Propagation of shear waves in an irregular boundary of an anisotropic medium.

Chattopadhyay, A. ; Gupta, S. ; Sharma, V.K. 等

Introduction

Because of its closeness to the natural situation, the study of the effect of irregular boundaries on the propagation of waves in an elastic medium has gained much in importance. As the analytical treatment of the irregularities of the surface in entails formidable mathematical difficulties, most of the research workers particularly in this branch, concentrated their effort with considerable success in considering the cases of slightly curved surfaces of different types. References may be made to Sato [1], De Noyer [2], Mal [3], Bhattacharya [4] and Chattopadhyay [5-7]. Sato studied the propagation of Love waves in a layer with sharp change in thickness while De Noyer considered the same in a layer over a half space with sinusoidal interface. Mal studied the problem when the thickness of the layer abruptly increases throughout a certain length of the path. Chattopadhyay [5] discussed the effects of irregularities and nonhomogeneities in the crustal layer on the propagation of Love waves. Chattopadhyay [6] also studied the propagation of SH Waves in a viscoelastic medium due to irregularities in the crustal layer. Chattopadhyay and Kar [7] investigated the effect on SH waves due to the presence of initial stress in the crustal layer and irregularity in the interface between the upper crust and lower semi-infinite medium with a source in it. Chattopadhyay [8] obtained the dispersion equation of Love waves in transversely isotropic layer of non-uniform thickness lying over an isotropic elastic material due to explosions by the use of Green's function technique. Wolf [9] discussed the propagation of Love waves in an isotropic layer with irregular boundary. Chattopadhyay and Pal [11] discussed the propagation of SH waves in an anisotropic layer with irregular boundary, and the displacement fields are obtained. They determined the reflected field in the anisotropic layer when an SH wave is incident on an irregular boundary in the shape of triangular notch.

In this paper we have discuss the propagation of shear waves in a layered monoclinic medium overlying a monoclinic half space. The irregularity has been taken in the layer over the half-space in the form of a triangular notch. The perturbation method is applied to find the displacement field. Finally, as an application the result obtained is used to get the reflected field in anisotropic layer when the shear wave is incident on an irregular boundary in the shape of triangular notch. It is observed that the amplitude of this reflected wave decreases as the length of the notch decreases and there will be no reflection if the length of the triangular notch is an integral multiple of the wavelength.

Formulation

The transmitted field due to horizontally polarized shear waves incident on an irregularity of a monoclinic layer has been considered. The interface between the layer and half space is given by y=0. The strain displacement relations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where u, v, w are displacement components in the directions x, y, z respectively and [S.sub.i] (i=1,2 ... 6) are the strain component. The stress strain relations for rotating y -cut quarts are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [T.sub.i] {i=1,2 ... 6} are stress components and [C.sub.ij] = [C.sub.ji] {i, j = 1,2...6} are elastic constants.

For waves propagating in the z-direction and causing displacements in the xdirection only (fig 1), we assume that

[FIGURE 1 OMITTED]

u = u (y, z, t), v = w = 0

Equations of motion without body forces are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Where [rho] is the density of the medium.

Using (1) and (3), the relation (2) becomes

[T.sub.1] = [T.sub.2] = [T.sub.4] = [T.sub.4] = 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solution of the problem

Using (5) in (4) one gets equation of motion as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

for the upper medium. Similarly for lower medium,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [u.sub.1], [u.sub.2] are the displacement components in the x-direction [C.sub.ij], [C'.sub.ij] are the elastic constants and [rho], [rho]' are the densities for the upper layer of thickness H and the lower half space respectively.

The upper boundary may be described by

y = -H + bh (z) (8)

where h (z) = 0 for z [less than or equal to] -s/2, z [greater than or equal to] s/2 = f (z) for -s/2 [less than or equal to] z [less than or equal to] s/2

b is maximum amplitude of the irregular boundary. For physical reason we require that the scattered field has only outgoing waves at z = [+ or -] and y = [infinity].

Boundary conditions

The boundary conditions are as follows.

The Shear stress vanishes at the free surface and

At the welded contact, the displacements and tractions are continuous.

The boundary condition (1) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Now at the welded contact, we have

[([T.sub.6]).sub.1] = [([T.sub.6]).sub.2] at y=0 which gives

[C.sub.56] [partial derivative][u.sub.1]/[partial derivative]z + [C.sub.66] [partial derivative][u.sub.1]/[partial derivative]y = [C.sub.56] [partial derivative][u.sub.2]/[partial derivative]z + [C'.sub.66] [partial derivative][u.sub.2]/[partial derivative]y at y = 0 (10)

Finally as the displacements are continuous at y = 0, we obtain

[u.sub.1] = [u.sub.2] at y = 0 (11)

Considering [u.sub.1] (y,z,t) = [U.sub.1] [(y, z)e.sup.i[omega]t] the equation (6) becomes

[C.sub.66] [partial derivative].sup.2][U.sub.1]/[partial derivative][y.sup.2] + [2C.sub.56] [[partial derivative].sup.2][U.sub.1]/[[partial derivative]y[partial derivative]z + [C.sub.55] [[partial derivative].sup.2][U.sub.1]/[[partial derivative][z.sup.2] = -[rho][[omega].sup.2][U.sub.1],

Again putting U1 = [u.sub.1] = [u.sub.1]'(y)[[e.sup.-i[alpha]z] in the above equation, we have

[d.sup.2][u.sub.1]/[dy.sup.2] - b [du'.sub.1]/dy + [a.sup.2][u.sub.1] = 0

where 2

[a.sup.2] = 1/[C.sub.66] ([rho][[omega].sup.2] - [[alpha].sup.2] [C.sub.55]), b = 2i[alpha][C.sub.56]/[C.sub.66] = [ib.sub.1] (say)

Solution for the differential equation (12) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where M = [square root of ([b.sup.'2] - [4a.sup.'2])]

Similarly for the lower medium

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where M' = ([b.sup.'2] - [4a.sup.'2])], b' = 2i[alpha]C'.sub.56]/[C'.sub.66]Now by boundary condition (11), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the boundary condition (10), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Scattered field that is satisfied the wave equation (6) and (7) in the form of a contour integral in the complex V plane is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We take positive sign only,

Where [k.sup.2.sub.1] = [rho][[omega].sup.2]/[C.sub.66], k.sup.2.sub.2] = [rho]'[[omega].sup.2]/[C'.sub.66]

The contour C is given by figure below

[FIGURE 2 OMITTED]

Now by boundary condition (11) for Scattered fields, we have

[u.sub.1] = [u.sub.2] at y = 0

which gives

D(u) = B(u) + C(u) (17)

Also by (10) we have ([T.sub.6]).sub.1] = ([T.sub.6]).sub.2] at y = 0, which gives

B(u)+(u)/B(u)-(u) = [gamma]

where [gamma] = i[sigma].sub.1] [C.sub.66]/[[sigma].sub.2] [C'.sub.66] + iu ([C'.sub.56] - [C.sub.56) (18)

C(u) = [gamma]-1/[gamma]+1 B(u)

and

D(u) = 2[gamma]/[gamma]+1 B(u) (19)

Using value of C(u) and D(u) the expression for the total displacement field in the Monoclinic layer and half space may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Similarly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

These equations will satisfy the wave equation and boundary condition (10) and (11). Now using boundary condition (9) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the boundary has a small irregularity ((b << 1)) then the perturbation technique may be applied for the evaluation of B(u) where

B(u) = b[B.sub.1] (u) + [b.sup.2][B.sub.2](u)+[b.sup.3](u)+ ... (23)

Using (23) in (22) and collecting the terms containing the 1st order in b i.e. using B(u) =b[B.sub.1](u), and finally inverting we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using this in (25) and (26) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

The contribution of above integrals are given by the roots of

Ui[C.sub.56] ([gamma]cos [[sigma].sub.1] H + I sin [[sigma].sub.1]H)+[[sigma].sub.1][C.sub.66](i cos [[sigma].sub.1] H - [gamma] sin [[sigma].sub.1] H) = 0

This equation may be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

where [gamma] is given by (18)

All the roots of above equation are real and denoted by um. The integral of type (26) has been evaluated by Sezawa. In order to evaluate the integral we choose the contour as real axis and a semi circle of large radius in the upper half plane with necessary cut as the branch points k1,k2. We need to consider only the contribution from the branch point for the surface SH-motion.

The branch line integral diminishes as [z.sup.-3/2] become negligible for large z. Also the integrals over the arcs vanish at infinity.

Now

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[sigma].sub.1m](I=1,2) are evaluated at [u.sub.m].

If we restrict our attention far from the irregular boundary then the contribution of the branch line integral are small compared to the residue terms.

Therefore, for x > z and um < 0. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

where P and Q are given by (28).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

where R denotes the number of roots.

Similarly for x<z with the roots um>0 we have closed the contour in the lower half plane and as before we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Using (31), (32) in (24) and (25) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

Using h (z) = 0 for, z [less than or equal to] - s/2, z [greater than or equal to] s/2

= f (z) for s/2 [less than or equal to] z [less than or equal to] s/2

the displacement fields for z << - s/2 are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

Similarly we can write displacement fields for z >> s/2.

Application. We will apply our result to get the reflected fields in anisotropic layer when the shear wave is incident on an irregular boundary in the shape of triangular notch (Fig 1).

For the triangular notch

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have considered the layer in which only the first mode can propagate that is, as given by the relation (27). The second term of the equation (33) gives the reflected field in the anisotropic layer.

Since [u.sub.1] = [alpha], and [[alpha].sub.11] = [[beta].sub.1], then the reflected fields for the triangular notch is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conclusion

It is interesting to observe that the amplitude of this reflected wave depends on

[sin.sup.2] ([alpha]s/2)/s.

This implies that the amplitude decreases as the length of the notch increases, that is there is less reflection as the slope of the notch decreases. There will be no reflection if sin([alpha]/2) = 0. Also [alpha] = 2[pi]/[lambda] where [lambda] is the wavelength. When the length of the triangular notch is an integral multiple of the wavelength then there will be no reflected fields.

Particular Cases

Case- I: When [C.sub.56] = 0 and [C'.sub.56] =0, then equation (20) takes the form [B.sub.1]H = [[mu].sub.2][[beta].sub.2]/[[mu].sub.1][[beta].sub.1]

In isotropic medium, we have

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This is well-known equation for Love waves in a homogeneous isotropic layer over a homogeneous isotropic half space. This shows that in a layered isotropic medium, the wave propagation is same as Love waves.

Case--II. When [C.sub.56] = 0 and [C'.sub.56] = 0, [C.sub.66] = [[mu].sub.1], [C'.sub.66] = [[mu].sub.2], then equation (32) takes the form

tan [[sigma].sub.1]H = [[sigma].sub.2][[mu].sub.2]/[[sigma].sub.1][[mu].sub.1]

This is the dispersion equation of SH waves in Anisotropic Layer with Irregular Boundary which was obtained by Chattopadhyay and Pal [11].

References

[1] Sato, Y., 1952, "Study on surface waves. VI Generation of Love and other types of SH waves," Bull Earthqu. Res Inst., 30, 101

[2] Noyer, J. De., 1961, "The effect of variations in layer thickness of Love waves," Bull Seis Soc. Amer., 51, 227

[3] Mal, A. K., 1962, "on the frequency equation for love wave due to abrupt thickening of the crustal layer," Geof Pure. E. Appl. 52, 59

[4] Bhattacharya, J., 1962, "On the dispersion curve for Love waves due to irregularity in the thickness of the transversely isotropic crustal layer," Gerl Beitr Z Geophys. 71 324

[5] Chattopadhyay, A., 1975, "On the dispersion equations for Love waves due to irregularity in the thickness of non homogeneous crustal layer," Acta Geophysica Polonica, 23 307

[6] Chattopadhyay, A., 1978, "Propagation of SH waves in a viscoelastic medium due to irregularity in the crustal layer," Bull Cal Math Soc. 70 303

[7] Chattopadhyay, A., and Kar, B. K., 1978, "On the dispersion curves of Love type waves in an initially stressed crustal layer having an irregular interface," Geophysical research Bull 16, 13

[8] Chattopadhyay, A.,1978, "On the strong SH motion in transeversely isotropic layer of non-uniform thickness lying over an isotropic elastic material due to explosions," Ind J. Pure appl. Math., 9, 528

[9] Wolf, B., 1970 "Propagation of Love waves in Layers with irregular boundaries," Pure and Appl. Geophysics, 78, 48

[10] Sezawa, K., 1935 "Love waves generated from a source of a certain depth," Bull Earthqu. Res Inst., 13, 1

[11] Chattopadhyay, A., and Pal, A.K., "SH Waves in Anisotropic Layer with Irregular Boundary, 1982, "Bulletin De L'academie Polonaise Des Sciences serie des sciences techniques. Volume-30, pp. 5-6

A. Chattopadhyay, S. Gupta, V.K. Sharma and Pato Kumari

Department of Applied Mathematics

Indian School of Mines University, Dhanbad--826004, India

E-mail: shishir_ism@yahoo.com