ z ]2 5 1 7 2 5 4 1 7 4 2 5 L 1 7 L 2 and
[ zx ]1 7 05
[ zx ]2 7 05
[ zx ]1 7 [ zx ]2 5 u 1 7 u 2 5
for unbonded interface5 for bonded interface.
Using equations (4) to (6), the requisite components of stresses can be written as 1i 7 15 22 2 2 i 2i 3 13i 3 27i 3 K i 2 2i 2 127i 3 K i 2 3 30i 4 i 5 2 x z xz
[ zz ]i
7 3i
[ zx ]i
7 127i 3 K i 2
[m zy ]i
7 i
Li 5 z
2 i 2i 2 i 3 17 3 K 2 2 Ki L i 5 2 7i i i xz x 2 z 2 [ z ]i 7 0i
4 i 5 z
Substituting equations (22) to (29) into the above boundary conditions, one obtains eight homogeneous equations in eight unknowns1 namely, A5 B5 C5 D5 E5 F5 G, and H . For non-trivial solution of these equations, the determinant of the coefficient matrix must vanish, that is
INTERFACE BETWEEN TWO MICROSTRETCH ELASTIC HALF-SPACES 1001 ai j 7 0
(30)
The non-vanishing entries of this determinantal equation are given by 7 8 232 2 3 132 3 272 3 K 2 2R22 3 302 a2 5 7 8 7 2 231 2 3 131 3 271 3 K 1 2R12 3 301 a1 5 7 8 7 232 2 3 132 3 272 3 K 2 2S22 3 302 b2 5 7 8 7 2 231 2 3 131 3 271 3 K 1 2S12 3 301 b1 5
a11 7 a12 a13 a14
a15 7 21 1272 3 K 2 2P2 5
a16 7 21 1271 3 K 1 2P1 5
a18 7 21 1271 3 K 1 2Q 1 5
a25 7 2 c2 P2 5
a28 7 1 d1 Q 1 5
a31 7 02 a2 R2 5
a34 7 01 b1 S1 5
a41 7 R2 5
a45 7 a47 7 21 5
a46 7 a48 7 1 5
a65 7 c2 5
a54 7 2b1 5
a26 7 1 c1 P1 5
a32 7 01 a1 R1 5
a42 7 R1 5
a66 7 2c1 5
a17 7 21 1272 3 K 2 2Q 2 5 a33 7 02 b2 S2 5
a43 7 S2 5
a51 7 a2 5 a67 7 d2 5
a27 7 2 d2 Q 2 5
a44 7 S1 5
a52 7 2a1 5
a53 7 b2 5
a68 7 2d1 5
the remaining entries for an unbonded interface are a72 7 21271 3 K 1 21 R1 5
a74 7 21271 3 K 1 21 S1 5
a76 7 71 3 171 3 2 K 1 c1 5 7 8 a78 7 71 2 3 171 3 K 1 2Q 21 2 K 1 d1 5 2
K 1 2P12
a81 7 1272 3 K 2 21 R2 5 a83 7 1272 3 K 2 21 S2 8 8 7 7 a85 7 72 2 3 172 3 K 2 2P22 2 K 2 c2 5 a87 7 72 2 3 172 3 K 2 2Q 22 2 K 2 d2 5 while those for a bonded interface are a71 7 1272 3 K 2 21 R2 5 a74 a76 a78
a72 7 1271 3 K 1 21 R1 5 a73 7 1272 3 K 2 21 S2 5 8 7 7 1271 3 K 1 21 S1 5 a75 7 72 2 3 172 3 K 2 2P22 2 K 2 c2 5 8 8 7 7 7 2 71 2 3 171 3 K 1 2P12 2 K 1 c1 5 a77 7 72 2 3 172 3 K 2 2Q 22 2 K 2 d2 5 8 7 7 2 71 2 3 171 3 K 1 2Q 21 2 K 1 d1 5 a81 7 a83 7 1 5 a82 7 a84 7 21 5
a85 7
P2 5
a86 7 P1 5
a87 7 Q 2 5
a88 7 Q 1
Equation (30) represents the period equation for Stoneley wave propagation at unbonded or bonded interfaces between two dissimilar microstretch solid half-spaces. This equation is an implicit function of phase velocity and wave number. Hence, Stoneley waves are dispersive in nature. Analytically, no definite conclusion can be drawn regarding the phase velocity of Stoneley waves and other characteristics from this equation. However, for very small 301 5 302 5 K 1 and K 2 , a definite conclusion regarding phase velocity of Stoneley waves can
1002 S. K. TOMAR and D. SINGH be obtained. Thus, neglecting the second and higher powers of the quantities 301 5 302 5 K 1 and K 2 , we obtain (see Midya, 2004) 9
Ri
Pi
9 39 i ji 62 2 231i 9 i 62 7 2 5 Si 7 2 2 5 6 0i 3i 3 27i 3 K i 9 9 9 i ji 62 2 2K i 9 i 62 2 7 2 5 Qi 7 2 2 5
i 7i 3 K i 2
and the entries ai j reduce to a11 7 1272 3 K 2 2 2 2 9 2 62 5
a12 7 2 1271 3 K 1 2 2 2 9 1 62 5
a13 7 1272 3 K 2 2 2 2 9 2 62 5
a14 7 2 1271 3 K 1 2 2 2 9 1 62 5
a15 7 21272 3 K 2 21 P2 5
a16 7 21271 3 K 1 21 P1 5
a17 7 21272 3 K 2 21 Q 2 5
a18 7 21271 3 K 1 21 Q 1 5
a25 7
2 c2 P2 5
a41 7
R2 5
a26 7
a42 7 R1 5
a46 7 a48 7 1 5
1 c1 P1 5
a31 7 02 a2 R2 5
a43 7 S2 5
a51 7 a2 5
a44 7 S1 5
a52 7 2a1 5
a32 7 01 a1 R1 5 a45 7 a47 7 21 5
a65 7 c2 5
a66 7 2c1 5
along with the following for an unbonded interface a72 7 21271 3 K 1 21 R1 5
a74 7 21271 3 K 1 21 S1 5
a76 7 1271 3 K 1 2 2 2 9 1 62 5 a81 7 1272 3 K 2 21 R2 5
a78 7 1271 3 K 1 2 2 2 9 1 62 5
a83 7 1272 3 K 2 21 S2 5
a85 7 1272 3 K 2 2 2 2 9 2 62 5
a87 7 1272 3 K 2 2 2 2 9 2 62 5
or (for a bonded interface) a71 7 1272 3 K 2 21 R2 5
a72 7 1271 3 K 1 21 R1 5
a73 7 1272 3 K 2 21 S2 5
a74 7 1271 3 K 1 21 S1 5 a75 7 1272 3 K 2 2 2 2 9 2 62 5 7 8 a76 7 2 1271 3 K 1 2 2 2 9 1 62 5 a77 7 1272 3 K 2 2 2 2 9 2 62 5 7 8 a78 7 2 1271 3 K 1 2 2 2 9 1 62 5 a81 7 a83 7 1 5 a82 7 a84 7 21 5
a85 7 P2 5
a86 7 P1 5
a87 7 Q 2 5
a88 7 Q 1
All other entries are zero. Expanding the determinant in equation (30) for an unbonded interface, the frequency equation for Stoneley waves yields
INTERFACE BETWEEN TWO MICROSTRETCH ELASTIC HALF-SPACES 1003 02 R2 3 01 R1 7 05
(31)
2 P2 3 1 P1 7 05
(32)
and 4 12
9 1 41 Z 1 1c2
c2 21 3 1 21
52182
4 3
9 2 42 Z 2 1c2
c2 12 2 2 3 2 22
52182 7 05
(33)
where 4 Z i 1c2 7 12 3 i 2 1 2 2
i2 7 13i 3 27i 289 i 5
c2 11 3 i 2 i2
5182 4 12
i2 7 7i 89 i 5
c2 2i 3 i i2
i 7 K i 87i 5
5182
4
c2 2 2 3 i 2 2 i
52 5
1i 7 15 22
i and i are the speeds of dilatational and shear waves (respectively) in medium Hi . Equation (31) shows a new velocity, which is not observed in micropolar elasticity and depends purely on microstretch elastic constants. Hence, the waves related to these modes may be called microstretch waves and refer to a hypothetical medium wherein only microstretch may occur. Similarly, equation (32) shows a new velocity, which is not observed in classical elasticity and depends purely on micropolarity constants. Hence, the waves related to these modes correspond to micropolar waves and refer to a hypothetical medium in which only rotation may occur. Next, expanding the determinant in equation (30) for a bonded interface, the frequency equation of Stoneley waves at a bonded interface between two microstretch elastic media yields equations (31), (32) and
9 2 22 M2 29 1 21 M1 29 2 22 12 3 2 2N2 29 1 21 12 3 1 2N1
M4 M3 21 1
7 05 2 2
9 2 2 12 3 2 2M4 9 1 2 12 3 1 2M3
29 M 9 M 2 2 2 1 1 1 2 1
21 1 N2 N1
(34)
where Mi
Ni
5 5182 4 4 c2 c2 2 3 i 2 2 5 Mi32 7 1 2 2 5 i i 3 i i2 5182 4 c2 7 12 5 1i 7 15 22 11 3 i 2 i2 7
The presence of equations (31) and (32) in both cases indicates that the modes corresponding to microstretch and micropolar waves are independent of the bonded or unbonded nature of the interface.
1004 S. K. TOMAR and D. SINGH
5. PARTICULAR CASES 5.1. Micropolar/Micropolar Unbonded Interface
If we neglect microstretch effects from both the half-spaces then we are left with the problem of Stoneley waves at a micropolar-micropolar interface. In the limiting case, when the quantities 30i 5 31i and 0i approach zero, we can see that in case of an unbonded interface, equation (31) is automatically satisfied and equations (32) and (33) represent the frequency equations for Stoneley waves at micropolar/micropolar unbonded interface. These equations are the same as those obtained by Tajuddin (1995) for the equivalent problem. 5.2. Micropolar/Micropolar Bonded Interface
It is easy to see that in absence of microstretch, equation (31) is automatically satisfied and equations (32) and (34) represent the frequency equations for Stoneley waves at micropolar/micropolar bonded interface. 5.3. Elastic/Elastic Unbonded Interface
If microstretch and micropolarity effects are neglected from both half-spaces then we are left with the problem of wave propagation at an unbonded interface between two uniform elastic half-spaces. For this case, making the quantities 30i 5 31i 5 0i 5 K i 5 i 5 i and i equal to zero in the frequency equations (31) to (33), we see that equations (31) and (32) are automatically satisfied, while equation (33) reduces to 9 2 42 9 1 41 Z 1c2 3 1 4 5 4 5182 Z 2 1c2 7 05 182 c2 c2 12 2 12 2 1 2
(35)
where 4 Z i 1c2 7
c2 22 2 i
52
3
c2 24 12 2 i
6182 3 6182 c2 12 2 5 i
1i 7 15 22
Equation (35) matches exactly with Murty (1975b)’s equation (1) for the relevant problem. 5.4. Elastic/Elastic Bonded Interface
Proceeding in a similar way as in Section 5.3., we can obtain the following frequency equation for Stoneley waves at a bonded interface between two uniform elastic half-spaces:
INTERFACE BETWEEN TWO MICROSTRETCH ELASTIC HALF-SPACES 1005
182 182
1 2 1 2 2
2 2 c2 2 c2
29 2 c2 2 2 22 2 2 229 1 2 1 2 9 c 29 2 2 1 2 2 1 1 1 2 1
182 182
c2 c2
1 2 1 2 1 1
22 21
7 0
29 2 1 2 c2 182 29 2 1 2 c2 182 9 2 2 2 c2
2 c2 9 2 2 2 2 2 2
2 2 1 2 1 1 2 1 2 1 2 1
182 182
c2 c2 1 2 2 21 1 2 1 2 2
2
1
Which exactly matchs with Stoneley (1924)’s equation (23) for the relevant problem 5.5. Rayleigh Waves in a Microstretch Elastic Half-space
If the upper half-space is totally neglected, then we are left with the problem of Rayleigh wave propagation at the free boundary surface of a microstretch elastic solid half-space. In this case, the relevant boundary conditions would be [ zz ]1 7 [ zx ]1 7 [m zy ]1 7 [ z ]1 7 0 at z 7 0 Using the requisite quantities and expressions in these boundary conditions, we obtain four homogeneous equations with four unknowns. The condition for non-trivial solution of these is b31 1b12 b23 2 b13 b22 2 2 b32 1b11 b23 2 b13 b21 2 7 05
(36)
where b11 7 21271 3 K 1 2 P1 5 b12 7 21271 3 K 1 2 Q 1 5 b13 7 [1271 3 K 1 2 2 2 9 1 62 ]11 2 b1 S1 25 b21 7 1271 3 K 1 2 2 2 9 1 62 7 b22 5 b23 7 2S1 1271 3 K 1 2 11 2 ba11 25 b31 7 a1 R1
1 c1 P1 5 b32 7 1 d1 Q 1 and P1 5 R1 5 Q 1 and S1 are those defined in equations (22)–(25). Equation (36) represents the frequency equation for Rayleigh waves at the free boundary of a microstretch half-space, and is the same as that given by Eringen (1999: p. 264, equation 6.6.201 replacing 31 with 31 83 and 30 with 30 83, as there is difference in the notation) 5.6. Rayleigh Waves in a Micropolar Half-space
If we neglect the microstretch effect in case Section 5.5., we are left with Rayleigh waves at the free surface of a micropolar elastic half-space. Thus when 311 7 301 7 01 7 0, equation (36) reduces to 1c1 P1 2 d1 Q 1 2[1271 3 K 1 2 2 2 9 1 62 ]2 2 P1 Q 1 S1 1271 3 K 1 22 2 1c1 2 d1 2 7 0
(37)
where S12 7 2 2
9 1 62 31 3 271 3 K 1
and P1 and Q 1 are as defined in equations (22) to (25). Equation (37) is the frequency equation for Rayleigh waves at free surface of a micropolar half-space. This equation matches with the equation given by Eringen (1999: pp. 179, 5.16.6) for the relevant problem.
0.56
Solid Curve - Stoneley waves at unbonded interface Dashed Curve - Rayleigh waves at free surface
Micropolar
Non-dimensional phase velocity
0.54
0.52
0.50
Microstretch
0.48 0.0
0.4
0.8
1.2
Non-dimensional wavenumber
1.6
2.0
Figure 1. Dispersion curves for Stoneley and Rayleigh waves.
0.52
Non-dimensional phase velocity
0.51
III
0.50 I
0.49 II
IV
0.48
0.47 0.0
0.4
0.8
1.2
Non-dimensional wavenumber
1.6
2.0
Figure 2. Effect of microstretch parameter on dispersion curves of Stoneley wave at unbonded interface between microstretch solid half-spaces (Curve I: O 01 = O 02 = 0.01, Curve II: O 01 = O 02 = 0.25, Curve III: O 01 = O 02 = 0.5, Curve IV: O 01 = O 02 = 0.6).
1006 S. K. TOMAR and D. SINGH
6. NUMERICAL RESULTS AND DISCUSSIONS In order to provide numerical examples, we recreated the model using the bisection method in the form of a FORTRAN program. The following values of relevant elastic parameters have been taken. In the elastic half-space H1 : 31 7 7583 6 1011 dyne8cm2 5 71 7 633461011 dyne8cm2 5 K 1 7 0014961011 dyne8cm2 5 301 7 003461011 dyne8cm2 5 311 7 0035 6 1011 dyne8cm2 5 01 7 0035 6 1011 dyne5 1 7 0289 6 1011 dyne5 j1 7 000625 cm2 5 9 1 7 12 gm8cm3 In the elastic half-space H2 : 32 7 6653 6 1011 dyne8cm2 5 72 7 582361011 dyne8cm2 5 K 2 7 0014061011 dyne8cm2 5 302 7 003261011 dyne8cm2 5 312 7 0032 6 1011 dyne8cm2 5 02 7 0034 6 1011 dyne5 2 7 0267 6 1011 dyne5 j2 7 000515 cm 2 and 9 2 7 11 gm8cm3 We have solved the frequency equation (30) for Stoneley waves at an unbonded interface for different values of non-dimensional wavenumber d, where d is an entity with the dimension of length. We have also solved frequency equations (36) and (37) to obtain the dispersion curves for Rayleigh waves at the free boundary of a microstretch solid half-space and at the free boundary of a micropolar solid half-space (respectively) using values of relevant elastic parameters given for half-space H1 above. It was found that both Stoneley and Rayleigh waves are dispersive in nature for certain initial ranges of parameter d. Figure 1 depicts the variation of the non-dimensional phase velocity c8V1 1V12 7 c12 3c32 2 with d. The solid curves depict to the dispersion curves for Stoneley waves, while the dotted curves are the dispersion curves for Rayleigh waves. It can be seen from this figure that the phase velocity of Stoneley waves at an unbonded microstretch/microstretch interface increase with the wavenumber in the range 01 d 20, beyond which it remains almost constant (i.e., independent of wavenumber). However, the phase velocity of Stoneley waves at an unbonded micropolar/micropolar interface first increases with d in the range 0 d 036 and then decreases with increasing d in the range 036 d 20, after which, it also remains almost constant. We also note that the dispersion curve for Rayleigh waves at the free surface of microstretch elastic half-space closely resembles the dispersion curve of Stoneley waves at an unbonded microstretch/microstretch interface. Similarly, the dispersion curve of Rayleigh waves at the free surface of a micropolar elastic half-space is similar to the dispersion curve of Stoneley waves at an unbonded micropolar/micropolar interface. Thus, we conclude that Stoneley waves are dispersive at unbonded microstetch/microstretch and unbonded micropolar/micropolar interfaces only for small values of the wavenumber. For higher values of the wavenumber, both Stoneley and Rayleigh waves are almost constant and hence almost non-dispersive. It can be seen that there is significant difference between the dispersion curves for Stoneley wave propagation at an unbonded micropolar/micropolar interface and at an unbonded microstretch/microstretch interface in the wavenumber range 01 d 20. This difference is due to the microstretch property of the half-spaces, which is responsible for lowering the phase velocity of Stoneley wave in this range of d. A similar conclusion can be infered about the Rayleigh wave dispersion curve. Figure 2 shows the effect of the microstretch parameters 30i on the dispersion curves of Stoneley wave at an unbonded interface between two microstretch elastic half-spaces. We observe that as the values of these parameters increase, the phase velocity of Stoneley waves decreases for a certain initial range of non-dimensional wavenumber. Curves I to IV indicate
INTERFACE BETWEEN TWO MICROSTRETCH ELASTIC HALF-SPACES 1007 the dispersion curves of Stoneley wave propagation at 30i 7 0015 30i 7 0255 30i 7 05 and 30i 7 06 respectively, showing that the microstretch property has significant effect on Stoneley waves.
7. CONCLUSION A mathematical treatment was used to study surface wave propagation at the free surface of a microstretch elastic half-space and at an unbonded/bonded interface between two dissimilar microstretch elastic half-spaces. Eringen’s theory was employed to derive the frequency equations of Stoneley waves in a linear homogeneous and isotropic microstretch elastic medium. The closed form of the frequency equations were derived for Stoneley wave propagation at both unbonded and bonded interfaces between two microstretch half-spaces when some parameters corresponding to microstretch and micropolarity are very small. We conclude that: 1. Stoneley waves at unbonded and bonded interfaces between two microstretch elastic halfspaces are dispersive. 2. Rayleigh waves at the free surface of a microstretch elastic solid half-space and also at the free surface of a micropolar elastic solid half-space are likewise dispersive. 3. Numerical results reveal that the phase velocity of Stoneley waves at an unbonded interface between two micropolar elastic half-spaces is greater than that at an unbonded interface between two microstretch half-spaces for a certain range of the wavenumber, showing that microstretch properties have a significant effect in this range. For higher wavenumber values, no effect of microstretch properties was observed on either Stoneley or Rayleigh waves. 4. Several particular cases have been deduced from the general formulation, including the results of Tajuddin (1995), Murty (1975b) and Stoneley (1924). Acknowledgement: One of the authors (Dilbag Singh) is thankful to Council of Scientif ic and Industrial Research, New Delhi for providing f inancial assistance in the form of JRF for completing this study.
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