Constructed Layered Systems: Measurements and Analysis

Copyright © 2012 by W. H. Cogill.

Library of Congress Control Number:        2012908805

ISBN:                  Hardcover                       978-1-4771-1283-0

ISBN:                  Softcover                        978-1-4771-1282-3

ISBN:                  Ebook                             978-1-4771-1284-7

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Contents

Chapter 1              SH-waves

Chapter 2             Single-layered inverse

Chapter 3             Inverse Problem

Chapter 4             Results of Measurements

Chapter 5             Computation

Chapter 6             Comments

Chapter 7             Program listings

Chapter 8             Experimental Results

Chapter 9             Computations

Chapter 10           Application of Vand’s formula

Bibliography

END NOTES

Chapter 1

SH-waves

1.1 Introduction

Horizontally-polarized shear waves (SH-waves) may be utilized for a purpose similar to that of waves of the Rayleigh type [3,10]. In this chapter, we consider the response to be expected from a system composed of two layers each having a constant thickness, and resting on an underlying medium which is semi-infinite in extent.

1.2 A system consisting of two layers overlying a semi-infinite medium

SH-fund

The system is defined in terms of cylindrical co-ordinates (r, θ, z) with the origin at the free surface and z measured positively downwards. The particle displacements are u and v in the r and θ directions respectively. The stresses are denoted by subscripts, for example, (pθθ)m, m = 1, 2, 3. The suffix 1 denotes quantities in the upper layer, 2 and 3 those in the lower layer and in the semi-infinite medium respectively.

From Nakano [18], the following expressions for the peak values of the shear stresses and displacements are obtained:

(a) In the surface layer,

(1.1)

(1.2)

(1.3)

(1.4)

(b) In the second layer,

(1.5)

(1.6)

(1.7)

(1.8)

(c) In the semi-infinite medium, we take s3 to be positive in order that the displacements may vanish as the value of z tends to infinity,

(1.9)

(1.10)

(1.11)

(1.12)

In writing equations (1.9) to (1.12) one boundary condition, that of zero displacement at infinite depth, has already been taken into account. The arbitrary constants A1, A′1, A2, A′2, and A3 can be eliminated using the conditions at the interfaces within the system. The boundary condition at the free surface, namely that the shear stresses are zero in the r, θ plane, is not required.

At the first interface, the stresses and displacements are continuous[7], i.e. when z = f

u1 = u2, v1 = v2, (1.13)

(pzr)1 = (pzr)2, (pθz)1 = (pθz)2

At the second interface, the stresses and displacements are continuous, i.e. when z = f + g

u2 = u3, v2 = v3, (1.14)

(pzr)2 = (pzr)3, (pθz)2 = (pθz)3

Owing to similarities between equations (1.1) to (1.4) and (1.5) to (1.8), the conditions relating to displacement and shear stress lead to only one equation.

The use of the boundary conditions (1.13), with equations (1.1) to (1.4) and (1.5) to (1.8), leads to the following equations:

(1.15)

and

(1.16)

The use of the boundary conditions (1.14) with equations (1.5) to (1.8) and (1.9) to (1.12), leads to the following equations:

(1.17)

and

(1.18)

As one of the arbitrary constants (A3) is known to be non-zero, A1, A′1, A2, A′2, and A3 can be eliminated from equations (1.15), (1.17) and (1.18), and the following wave velocity equation is obtained [21]:-

(μ3s3 cosh s2g + μ2s2 sinh s2g) μ2s2 cosh s1f

+ (μ3s3 sinh s2g + μ2s2 cosh s2g) μ1s1 sinh s1f = 0 (1.19)

Several forms of equation (1.19) are possible. These depend on whether the displacements are considered to vary exponentially or in an oscillatory manner with the depth. From (1.19), there is no possibility of oscillatory variation in all three media, as the equation yields no solution if s1, s2, and s3 are all considered imaginary.¹

1.3 Forms of Equation (1.19)

Four forms of equation (1.19) are considered.

Cases14

1.3.1 Case 1

Write Image14977.EPS . From equation (1.19) there is obtained

Put Image14989.EPS , then

.

On writing Image15001.EPS , s2, and s3 in full and re-arranging, there is obtained

(1.22)

1.3.2 Case 2

Write Image15013.EPS . From equation (1.19) there is obtained

(1.23)

Put Image15025.EPS , then

.

On writing s1, Image15037.EPS and s3 in full and re-arranging, there is obtained

(1.24)

1.3.3 Case 3

Write Image15049.EPS , Image15055.EPS . From equation (1.19) there is obtained

. (1.25)

Put Image15067.EPS , then

.

On writing Image15079.EPS , Image15085.EPS , and s3 in full and re-arranging, there is obtained

(1.26)

1.3.4 Case 4

From equation (1.19) there is obtained

. (1.27)

Put Image15103.EPS , then

.

On writing s1, s2, and s3 in full and re-arranging, there is obtained

(1.28)

1.4 Matrix Methods

Most systems which are of practical importance are more complex that those considered in the first part of this chapter. They contain layers of materials having a wide range of stiffnesses. The layers may be interspersed in any order of stiffness. The deepest layer of which the system is composed is usually regarded as extending to infinite depth and is represented by a semi-infinite medium. A method is required of calculating the phase velocities of SH waves at the free surfaces of these structures. It is expected that the phase velocities will vary with the frequency. The phase velocities will be complex quantities if leaking modes are considered (Su and Dorman [25]).

The method proposed by Thomson [26] will be followed, and use will made of the notation employed by Thrower [27]. The component materials are considered as purely elastic, and leaking modes are not considered. The vector representing the shear stress pθz and the displacement v in the m-th medium from the free surface can be written

Image15121.EPS (1.29)

The peak value of the shear stress can be written (Nakano [18])

(1.30)

, (1.31)

where S = A′ + A and S′ = A′-A, and the peak value of the displacement can be written

(1.32)

. (1.33)

Here C′ is some circular function, the exact form of which is not important as it cancels when the boundary conditions are applied. As the trajectory of the particle motion is horizontal and purely tangential to the direction of propagation, only one shear stress and one displacement need be considered. This suffices to determine the stresses and displacements throughout the system.

In matrix notation

Image15151.EPS (1.34)

where

Image15157.EPS and (1.35)

(1.36)

where the suffix m denotes that the quantities are those which apply to the m-th medium below the free surface of the structure. Placing the origin of the coordinates at the (m-1)-th interface, the expression for the stress-displacement vector Image15169.EPS , within the m-th layer at the (m-1)-th interface is

(1.37)

where Image15181.EPS is derived from Image15187.EPS by putting z = 0. The stress-displacement vector Image15193.EPS within the m-th layer at the m-th interface is

(1.38)

where Image15205.EPS is derived from Image15211.EPS by putting z = Hm, the thickness of the m-th layer. Equation (1.37) can be written

(1.39)

The vectors Image15223.EPS and Image15229.EPS , representing the stresses and displacements at the top and bottom of the m-th layer, are therefore related by

(1.40)

It is assumed that there is