Foundations of Engineering Mechanics

Foundations of Engineering Mechanics B. Grigori Muravskii, Meehanies of Non-Homogeneous

and Anisotropie Foundations

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B. Grigori Muravskii

Meehanies ofNonHomogeneous and Anisotropie Foundations Translated by Boris Krasovitski

With 149 Figures

,

Springer

Series Editors:

J. Wittenburg Universität Karlsruhe (TH) Institut für Technische Mechanik Kaiserstr. 12 76128 Karlsruhe Germany

Vladimir I. Babitsky, DSc Louborough University Department of Mechanical Engineering LEII 3 TU Loughborough Leicestershire United Kingdom

Author: Grigori B. Muravskii Geotechnical Department Faculty of Civil Engineering Technicon 32000 Haifa

Israel Translator: Boris Krasovitski POB7843 36811 Nesher Israel

ISBN 978-3-642-53602-1 Library of Congress Cataloging-in-Publieation Data Muravskii, Grigori B: Meehanics of non-homogeneous and anisotropie foundations I B. Grigori Muravskii. Translated by Boris Krasovitski. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001 (Foundations of engineering meehanics) ISBN 978-3-642-53602-1 ISBN 978-3-540-44573-9 (eBook) DOI 10.1007/978-3-540-44573-9

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Preface

Although realistic soil and rock foundations reveal noticeable deviations in their properties from homogeneity and isotropy, the model of the homogeneous isotropie elastic half-space is widely used when studying static and dynamie interactions between a defonnable foundation and structures. This is explained by significant mathematieal difficulties inherent in problems conceming mechanies of anisotropie and heterogeneous elastic bodies. Solving the basic static and dynamie problems for heterogeneous and anisotropic half-spaces, such as different contact problems and problems of constructing Green's functions, has become possible in the last few decades due to the development of computer engineering techniques and numerical methods. This book contains the results of investigations in the area of statics and dynamies of heterogeneous and anisotropic foundations, carried out by the author in the last five years while working in the Faculty of Civil Engineering at Technion - Israel Institute of Technology. The book is directed at engineers and scientists in the areas of soil mechanics, soil-structures interaction, seismology and geophysics. Some characteristic features of the book are: i) Constructing (Chap.l) solutions in a general fonn for the heterogeneous (in the depth direction) transversely isotropic elastic half-space subjected to different loadings, hannonic in time. Characteristics of the given half-space have an influence on functions (of depth z and parameter k of Hankel's transfonns), which are detennined from a system of ordinary differential equations. ii) New dynamic solutions relating to the homogeneous transversely isotropic elastic half-space subjected to the action of vertical and horizontal forces applied to the half-space surface or below the surface. Solutions are presented relating to basic contact problems for this kind ofhalf-space. iii) Dynamic and static solutions for the linearly heterogeneous half-space, which, as applied to static problems, was intensively studied in works by R. E. Gibson and his coworkers. Solutions are presented for a half-space with exponentially varying stiffuess. In these cases it is possible to find analytical solutions for the above-mentioned ordinary differential equations; thus the required amplitudes of vibrations are expressed in the fonn of integrals containing the detennined functions. iv) Development ofnumerical-analytical methods for the considered problems, which consist of a combinations of integral representations of the sought functions and numerical treatment of ordinary differential equations for the

VI

Preface

corresponding Fourier-Bessel transfonnations. The piecewise constant approximation for varying coefficients of these differential equations results in the well-known method of thin layers. Employing the general representations developed in Chap.l and using simple solutions in the intervals (elements) with constant properties allows the construction of effective solutions. Another appropriate treatment is the use of Runge-Kutta method with additional application of Gogunov's method to eliminate computational difficulties connected with large values of the transfonnation parameter k. The book presents, graphically, a large number of results of computations, which give a clear pieture of the behavior of the mechanical systems considered. These results can serve for estimating accuracy of different simplified methods, e.g. combining the division of a half-space into thin layers with setting a fonn of displacements distribution within the layers. I express my gratitude to my colleagues - Professors ofTechnion R. Baker and s. Frydman for useful discussions of geotechnical problems connected with the material of the book. Haifa, April 2001.

G. B. Muravskii

Contents

Introduction ................................................................................. 1 Chapter 1. General Solutions of Harmonie Vibrations in Heterogeneous Transversely Isotropie Half-Spaee .................•.................................... 5 1.1 Basic Relationships of Theory of Elasticity for Transversely Isotropic Body ......................................................................................... 5 1.2 Particular Solutions for Transversely Isotropie Heterogeneous Elastic Medium ..................................................................................... 8 1.3 Statement of Boundary Conditions for Planes z = const ....................... 13 1.4 General Solution for Harmonie Vibration ofHalf-Space Subjected to Surface Loads ......................................................................... 18 1.5 General Solution for Harmonie Vibrations of Half-Space Subjected to Loads Applied below its Surface .................................................. 31 1.5.1 Action ofVertieal Force ................................................... .31 1.5.2 Action ofHorizontal Force ................................................ .34 1.6 Application ofFunctions Related to Dilatation and Rotation ofDisplacement Field .................................................................. .37 1.6.1 Vertieal Force Applied within Half-Space ............................... .40 1.6.2 Horizontal Force Applied within Half-Space ............................ .42 1.7 Application ofSuperposition Principle to Loadings ofRectangular and Circular Domains .................................................................. .44 1.7.1 Vertical Load Distributed Uniformly over Rectangular Domain ...... .44 1.7.2 Horizontal Load Distributed Uniformly over Rectangular Domain ... 50 1.7.3 Vertical Load Distributed Uniformly over Circular Domain ........... 53 1.7.4 Horizontal Load Distributed Uniformly over Circular Domain ....... 54 1.7.5 Axisymmetric Radial Load Applied to Circular Domain ............... 57 1.7.6 Antisymmetric Vertical Load Applied to Circular Domain ............. 58 1.7.7 Horizontal Load Acting in Tangential Direction ......................... 61 1.7.8 Self-Balanced Horizontal Load ............................................ 62 1.7.9 Loading ofInfmite Strip ................................................... 65 Chapter 2. Statie and Dynamic Problems for Homogeneous Transversely Isotropie Half-Space ..•......•.•........••..•.......•.•.....................••.••....•..... 68 2.1 Vibrations ofHalf-Space Subjected to Vertical Force Applied to Half-Space Surface .................................................................. 68 2.1.1 Static Action ofVertical Force ............................................. 72

VIII

Contents

2.1.2 Free Vibrations ofHalf-Space .............................................. 76 2.1.3 Forced Vibrations ofHalf-Space ........................................... 77 2.2 Model ofDefonnable Foundation as Limiting Case ofTransversely Isotropic Half-Space .................................................................... 88 2.3 Vibrations ofHalf-Space Subjected to Action ofHorizontal Force Applied to Half-Space Surface .................................................................. 90 2.3.1 Static Action ofHorizontal Force .......................................... 92 2.3.2 Analysis of Amplitudes ofVibrations ..................................... 94 2.4 Torsional Vibrations ofHalf-Space .............................................. 96 2.5 Action ofForce Applied in Infinite Space ....................................... 99 2.5.1 Action ofVertical Force .................................................... 99 2.5.2 Action ofHorizontal Force ................................................ 102 2.6 Vibrations ofTransversely Isotropic Half-Space under Action ofForce Applied within Half-Space ................................................. 106 2.6.1 Action ofVertical Force ................................................... 106 2.6.2 Action ofHorizontal Force ................................................ 111 2.7 Contact Problems for Transversely Isotropic Half-Space .................... 116 2.7.1 Static Stiffnesses for Circular Disk on Transversely Isotropic Half-Space .......................................................................... 116 2.7.2 Dynamic Stiffnesses for Circular Disk on Transversely Isotropic Half-Space .......................................................................... 135 2.8 Plane Problems for Transversely Isotropie Half-Space ....................... 147 2.8.1 Action of Vertical Load Distributed Unifonnly along Infinite Line on Half-Space Surface ...................................................... 148 2.8.2 Action ofHorizontal Load Distributed Unifonnly along Infmite Line on Half-Space Surface ...................................................... 151 2.8.3. Contact Problems for Strip Stamp ....................................... 154

Chapter 3. Mechanics ofIsotropic Half-Space with Sbear Modulus Varying Linearly with Depth .•.•....•.•.•••••...••••...••••.•••••.•.•.•.•••••.••••.•.•..•..••••..• 168 3.1 Fundamental Solutions ofSystem ofEquations (1.l30}--(1.133) ........... 168 3.2 Vibrations ofIsotropic Linearly Heterogeneous Half-Space under Action ofVertical Force Applied to Half-Space Surface .......................... 174 3.3 Vibrations ofIsotropic Linearly Heterogeneous Half-Space under Action ofHorizontal Force Applied to Half-Space Surface ...................... 179 3.4 Determining Properties ofLinearly Heterogeneous Half-Space Using Characteristics ofSurface Waves ..................................................... 183 3.4.1 Application ofSolution Related to Vertical Vibrations of Half-Space Surface under Action of Vertical Force ....................... 184 3.4.2 Application ofSolution Related to Horizontal Vibrations ofHalf-Space Surface under Action ofHorizontal Force .................... 191 3.5 Some Static Problems for Linearly Heterogeneous Half-Space ............. 194 3.5.1 Displacements ofHalf-Space Surface under Action of Surface Concentrated Forces .................................................. 194 3.5.2 Static Stiffnesses for Circular Disk Resting on Isotropic

Contents

IX

Linearly Heterogeneous Half-Space ............................................. 206 3.6 Dynamic Stiffness ofCircular Disk Resting on Linearly Heterogeneous Half-Space ............................................................ 214 3.7 Vibrations ofLinearly Heterogeneous Half-Space Subjected to Force Applied within Half-Space .................................................230 3.7.1 Action ofVertical Force ................................................... 230 3.7.2 Action ofHorizontal Force ................................................ 238 3.8 Plane Problems for Linearly Heterogeneous Half-space ..................... 244 3.8.1 Action ofVertical Load Distributed Uniformly over Infinite ........... . Straight Line on Half-Space Surface .. , .................................... '" ...244 3.8.2 Action ofHorizontal Load Distributed Uniformly over Infmite Straight Line on Half-Space Surface ............................................ 246 3.8.3 Static Surface Green's Functions in Plane Problems for Linearly Heterogeneous Half-Space ........................................ 249 Chapter 4. Mechanies of Transversely Isotropie Half-Spaee with Stiffness Varying Exponeotially with Depth •.••••.•..•••••••••••••••••.•.•.•••.....•.••.••..•••257 4.1 Vibrations ofTransversely Isotropie Half-Space Having Elastic Coefficients Bounded at Infinite Depth ...........................257 4.1.1 Variation ofElastic Parameters ofHalf-Space with Depth .......... .257 4.1.2 Construction of Fundamental Solutions .................................. 261 4.1.3 Vibrations ofHalf-Space under Action ofVertical Force Applied to Half-Space Surface ......... '" ....................................... 265 4.1.4 Vibrations ofHalf-Space under Action ofHorizontal Force Applied to Half-Space Surface ................................................... 268 4.2 Vibrations ofTransversely Isotropie Half-Space with Stiffness Increasing without Bounds ............... '" ...............................270 4.2.1 Varying ofElastic Parameters ofHalf-Space with Depth ............. 270 4.2.2 Construction ofFundamental Solutions ..................................272 4.2.3 Vibrations ofHalf-Space under Action ofVertical Force Applied to Half-Space Surface ................................................... 276 4.2.4 Vibrations ofHalf-Space under Action ofHorizontal Force Applied to Half-Space Surface ...................................................280 Chapter 5. Applieatioo of Numerieal--Analytieal Methods to Statie and Dyoamie Problems for Heterogeneous Half-Spaee .•.••••••••.•.••••.••••••••..•.•281 5.1 Introduction ........................................................................ 281 5.2 Heterogeneous Half-Space Subjected to Vertical Force Applied to Half-Space Surface ........................................................284 5.2.1 Application ofThin Layer Technique for Numerical Solution of Differential Equations ........................................ '" ....285 5.2.2 Application ofRunge-Kutta Method for Numerical Solution ofDifferential Equations .............................................. .297

X

Contents 5.2.3 Parameter Determination for Isotropie Half-Space with Shear Modulus Varying by Power Law (Action ofVertical Force) ............... 299 5.3 Heterogeneous Half-Space Subjected to Horizontal Force Applied to Half-Space Surface ............................................ , ......... 306 5.3.1 Application ofThin Layer Technique for Numerical Solution ofDifferential Equations ............................................. .307 5.3.2 Application ofRunge-Kutta Method for Numerical Solution of Differential Equations ......................................................... 31 0 5.3.3 Parameter Determination for Isotropie Half-Space with Shear Modulus Varying by Power Law (Action ofHorizontal Force) ............. 311 5.4 Vibration Problems for Heterogeneous Half-Space Subjected to Force Applied within Half-Space ................................................ 315 5.4.1 Action ofVertical Force. Application ofThin Layer Technique for Numerical Solution ofDifferential Equations ............... 315 5.4.2 Action ofVertical Force. Application ofRunge-Kutta Method for Numerical Solution of Differential Equations .................. 321 5.4.3 Action ofHorizontal Force. Application ofThin Layer Technique for Numerical Solution ofDifferential Equations ............... 324 5.4.4 Action ofHorizontal Force. Application ofRunge-Kutta Method for Numerical Solution ofDifferential Equations .................. 327 5.5 Static solutions ..................................................................... .328 5.6 Contact Problems for Circular Disk Resting on Half-Space with Stiffness Increasing with Depth by Power Law ................................... .341 5.6.1 Static Stiffnesses for Circular Disk. ..................................... .341 5.6.2 Dynamic Stiffnesses for Circular Disk ................................. .350

References ..................................................................................356 Index ....................................................................................... 362

Introduction

The need to consider the heterogeneity and anisotropy of materials when solving various static and dynamic problems is common in engineering practice, especially when considering the properties of soil foundations. Numerous experimental results prove that the in-situ properties of soil and rock foundations differ from those of isotropie homogeneous media [25, 32, 48, 65, 88, 92, 96, 119]. The influence of the heterogeneity and anisotropy of soils on their strains and stresses has been a subject of concern, primarily in static problems [30, 44, 51, 126]. At the same time, the first reports appear, which deal with the dynamies of the heterogeneous isotropie elastic half-space (the medium is supposed to be heterogeneous in the vertical direction, i.e. in the direction normal to the boundary ofthe half-space). A comprehensive review ofthese reports is presented in [29]; in most cases, the authors deal with the free vibrations of a half-space, especially with the influence of different types of heterogeneity on characteristics of the Rayleigh and Love waves. A commonly accepted approach that enables heterogeneity to be incorporated into dynamic problems has been developed in [50, 111]. This approach is based on considering the elastic foundation as a set of homogeneous layers ([49, 62, 68] and others). For each layer, the solutions are relatively simple, while the complete solution of the problem is derived by employing contact conditions between the layers. Possible complications of this method are due to some terms in the solution for separate layers that grow exponentially, which can lead to an unacceptable loss of accuracy in the constructed total solution. However, these complications may be prevented following some special techniques [4, 9, 21, 27, 41, 62, 70, 83, 131]. The method using by stiffness matrices was shown to be effective in preventing the above-mentioned computational difficulties [11-13, 57]. The author of [11-13] presents abasie technique for solving dynamic problems in transversely isotropie layered half-spaces, considering initial stresses in the material. Following a commonly accepted approach to the problems of the mechanies of layered foundations, the equations of motion (in Fourier's or Hankel's transformation space) are treated in the vertical direction using a fmite-element technique ([58, 59, 60, 66, 72, 73, 85, 102, 108, 122-124] and others). These methods have been reported to be suitable for determining Rayleigh and Love wave parameters, when applied to the problems with free vibrations, and in some illustrative examples for vibrations under a given loading. However, the question concerning the range of where the finite-element method could be applied in the problems that require integrating by the transformation parameter (wave number) is still an open question. As wave numbers increase, the approximation of the

B. G. Muravskii, Mechanics of Non-Homogeneous and Anisotropic Foundations © Springer-Verlag Berlin Heidelberg New York 2001

2

Introduction

displacements field by linear or parabolic functions deteriorates; this fact, for example, results in a singularity of Green's functions at those points where the functions are finite for the corresponding exact solution. Compared with the problems for layered foundations, there is a significantly smaller number of reports concerning continuous heterogeneity. This is due to mathematical complications that appear in this case. Differential equations that contain the parameter of Fourier's or Hankel's transformation for functions of z-argument (that varies in the direction of the foundation depth) have explicit analytieal solutions only for limited types of heterogeneity, even in the isotropie cases, but especially in dynamic problems. Intensive studies of the elastic foundations with the shear modulus, which depends linearlyon the depth of foundation, commenced with the static problems solved by Gibson [39], and continued in the following publications ([8, 14-16,20,22-24, 55, 56]). Dynamic problems have been solved for a linearly heterogeneous half-space with Poisson's ratio v = 0.5 [6, 7, 78, 79, 106, 112], and for all acceptable values ofv [82, 114]. Note that in [106, 112, 114], the authors consider free vibrations of a half-space. Free shear vibrations are a subject of consideration in [112, 113]. Static problems have been solved for power-Iaw behavior of shear modulus versus depth of half-space (G = G1Z U ), beginning with the zero value at the surface of the half-space [93, 98-100], [51,55,63,64, 76]. In the latter group ofreferences, the following assumption has been employed: the value of a and Poisson' s ratio v are related as av = I-2v. This relationship provides a radial distribution of stresses in the half-space and enables a solution to be constructed for an axially symmetric case even with a non-zero shear modulus at the half-space surface [22, 56]. Some alternative relationships between v and a that lead to relatively simple solutions were suggested in [89, 90]. In dynamic problems, the power-Iaw function for shear modulus versus depth enables analytical solutions to be buHt for corresponding the Fourier's or Hankel's transformations only for a = 1, or for a = 2, v = 1/4 [46,47] (for shear vibrations, analytical solutions exist, in addition, for a = 0.5 [113] and for some other values). For a = 2, v = 114, aseries of static and dynamic problems have been investigated in the 2-D framework [31, 33]. An axially symmetrie dynamie eontact problem has also been solved [47]. Note that, in reality, the shear modulus increases with depth slower than by the quadratic law. Nevertheless, these results are important, as they provide a valuable estimate of the qualitative and quantitative effects that occur due to heterogeneity. In addition, they lead to the possibility of modeling an arbitrary heterogeneous foundation as a layered foundation with a parabolic approximation of the shear modulus within each layer. This option enables the discontinuity in material properties to be avoided at the boundaries of separate layers, which appears as a result of a rougher approximation with homogeneous layers [31]. An exponential relationship is an alternative type of function for shear modulus versus depth, wh ich makes it possible to derive efficient analytical solutions. In dynamic problems, an exponential relationship has been chosen to describe the variation of shear modulus with depth in [127], where the author considers Lovetype waves in the half-space; for the problem of anti-plane vibrations of a

Introduction

3

semi-plane with shear modulus, exponentially deereasing with depth [38]; and for the problem of vibrations of the half-spaee, subjeeted to a eoneentrated force (vertieal and horizontal) at the surfaee ofthe half-spaee [80]. A relationship for the shear modulus and for the density of the medium that contains a deereasing exponent, and, therefore, provides limited values of these parameters with inereasing depth, was suggested in [94, 95] (in one partieular ease, density ean be constant). For the type of heterogeneity considered, analytical solutions may be derived in the form of apower series; solutions of numerous problems are presented in reports [81, 116-121]. A speeifie ease ofseparation ofthe equations ofmotion with a = 2, v = 1/4 [46, 47] is not unique. Other eases have been the subjeet of investigations elsewhere [3, 52,54], although, in order to provide the possibility of separation of equations, a special form of relationship between the density and elastie parameters of the medium was required. In some eases, these relationships were found to be unrealistie. The purpose of this book is to present aseries of solutions (previously unpublished mostly) for the statie and dynamie problems of anisotropie and heterogeneous elastie half-spaees. The investigation is limited to the ease of transversal isotropy, and to the ease where the eharaeteristies of the half-spaee only in the direetion of its depth vary. Special emphasis is laid on the studies of steady-state harmonie vibrations of half-spaees under given loads, aeting at the surfaee of the half-space, or at a certain depth. In addition, a study of harmonie vibrations of a eireular stiff disk, whieh is in contaet with the surfaee of the half-spaee, is presented. When the solutions of the problems of harmonie vibrations are available, it is possible to eonstruet appropriate solutions for arbitrary time-dependent loads. The book contains five ehapters. In Chap. 1, general solutions are presented for the problem of vibrations in the transversely isotropie half-spaee subjeeted to specifie loads. Mueh attention has been paid to the ease of eoneentrated vertieal and horizontal forces, and thus the solutions are available for any arbitrary distribution of loads. Solutions whieh are derived in integral form eontain funetions that satisfy ordinary differential equations. The problem leads to the eonstruetion of the fundamental solutions of these differential equations, whieh eontain the parameter k of Hankel's transformation of the required funetions. In addition in Chap. 1, we present formulas, based on the prineiple of superposition, that allow ealculation of linearly deformed foundations, subjeeted to loads whieh are distributed over the eireular and reetangular domains. These formulas eontain Green's funetions (without further speeifieation of their form), and they are presented as single integrals, instead of original double ones. In the following three ehapters, we eonsider various problems dealing with the homogeneous transversely isotropie half-spaee (Chap. 2), an isotropie linearly heterogeneous half-spaee (Chap. 3), and a transversely isotropie half-spaee varying exponentially with depth shear modulus (Chap. 4). In the latter ease, two options are eonsidered: first, the half-spaee with an infmite inerease of shear modulus with depth, and, seeond, one with a fmite limit for the shear modulus. Chapters. 2-4 eontain the results of a vast number of eomputations needed for

4

Introduction

construction of exact solutions of the above-mentioned key differential equations. The efficient choice of contour of integration and the appropriate computational methods provide a precise determination of the amplitude of vibrations of points of the half-space. The solutions presented could serve as a standard for estimating the accuracy of numerical-analytical solutions, for which the differential equations (for Fourier's or Hankel's transformation) are solved numerically. One of the widely used numerical methods employs a multilayer half-space with piecewise constant characteristics within each layer as an approximation of the given continuously heterogeneous half-space. In Chap. 5, numerical-analytical methods of constructing solutions for static and dynamic problems in a heterogeneous half-space are presented. Two approaches are employed in order to construct the solutions of the differential equations: fIrst, a piecewise constant approximation of the coefficients of the equations, and, second, the Runge-Kutta method. Solutions of the differential equations for high values of k (the parameter in Hankel's transformation) should decrease rapidly, as the distance from the horizontal plane subjected to the load increases. Therefore, the bounds of the domain of interest are moved (as k-parameter grows) towards this plane. This method results in the saving of the computational resources. It enables us to construct the required solutions for the force applied to the surface ofthe half-space, without any additional measures. In those cases where the force is applied to an inner point of the half-space, an orthogonalization method due to Godunov [42] is used, in addition to the Runge-Kutta method. Similarly, when using the piecewise constant method of approximation, we apply a transformation of solutions to eliminate its increasing parts. The solutions obtained are verifIed by comparing them with those of corresponding problems developed for the linearly heterogeneous half-space in Chap. 3. Note that the determination of the integration contour when employing inverse Fourier's and Hankel's transformations, calculation of the appropriate integrals, and the algorithms of solutions of various contact problems are discussed in detail in Chap. 2, where these items are introduced for the fIrst time.

Chapter 1. General Solutions of Harmonie Vibrations in Heterogeneous Transversely Isotropie Half-Spaee

1.1 Basie Relationships of Theory of Elastieity for Transversely Isotropie Body Generally, Hooke's law for an anisotropie elastic body contains 36 coefficients Ag [65]: (JX

= All&x

+ A12 &y + A13 &z + A14 y yz + AlsY xz + A16 y xy'

(J Y

= A21 &x

+ A22 &y + A23&z + A24 y yz + A2s Yxz + A26 y X)', (1.1)

where the components of the stress tensor homogeneous functions ofstrains

&r"'"

(Jr"'"

Txy

are presented as linear

Yxy' Assuming the existence ofan elastic

potential, we reduce the number of coefficients to 21 (Aij = Aji)' Further reduction of their number is possible if the elastic properties are symmetrical. In the following, we consider a case with transverse isotropy, when the material is isotropie in the "horizontal" planes (in X; Y-axes), while its elastic properties in the "vertical" planes are not related to those in the horizontal ones. In this case, the strain properties ofthe material are stated with 5 constants [65]. Using cylindrical coordinates r,S,z (Fig. 1.1), where coordinates r,S vary in the plane ofisotropy, we present relationships (1.1) for the transversely isotropie material as folIows: (1.2)

G=AE,

where the stress vector, G, the strain vector, coefficients, A, are given as

E,

and a matrix of elastic

B. G. Muravskii, Mechanics of Non-Homogeneous and Anisotropic Foundations © Springer-Verlag Berlin Heidelberg New York 2001

6

I. General Solutions ~~~.-__________X

Sutface of half-space

I Zo

I I

~---+--

y

L..--z

; M (r, S , z)

z Fig. 1.1. Half-space referred to cylindrical coordinates

(1.3a) (1.3b) A rr ArS Arz ArS A rr Arz

A=

Arz

An

o

A zz

(1.3e)

o

and (1.4)

Here, GrS and Grz are shear modules for the plane of isotropy and for any plane normal to the plane of isotropy, respeetively. In the specifie ease with isotropie material, only 2 of 5 eonstants remain independent, (1.5)

where G is the shear modulus ofthe material, and J.... is Lame's eonstant. A relationship inverse to (1.2) may be presented as &

=

where

ao,

(1.6)

1.1. Basic Relationships

I

arS arz a rr arS a rr a rz

0

a zz

arz arz 3=

7

(1.7)

G-rz1 G-rS1

0

G-1 rz

G;d

=

(1.8)

2(arr - arS)·

The elements of matrix 3 may be derived by using engineering parameters [65]; E, E' are Young's modulus for the plane of isotropy and for any direction normal to this plane, v is Poisson's ratio, which characterizes a transverse constriction in the plane of isotropy (r, 3 ) for the case of stretching in this plane, and v' is Poisson' s ratio for the case of stretching in the z-direction: 1

a rr =

E'

arS

=-

v 1 v' E' a zz = E' arz = - E"

G rS

E

= 2(1 + v)

(1.9)

The elastic coefficients Aij, which constitute matrix A (1.3c), may be represented in terms of aij in (1.7): A A

2

= arrazz - arz rr (a rr -ars)m ' 2

- _ arSazz - arz

rS -

(a rr - ars)m

Arz = _ a rz

m

A

(1.10a)

zz

,

,

= a rr + arS , m

(1.1 Ob) (1.1 Oc)

(l.lOd)

where (1.11)

A positively defined elastic potential leads to certain requirements for the coefficients of matrices A and 3. Thus, the following conditions follow from a positive definition ofthe matrices A and 3: (I. 12a) (1.12b)

Employing relationships (1.10), we obtain, in addition,

8

1. General Solutions

m > 0, a rr - ars> 0, a rr + arS> 0 .

(1.13)

In what folIows, we need relationships between the strains (1.3b) and the displacements U" Uz,U S along the coordinate lines of the cylindrical system of coordinates. In matrix form, these relations may be presented as follows: (1.14)

&=D u,

where

u=[u"uz,uS]T,

a ar 1

r 0 D=

a az

I a ras

0

(1.15)

0

0

0

1 a ras

a az a ar 0

1 a ras

0 (1.16) 0

1 a --+-

r ar a az

1.2 Partieular Solutions for Transversely Isotropie Heterogeneous Elastie Medium Consider a half-space ( z > Zo ), or a layer (zo < z < Zo + H) with boundary planes that are infinite in directions normal to the Z-axis (Fig. 1.1). A time dependence for all considered values is assumed to have an exponential form exp(i ro t), where

n.

ro is a circular frequency of vibrations, and i = As a rule, the multiplier exp(i ro t) is reduced in most of the equations considered below; we shall deal with the amplitudes of the studied parameters. We also assume that the planes, which are parallel to the coordinate planes (x, y), or (r, S), are planes of isotropy. Properties of material, i.e. elastic coefficients (elements of matrix A) and density p, may vary in the direction of the Z-axis (the vertical axis). Application of Hankel's transformation to the functions in cylindrical coordinates with respect to the variable r, and Fourier series with respect to S, are efficient measures that enable us to solve problems of vibrations of half-spaces and layers under the conditions described above.

1.2. Particular Solutions

9

For the case of an isotropie and homogeneous medium, application of the vector and scalar potentials (Lame - type solutions) has become common practice in solving dynamic problems. This approach leads to the wave equations that, as a rule, are interconnected through boundary conditions of the problem. For an anisotropie homogeneous body, such aseparation of equations requires application ofsome additional constraints on the elastic parameters ofthe material (e.g. Carrier's condition [19] that was used [61] to solve the problem ofvibrations of a stiff disk on a transversely isotropie half-space). For the isotropie heterogeneous half-space, transformation of the problem into separated equations for the potentials is possible for some specific types of heterogeneity (e.g. for a quadratic variation of shear modulus G(z) and Poisson's ratio v = 0.25, potentials were employed to solve a plane problem in [31, 33, 46], and for an axisymmetric problem in [47]). Various aspects conceming the separation of equations of motion have been discussed in detail in [3, 52, 54]. Note that in numerous cases, a successful solution of dynamic problems for an elastic medium is possible even by using non-separated equations. As shown in [124], Sezawa's representation [103] for displacements, whieh was used for an isotropie homogeneous medium, may be generalized for the case of trans verse isotropy and heterogeneity along the Z-axis (depth direction). Note that Sezawa's solutions were employed in numerous works to solve dynamie problems for an elastie isotropie homogeneous half-space and for a set oflayers (e.g. [17, 57, 58, 128]). For the transversely isotropie medium which is heterogeneous with depth, the following representation is used for the amplitudes of displacements, Ur' U z'

U s' along the coordinate lines ofthe cylindrieal system of coordinates [124]:

Ur = -[q(Z,k) dJJrü + dTl

P(Z,k)~Jncrü]rl' Tl

U z = w(z,k) Jn (Tl)r\ , Us

(1.17b)

= [P(Z,k) dJn(Tl) +q(Z,k)~Jn(Tl)]r2' dT)

(1.17a)

T)

(1.17c)

where Tl = kr,

(U8a)

r

(1.l8b)

\

=

[cos(ns)] sin(nS) ,

r 2 = [ sin(nS)

] - cos(nS) .

(1.l8c)

( n = 0,1,2, ... ). Representations (1.17) contain Bessel's functions ofthe order n, and functions q, w, p that depend on coordinate z and parameter k; apparently, two options of the

1. General Solutions

10

trigonometrie dependenee on the angle S are possible. For a more general dependenee on S, the problem's solution is presented in the form of a Fourier series, i.e. n-summation of expressions (1.17) is performed. Note that numerous important solutions may be derived at speeifie values of n. For example, at n = 0, we obtain an axisymmetric problem. When ehoosing the upper line in r l and r 2, funetion p is excluded from the solution, while the ehoice of a lower line leads to the problem of torsional vibrations. In the last ease, the only non-zero funetion is U 8 , expressed in terms of the funetion p. At n = I, solutions may be obtained, first, for the ease with a horizontal load distributed uniformly over a eireular domain on the surfaee of a half-spaee or a layer, and, seeond, for the ease of a vertical momentum load, which tends to rotate a cireular domain about its horizontal Y-axis. While the form of dependence on the coordinates rand S is known apriori (it corresponds to Sezawa's representation), a Z-dependence should be found based on the following requirement: functions (1.17) and the corresponding stresses obtained from equations (1.2) and (1.14) should satisfy the equations ofmotion for the elastic medium (assuming the absence ofvolume forces):

(1.19)

At this stage, functions (1.17) are considered as particular solutions containing the parameter k. In the following, our intention is to integrate these solutions with respect to k from zero to infmity, taking into aecount possible points of singularities. As shown below, using the properties of Hankel's transformation enables us to satisfy the given conditions specified for the planes z = const. Note that for a homogeneous isotropie medium functions q(z,k), w(z,k), p(z,k) are expressed through the exponential functions containing radicals:

at 2

[ = k

- C~ J

1/2

2

(02

,

(1. 20)

where C S

= ( G )112 C = (~/\'+2G )112 P

,

p

P

(1.21)

Here, Cp and C. eorrespond to the veloeities of propagation of eompression and shear waves, respectively.

1.2. Particular Solutions

11

Having found the strains corresponding to the displacements (1.17) by using relationship (1.14), we now construct the stress tensor following (1.2) for the particular solution (1.17). When calculating derivatives, we keep in mind the following relationships: drl

da

(1.22a)

=-nr 2 ,

(1.22b)

(1.22c) Expression (1.22c) corresponds to Bessel's equation [1]. The component amplitudes of the stress tensor are presented in the following form (instead of cr r , crs' cr z ' 'trz' 't rS ' 't Sz ' we use for the particular solution considered Sr' Ss, Sz' T',.z' TrS ' TSz ' respectively): Sr

Ss

=

=

{J

n (l1{ kn(A rr

+

dJ n (l1) k(A rr - ArS)(q - np)}r I' d 11 11

{J

n (l1{ -

-~~)(p-nq) + Arrkq+ An

kn(Arr -

!;]

~~)(p-nq) + Arskq+ An

_ dJ n (l1) k(Arr -Ars)(q-np)}r d11 11 l'

!;] (1.23)

12

I. General Solutions

In order to obtain equations whieh are satisfied by the previously introdueed funetions q(z,k), w(z,k), p(z,k) , we apply the equations of motion (1.19); for the steady-state harmonie vibrations, whieh are the subjeet of our study, these relationships are transformed into the equations for amplitudes. Substituting solutions (1.17) and the eorresponding amplitudes of stresses (1.23) into these equations, we apply the formulas (1.22) again. Note that in the first two equations, a11 terms have the eommon multiplier r l' and in the third one, r 2. In eaeh equation, a11 terms are transformed to the form whieh eontains J n (TI), or dJ n ( TI) / d TI . The requirement of a vanishing sum of terms eontaining J n (TI) in the first equation (1.19) leads to

p +_ dG_ dp 2 -k 2G )p=O. d 2_ G _ rz_+(pro rz dz2 dz dz rS

(1.24)

In turn, eonsideration of the terms eontaining dJ n (TI) /d TI yields

G d2q+dGrzdQ+(pro2_k2A )q-k(A +G )dw_kdGn:w=O. rz dz2 dz dz rr rz rz dz dz (1.25) Studying the seeond equation (1.19), we ean see that the terms with the derivative of Bessel's funetion eaneel, and equating the sum of the terms with J n (TI) to zero, we obtain d 2 w dAzz dw 2 2 dq dArz AZZ - 2 +----+(pro -k Grz)w + k(Arz +Grz)-+k--q=O. (1.26) dz dz dz dz dz The last equation of (1.19) leads to the previously obtained equations (1.24) and (1.25). Thus, displaeements (1.17) satisfy a11 the requirements that define the behavior of a linearly elastie transversely isotropie heterogeneous medium, under the following eondition: funetions q(z,k), w(z,k), p(z,k) satisfy the system of equations (1.24), (1.25), (1.26). These equations were developed in [124]. Note that the first equation does not depend on the other two, whieh are intereonneeted. The equations do not eontain parameter n (the number of harmonies with respeet to the angle eoordinate 3). Next, we transform equations (1.24)-(1.26) by using the fo11owing dimensionless variables:

G rS -

G rS Grz(zo) '

A

zz

=

Azz Grz(zo) '

A

"

=

A" Grz ( Zo )'

A

rz

=

Arz Grz ( Zo )'

(1.27)

1.3 Statement of BoundaJ)' Conditions

13

where a reference length, z" has been introduced; all elastic coefficients and density are related to the shear modulus and density in the plane Z = ZO, respectively. Let us take into account the dissipative properties ofthe medium. In order to incorporate the dissipation of energy in the material, we apply, keeping in mind processes harmonic in time, complex elastic coefficients. In its simplest form, the friction within the material may be accounted for by considering the value Grz(zo) as the complex one: (1.28) where a small, constant, positive parameter E enables us to account for the dissipation of energy in the material of the half-space, and GrzO is the shear modulus of the material at z = Zo, for the case of an ideally elastic material. Assume that the dimensionless coefficients of elasticity denoted with "~,, in (1.27) are real. Hence, the frequency-independent mechanism of energy dissipation is taken to be identical for various types of deformations. Taking into account relationships (1.27) and (1.28), we rewrite equations (1.24)-(1.26) as follows:

Gd2p+dGdp+Cß2e2_PG )~=O dZ'2

dZ' dZ'

P

~d2q

dGdq

~ 2 2

G-+--+(pß dZ'2 dZ' dZ'

r3 P

e

(1.29)

,

~2~ ~ ~~ ~ dw ~dG~ -k A )q-k(A +G)--k-w=O rr rz dZ' dZ' '

(1.30)

where

e = ffiZ ~P(Zo) r

ß=

(1.32a)

GrzO '

/I.

(1.32b)

V~

For parameter ß, a radical with a positive real part has been chosen. A frequency parameter, e, may reflect the influence of heterogeneity via value zr' if the function that characterizes heterogeneity contains

Zr

as a parameter.

1.3 Statement of Boundary Conditions for Planes z = const Integration of expressions (1.17) for the amplitudes ofvibrations and (1.23) for the stress amplitudes with respect to parameter k from zero to infinity yields integral

14

l. General Solutions

representations which satisfy all the equations of the theory of elasticity in the domains, where no volume forces exist. If stresses or displacements are specified for some planes z = const, they result in the generation of the corresponding boundary conditions for functions q(z,k), w(z,k),p(z,k). Consider stresses Sz' Tm T3z which occur in the planes of z = const. Following equations (1.23) we have: (1.33a) (1.33b) (1.33c) (1.33d)

(1.33e) (1.33t)

In order to use the properties of Hankel's transformation for studying the above-mentioned integrals with respect to parameter k, we have to obtain the combinations of stresses which contain Bessel's function only as a multiplier (value Sz already has the sought form). The last two equations (1.33) yield dq d P ) Jn+1(Tl), Trz + T3z = Grz -kw+--dz dz

(1.34a)

P ) Jn-1(Tl), dTrz -T3z =Grz k wdq --dz dz

(1.34b)

A

A

A

A

(

(

where the following recurrent relationships for Bessel's functions [1] have been accounted for:

n dJn(Tl) Jn+1(Tl) = Jn(T])---d- , Tl Tl

(1.35a)

n dJ n(Tl) Jn-1(Tl) = Jn(Tl)-+-- . Tl dTl

(1.35b)

Suppose that for some value z = z(J the amplitudes of stresses have the following form: (1.36a)

1.3 Statement ofBoundary Conditions

15

'rz =irz(r}rl'

(1.36b)

'sz = isz(r)r 2·

(1.36c)

The integral representations of the solution of the problem satisfy condition (1.36), ifat z = z" (1.37a)

('Szdk=o-z,

(1.37b)

(1.37c) or

fl

Arzkq+ Azz

~; }n(kr)dk = o-z(r),

(l.38a)

Grz fOO(-kw+ dq - dP)Jn+1(kr)dk = irz(r)+isz(r), o dz dz

(1.38b)

Grz fOO(kw- dq - dP)Jn_l(kr)dk=irz(r)-iszCr). o dz dz

(1.38c)

Using the formulas relating to the direct and inverse Hankel's transformation, iOOrf(r)Js(kr)dr = F(k) ,

(l.39a)

iookF(k)Js(kr)dk = f(r) ,

(1.39b)

we obtain the conditions for functions q(z,k), w(z,k), p(z,k), which must hold at

(1.40)

(1.41a)

16

1. General Solutions

kw- dq _ dp dz dz

=

k r"'r[irz(r)-iaz(r)]Jn_l(kr)dr. Grz(zcr) Jo

(1.41b)

Equations (1.41) give dp = dz

k 2Grz (zcr)

[1

00

0

" r[Trz(r) + Taz(r)]Jn+l( krd ) r

+ lOOr[irz(r)-iaz(r)]Jn_l(kr)dr].

(l.42a)

dq_kw= k [[r[Trz(r) + Taz(r)]Jn+l(kr)dr dz 2Grz (zcr) 0 -IOOr[irz(r)-iaz(r)]Jn_l(kr)drj.

(1.42b)

These equations, in addition to equation (l.40), may serve as boundary conditions for the functions q(z,k), w(z,k),p(z,k) when stresses are specified on the plane =za. In order to determine the boundary conditions for these functions, in the case with known amplitudes of displacements in the plane z = Zu, we develop from equations (1.17): Z

(l.43a) (1.43b) (1.43c)

(l.43d) (1.43e) U z = wJn(rJ).

(1.43t)

Let the amplitudes of displacements in the plane z = Zu have the form (1.44a)

Ur = ur(r)r1 ,

(1.44b)

Ua=Ua(r)r 2

•

Analogously to equations (1.37),

(1.44c)

1.3 Statement of Boundary Conditions

17

or f'WJn(kr)dk = uz(r),

(1.46a)

f(q- p)Jn+l(kr)dk = ur(r)+ ul}(r) ,

(1.46b)

- f(q+ p)Jn_l(kr)dk = ur(r)-ul}(r).

(1.46c)

Further employment of the direct and inverse Hankel's transformations (1.39) results in the following conditions for the functions q(z,k), w(z,k), p(z,k) at

q- p = k f'r[ur(r)+Ul}(r)]Jn+l(kr)dr,

(1.47a)

q+ p = -k f'r[ur(r)-Ul}(r)]Jn_l(kr)dr,

(1.47b)

W =

(1.47c)

k f'rUz(r)Jn(kr)dr .

From the ftrst two equations (1.47) we obtain: q=

~{ fOr[ur(r) + ul}(r)]J

n+1 (kr)dr

- f'r[ur(r) - ul}(r)]J n_l(kr)dr},

(1.48a) p

= -~{

f'r[ur(r) + ul}(r)]Jn+l(kr)dr + f'r[ur(r)-Ul}(r)]Jn-l(kr)dr}

(1.48a) Boundary conditions (1.42a) and (1.48b) for the function p do not contain functions q and w. Accounting for the form of equations (1.24), (1.25), and (1.26), we come to the conclusion that the problems of determining the function p, and the functions q and w, are completely independent.

18

I. General Solutions

1.4 General Solution for Harmonie Vibrations of Half-Spaee Subjected to Surfaee Loads In this section we consider some cases of vertical and horizontal loads applied over the circular domain of radius R on the surface of the half-space. In particular, we shall obtain the solutions for the case of concentrated forces, which enable us to construct solutions for various forms of distribution of applied loads.

1.4.1 Axisymmetrie Vertieal Load Applied over Cireular Domain Consider an axisymmetric vertical load harmonie in time applied to the surface of a half-space. An example of a load is shown in Fig. I.2a: the force Po exp(i rot) is uniformly distributed over the circle ofradius R at the surface ofthe half-space. Assuming n = 0 (the upper line in r j ), we present the amplitudes of vibrations as a result of integrating expressions (1.17): Ur

=

Uz

=

L~(Z,k)JI(kr)dk , L~Z,k)Jo{kr)dk,

Us =0.

(1.49a)

(1.49b) (1.49c)

At the surface of the half-space ( Z = zo), tangential stresses vanish, and

o-z = -Po /(nR 2 )

for r < R (for the example presented in Fig. 1.2a) and o-z = 0 for r>R. The boundary conditions at z = zo, in accordance with equation (1.40) at z(J = zo, and with equation (1.42b), have the following form: kPo Po [ rJo{kr)dr=2 J1{kR) , nR G",{zo) 0 nRG",{zo)

(1.50a) dq -kw=O .

dz

(1.50b)

Next, we consider equations (1.25) and (1.26) (or (1.30), (1.31», which should be satisfied by the functions q(z,k) and w{z,k). This system of equations has four fundamental solutions, two of which (ql' WI and ql' w\ ) satisty the condition of absence of sources at infinity (z ~ oe). These solutions are not necessarily decreasing, but, for sufficiently high values of parameter k, solutions corresponding to the condition of absence of sources at infmity decrease as

1.4 Ha1f-Spaee Subjeeted to Surfaee Loads

o

o

/1 y/ I

x

Surface ofhalf-space

Surface ofhalf-space

y/

Zo

l

~,.I..JlL

19

__.

~.

I,

x

Zo

Qo '/IR'

~

~rts;4Jr--'I,-/

I

z

(a)

(b)

z

Fig. 1.2a,b. Harmonie vertieal (a) and horizontal (b) forees applied to eireular area on half-spaee surfaee

z ~ 00. For example, the sought solutions for the case of an isotropie homogeneous half-space with elastic properties, determined by (1.5), take the form:

ql = al exp( -a)z) , kexp(-a)z),

(1.5lb)

=kexp( -a2 z ) ,

(1.5lc)

w) =

q2

(I.51a)

(1.5Id) where al and (l2 are calculated from (1.20). Introducing two arbitrary coefficients A) and A2 we present functions q(z,k) and w(z,k) as a linear combination of the above-mentioned fundamental solutions: q(z,k) = A)(k)q)(z,k) + A2(k)q2(z,k) ,

(1.52a)

w(z,k) = A)(k)w)(z,k) + A2(k)w2(z,k).

(1.52b)

Substitution of these expressions into the boundary conditions (1.50) yields the following system of equations for the coefficients A) and A2 : (1.53) where (1.54a)

20

1. General Solutions

(1.54b) (1.54c) (1.54d) Here, the fundamental solutions q/z,k), w/z,k) (j = 1,2) and the elastie parameters are taken for the surface of the half-space (z = zo). A complete solution ofthe problem is obtained from (1.49), (1.52), (1.53) in the form Ur

=

Po IOCJC22q)(Z,k)-C2)q2(Z,k) J)(kR)J)(kr)dk, 1tRGrz(zo) 0 D

(1.55a)

(1.55b)

Uz =

where (1.56) The solution corresponding to the concentrated vertieal force is constructed from (1.55), after replacing J) (kR)/ R with k!2: (1.57a) Uz

=

Po IOCJC22W)(z,k)-C2IW2(z,k) kJo(kr)dk. 21tGrz (zo) 0 D

(1.57b)

A contour of integration in the expressions (1.55), (1.57) is not specified. For statie problems and for some cases with dynamic problems dealing with the heterogeneous half-space vibrating at low frequencies, integration along the real axis (with respect to the real values of parameter k) is possible. As a rule, in dynamic problems, a value of D (the determinant of the system of equations (1.53)) may become zero at some positive values of k when damping is absent. This fact leads to the appearance of simple poles for the integrands in the expressions (1.55), (1.57). For these values of k, a non-zero solution of (1.53) exists, when the right side is equal to zero, and, respectively, a non-zero solution of the form (1.52) exists too. From a physical point of view, this means that free harmonie vibrations of the half-space (without any external loads) are possible. For the case of the homogeneous isotropie half-space, the single pole, whieh corresponds to the Rayleigh waves, exists on the real axis. On the contrary, for the heterogeneous half-space, numerous poles, or even an infinite number of poles, may exist. Complications arising from the singularities in expressions (1.55),

1.4 Half-Space Subjected to Surface Loads

21

(1.57) may be overeome either by aeeounting for the dissipative properties ofthe medium (this results in the shifting of the poles from the real axis), or by applieation of an appropriate eontour of integration. An interesting faet in this eonneetion is the following: for the ease of harmonie vibrations of the isotropie half-spaee with a shear modulus, whieh grows exponentially in the Z-direetion, eoupled zero points of the value D may oceur at eertain frequeneies of vibrations. These frequeneies eorrespond to the resonanee: when damping is negleeted, they result in infmite amplitudes of vibrations. Additional questions eonneeted with the possibility of a zero value of D at real values of parameter k are diseussed below in more detail for specifie forms ofhalf-spaees. Integration of expression (1.23) with respeet to parameter k by taking (1.52) and (1.53) into aeeount yields the amplitudes of stresses eorresponding to the eonsidered load. For example, expressions for the stresses Cl" and 'trz take the form

(l.58a)

(1.58b) A similar teehnique is used in order to eonstruet solutions for different types of load. For example, let the force Po exp(iro() be distributed over the eirele of radius R, aeeording to the statie problem for a eireular stamp with the smooth base pressed against an isotropie homogeneous half-spaee with foree Po. Then, at z == Zo and r < R we have

cr == z

Po 21tR(R2 _r2 /2 .

i

(1.59)

Aeeording to equation (1.40), equation (l.50a) beeomes Arzkq + A- zz -d w == dz

Po sm·(kR). 21tRGrz(zo)

(1.60)

As a result, in solution (1.55), (1.58), J1(kR) should be replaeed with sin(kR)/2.

22

1. General Solutions

o.f········ ,.' .'., .._ ... ,.,. ,......

/[

Y

i

x

.......... ...........

~./ Zo

Surface ofhalf-space

~ ~;:;--z Fig. 1.3. Torque applied to circular area on half-space surface

1.4.2 Tangential Axisymmetric Load Applied to Circular Domain An example for this type of load is shown in Fig. 1.3: a tangential load qs (r )exp(i rot) is applied to the circ\e of radius R causing a torque with respect to the Z-axis with amplitude R

(1.61)

M z = 2n Lqs(r)r 2 dr .

Consider the following case: (1.62) Here, we have M z

qo

= nqoR

4

(1.63a)

2

2M.

=--t.

(1.63b)

nR

In solution (1.17), we assume n = 0 (a lower line in Vs

dJo(r)) = -p(z,k)-= p(z,k)J](r)), dTl

r j ): (1.64a) (1.64b)

1.4 Half-Space Subjected to Surface Loads

Uz =

o.

23

(1.64e)

Amplitudes of vibrations of the points in the half-spaee in a tangential direetion are presented in the form ofthe following integral: "8

=

f~(z,k)JI(kr)dk.

(1.65)

o

The funetion p(z, k) satisfies equation (1.24) and the following boundary eondition, whieh follows from equation (1.42a) at za = zo: (1.66) Here, we take into aeeount that at z =

Zo

(1.67a) (1.67b) Equation (1.24) has two fundamental solutions, one of whieh, PI (z, k), satisfies the eondition of absence of sources at infinity. For example, in the ease of an homogeneous isotropie half-spaee, this solution has the form exp( -a.Iz) where is found from (1.20). Using the solution PI(z,k) we present the sought funetion, p(z,k) , in the following form: 0.1

(1.68) Constant Cl is determined from the boundary eondition (1.66). For loads of the type (1.62), we obtain, by using the properties of Bessel's funetions [1], the following expressions: Cl dPI(zo,k) dz

=

2Mz k G rz (zo)nR 4

r>

Jo

JI(kr)dr =

2Mz J 2 (kR). (1.69) Grz (zo)nR 2

Following (1.65), the amplitudes of displacements beeome

"= 8

2Mz G rz (Zo) R n 2

1

(z k)[dPI(Zo,k)]-I J (kR)J (kr)dk . PI' dz 21

00

0

(1.70)

For the homogeneous isotropie half-spaee, aeeounting for the above-mentioned expression for PI(z,k) yields

24

1. General Solutions

Us

=

2M z G rz (zo)1tR 2

[exp[- R, the second integral equals [1] r"'J 2 (kR)J I (kr) dk = R 2

Jo

k

8r 2

F.(l ..!...3.

2 I

R2 )

2' 2' 'r 2

•

(1.73)

Tbe hypergeometric function entering (1.73) reaehes unity as R ~ o. Amplitudes of vibrations of the surfaee of the homogeneous isotropie half-space, subjected to the concentrated torque, may be presented as (1.74) Accounting for the expression for + ±sin 2q> ) + WhS(h{ q> - ±sin2q> )} ds

r-R

J[

TJ+I

=qoR 2

WhrO"R)( q>+±Sin2q> )+WhS(AR)( q>-±Sin2q> )]AdA. (1.215)

TJ-I

This result corresponds to the case of r ~ R. If r < R, the lower integrals in equation (1.215) should be replaced by R - r or (1 integral similar to those in (1.215), but between the limits of zero 1- 11 and with q> = 1t, should be added. A general expression for

limit of the 11), and an to R-r or ölx may be

I. General Solutions

56

2R

x r

r

M(r,S,z)

Fig. 1.10. Unifonnly distributed horizontal load applied to circular area written as folIows:

J[WhrO\.R)( q>+~Sin2q>

Tl+1

DlxCr ) = qoR

2

ITl-11

+

{q" nR'

'![

Wh' (All) +

)+Whl}(AR)(

w" (AR)JA dA

o

(r

~

q>-~Sin2q> )]AdA

(r < R)

(1.216)

R).

Similarly, amplitudes ö 2x are detennined; in the expression for ö lx , values and

whS

whr

interchange:

f

Tl+1

D2x

= qoR2 Whr(AR)[( q>-~Sin2q> )+Whl}(AR)( q>+~Sin2q> )}dA !Tl-I! +

{q"nR' '![

Wh, (All) +

o

w" (AR)]A d A (r

~

(r < R)

(1.217)

R).

For the amplitudes Dlz , the corresponding result is similar to Dx from (1.210):

1.7 Application ofSuperposition Principle

57

Tl+I

0lz =

2qoR 2 fWhZ(AR)Sin(T] +

~Sin(2q»]

'1-\

(1.230) For generalization, one should replace the lower limit in the last integral in relationship (1.230) with I T] -I I, and add the term equal to the last summand in equation (1.220).

1.7.8 Self-Balanced Horizontal Load Consider a horizontalload having the radial component qr = qop2 cos(S) and the tangential component qs

= qop2 sin(S)(Fig. 1.14). This form of load is self-

1.7 Application of Superposition Princip\e

63

balanced: the resultant force and the moment corresponding to this load vanish. Using a horizontal uniformly distributed load applied to the elementary ring-shaped domains (this type of load is considered for a circular area in section 1.7.4) is not sufficient for solving a contact problem for a circular stamp. In order to provide specified horizontal displacements over the entire contact domain, additional loadings are required. A considered self-balanced load is suitable for this purpose. As shown in the previous subsections, in order to fmd a solution for an arbitrary point, one has to calculate the amplitudes of vibrations 81z ' 8lx and 8 2x at points MI and M 2 , with further application of relationships (1.214).

Consider amplitudes 8 b

(

r ~ R ):

r+Rcp

8 lx = 2qo f f {p2Whr(s)[cos(3)cos(\II) - sin(3)sin(\II)]cos(y) r-R 0

JJ

r+Rcp

{Whr(S)[(r 2 +s2)cos(y)-2rs]cos(y)

= 2Qo

r-R 0

+wh&(s)(r 2 -s2)sin 2(y)}sdyds lI+ I

= QoR 4

f {WhrO"R)[O'? +T)2{ IlI-II

+WhS(AR)(T)2

q>+~Sin(2 a;z' which were mentioned in (1.12) and (1.13), yield the following constraints, respectively: ZZ

I-v ~ < 2V,2 ' 1

~ Svz versus parameter ~ at v = v' , ~'= 2(1 + v') (the degree of anisotropy is only determined by deviation of parameter ~ from unity) and for Poisson's ratios v'= v = 0,1/3,1/2. According to relationsips (2.27), in the first

2.1 Action ofSurface Vertical Force

75

........ . v=1/3

-

- _112

v-o 1.2

Sw 1,0

0 .8

0 ,6 0 .0

0 .5

1.5

1.0

2.0

2.5

3 .0

~

(a)

0.0 +--L--L--'--+---'--'--'--'---'--'--'--+

1.4 +--'--'--'---'--'--'--'---'---'--'---'-+

- - - - - v=113

~. 2

----.- v- 1/2 -

.

v..()

1.2

~

s.,

s~

. • ..... . v-113

~. 6

1.0

---\1=1/2 ----.- v=O ~. 8

O~ +-,-,-.-.--r-r-.-r~-.-.-+

(b)

0.0

0.5

1.0

1.5

S

2.0

2.5

3.0

0 ,0

0.5

1.0

1.5

2.0

2.5

3.0

(c)

Fig. 2.1a,b,c. Normalized static displacements of points of transversely isotropie half-space subjected to vertical force

ease parameter ~ has no upper bound, while in the seeond and third eases the bounds are 3 and 1, respeetively. At ~ = 1, the values of Sw' Svz are equal to unity, and Svh = -0.5 , -0.25, 0 at v' = 0, 1/3, 1/2 , respeetively, being in agreement with the results for the isotropie case. As expeeted, increasing medium stiffness in the horizontal direetions results in a reduetion of displaeements. The influence of anisotropy is more notieeable in the ease of horizontal displacements.

76

2. Statie and Dynamie Problems fOT Transversely Isotropie Half-Space

The value of Svz is practically independent ofPoisson's ratio within the domain of definition of Svz . 2.1.2 Free Vibrations ofHalf-Space

Next, we consider the possibility ofthe occurrence ofundamped free vibrations of a half-space. In order to answer this question, one should study the value of D that is a determinant of the system of equations (1.53) for coefficients Aj • Zeros of this value that correspond to the occurrence of free vibrations (Rayleigh waves) result in the poles of integrands in solution (2.1) of the problem. The roots kR of the equation D = 0 at ß = 0 are calculated for the case of relationships (2.26) at v = v', 1;' = 2(1 + v'), similar to the calculations performed in the previous example. Parameter

kR may be called a dimensionless wave number for Rayleigh

waves (corresponding to dimensionless distances r in (2. 19a». In Fig. 2.2, parameter kR is presented versus I; for three different values v = 0, 1/3, 1/2 . At

I; = 1 (the isotropic case), we obtain known results: kR =1.1441, 1.0724, 1.0468 for v = 0,113,1/2, respectively. Note that in the study offree vibrations, in order to construct a solution in the form of propagating waves, one should consider the dependence of solutions on coordinate r in the form of Hankel's functions H~2)(kRr) or H~2)(kRr), instead of in the form JO(kRr) or J) (kRr) (as in the integrands (2.1»; in 2-D problems, the function exp( -ikRx') corresponding to solution (1.91) is used. Following (2.19), the spatial wave number with respect to variable r (or x) may be written as k = kRO} R

(2.28)

C

/%

with the corresponding velocity ofRayleigh waves: VR

=~= ~/% kR

kR

(2.29)

•

The results presented in Fig. 2.2 indicate that a reduction in material stiffness in the horizontal direction is followed by a decreasing velocity ofRayleigh waves, as compared with the velocity of shear waves Crz. For all values of 1;, we have kR > 1; at v = 0, the value of

kR

tends to unity, as I; increases.

2.1 Action of Surface Vertical Force 1.6

77

+--...l..---'----'-----'---tv = v', e=O

1.4

1.2

1.0

+ - -....----,.-- --,---,-- - t -

o

2

4

3

Fig. 2.2. Dimensionless wave numbers

5

kR

versus anisotropy parameter

S

2.1.3 Forced Vibrations of Half-Space In order to caIculate the amplitudes of vibrations by using formulas (2.1), one

should perform the integration by applying a corresponding path of integration because of singularities in the integrands. If one takes into account the dissipative properties of the medium by specifying some value E > 0 in equations (1.28) and (1.32b), then a zero of D moves down from the real axis in the complex plane k.

kR is a root of the equation D = 0 at E = 0 (13 = 1), then the complex value 13 kR (that corresponds to a point in the complex plane k, which is located below the real axis) makes D vanish at E > O. Indeed, in going from E = 0, k = kR to E > 0, k = ßkR , values B and C gain multipliers 13 2 and 13 4 , respectively, and, If

following equation (2.7), new roots l"I j are obtained from the previous ones by multiplying the latter by

ß.

Furthermore, following (2.14), coefficients CWj

remain unchanged. Coefficients ci}' determined from (1.54), gain the multiplier 13, while a new value of D in the given transition becomes equal to its previous value, multiplied by 13 2 , i.e. zero. Taking into account the location ofthe roots of D in the complex plane at E > 0, application of the path of integration shown in Fig. 2.3

78

2. Statie and Dynamie Problems for Transversely Isotropie Half-Spaee

" ,....

Re(k) B Fig. 2.3. Integration eontour for dynamie problems

may serve as a reasonable approach to integrating expressions of the form (2.1). Here, h is a small positive value (h = 0.03-0.05), and a point B is located to the right from possible poles of the integrand at e = 0 (in the considered case, a single pole kR exists). In computations, an abscissa of point B was taken to be equal to 20. Tbe convergence of the integrals at z > 0 is provided by exponentially decreasing multipliers entering solutions ql' wl and q2' w 2 (see (2.5)). For the points on the surface of the half-space, the integrals in (2.1) converge only due to decreasing oscillations of Bessel's functions. For the case with z = 0, the following procedure may be recommended. If the value k = kB does not provide sufficient precision in the calculation of Bessel's functions by using the principal term in their asymptotic representation [1] (2.30)

kBI > kB • In our caIculations, we exceeds kB, then one should add the

we introduce a second point, BI' with abscissa assume

kBI = 90/ r.

If the latter value

integral between the limits kBand kBI to the integral taken along that part of the integration path located to the left of point B. Finally, integrals between the limits km = max(kB,kBI ) and 00 are determined approximately with the help of the following formula, derived by integration by parts and using representation (2.30):

r -( Im

(k)

1t) -

2__ )1/2 co{kr - mt - - - dk 7tkr 2 4

2.1 Action ofSurface Vertical Force

6 ~

-+

__- L_ _- L__- i_ _~_ _~_ _

79

0

V'=V=\l3,8=O ·10

5

·20 -30

0"" .t) applied in the plane z = 0 and directed along the vertical Z-axis; at first, the force is assumed to be uniformly distributed over the area of a cirele of radius R. As for the case of a force applied to the surface of a half-space, we use a representation of the solution in the form (1.49). Dimensionless variables given in (1.27) are used. Taking into account the symmetry of the problem, one may rewrite functions q(z,k) and w(z,k) for z ~ 0 in the following form: q(z,k) = A(k)[q1(Z,k)-q2(Z,k)] ,

(2.91a)

w(z,k) = A(k)[w\(z,k)-w2(z,k)] ,

(2.91b)

where coefficient A(k) is introduced; functions qj'

w j

are determined by (2.5)

with 11 = 111,11 2 ; Cq = 1 and Cw = CWj by (2.14). We eonstruct the solution for

z < 0 by continuing functions W and

q

in the symmetrie and anti symmetrie

ways, respectively. Since q = 0 at z = 0, function q(z,k) and its first derivative are continuous over the entire spaee; function w(z,k) is also continuous. Thus, the assumed form of the solution provides continuity of displacements when erossing the plane of application of the external load. Moreover, following equation (1.42b), the requirement ofeontinuity ofthe value on the left side ofthis equation is satisfied (this requirement is related to the absence of horizontal external loads in the plane z = 0). Therefore, we have to consider only one requirement, related to condition (1.40), which determines discontinuity in the value of derivative, by using an analogy with equation (1.100). Taking into account the evenness of function w, the following boundary condition for solution (2.91) may be written at z = 0:

100

2. Statie and Dynamie Problems for Transversely Isotropie Half-Space

2 dw =-b W d Z-

=-~J (kR). RA 1 1t

zz

(2.92)

Hence, we find coefficient A in solution (2.91): (2.93) Tbe amplitudes of vibrations for the domain z ~ 0 may be found from (1.49) by using (2.91) and (2.93). For the case ofthe concentrated force, we obtain Ur =

Po 41tGrzo r

(2.94a)

SVh'

(2.94b) (2.95a)

*"

At r = 0, Z 0, the following transformation of these expressions is reasonable: one should divide by z, rather than by r, in the multipliers of Svh and Svv , and replace with before the inte~ls. Here, as in previous sections, we assurne for dynamic problems that e = 1 and apply equations (2.19).

r

z

Particular Case: Isotropie Spaee

Consider the case of an isotropie spaee when roots TJ j and coefficients CWj have the form as given in (2.15), (2.16). Integrals (2.94b), (2.95b) are rewritten as folIows:

(2.97)

2.5 Force Applied in Infinite Space

101

The integrals in equations (2.96), (2.97) may be expressed in finite form by employing tabulated integral (2.49) (u = 1", a = Z, b = i ß or b = i ß /(1zj/2). In order to obtain the integrals in (2.96), one should apply (2.49) at v = 0 and then differentiate both parts ofthis equations with respeet to a and u. This resuIts in Sm

~ "z[(! d~)'(! [expH ßaHxpH ßal K)])

l

(2.98)

where (2.99) One may rewrite the first summand in square braekets in integral (2.97) as follows:

~ k

e _ß2

~ [~P _ß2 +~~~ß2 exp(-z~e-ß2)= k

k

e _ß2

1exp(-z~e-ß2),

(2.100) so that the problem of determination of Svv reduees to the evaluation of three integrals: the first two integrals, obtained from (2.49) at v = 0 by double differentiation with respeet to a, are identical; the last integral is determined direetly from (2.49). The expression for Svv takes the form Svv = 1"

{~ ß2 exp( -ißa)

+[! a +z'(! d~n >xp(-;ßa)-exp(-;ßa/K)[)} dd

(2.101)

The derived solution (2.98), (2.101) for an isotropie half-spaee agrees with the known solution by Stokes [29]. Static Action ofVertical Force

Consider the statie solution for a transversely isotropie spaee subjeeted to a vertical force. Employing relationships (2.20), (2.21), we rewrite the expressions for normalized displacements in (2.94b), (2.95b) in the following form (z ~ 0): +1) [ [exp(- k~~~) S vh =- 1"(1 ~; ~2 1hz -exp (k~~~)] - 1hz J 1 (k~~)dk~ r , Azz< Th - "h ) 0 Svv =-

(2.102)

1"(1 + 1) [~ ~ ~ ~ ~ ~ ; 2 [Cwlexp(-kTllz)-Cwzexp(-kTlzz)]Jo(k1")dk (2.103)

A zz (1]1 - 1]z)

0

or using tabulated integrals:

102

2. Statie and Dynamie Problems for Transversely Isotropie Half-Spaee

(2.104)

Svv =

(2.105)

These expressions also hold for negative values of z. Clearly, nonnalized coordinates and may be replaced by coordinates z and r, respectively. The indetenninacy of expressions that represent the solution of the problem at Tl I = Tlz has been discussed in section 2.1.1. One may obtain the components of the stress tensor by differentiating the expressions for the displacements.

z

r

2.5.2 Action of Horizontal Force Here, we consider the action of a horizontal force Qo exp(i rot) applied to the plane z = 0 within a transversely isotropie space. In the following, the representation of the solution given in equation (1.76) is used. Instead of (2.91), we search for a solution at z > 0 in the following fonn (function p(z, k) has been added): q(z,k) = B(k)[Cwz(Mz,k)- Cwlqz(z,k)],

(2.106a)

w(z,k) = B(k)[Cwzwl(z,k) - Cwlwz(z,k)] ,

(2.106b)

p(i:,k) = C(k)PI(z,k) ,

(2.106c)

where the same fundamental solutions qj'

wj

as in equations (2.91) are used, and

function PI has the fonn (2.57). The proposed fonn of solution for q and selected in order to satisfy the condition w(O, k)

w is

=

0 , which sterns from symmetry

considerations. For z < 0, functions q(z,k)

and p(z,k) are continued

symmetrically, and function w(z,k) antisymmetrically. Condition w(O,k) = 0 resuIts in the continuity of function w(z,k) and its derivative with respect to z over the entire space; functions q(z,k) and p(z,k) are also continuous. Thus, continuity of displacements, and, in accordance with equation (1.40), of stresses cr z' is provided when passing the plane of application of the load. One may caIculate coefficients B(k) and C(k) by providing the required jump in the value of derivatives with respect to z of functions q(z,k) and p(z,k) when passing the plane z = 0 . Taking into account both the evenness ofthese functions

2.5 Force Applied in Infinite Space

103

and equations (1.110), (1.114), one may fonnulate the following boundary condition at z = 0 for functions q(z, k) and j5(z, k) in the case of the unifonn distribution ofthe force over a circular area ofradius R: 2 dq =b q d~ z

=~J (kR) , RG [

(2.107)

rz

1t

(2.108) Hence we detennine coefficients B and C in solution (2.106): B=-

QoJ[(kR) , 21tRGrz(Cw2 TJ[ -Cw [TJ2)

C=-

QoJ[(kR)

(2.109)

.

(2.110)

21tRGrz~Grsk2 - ß e 2

2

The amplitudes of vibrations at z ~ 0 are presented in the fonn of integrals (l. 76) taking into account (2.1 06), (2.109), (2.110). In the case of a concentrated force, we obtain (2.111) (2.112a)

(2.112b)

(2.113a)

ShS

=

uz = -

where

_ß2 r f[[lJ(z,k) Qo

S

41tG hz' rzOr

p(z,k)]

J[~;) + P(Z,k)Jo(kr)]dk,

(2.113b) (2.114a)

104

2. Statie and Dynamie Problems for Transversely Isotropie Half-Spaee

(2.115a)

p(z,k) =

~~ ~ k k 2 G rS

_ß 2 e2

exp[-z~eGrs -ß e ]. 2

(2.115b)

As in previous sections, we take e = 1 when solving dynamic problems and employ dimensionless coordinates (2.19). Replacing r with z in the coefficients of the normalized amplitudes and replacing with in the multipliers before the integrals are reasonable when calculating displacements on the Z-axis (r = 0).

r

z

Static Action 0/ Horizontal Force Consider static displacements that occur due to the action of a horizontal force. Taking ß = I, e = 0 , we apply relationships (2.20). Normalized displacements are expressed by using the same integrals as in equations (2.62}-{2.64):

(2.116)

r - -;::Tlo

r

exp (k~~ - TloZ~{Jl(kr) --=::::::-- - J 0 (k~~)ldk~ r ,

0

kr

(2.117)

(2.118) The subsequent integration yields

2.6 Force Applied within Half-Space

105

(2.119)

(2.120)

S hz

=

Cw ]CW2 z

Cw2 11] -Cw]112

[_

11]

F3 (r, zTh)

+

112

]

F 3 (r, zTh) ,

(2.121)

where the nomenclature given in (2.68}-(2.70) is used. Consider a possible case of coincidence of values 11] and 112 when calculated static displacements (2.119}(2.121) and (2.104), (2.105) become indetenninacies. As in section 2.1.1, we recommend applying (instead of evaluating the indetenninities) a small variation of elastic parameters to avoid the equality of roots 11] and 112 . The solutions of dynamic and static problems conceming the action of a concentrated force on an infinite transversely isotropie space are significant for applications of the method of boundary elements in dynamic and static problems for a transversely isotropie body.

2.6 Vibrations of Transversely Isotropie Half-Space under Action of Force Applied within Half-Space

2.6.1 Action ofVertical Force Let a vertical force Po exp(i rot) be applied in the plane Z = z] within a transversely isotropie half-space. Instead of constructing a solution for the homogeneous half-space in its general fonn (given in relationships (1.109)), one could simplify the construction of the solution by employing a solution for the infmite space, with the addition of corresponding corrections, to satisfy the conditions of vanishing stresses on the surface of the half-space. This technique has been employed by Mindlin [75] to solve the problem dealing with the static action of a force within an isotropie homogeneous elastic half-space. For the sake of generality, we apply the fonn of solution given in relationships (1.109). Parts of the solution, which correspond to the action of a force in an infinite space, are separated out in the process ofbuilding the solution, so that a reliable check ofthe perfonned transfonnations can be provided. The application of the dimensionless parameters given in (1.27) yields, for the case of the concentrated force,

106

2. Statie and Dynamie Problems for Transversely Isotropie Half-Spaee (2. I22a)

(2.123a)

Here, in accordance with the technique presented in section 1.5.1, we employ the particular solution (denoted by index a), which vanishes at z > z\ and may be expressed as a linear combination of fundamental solutions at z < z\

(z < z\ = z\ / zr):

(2.124a) 4

W: = LBjWj.

(2.124b)

j

Following (2.5), the fundamental solutions have the form

qj

wj

= C qj exp(-Tljz), = C Wj exp(-Tlß) ,

(2.125a) (2.125b)

where coefficients C qj = I, while coefficients CWj are determined by (2.14), in accordance with the number of roots Tl j of the characteristic equation (2.7). The frrst two roots, TI\ and Tl2' are found according to (2.9); Tl3 = -Tl\, Tl 4 = -Tl2 . Next, coefficients Bj are determined by employing the system of equations (1.105) with dimensionless values qj' wj ' z\,k. The last transition has no impact on the form ofthe system of equations. In order to remove exponential multipliers in the coefficients of the system of equations, one could introduce modified coefficients

B; :

B; = Bj exp(-Tlß\). Following (1.105), we obtain

(2.126)

2.6 Force Applied within Half-Space

107

(2.127)

Employing (2.14) and the relationship between the roots l]j' we have (2. 128a) (2.128b) Keeping in mind relationships (2.128), we obtain the solution of system (2.127):

k(Arz + 1) 2(l]~-";)'

Coefficients

A;, ~

(2.129)

are determined from the system of equations (1.53) with the

right sides equal to the following (according to relationships (1.107) at values:

Zo

= 0)

(2. 130a)

(2.l30b)

where the values of

cij

are calculated in accordance with equations (1.54). Note

that the following relationships hold: (2.131) In order to clarify the behavior ofthe integrands in solution (2.122), (2.123), it is reasonable to group summands in brackets (z < Zl ):

A; (k)iMz,k) + A; (k)q2 (z,k) + q; (z,k) 2

=

4

L>;(k)qj(z,k) + LBj(k)q/z,k) j=1 j=3

(2. 132a)

A;(k)w1(z,k) + A; (k)w 2 (z,k) + w:(z,k) 2

=

4

LA; (k)wj(z,k) + LB/k)w/z,k) j=1 j=3

(2. 132b)

108

2. Statie and Dynamie Problems for Transversely Isotropie Half-Spaee

where (2.133) Taking ioto account the fonn of the system of equatioos (1.53) with right sides (2.130), we obtain the following system of equations for the new coefficients

A;:

L

clßj

= -B;[cll exp( -TJIZ;)-C\2 exp( -rbZ;)],

L

c2ßj

= B;[c21 exp(-TJI ZI)-c22 exp(-TJ2ZI)]'

4

cllA; + c12 A;

=-

j=3

4

c21 A; +C22 A;

=-

(2.134)

j=3

resulting in A; = B;[K1 exp(-TJ\Z\)+ K 2 exp(-TJ2zl)]'

(2.135) A; = B;[K 3 exp( -TJIZI) + K 1 exp( -TJ2ZI)]

with (2. 136a) K = 2Cl2 C22 2 D'

= 2Cll C21

K 3

D'

(2. 136b) (2. 136c) (2. 136d)

Coefficients

A; decrease exponentially at high values of k and ZI > 0 .

The oonnalized amplitudes of vibrations (2. 122a), (2.l23a) at Z ~ ZI are presented as

+B;[exp(-TJI(zl -z»-exp(-TJ2(zl -z»)]}dk Svv =

f232 Azz

(2. 137a)

[ k Jo(kr){A1A(k)Cw \ exp( -TJ\z) + A;(k)CW2 exp(-TJ2z) 0

(2. 137b)

2.6 Force Applied within Half-Space

109

For the part of the half-space Z > ZI' we apply representation (2.122), (2.123) with (ja = Wa = O. Using for relationship (2.133), one may rewrite the expressions for normalized amplitudes for

Svr =

~:r zz

[k

Z

> ZI as follows:

JI(kr){AI~(k)exp(-Thz)+ A;(k)exp(-112 Z)

0

(2. 138a)

- B;[exp( -111 (z - ZI)) - exp( -112(Z - zl))]}dk Svv = r!2 [ k Jo(kr){A1" (k)C w1 exp( -11lz) + A;(k)CW2 exp( -112Z) Azz 0

(2. 138b) The last summands in expressions (2.137) and (2.138) (containing multiplier B;) represent the solution for infinite space given in equations (2.94), (2.95) for Z ~ 0 ; summands containing coefficients AI~' A; express a correction, required in order to satisry the condition of vanishing stresses in the plane Z = O. The resuIts obtained indicate that at Z ZI the integrands contain a multiplier which

*

decreases exponentially with increasing

k.

At Z = ZI' the multiplier of Bessel's

function in expression Svv tends to a finite limit as

k ~ 00, and only decreasing

and oscillation ofBessel's function Jo(kT=') provide arelatively slow convergence of the integrals. It is reasonable to take parameter e equal to unity (in dynamic problems) and use values (2.19) as dimensionless coordinates. Considerations employed for the calculation of integrals, which express the amplitudes of vibrations of a half-space subjected to a force applied to its surface, still hold for indicates that the given case (see section 2.1). The structure of coefficients Rayleigh waves, re1ated to the vanishing of determinant D, develop progressively less with increasing depth ZI ofthe application point ofthe force.

A;, A;

Static Action 0/ Vertical Force Consider the static problem (e = 0, ß = 1 ). By employing values (2.20), (2.21), one may express displacements of the points in a half-space in finite form, in the same way as for solving other static problems dealing with the transversely isotropie half-space. For 0 ~ Z ~ ZI :

110

2. Statie and Dynamie Problems for Transversely Isotropie Half-Spaee

+K2 [J1(kr)eXP(-k(TJIZ +TJ2ZI»dk +K3 [J1(kr)eXP(-k(TJ2Z +TJlz.»dk +K1 [J 1(kr) exp(-kTJ2(Z +zl»dk + [J1(kr)eXP(-kTJ1(ZI

-z»dk -

[J1(kr)eXP(-kTJ2(Z.

l

-z»dk

(2.1 39a)

Svv =

~~rz ~?r_

2(Th - Th)A zz

[K1Cw1 [Jo(kr)eXP(-kTJ1(Z + z.»dk 0

+K2CW1 [J o(kr)eXP(-k(l1IZ + 112z.»dk + K3CW2 [Jo(kr)eXP(-k(TJ2Z + TJlzl»dk

+K1CW2 [Jo(kr) exp(-kTJ2 (z +zl»dk

-CW1 [Jo(kr)eXP(-kTJ1(ZI -z»dk, +CW2 [Jo(kr)eXp( -kTJ2(ZI - Z»dk]

(2. 139b)

where the following constants are introduced:

(2. 140a) (2. 140b)

-3 K

=

C21 2C;I_ . D

The values of Ci}' jj are given in (2.21). Integrating yields [43]

(2. 140c)

2.6 Force Applied within Half-Space

111

(2.141)

(2.142) where symbol F 3 given in (2.70) is used. Expressions for Svr' Sw at z > z! have the same form (2.141), (2.142). In order to obtain the statie solution for the isotropie ease, when the roots Ti! and Ti z are equal, one should evaluate an indeterminaey ofthe form 0/0, resulting in a known Mindlin's solution, or perform ealculations by using formulas for a transversely isotropie half-spaee with a small deviation of elastie properties from those of an isotropie body. 2.6.2 Action of Horizontal Force Let a horizontal force Qo exp(i rot) be applied in the plane z = z! loeated within a transversely isotropie half-spaee. We shall employ the solution presented in relationships (1.120), (1.121). A transition to the ease of the eoneentrated force and using the dimensionless variables given in (1.27) yields

ur = Ur eos3, U a = Ua sin3, Uz = Uz eos3,

Shr =

A

Ua

ß2 y

=-

[[[Ct (z,k)- p* (z,k)] Jlk~) -q* (Z,k)Jo(kY)]k dk, Qo

21tGrzo r

Sha,

(2.143)

(2.1 44b)

(2.145a)

112

2. Statie and Dynamie Problems for Transversely Isotropie Half-Space

(2. 146a)

(2. 146b) where

1/Ci,k) = A;(k)"(Mz,k) + A;(k)q2(Z,k) + q;(z,k) ,

(2.147)

w'(z,k) = A;(k)w\(z,k) + ~(k)W2(Z,k) + w:(z,k) ,

(2.148)

p'(z,k) = C'(k)p\(z,k) + p:(z,k).

(2.149)

For z > z\, the summands with index a are dropped. Here, we keep the notations introduced when considering action of the vertical force for new values, and use representations (2.124}-(2.126) as before. In contrast to the system of equations (2.127), the following system of equations takes place, according to (1.113):

B; + B; + B; +

B: = 0 ,

(2.150)

TJ\Cw\B; + TJ2 CW2 B; + TJ3CW3B; + TJ4CW4B: = 0 . By using the previously stated relationships between the roots of the characteristic equation and between coefficients CWj ' the derivation of the solution for system (2.150) is straightforward:

(2.151)

Transformation of (2.147) and (2.148) in accordance with (2.132), (2.133) and application ofthe system of equations analogous to (2.134) yields cllA\' + c12 A;

=-

!

c\jBj

= B;(Cll exp( -TJ\Z;) -

~w\ c

j~

w2

12

eXP(-TJ\Z;»), (2.152)

c2\A; + c 22 A;

= - ~ c2ß j = - B; ( c2\ exp( -TJ\ Z;) -

~:: C22 exp(-TJ2Z;)) .

2.6 Force Applied within Half-Space

Hence we obtain coefficients Al' =

1 \3

A; :

-B;[K exp(-rhZI)+ CCWl KeXP(-lhZI)]' I

2

w2

(2.153)

where the values of K j from (2.136) are used. Now, we may rewrite functions q*(z,k) and w*(z,k) given in (2.147), (2.148) in a more convenient form for our calculations. For

Z

~ Zl

q* (z,k) = Al' (k)exp( -Thz) +

A; (k)exp(-Thz)

-B;[eXp(-rh(Z; -z»- ~:: exp(-lh(ZI -Z»],

(2. 154a)

w· (z,k) = Al" (k)Cwl exp( -Thz) + A; (k)Cw2 exp( -1hz) (2. 154b) For Z ~ zp we apply (2.147), (2.148) where the last summand is dropped. Using (2.133), we obtain q* (z,k) =

A; (k)exp(-1hz) + A; (k)exp( -1hz)

~:: exp(-1l2(z -Zl»] w* (z,k) = A; (k)CWl exp( -lllz) + A; (k)CW2 exp( -1l2z) -B;[eXP(-lll(Z -Z;»-

(2.155a)

(2.155b) Next, we consider function p·(z,k) determined in (2.149), (1.116)-(1.119). FOT

Z::O;ZI:

p. (z, k) = C· (k)Pl (z,k) + ClPl (z,k) + C2P2(Z, k) (2.156) where a notation analogous to (2.133) is used: C'(k) = C·(k)+Cl(k).

(2.157)

Functions Pj(z, k) (j = 1, 2) represent fundamental solutions of equation (2.56):

114

2. Statie and Dynamie Problems fOT Transversely Isotropie Half-Spaee

PI (z,k) = exp( -TJoz) ,

(2.158a)

P2(Z,k) = exp(TJoz),

(2.158b)

with the value of TJo given in (2.58). Coefficients CI' C2 satisfy the system of equations (1.117), in which dimensionless variables PI (ZI Jh 152 (ZI' k), Z should be introduced. From a computational point of view, it is convenient to use the following variables analogous to (2.126): C; = CI exp( -TJOZI) ,

(2.1 59a)

C;

(2.1 59b)

=

C 2 exp(TJozl) .

The system of equations for these coefficients follows from system (1.117): C; +C; =0, (2.160) Henee, l_ CI• =-C'2__ -

(2.161)

2TJo

According to (1.119) and (2.157), we have

C A(k~)-

-

C 2 (k~)dp2(O,k)/dpI(O,k) -C (k~)-- --exp I ( -TJoz~) 2 j

dz

dz

2TJo

Taking into account (2.156), (2.159), (2.162) gives the expression for

•

(2.162)

p' (z, k) in

its final form. For Z ~ ZI : (2.163) For z;::: ZI , one should omit the second term in (2.149). Thus, using (2.157), we obtain

p' (z,k) = __I_[exp( -TJo(z + ZI)) + exp( -TJo(z 2TJo

ZI ))].

(2.164)

The substitution ofthe obtained expressions (2.154), (2.155), (2.163), (2.164) into (2. I 43}-(2.146) results in the form of solution suitable for computations. Note that the last terms in (2.164) and (2.155) (having coefficient B;) correspond to the action of the force in an infmite space; the contribution of these terms to the amplitudes of vibrations leads to results which agree with those obtained from (2.111}-(2.115).

2.6 Force Applied within Half-Space

115

Static Action 0/ Horizontal Force

In the following, we take in results of the previous formulas 8 = 0, ß = 1 and employ the values given in relationships (2.20), (2.21), (2.61). The displacements of a half-space are reduced to integrals of the same form as those in relationships (2.62}-(2.64), but with other powers in the exponential functions. Using notations (2.68}-(2.70), we obtain the following expressions for the normalized displacements of points in a transversely isotropie half-space subjected to the action of a horizontal force within the half-space (z ::; Zl ):

s _ hr -

Cw2 r [R\rh(z+z\) 2(ih Cw2 -rbC w\) F\(r,rh(z+z\»

Cw\ - Cw2 F2 (r,r'b (z\

-z»

1

Cw \K2 (zrh +z\r'b) + Cw2 F\(r,zrh +z\l12)

r(z+z\)

r(z\ -z)

+ 2l]oF\(r,l]o(z+z\» + 2l]oF\(r,l]o(z\ -z»' (2.l66)

11 2(zl -z) ]} F3 (r,112(zl-z» . (2.167)

116

2. Statie and Dynarnie Problems for Transversely Isotropie Half-Spaee

In order to obtain expressions for the displacements in the case Z ~ z\ , one should modify, in (2.165) and (2.166), only terms eontaining z\ -z by replaeing (in accordance with (2.155) and (2.164»

Iz -

z\

I,

z\ - z with z - z\. With substitution

formulas (2.165) and (2.166) hold for any relationship between z and

Expression (2.167) for Shz is still valid for z ~ Zl. A comparison of the obtained results with the solution for an infmite space presented in equations (2.119)-{2.121) shows complete agreement between the part of solution for the half-space, which contains z - Zl' and the solution for the infinite space (an "extra" 2 in the denominators ofrelationships (2.165)-{2.167) is explained by the fact that the coefficients of normalized displacements in the solutions for the half-space are half as great as those for the space). Zl.

2.7 Contact Problems for Transversely Isotropie Half-Spaee

2.7.1 Statie Stiffnesses for Cireular Disk on Transversely Isotropie Half-Spaee Consideration of displacements of the surface of a transversely isotropie half-space, subjected to the static action of vertical and horizontal forces applied to the surface, leads to the following conclusion: the relationship between the displacements and distance r between the considered point on the surface and the force application point has a form similar to that for the case of an isotropie half-space. Hence, some formulas for stifIness taken from the theory of the isotropie half-space may be easily generalized for the ease oftransverse isotropy. This holds for the vertical and rocking (about the horizontal axis) stifIness for a eireular disk having a relaxed eontact with the half-spaee and for the torsional stifIness. For the isotropie case, the vertical stifIness takes the following form [129]: K . = 4GR Zl I-v'

(2.168)

where R is the radius of the disk, and G and v are the shear modulus and Poisson's ratio of the isotropie elastic medium, respectively. In the ease of the relaxed contact, the vertical stifIness is determined by Green's function, which expresses vertical displacements ofthe surface ofthe half-space resulting from the vertical unit force applied to the surface. In aecordance with equation (2.1 b), this Green's function has the following form: I-v' wvv(r) = Svv' (2.169) 2Gn01tr

2.7 Contact Problems for Transversely Isotropie Half-Space

117

where for the statie ease eonstant Svv is given in (2.24b). The Green's funetion for the isotropie half-spaee is expressed as I-v ww(r)=--. 2Grr.r

(2.170)

As follows from a eomparison of expressions (2.169) and (2.170), the formulas for the stiffuess K z in the ease of trans verse isotropy may be obtained by multiplying the value in equation (2.168) by the ratio of Green's funetions in (2.170) and (2.169). This results in

K = z

4GrzoR = ~GrzoDR_ (1- VI )Sw c2! CW2 - C22 CW!

(2.171)

Analogously, employing the expression for the roeking stiffuess for an isotropie half-spaee [129]

K

8GR 3 3(1-v) ,

(2.172)

.=---

'Py '

we obtain the eorresponding relationship for the transversely isotropie half-spaee in the ease of relaxed eontaet (i.e. negleeting the influenee of tangential stresses in the eontaet area on the vertieal displaeements ofpoints): K

= 'Py

8GrzoR 3 3(1- VI )SV>

8GrzoDR 3 3(C2l~W2 - C22 CW!)

(2.173)

The torsional statie stiffuess for the isotropie half-spaee may be written as [129] . = 16GR 3

K

1),

(2. 178a)

J{

'1+ 1

81x (Tl, R) =

1'1-11

Shr[ q>Tl +

~ sin(2q»

- 2A sin( q»]+ (1- VI )ShS Tl[ q> - Sin;2q» ]} d A

120

2. Statie and Dynamie Problems for Transversely Isotropie Half-Space

+

{7

"'1IS" +(1-

v')s~ldA

o

(" 51)

(2. 178b)

(Tl> 1),

(2.1 79b) The expression for 8Iz (r,R) is derived by taking into account the relationship whz(r) = -wvh(r), whieh holds due to the principle of reciprocity. In statie

problems, one should recall that values of Svv, Svh, Shr> Sh3 are constant (see formulas (2.24), (2.71)--(2.73» and integrate the complementary terms in the equations for displacements. The second argument, R, denoted by "-" in normalized displacements, should be dropped in the static case; it was written in order to preserve generality. In dynamic problems and in problems dealing with the heterogeneous half-space argument R enters functions Svv, Svh, Shr> Sh3, which contain, in addition, integration variable A. Recall that in the final formulas constructed in sections. 1.7.3-1.7.8 for the circular loading area, argument r in Green's functions is replaced with AR. Let the disk radius be equal to R. Denoting the radii of ring-shaped elements as Rj (Ro =0, RN =R ), the radii of the mean points of the elements as following relative variables: -

1:

•

1j =

(Rj + Rj _ I )/2 (i = 1,2, ... ,N), we introduce the

1j =-

-

(2. 180a)

R' Rj

(2. 180b) R·=} R The system of equations, wh ich serves for the determination of unknowns Pj,qOj' may be written in the following form: N

N

LKijPj + LLijqOj = I, j=1

j=1

L

L

N

j=!

(2.181)

N

MijPj +

j=!

Nijqoj = 0 (i = I, ... ,N),

2.7 Contaet Problems for Transversely Isotropie Half-Spaee

121

1.3 - t - - - ' - - - ' - - - - - - ' - - . . . L . . . - - - L - - t _

1.2

K

____ /._ 'l' y

-'--_ Kz K.'I' ---.--- --/ ----

1. 1

-- -__

y

-

Kx

-- --- - -. - - /. ------ -- -.

1.0

- - - \1=1/3

v=o 0 .0

0 .5

1.5

1.0

2 .0

1), (2. 192a)

~+I

8~;( 1'), R) = j{Shr[

1),

(2. 193a)

124

2. Statie and Dynarnie Problems for Transversely Isotropie Half-Space

J

TI+I

(2. 193b)

2(1- V')Svh sin(cp)dA .

'ö}!>(TJ,R) = -

ITI-II

For the case ofthe horizontal self-balanced load: (2. 194a) '1+\

Ö\ 1).

For the case ofthe vertical antisymmetrie load: Po(l-v')R 2 -(3) Ö1x (TJ,R) , 2G rzo 1t

ö1x(r,R) =

'51~) (TJ, R) =

(2. 197a)

'1+1

fSvh [2TJ sin(cp) - CPA - 0.51.. sin(2cp)] d I..

1'1-11

_{7S~""d)' (~~I) o

(2. 197b)

(TJ>I),

15 (r R) = Po(l- v')R 2 '5(3)(n R) 2x , 2G 2x .• ' ,

(2. 198a)

rz0 1t

'5~!)(TJ,R) = -

T)+I

fSvh[CP - 0.5sin(2cp)]AdA 1..-11

-

{.!.2 (1- v' )Svh1t(1- TJ)2 o

(TJ ~ I)

(2.198b)

(TJ > I),

15 (r R) = Po(1- v')R 2 '5(3)(n R) Iz , 2G Iz .• ' ,

(2. 199a)

rz0 1t

(2. 199b)

The system of equations for unknowns qj' qOj' POjmay be written in the following form ( i = I, ... , N for each of the three groups of equations):

126

2. Static and Dynamic Problems for Transversely Isotropie Half-Spaee N

N

N

~C(l)~C(2)~C(3)--1 ~ ij qj + ~ ij qOj + ~ ij POj - , N

N

~ C(4)-

L...J

i}

j=1 N

i}

~C(6)-

qOj + L...J j=1

N

~C(7)-

L...J

N

~ C(5)-

qj + L...J j=1

y qj +

j=1

i}

POj =

N

~C(8)~C(9) L...J i} qOj + L...J i} POj

j=1

1

0

-

-

,

(2.200)

,

j=1

where

-

qß

(2.201a)

qj=-2G ' rz0 1t

-

qoß

3

(2.20Ib)

qOj=-2G ' rz0 1t

_ POj

=

Po/I- v')R 2

2G

(2.20Ie)

rz0 1t

(I) _ - -(1) Ci} - R/O\X (r;

(2) _ -3-(2) -

-

-

-(1) -

-

I Rj,R j )- Rj_\ö\X (r; I Rj_I,R j _I ), -

-3 -(2) -

-

(2.202)

-RjÖ IX (r;IRj,Rj)-Rj_IÖIX (r;IRj_I>R j _I ),

(2.203)

(3) _ -2-(3) -2 -(3) Ci} -RjÖ lx (rjIRj,Rj)-Rj_\ölx (rjIRj_I,Rj_I),

(2.204)

Ci]

(2.205) (2.206) (2.207) (2.208) (2.209) (2.210) The first N equations (2.200) for i = I, ... ,N express the faet of equality to unity for displaeements of the points that belong to the X-axis along this axis; the next N equations are the same for the points that belong to the Y-axis; and the last N equations allow vertieal displaeements to vanish for the mean points of the ring-shaped elements.

2.7 Contaet Problems for Transversely Isotropie Half-Space

127

Next, we determine the horizontal stiffness for the disk by summing the stresses qj' employing an analogy with relationship (2.188): N

~

2

2

2

N

~-

-2

-2

(2.211)

K x =1t ~qlRj -Rj _I )=21t GrzoR ~qlRj -Rj _I )· j=1 j=1

Weakening ofthe contact conditions results in the simplification of equations. By neglecting the influence of vertieal stresses on the horizontal displacements of the points in the contact area, we obtain the following system of equations: N

L Cbl)qj + j=1

L Cb )qoj = 1, N

2

j=1

N

N

j=l

j=l

(2.212)

L Cb4 )qj + L C~5)qOj = 1. Considering only contact stresses that are uniformly distributed within the ring-shaped elements, and requiring that the contact conditions hold only for the points on the X-axis, we obtain N

LC~l)qj =1.

(2.213)

j=1

A statement ofthe problem similar to (2.213) has been applied to solve the static problem [36], and to solve the dynamic problem [61]; in both cases, the solution was constructed by using dual integral equations. Note that the application of equations (2.212) or (2.213) in static problems leads to identical results, since the load parallel to the X-axis and varying with radius according to the law q

= x

Qo 21tR~R2

(2.214)

_r2 '

ensures the equality of displacements along the X-axis for all points in the contact area and the vanishing of displacements along the Y-axis. In other words, the system (2.212) should yield practically zero values of qOjand the same values of qj as in the case of system (2.213). Indeed, calculations prove that this statement

holds. Calculations also indieate that the parts of matrices C~l) and

cb

4) ,

which

are an outcome of the terms in the expressions for ~~) and ~~ containing sin(2