Lai_Rix_98

Georgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of TechnologyGeorgia Institute of Technology

Simultaneous Inversionof Rayleigh PhaseVelocity and Attenuationfor Near-Surface SiteCharacterization

Carlo G. Lai, PhD

Glenn J. Rix, PhD

National Science Foundation and

U.S. Geological Survey

July 1998

School of Civil andEnvironmental Engineering

iACKNOWLEDGMENTS

This research was supported by the National Science Foundation under Grant No.CMS-9402358 and the U.S. Geological Survey under Award No. 1434-95-G-2634. Anyopinions, findings, and conclusions or recommendations expressed in this material arethose of the authors and do not necessarily reflect the views of the National ScienceFoundation and the U.S. Geological Survey. The authors are grateful to Dr. Clifford J.Astill of the National Science Foundation and Dr. John D. Sims of the U.S. GeologicalSurvey for their support and encouragement.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS i

LIST OF TABLES vii

LIST OF ILLUSTRATIONS ix

SUMMARY xvii

CHAPTER

1 INTRODUCTION 11.1 Motivation......................................................................................................... 11.2 Research Objectives......................................................................................... 41.3 Dissertation Outline ........................................................................................ 8

2 DYNAMIC BEHAVIOR OF SOILS ........................................................................92.1 Introduction.......................................................................................................92.2 A Survey on Modeling Soil Behavior.......................................................... 10

2.2.1 Overview............................................................................................ 102.2.2 The Continuum Mechanics Approach ......................................... 102.2.3 The Discrete Mechanics Approach............................................... 13

2.3 Phenomenological Modeling of Soil Behavior.......................................... 152.4 Experimental Observations.......................................................................... 16

2.4.1 Overview............................................................................................ 162.4.2 Threshold Strains ............................................................................. 162.4.3 Stiffness Degradation and Entropy Production.......................... 20

2.5 Constitutive Modeling and Model Parameters.......................................... 262.5.1 Overview............................................................................................ 262.5.2 Linear Viscoelastic Constitutive Models....................................... 262.5.3 Low-Strain Kinematical Properties of Soils (LS-KPS)............... 352.5.4 Experimental Measurements of LS-KPS ..................................... 50

3 RAYLEIGH WAVES IN VERTICALLY HETEROGENEOUS MEDIA... 573.1 Introduction.................................................................................................... 573.2 Rayleigh Eigenvalue Problem in Elastic Media ......................................... 58

3.2.1 Solution Techniques ........................................................................ 633.3 Effective Rayleigh Phase Velocity in Elastic Media.................................. 673.4 Rayleigh Greens Function in Elastic Media.............................................. 713.5 Rayleigh Variational Principle in Elastic Media......................................... 76

3.5.1 Modal Rayleigh Phase Velocity Partial Derivatives..................... 783.5.2 Effective Rayleigh Phase Velocity Partial Derivatives ................ 833.5.3 Attenuation of Rayleigh Waves in Weakly Dissipative

Media.................................................................................................. 87

iv

3.6 Rayleigh Eigenvalue Problem in Viscoelastic Media.................................903.6.1 A Solution Technique ......................................................................91

3.7 Effective Phase Velocity and Greens Function in ViscoelasticMedia ................................................................................................................99

3.8 Modal and Effective Partial Derivatives in Viscoelastic Media .............101

4 SOLUTION OF THE RAYLEIGH INVERSE PROBLEM...........................1054.1 Introduction ..................................................................................................1054.2 Ill-Posedness of Inverse Problems ............................................................1064.3 Coupled Versus Uncoupled Analysis ........................................................1074.4 Inversion Strategies ......................................................................................1084.5 Occams Algorithm ......................................................................................1104.6 Uncoupled Inversion ...................................................................................118

4.6.1 Overview ..........................................................................................1184.6.2 Uncoupled Fundamental Mode Analysis ....................................1204.6.3 Uncoupled Equivalent Multi-Mode Analysis..............................1214.6.4 Uncoupled Effective Multi-Mode Analysis.................................122

4.7 Coupled Inversion........................................................................................1234.7.1 Overview ..........................................................................................1234.7.2 Coupled Fundamental Mode Analysis.........................................1264.7.3 Coupled Equivalent Multi-Mode Analysis ..................................1264.7.4 Coupled Effective Multi-Mode Analysis .....................................127

5 RAYLEIGH PHASE VELOCITY AND ATTENUATION MEASUREMENTS ..................................................................................................1295.1 Overview........................................................................................................1295.2 Conventional Measurements Techniques.................................................130

5.2.1 Phase Velocity Measurements.......................................................1315.2.2 Attenuation Measurements ...........................................................134

5.3 New Measurements Techniques ................................................................1385.3.1 Uncoupled Measurements .............................................................1385.3.2 Coupled Measurements..................................................................142

5.4 Statistical Considerations.............................................................................1435.4.1 Overview ..........................................................................................1435.4.2 Statistical Aspects of Conventional Measurements...................1455.4.3 Statistical Aspects of New Measurements Techniques.............147

5.4.3.1 Uncoupled Analysis ........................................................1475.4.3.2 Coupled Analysis.............................................................148

5.4.4 Statistical Aspects of Uncoupled Rayleigh Inversion................1525.4.5 Statistical Aspects of Coupled Rayleigh Inversion.....................154

6 VALIDATION OF THE ALGORITHMS..........................................................1576.1 Overview........................................................................................................1576.2 Lambs Problem............................................................................................1576.3 Numerical Simulations.................................................................................162

v6.3.1 Uncoupled Analyses....................................................................... 1716.3.1.1 UFUMA Inversion Algorithms.................................... 1716.3.1.2 UEQMA Inversion Algorithms................................... 178

6.3.2 Coupled Analyses ........................................................................... 1886.3.2.1 CFUMA Inversion Algorithms.................................... 1886.3.2.2 CEQMA Inversion Algorithms ................................... 192

6.3.3 Results and Discussion .................................................................. 199

7 EXPERIMENTAL RESULTS............................................................................... 2077.1 Overview ....................................................................................................... 2077.2 Treasure Island Naval Station Site ............................................................ 2077.3 Uncoupled Inversion................................................................................... 2097.4 Coupled Inversion ....................................................................................... 2167.5 Results and Discussion................................................................................ 217

8 CONCLUSIONS AND RECOMMENDATIONS........................................... 2218.1 Conclusions................................................................................................... 2218.2 Recommendations for Future Research................................................... 227

APPENDIX A - Elliptic Hysteretic Loop in Linear Viscoelastic Materials ................. 229A1 Harmonic Constitutive Relations ......................................................................... 229A2 Energy Dissipated in Harmonic Excitations ...................................................... 230A3 Principal Axes of the Elliptic Hysteretic Loop .................................................. 231

APPENDIX B - Effective Rayleigh Phase Velocity Partial Derivatives ........................ 233

APPENDIX C - Description of Computer Codes ........................................................... 241C1 UFUMA (Uncoupled-Fundamental-Mode-Analysis) ....................................... 241C2 UEQMA (Uncoupled-Equivalent-Multi-Mode-Analysis)................................ 243C3 CFUMA (Coupled-Fundamental-Mode-Analysis)............................................ 244C4 CEQMA (Coupled-Equivalent-Multi-Mode-Analysis) .................................... 245

BIBLIOGRAPHY.................................................................................................................. 247

vii

LIST OF TABLES

Number Page

2.1 Phenomenological Soil Responses to Cyclic Excitation as a Function ofShear Strain Level ............................................................................................................. 19

2.2 Measurement of Low-Strain Dynamic Properties of Soils (LS-DPS)Comparison between In-Situ and Laboratory Techniques........................................ 51

6.1 Medium Properties and Frequencies Used for Validation of the ElasticLambs Problem.............................................................................................................. 159

6.2 Medium Properties and Frequencies Used for Validation of theViscoelastic Lambs Problem........................................................................................ 161

6.3 Medium Properties Used for the Validation of the Inversion Algorithms(Case 1).............................................................................................................................. 163



6.6 Inversion Algorithms RMS Error Misfit for Case 1 Soil Profile ............................. 200



6.9 Inversion Algorithms Performance in Terms of RMS Error Misfit ...................... 204

ix

LIST OF ILLUSTRATIONS

Number Page

1.1 Seismic Energy Path in Ground Response Analysis (Modified fromEPRI, 1993) ..........................................................................................................................1

1.2 Influence of Gmax on the Acceleration Response Spectrum.........................................2

1.3 Influence of DSmin on the Acceleration Response Spectrum........................................3

2.1 Cause-Effects Relationships in Soil Response to Dynamic Excitations .................. 17

2.2 Dependence of Threshold Shear Strains from Plasticity Index(After Vucetic, 1994) ........................................................................................................ 20

2.3(a) Effect of Mean Effective Confining Stress on Modulus DegradationCurves for Non-Plastic Soils (PI = 0) (After Ishibashi, 1992) .................................. 21

2.3(b) Effect of Mean Effective Confining Stress on Modulus DegradationCurves for Plastic Soils (PI = 50) (After Ishibashi, 1992) .......................................... 22

2.4 Modulus Degradation Curves for Soils of Different Plasticity(After Vucetic and Dobry, 1991) ................................................................................... 23

2.5 Dependence of Energy Dissipated within a Soil Mass on Cyclic ShearStrain for Soils of Different Plasticity (After Vucetic and Dobry, 1991)................. 23

2.6 Frequency Dependence of the Energy Dissipated Within a Soil Mass(After Shibuya et al., 1995) .............................................................................................. 25

2.7 Typical Relaxation and Creep Functions for a Viscoelastic Solid............................. 28

2.8 Graphical Representation of the Components of the ComplexModulus.............................................................................................................................. 39

2.9 Stress-Strain Hysteretic Loop Exhibited by a Linear ViscoelasticModel during a Harmonic Excitation............................................................................ 40

2.10(a) Influence of Frequency on Phase Velocity of Viscoelastic Waves asPredicted by the Dispersion Relation Eq. (2.35) ......................................................... 48

2.10(b) Influence of Damping Ratio on Phase Velocity of Viscoelastic Wavesas Predicted by the Dispersion Relation Eq. (2.35)..................................................... 49

2.11 Ranges of Variability of Cyclic Shear Strain Amplitude in Laboratoryand In-Situ Tests (Modified after Ishihara, 1996)........................................................ 52

2.12 Fixed-Free Resonant Column Apparatus (Modified after Ishihara, 1996) .............. 53

3.1 Rayleigh Waves in Vertically Heterogeneous Media ................................................... 61

3.2 Rayleigh Waves Dispersion Curves in Vertically Heterogeneous Media ................. 66

x3.3 Rayleigh Displacement Eigenfunctions in Vertically HeterogeneousMedia ...................................................................................................................................66

3.4 Geometric Spreading Function for Different Types of Media..................................75

3.5 Partial Derivatives of Rayleigh Phase Velocity with Respect to VP and VSfor an Homogeneous Medium........................................................................................82

3.6 Rayleigh Waves in Viscoelastic Multi-Layered Media ..................................................91

3.7(a) Roots of Rayleigh Secular Function in the Region C of the wR-Plane......................95

3.7(b) Roots of Rayleigh Secular Function in the Region D of the zR-Plane ....................95

4.1 Algorithms for the Solution of the Rayleigh Inverse Problem................................109

4.2 Flow-Chart of Rayleigh Simultaneous Inversion Using OccamsAlgorithm..........................................................................................................................117

4.3 Algorithms for the Solution of the Uncoupled Rayleigh Inverse Problem ...........119

4.4 Algorithms for the Solution of the Strongly Coupled Rayleigh InverseProblem.............................................................................................................................125

5.1 Typical Configuration of the Equipment Used in SASW Testing...........................131

5.2 Source-Receivers Configuration in SASW Phase VelocityMeasurements ..................................................................................................................132

5.3(a) SASW Arrangement Using Common Receiver Midpoint Array .............................133

5.3(b) SASW Arrangement Using Common Source Array..................................................133

5.4(a) Attenuation Coefficient Computation at Treasure Island Site.................................136

5.4(b) Attenuation Coefficient Computation at Treasure Island Site.................................137

5.5(a) Geometrical Interpretation of Effective Rayleigh Phase Velocity ..........................140

5.5(b) Geometrical Interpretation of Effective Rayleigh AttenuationCoefficient ........................................................................................................................141

6.1(a) Comparison of Solutions for the Elastic Lambs Problem (Case 1)........................159

6.1(b) Comparison of Solutions for the Elastic Lambs Problem (Case 2)........................160

6.1(c) Comparison of Solutions for the Elastic Lambs Problem (Case 3)........................160

6.2(a) Comparison of Solutions for the Viscoelastic Lambs Problem (Case 1) ...............161

6.2(b) Comparison of Solutions for the Viscoelastic Lambs Problem (Case 2) ...............161

6.2(c) Comparison of Solutions for the Viscoelastic Lambs Problem (Case 3) ...............162

6.3 Rayleigh Dispersion Curves for Case 1 Soil Profile....................................................164

6.4 Rayleigh Effective Dispersion Curve for Case 1 Soil Profile ....................................165

6.5 Rayleigh Attenuation Curves for Case 1 Soil Profile ..................................................166

xi

6.6 Rayleigh Dispersion Curves for Case 2 Soil Profile ................................................... 166

6.7 Rayleigh Effective Dispersion Curve for Case 2 Soil Profile.................................... 168

6.8 Rayleigh Attenuation Curves for Case 2 Soil Profile ................................................. 168

6.9 Rayleigh Dispersion Curves for Case 3 Soil Profile ................................................... 169

6.10 Rayleigh Effective Dispersion Curve for Case 3 Soil Profile.................................... 170

6.11 Rayleigh Attenuation Curves for Case 3 Soil Profile ................................................. 170

6.12 Fundamental Mode Theoretical and Synthetic Dispersion Curvesfor Case 1 Soil Profile ..................................................................................................... 171

6.13 Shear Wave Velocity Profile from UFUMA Inversion Algorithmfor Case 1 Soil Profile ..................................................................................................... 172

6.14 Convergence of UFUMA Inversion Algorithm for Case 1 SoilProfile ............................................................................................................................... 172

6.15 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UFUMA Inversion Algorithm for Case 1 Soil Profile..................................... 173

6.16 Attenuation Curves RMS Misfit Error using UFUMA InversionAlgorithm for Case 1 Soil Profile.................................................................................. 173



6.19 Convergence of UFUMA Inversion Algorithm for Case 2 SoilProfile ............................................................................................................................... 175





6.24 Non-Convergence of UFUMA Inversion Algorithm for Case 3 SoilProfile ............................................................................................................................... 178



xii

6.27 Effective Theoretical and Synthetic Dispersion Curves for Case 1Soil Profile ........................................................................................................................180

6.28 Shear Wave Velocity Profile from UEQMA Inversion Algorithmfor Case 1 Soil Profile......................................................................................................180

6.29 Convergence of UEQMA Inversion Algorithm for Case 1 SoilProfile ................................................................................................................................181

6.30 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UEQMA Inversion Algorithm for Case 1 Soil Profile ....................................181

6.31 Attenuation Curves RMS Misfit Error using UEQMA InversionAlgorithm for Case 1 Soil Profile ..................................................................................182











6.42 Fundamental Mode Theoretical Dispersion and AttenuationCurves for Case 1 Soil Profile ........................................................................................188

6.43 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm for Case 1 Soil Profile................................................189

6.44 Convergence of CFUMA Inversion Algorithm for Case 1 SoilProfile ................................................................................................................................189

xiii

6.45 Fundamental Mode Theoretical Dispersion and AttenuationCurves for Case 2 Soil Profile........................................................................................ 190

6.46 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm for Case 2 Soil Profile ............................................... 191

6.47 Convergence of CFUMA Inversion Algorithm for Case 2 SoilProfile ............................................................................................................................... 191

6.48 Fundamental Mode Theoretical Dispersion and AttenuationCurves for Case 3 Soil Profile........................................................................................ 192

6.49 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm for Case 3 Soil Profile ............................................... 193

6.50 Convergence of CFUMA Inversion Algorithm for Case 3 SoilProfile ............................................................................................................................... 193

6.51 Effective Theoretical Dispersion and Attenuation Curves for Case 1Soil Profile........................................................................................................................ 194

6.52 Shear Wave Velocity and Shear Damping Ratio Profile fromCEQMA Inversion Algorithm for Case 1 Soil Profile .............................................. 195

6.53 RMS Error Misfit of CEQMA Inversion Algorithm for Case 1 SoilProfile ............................................................................................................ .................. 195



6.56 RMS Error Misfit of CEQMA Inversion Algorithm for Case 2 SoilProfile ............................................................................................................................... 197



6.59 RMS Error Misfit of CEQMA Inversion Algorithm for Case 3 SoilProfile ............................................................................................................................... 198

6.60 Inverted Shear Wave Velocity Profiles for Case 1 Soil Stratigraphy ....................... 199

6.61 Inverted Shear Damping Ratio Profiles for Case 1 Soil Stratigraphy...................... 200





xiv

7.1 Treasure Island National Geotechnical Experimentation Site (AfterSpang, 1995) .....................................................................................................................207

7.2 Soil Profile and Properties at the Treasure Island NGES(After Spang, 1995) .........................................................................................................208

7.3 Fundamental Mode Theoretical and Experimental DispersionCurves at Treasure Island NGES .................................................................................210

7.4 Shear Wave Velocity Profile from UFUMA Inversion Algorithm atTreasure Island NGES ...................................................................................................210

7.5 Convergence of UFUMA Inversion Algorithm at Treasure IslandNGES................................................................................................................................211

7.6 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UFUMA Inversion Algorithm at Treasure Island NGES ..............................212

7.7 Attenuation Curves RMS Misfit Error using UFUMA InversionAlgorithm at Treasure Island NGES ...........................................................................212

7.8 Effective Theoretical and Experimental Dispersion Curves atTreasure Island NGES ...................................................................................................213

7.9 Shear Wave Velocity Profile from UEQMA Inversion Algorithm atTreasure Island NGES ...................................................................................................213

7.10 Convergence of UEQMA Inversion Algorithm at Treasure IslandNGES................................................................................................................................214

7.11 Shear Damping Ratio Profile and Theoretical Attenuation Curvefrom UEQMA Inversion Algorithm at Treasure Island NGES .............................214

7.12 Attenuation Curves RMS Misfit Error using UEQMA InversionAlgorithm at Treasure Island NGES ...........................................................................215

7.13 Fundamental Mode Theoretical and Experimental Dispersion andAttenuation Curves at Treasure Island NGES...........................................................215

7.14 Shear Wave Velocity and Shear Damping Ratio Profile fromCFUMA Inversion Algorithm at Treasure Island NGES ........................................216

7.15 Convergence of CFUMA Inversion Algorithm at Treasure IslandNGES................................................................................................................................217

7.16 Comparison at Treasure Island NGES of Shear Wave Velocityfrom Surface Wave Test Results with Other IndependentMeasurements ..................................................................................................................218

7.17 Comparison at Treasure Island NGES of Shear Damping Ratiofrom Surface Wave Test Results with Other IndependentMeasurements ..................................................................................................................219

xv

SUMMARY

Surface wave tests are non-invasive seismic techniques that can be used to determinethe low-strain dynamic properties of a soil deposit. In the conventional interpretation ofthese tests, the experimental dispersion and attenuation curves are inverted separately todetermine the shear wave velocity and shear damping ratio profiles at a site. Furthermore,in the inversion procedure, the experimental dispersion and attenuation curves are matchedwith theoretical curves, which include only the fundamental mode of propagation.

The only approach available in the literature that accounts for multi-mode wavepropagation is based on the use of Greens functions where the partial derivatives ofRayleigh phase velocity with respect to the medium parameters required for the solution ofthe inverse problem are computed numerically, and therefore very inefficiently.

This study presents a new approach to the interpretation of surface wave testing. Thenew approach is developed around three new ideas. First, the definition of the low-straindynamic properties of soils and the Rayleigh wave eigenproblem are revisited andreformulated within the framework of the linear theory of viscoelasticity. Secondly, anexplicit, analytical expression for the effective Rayleigh phase velocity has been derived.

The effective phase velocity concept forms the basis for the development of a newsurface wave inversion algorithm based on multi-mode rather than modal dispersion andattenuation curves. Closed-form expressions for the partial derivatives of the effectiveRayleigh phase velocity with respect to the medium parameters have also been obtained byemploying the variational principle of Rayleigh waves.

Thirdly, a numerical technique for the solution of the complex-valued Rayleigheigenproblem in viscoelastic media has been implemented. An immediate application ofthis solution is the development of a systematic and efficient procedure for simultaneouslydetermining the shear wave velocity and shear damping ratio profiles of a soil deposit fromthe results of surface wave tests. The simultaneous inversion of surface waves data offerstwo major advantages over the corresponding uncoupled analysis. First, it explicitlyrecognizes the inherent coupling existing between the velocity of propagation of seismicwaves and material damping as a consequence of material dispersion. Secondly, thesimultaneous inversion is a better-posed mathematical problem (in the sense ofHadamard). The new approach to surface wave analysis is illustrated using severalnumerical simulations and experimental data.

11 INTRODUCTION

1.1 Motivation

Geotechnical earthquake engineering is a well-established discipline concerned withunderstanding the role-played by soils in the effects induced by earthquakes. An essentialpart of geotechnical earthquake engineering is ground response analysis. The objective ofground response analysis is the prediction of the free-field site response induced by acatastrophic event, which may be an earthquake or an explosion, occurring in the interiorof the earths crust. A correct implementation of a ground response analysis requires aproper modeling of several aspects of the problem including the rupture mechanism at thesource and all the phenomena associated with the propagation of seismic waves from thesource to the desired site at the free-surface. The latter includes transmission of seismicenergy within the continental and oceanic structures of the earth, as well as wavepropagation within the soil mass overlaying the bedrock. Figure 1.1 is a schematicrepresentation of the spread of seismic energy once it is released from the source. Groundresponse analysis has important applications in several areas of geotechnical earthquakeengineering and soil dynamics. Some of the most common include local site response

Source

Regional Geology

t&& ( )y tF

Local Site Conditions

t&& ( )y tL

Figure 1.1 Seismic Energy Path in Ground Response Analysis (Modified from EPRI,1993)

2 Introduction

analyses for the development of design ground motions and response spectra, studies ofsoil liquefaction potential, seismic stability analyses of slopes and embankments, and studiesof dynamic soil-structure interaction.

A crucial step in implementing a ground response analysis is the selection of theconstitutive models and their associated parameters used to simulate the dynamic behaviorof the soil. Studies have shown that the strain level(s) induced by an earthquake can rangeanywhere from 10-3% up to 1+% depending on several variables including the magnitude ofthe event, the source mechanism, the distance from the epicenter, and the properties ofthe medium (Kramer, 1996). Therefore, an appropriate constitutive model requires adefinition of the model parameters over a broad range of strain levels, ranging from verysmall strain levels (below the linear cyclic threshold strain), where the response of themedium can be considered linear but not necessarily elastic, to intermediate and large strainlevels where non-linear behavior dominates. In many cases, the very small-strain dynamicproperties of soils are sufficient, since there are often circumstances in seismology and soil-dynamics where the assumption of linearity is an acceptable approximation.

The following example illustrates the crucial role played by the very small-straindynamic properties of a soil deposit in controlling the amplification or de-amplification ofan input motion applied at the bedrock. Figure 1.2 and Figure 1.3 illustrate the results of alocal site response analysis performed using the computer program SHAKE91 (Idriss andSun, 1991). This code solves the initial-boundary value problem associated with the one-

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Period [sec]

Spec

tral

Acc

eler

atio

n [g

] Gmax = 150.8 MPa

Gmax = 67.0 MPa

Gmax = 16.8 MPa

Input Motion

z = 5 %

Figure 1.2 Influence of Gmax on the Acceleration Response Spectrum

Introduction 3

dimensional wave propagation in layered viscoelastic media using an equivalent linearanalysis. The input motion used in the numerical simulation was the N90E accelerationrecord of the 18 May 1940 El Centro earthquake scaled to a maximum acceleration of0.15g. This acceleration time history was applied at the base of a homogeneous soil depositoverlaying the bedrock and having a thickness m30H = .

Figure 1.2 shows the influence of the initial tangent shear modulus Gmax (or the shearwave velocity VS ) on the acceleration response spectrum. As expected, the maximumresponse of the spectrum is attained at periods close to the fundamental period of the site,calculated with the well-known expression 4H VS/ .

The influence of the initial shear damping ratio DSmin (value of shear damping ratioassociated with a strain level below the linear cyclic threshold strain) on the accelerationresponse spectrum is shown in Figure 1.3. Low values of damping ratio results in a largeamplification of the input motion at the bedrock, particularly at periods close to the naturalperiod of the site. In both response spectra the structural damping x was assumed equal to5%.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Period [sec]

Spec

tral

Acc

eler

atio

n [g

]

Dsmin = 0.5 %

Dsmin = 5.5%

Input Motion

z = 5 %

Figure 1.3 Influence of DSmin on the Acceleration Response Spectrum

The above figures illustrate the important role played by the low-strain dynamicproperties of a soil deposit in determining the dynamic response of a single degree offreedom system. The low-strain dynamic properties of soil deposits can be measured with avariety of techniques. They are generally classified into laboratory techniques and in-situ or

4 Introduction

field techniques. At the end of Chapter 2, will be presented a summary with the mostimportant advantages and disadvantages of these techniques.

The main focus of this research effort was on the determination of the very small-strain dynamic properties of a soil deposit from the interpretation of the results of surfacewave tests. The use of surface (Rayleigh) waves for geotechnical site characterization hasseveral advantages over more conventional seismic methods such as cross-hole and down-hole tests. The most attractive feature of surface wave tests is that they are non-invasiveand hence they do not require the use of boreholes, which permits the tests to beperformed more rapidly and at lower cost than most invasive methods.

Furthermore, at sites where subsurface conditions (e.g. gravelly soils) or environmentalconcerns (e.g., solid waste landfills) hinder the use of boreholes and probes, surface wavetests may constitute the only possible choice for an in-situ site investigation. The nextsection describes the most important research objectives pursued during this study.

1.2 Research Objectives

Three primary objectives were envisioned at the beginning of this research effort. Thefirst objective was the development of a systematic and efficient procedure forsimultaneously determining the low-strain values of VS and DS from the results of surfacewave tests. The most common application of surface wave methods is the determinationof the shear wave velocity profile at a site (Nazarian, 1984; Snchez-Salinero, 1987; Rix,1988; Stokoe et al., 1989). Recently, Rix et al. (1998a) developed a procedure to calculatenear-surface values of material damping ratio from measurements of the spatial attenuationof Rayleigh waves. However, until now the two problems of determining the shear wavevelocity and the shear damping ratio profiles at a site have been considered separately andtherefore uncoupled.

One of the goals of this study was to present a different approach to the problem,where Rayleigh wave phase velocity and attenuation measurements are invertedsimultaneously. The simultaneous inversion of Rayleigh wave phase velocity andattenuation measurements has two major advantages over the corresponding uncoupledanalysis: it is an elegant procedure to account for the coupling existing between phasevelocity of seismic waves and material damping and the simultaneous inversion is a better-posed mathematical problem (in the sense of Hadamard).

The numerical solution of a non-linear inverse problem is obtained in most cases fromthe iterative solution of the corresponding forward problem, which in this case is theboundary value problem of Rayleigh waves in dissipative media. In developing the solutionof the Rayleigh forward problem, extensive use was made of the powerful and elegantmethods of complex variable theory, more precisely of the theory of analytic functions.

Introduction 5

Subsequent chapters of this dissertation will provide a description of the theoretical basisof the simultaneous inversion and will illustrate its applications to some experimental data.

The second objective was, in a sense, motivated by the first objective of thisdissertation. The problem of determining the very small-strain dynamic properties of soilsraises fundamental questions about the meaning of words like properties of soils. Implicit tothe definition of such a term is assumptions of material behavior to which ascribe certainbehavioral properties. As a result, different idealizations of material behavior will require thedefinition of different types of material properties. It is unfortunate that often in thegeotechnical literature, it is customary to take for granted certain definitions of materialbehavior without ever questioning the validity or the appropriateness of these definitions.One remarkable example is constituted by the so-called dynamic properties of soils a term usedto collectively denote stiffness and material damping ratio of soils. Chapter 2 of thisdissertation attempts to revisit the definition of these parameters within the framework ofa consistent theory of mechanical behaviour. It is shown that whereas it is not a trivial taskto construct a mathematical model describing the behavior of complex materials such assoils, it is still possible to formulate relatively simple and accurate phenomenologicalmodels by restricting the formulation to the low strain spectrum. These and other issuesrelated with constitutive modeling of soils are addressed in this chapter, from a perspectivethat is relevant to problems of geotechnical earthquake engineering.

Finally, the third objective of this research effort was developing a better understandingof the theoretical aspects associated with the interpretation of surface wave measurements.In the current procedure the shear wave velocity and shear damping ratio profiles aredetermined from the application of an inversion algorithm to an experimental dispersionand attenuation curve. Minimization of the distance (specified by an appropriate definitionof norm) between these curves and those predicted theoretically from an assumed profileof model parameters is the most common criterion used for the solution of the inverseproblem of surface waves. This procedure has an important limitation: the simulated(theoretical) dispersion and attenuation curves are defined as modal response functions, i.e. theyare referred to a specific mode of propagation of Rayleigh waves. Conversely, theexperimental dispersion and attenuation curves reflect, in general, the contributions ofseveral modes of Rayleigh wave propagation and also of body waves in the near field.

There are currently two procedures used to overcome this limitation. The first andmost common one is based on comparing the experimental dispersion and attenuationcurves with those of the fundamental mode obtained theoretically. This method is referredto in the literature as a 2-D analysis of surface waves (Rosset et al., 1991;). The resultsprovided by the 2-D analysis are generally satisfactory for normally dispersive (i.e. regular)shear wave velocity profiles (Gucunski and Woods, 1991; Tokimatsu, 1995). The secondmethod of interpretation of surface wave data referred in the literature as a 3-D analysisconsists of reproducing with a numerical simulation the actual set-up of the experiment.The theoretical phase velocities, for instance, are computed from the phase differences

6 Introduction

between theoretical displacements, and the latter are calculated at locations that emulatereceivers spacings used in the experiment. This method is exact, however it has thedisadvantage of requiring the use of a Greens function program for computing thedisplacement field, which is difficult and time-consuming if one wants to include the bodywave contributions in the near field. Furthermore, in this approach the partial derivativesrequired for the solution of the non-linear inverse problem of determining an unknownshear wave velocity profile that corresponds to a given experimental dispersion curve, arecomputed numerically. Computation of numerical partial derivatives is notoriously an ill-conditioned problem, and in this case is also computationally expensive (if compared withother methods).

This study attempts a new interpretation of surface wave measurements, whichcombines the simplicity of a 2-D analysis with the robustness of a 3-D analysis. This isachieved by deriving an explicit expression for the effective phase velocity of Rayleighwaves in vertically heterogeneous media (in the literature this quantity is often referred toas the apparent phase velocity). This is the phase velocity measured experimentally insurface wave tests if the contribution of the body wave field is neglected. As expected, theeffective phase velocity is a local quantity in the sense that its value varies continuously withthe distance from the source at a given frequency. The effective phase velocity arises fromthe superposition of several modes of propagation of Rayleigh waves, each traveling at adifferent phase velocity, which is denoted as the modal velocity. In dissipative media, theeffective wave propagation leads naturally to the concept of effective attenuationcoefficient, which is also a local quantity. In light of these results, the commonly usednotions of dispersion and attenuation curves should be more properly replaced by those ofdispersion and attenuation surfaces.

Closed-form analytical expressions for the partial derivatives of the effective phasevelocity with respect to the medium parameters (shear and compression wave velocities)were also obtained by employing the variational principle of Rayleigh waves. These partialderivatives are essential for an efficient and accurate solution of the Rayleigh inverseproblem.

Finally, in this attempt to re-formulate the current interpretation of surface wavemeasurements, a new approach is proposed which is based on the replacement of thedispersion and attenuation curves with a different type of response function: thedisplacement spectra. The motivation for introducing this new procedure was largelymotivated by the observation that in surface wave tests the primitive quantities measuredexperimentally are the displacement phase and amplitudes, and not the Rayleigh phasevelocities and attenuation coefficients. In fact, the effective Rayleigh phase velocity isnothing but the partial derivative, at constant frequency, of the displacement phase withrespect to the source-receiver distance. A similar interpretation holds for the Rayleigh waveattenuation coefficient if the notion of displacement phase is replaced by that ofdisplacement amplitude.

Introduction 7

In their efforts to identify the structure of the Earth, seismologists use time historyrecords combined with digital signal processing techniques to obtain modal dispersion andattenuation curves generated by seismic events. Geotechnical engineers use the dispersiveproperties of surface waves generated by active sources for near-surface sitecharacterization. In attempting to find a solution to their respective problems,seismologists and geotechnical engineers face similar problems and difficulties, therefore itis natural that they often come up with similar solution strategies. However, there are twomajor differences that profoundly distinguish the problems faced by seismologists andgeotechnical engineers.

The first and most important difference is the scale factor. Whereas for seismologiststhe layer thickness of their stratified Earth is on the order of kilometers, geotechnicalengineers deal with layers whose size is two or even three order of magnitude smaller. Alsothe frequencies involved in seismology and geotechnical engineering are very different.Most of the energy contained in a seismic record has a frequency range on the order of 0.1to 10 Hz. Geotechnical engineers analyze surface waves having frequencies up to 200 Hzor more. Furthermore, there is a substantial difference in seismology and geotechnicalengineering, concerning the distances over which surface waves are detected and recordedwith seismometers and geophones.

As a result of different spatial and temporal scales involved in seismology andgeotechnical engineering, the phenomenon of surface wave propagation will assume inthese two disciplines certain unique and distinctive features. In seismology for example, themodes of propagation are in most cases well defined and separated from each other, andseismologists can determine them from the interpretation of time-history records. On thecontrary, in geotechnical engineering surface wave modes generated by harmonicoscillators are mostly superimposed rather than separated to each other. It is thereforenatural to expect based on these observations, different methods of interpretation inseismology and geotechnical engineering.

The second difference between the problems faced by seismologists and geotechnicalengineers is that seismologists do not have control over the source of wave energy:earthquakes occur at times, locations and with characteristics (duration, frequency content,source mechanism, etc.) that to this date are not predictable. Conversely, not only cangeotechnical engineers select the source type, but they can also choose its spatial location.As a result, the task of geotechnical engineers in interpreting surface wave data isenormously simplified if compared with that of seismologists, as long as the former canturn to their advantage their ability of control over the source.

In summary, the objectives of this research effort were to reformulate the conventionalinterpretation of surface wave tests by developing a technique to simultaneously invertRayleigh phase velocity and attenuation data, while accounting for the multi-mode nature

8 Introduction

of Rayleigh wave propagation in vertically heterogeneous media. These goals were achievedby first constructing a consistent model of soil dynamic behavior at very-small strain levels.

1.3 Dissertation Outline

The dissertation is organized into eight chapters and three appendices. Chapter 2 is anintroduction to the fundamental problem of modeling soil behavior at low-strain levelsunder dynamic excitation. Objective of this chapter is to provide experimental evidence forsupporting the assumption of linear viscoelasticity as an appropriate constitutive law formodeling dynamic soil behavior at low-strain levels. After a critical overview of theavailable models of soil behavior, the viscoelastic constitutive model is introduced and thecorresponding model parameters are rigorously defined. Chapter 3 reviews the theory ofRayleigh waves propagation in elastic and viscoelastic vertically heterogeneous media. Afterillustrating well-known results, an explicit analytical expression for the effective phasevelocity is derived. The variational principle of Rayleigh waves is then used to obtainexplicit relationship for the partial derivatives of the effective Rayleigh phase velocity withrespect to the shear and compression wave velocity of the medium. An important resultpresented in this chapter is a new numerical technique for the solution of the complexRayleigh eigenproblem in linear viscoelastic media. The technique is quite general and it canalso be applied to strongly dissipative media. Chapter 4 illustrates the main aspectsassociated with the solution of the Rayleigh inverse problem, and presents the inversionalgorithms developed in this study. Chapter 5 reviews the conventional techniques used insurface wave measurements, and introduces a new methodology aimed to improveconsistency, in surface wave testing, between measurement procedures and interpretationof the results. Some statistical considerations related with surface wave measurements arealso analyzed. Chapter 6 presents the results of a systematic numerical simulation for thevalidation of the algorithms developed in this study. Chapter 7 illustrates an example ofapplication of these algorithms to a real site. Finally, Chapter 8 presents the conclusions ofthis research study and illustrates some recommendations for future research.

92 DYNAMIC BEHAVIOR OF SOILS

2.1 Introduction

Scientific understanding proceeds by way of constructing and analyzing models of the segments oraspects of reality under study. The purpose of these models is not to give a mirror image of reality, not toinclude all its elements in their exact sizes and proportions, but rather to single out and make available forintensive investigation those elements which are decisive. We abstract from non-essentials, we blot out theunimportant to get an unobstructed view of the important, we magnify in order to improve the range andaccuracy of our observation. A model is, and must be, unrealistic in the sense in which the word is mostcommonly used. Nevertheless, and in a sense, paradoxically, if it is a good model it provides the key tounderstanding reality. (From Baran and Sweezy, 1968).

Another feature that adds its contribution to the complexity of soil behavior, is thecoupling effect of soil responses. Thermomechanical coupling is one example of a responseinteraction effect, which is usually negligible in soils. However, soils may exhibit othercoupling effects, which may be more important including piezo-electric and chemico-mechanical coupling (Fam and Santamarina, 1996). Accounting for these responseinteraction phenomena may lead to unexpected consequences such as the reformulation ofthe principle of effective stress of classical soil mechanics. Newer formulations of thisprinciple (Mitchell, 1976) recognize that mechanical effects (i.e. change of the effectivestress) may be obtained not only by variations of the gravitational fields (i.e. total stressand/or hydrostatic pressure), but also by means of electro-chemical perturbations (doublelayer theory).

To date, a comprehensive constitutive model able to account in a unified framework forall these phenomena is not available. Even if such a model existed, its complexity would beformidable, and most likely not suitable for applications to real-world engineering problems.However, despite the complexity of soil behavior, soils do not exhibit all their features withthe same degree of importance. Depending on the nature of the problem underinvestigation, which includes its intrinsic spatial and temporal scales, the strain levelsinvolved, and the dominant external fields, many of the features that characterize soilbehavior may be regarded as secondary. They may be interpreted as second or even higherorder effects, and in most cases they may be neglected without appreciable changes in theresulting analyses.

This is true in many other engineering disciplines and applied sciences, and echoing thepreface of Baran and Sweezy, (1968), it may be said that the art of good engineering oftenidentifies with the ability of transforming a difficult problem into a simpler one byattentively discerning what is important from what is superfluous or unessential.

10 Dynamic Behavior of Soils

2.2 A Survey on Modeling Soil Behavior

2.2.1 Overview

Two approaches or philosophies are currently used to model the mechanical behaviorof soils and of solids in general. The classical approach is that of continuum mechanics,which is based on the identification of a deformable medium, in this case soil, with regionsof the three-dimensional Euclidean space. In this approach the mass distribution as well asall the pertinent field variables (deformation gradient tensor, stress tensor, displacementvector, etc.) are assumed to be continuous function of the coordinates.

An alternative to continuum mechanics which is gaining popularity is discretemechanics, which has its roots in explicitly recognizing the discrete nature of soils (and ofmatter in general) and hence modeling them as an aggregate of interacting rigid ordeformable discrete particles.

2.2.2 The Continuum Mechanics Approach

Despite its limitations, continuum mechanics is a formidable tool in the solution of aninnumerable class of practical problems. Most of the strengths of continuum mechanicscome from the consequences of the continuity assumption such as the availability of thepowerful tools of differential and integral calculus (Malvern, 1969). It is by using theconcepts of calculus that the fundamental concepts of stress and strain at a point may bedefined.

Classical continuum mechanics was originally conceived to describe the mechanicalbehavior of bodies composed of one constituent which could be solid, liquid or gas as longas the continuity assumption of the field variables holds (within an acceptable accuracy). Butthe assumption of continuity itself does not prevent the possibility of describing themechanics of heterogeneous materials. The extension of one-constituent continuummechanics to bodies composed of more than a single substance leads naturally to the socalled theories of mixtures (Truesdell, 1957). Although the origin of such theories may be datedback at the beginning of the century throughout the work of notable chemists andphysicists working on the kinetic theories of gases, the first systematic attempt to constructa multi-component theory of continuum mechanics is due to Truesdell (1957). Since thenthis theory has been extended to include several other features including chemical reactionsoccurring among the constituents and electromagnetic effects (Eringen, 1976).

However, classical mixture theories are based on the fundamental postulate that amixture is represented by a sequence of continuous bodies all of which occupy the sameregions of space simultaneously (Truesdell, 1957). This assumption of intermiscibility maybe appropriate to model mixtures of fluid-like components; however there are physicalsituations where this assumption is not appropriate. A few examples include soils, porousrocks, granular materials, and multiphase suspensions where the mixture consists ofidentifiable solid particles or a matrix surrounded by one or more fluids. These types of

Dynamic Behavior of Soils 11

materials lead to the important distinction between multiphase immiscible mixtures andmiscible mixtures (Goodman and Cowin, 1972). The continuity assumption may still be usedbut an additional continuous field variable must be introduced: the volume fraction whichcorresponds to the proportion of volume occupied by each component of the mixture.This scalar field reflects important microstructural features of the mixture subjected to athermomechanical process.

Applications of the theory developed by Goodman and Cowin (1972) to model thebehavior of particulate materials have produced interesting results. In their formulation thebalance laws are essentially the same as those proposed by Truesdell (1957) with theexception that a new equation of balance is included to account for the role-played byvolume fraction changes. This equation is called the balance of equilibrated forces and itdescribes the distribution of microstructural forces which is effective in a multiphasemixture. In essence, the balance of equilibrated forces states that the internal distribution offorces among the constituents of the mixture is directly related to the changes of theirvolume fractions. It can be viewed as a generalization of the principle of effective stress ofclassical soil mechanics. One of the most attractive features of this theory is its ability tomodel dilatancy, a phenomenon that cannot be modeled with classical continuummechanics. Nevertheless, it should be emphasized that although volume fraction is animportant field variable, it is not sufficient to discriminate between two mixtures withuniform distributions of grains, one with large grains and the other with small grains of thesame material density. In other words, volume fraction alone cannot take into account thegrain size distribution of the constituents (Passman, Nunziato and Walsh, 1984) and in thissense is a scaleless theory. An interesting new approach to the construction of a multi-component theory of immiscible mixtures has been proposed recently by Wilmanski (1996).One of the main features of this theory is the replacement of the equation of balance ofequilibrated forces of Goodman and Cowin with a balance equation for porosity. Theintroduction of this new law of balance is motivated by microscopic considerations of thetime rate of change of the geometry of the solid phase of the mixture with respect to thefluid phase.

The application of the theory of immiscible mixtures to multi-phase media composed ofa solid phase and one or more fluid phases leads naturally to the theories of porous media (Biot,1955; Bowen, 1982). Such theories, which are special cases of the more general theoriesof mixtures, have been the subject of considerable interest over the last 35-40 years. Thisinterest continues today in the form of different formulations and/or assumptions (DeBoer, 1996). Theories of porous media constitute a possible mathematical framework formodeling the mechanical behavior of complex multi-phase materials such as soils.

A different line of thought for modeling particulate materials within the realm ofcontinuum mechanics was developed during the late 1970s and early 1980s by Oda (1978),Rothenburg (1980), Nemat-Nasser (1982), and Satake (1982), just to mention few of theearly investigators. Their approach was to supplement classical continuum mechanics with


concepts derived from studies of micromechanics. An important aspect of this theory isthat it provides the link between macroscopic quantities such as the stress tensor and themicrostructural parameters describing the internal arrangement of the particles (such asparticle orientation, orientation of contacts, distribution of inter-particle contact micro-forces, etc.). This link is obtained by means of appropriate averaging procedures of theabove microstructural variables over a representative elemental volume.

As the studies in this area of micromechanics continued, it became apparent the needfor introducing a new quantity able to describe the spatial arrangements of particulatematerials. This new quantity was introduced with the name of fabric tensor (Oda, 1978), andsince then the use of the fabric tensor as a descriptor of the packing of granular materialshas increased. The fabric tensor is defined as a second rank symmetric tensor and itdescribes a continuous field variable. Its use in the mechanics of granular materials has leadto the important definitions of solid phase and void phase fabric tensors.

These quantities, in particular the void phase fabric tensor, have played a major role inthe applications of the concepts of micromechanics to critical state soil mechanics(Muhunthan and Chameau, 1996), particularly because it has been shown how to determinethem experimentally (Muhunthan, 1991; Frost and Kuo, 1996; Kuo and Frost, 1997). Thework in this area of soil modeling has been very intense in the recent years, and realisticconstitutive equations relating micro-scale variables and the macro-scale variables have beenproposed for both granular materials (Christoffersen et al., 1981) and cohesive soils (Masadet al., 1997). The results obtained thus far are encouraging, but more research is required fora definitive validation of these theories.

This brief survey of the use of a continuum mechanics framework to model soilbehavior is concluded with a short introduction to the so-called polar or generalized theories ofcontinuum mechanics. It is an interesting subject, which is appealing for its inherent capabilitiesof modeling continua having an inner microstructure. The first theory on polar continuawas that of the Cosserat brothers in 1907 who laid down the foundations of what today isknown as Cosserats elasticity to be distinguished from the classical theory of elasticity alsocalled Cauchys elasticity. Since then, there have been a large number of contributors (Greenand Rivlin, 1964; Eringen and Suhubi, 1964, Mindlin, 1964, Eringen, 1976).

Polar materials are defined as those that admit the existence of couple stresses and bodycouples (Truesdell and Noll, 1992). Such a possibility, which is disregarded in classicalcontinuum mechanics, leads to the construction of an alternative continuum mechanicswhere the geometrical points of the continua may possess properties similar to those ofrigid or deformable particles. Thus the geometrical points of classical continuum mechanicswhich possess three degrees of freedom are extended to include additional degrees offreedom which may be the three independent rotations (micropolar continua). In practice thisgeneralization may continue by simply ascribing additional degrees of freedom to thematerial point (Eringen, 1976). Micromorphic continua are defined as media having geometrical


points with a total of twelve degrees of freedom: three translations, three rotations, and sixmicrodeformations. Now the material point not only can translate and rotate like a rigidparticle, but it may behave as if it were a deformable particle.

Obviously, the kinematics of polar continua is inherently much more complicated thanthe kinematics of classical continuum mechanics, particularly in the case of micromorphiccontinua. Non-locality is the peculiar feature of polar continuum mechanics, which essentiallymeans that this theory is able to account implicitly for the scale effects induced by the innermicrostructure of the continua (Granik and Ferrari, 1993). In this sense classical continuummechanics is clearly a scaleless theory.

Polar continuum mechanics is not the only type of non-local continuum theory ofmechanics. Others include the so called materials of grade N (Ferrari et al. 1997) which aredefined as those deformable media whose kinematics are described not only by thedeformation gradient of classical continuum mechanics, but also by higher gradientmeasures (Truesdell and Noll, 1992).

Although generalized theories of continuum mechanics have been advancedconsiderably in recent years, very few applications have been implemented, particularly insoil mechanics. Possible explanations include the fact that, despite their elegance, thesetheories are complex (Ferrari et al. 1997). Furthermore, there are additional difficultiesassociated with the physical interpretation and experimental determination of theirconstitutive parameters.

2.2.3 The Discrete Mechanics Approach

The final part of this section is dedicated to alternatives to continuum mechanics as aframework to model the mechanical behavior of soils. As mentioned at the beginning ofthis section, in recent years the popularity of discrete mechanics has increased among soilmechanicians. The attractive feature of discrete mechanics is the explicit recognition by thistheory of the discrete nature of matter, even though it is clear that the manifestation of thisnature is scale dependent.

By analogy to continuum mechanics, there are several classes of discrete mechanicstheories or techniques. Among them, the two mentioned here are Doublet Mechanics (DM)and the Discrete (or Distinct) Element Method (DEM). The most popular theory, at leastwithin the geotechnical community, is certainly the DEM. Its original formulation datesback to the work of Cundall and Strack in 1979, and since then DEM has been usedextensively in studying the mechanical responses of granular materials (Cundall and Strack,1979; Ting et al., 1989).

Recently, DEM has also been applied to study the constitutive behavior of water-saturated cohesive soils (Anandarajah, 1996). The essential feature of DEM is modeling asoil element as a discrete assemblage of interacting rigid or deformable particles. The


interaction among particles is governed by appropriate constitutive laws, which specify themagnitude and the direction of the contact forces. The overall system is subjected to thelaws of dynamics with forces and moments due to the self-weight of the particles and toparticle-to-particle interaction.

The computational procedure of DEM involves an explicit time-integration scheme ofthe equations of motion. DEM simulations are computationally very expensive and thislimits the size of the problems (number of particles) that can be analyzed. Sometimes forthe purpose of reducing the complexity of the simulations, it has been found convenient toanalyze two-dimensional problems using circular or elliptical disks (Ting et al., 1993). Parallelcomputing seems to be the answer for the future of DEM (Kuraoka and Bosscher, 1996),but more research is needed on modeling the particle-to-particle interactions.

A theory that has been recently proposed and that, in view of the authors (Ferrari et al.,1997; Granik and Ferrari, 1993), should bridge the gap between continuum and discretemechanics (DEM) is Doublet Mechanics (DM). The essential feature of DM is its buildingblock, which is constituted by a pair of geometric points separated at a finite distance (adoublet). This elementary unit replaces the differential volume element of continuummechanics and the discrete particle or grain of DEM. In the kinematics of DM, thegeometrical points or nodes of a doublet have the following degrees of freedom: they canmove relative to each other in both the axial and the normal directions to their commonaxis; moreover, they may rotate about their common axis. DM can be constructed withdifferent degrees of approximation (Ferrari et al., 1997).

In its simplest form it is a scaleless theory which reduces to classical continuummechanics. However, higher order approximations of DM are non-local theories and hencethey may account for the scale effects caused by the discrete nature of the medium.Preliminary results from the application of this theory are promising. It has been shown forinstance (Granik and Ferrari, 1993), that DM may solve the well-known Flamants paradox(the Flamants problem is the two dimensional equivalent of the Boussinesqs problem) ofthe classical theory of elasticity. However, additional studies and further applications(supported by experimental data) are required for the ultimate validation of the theory.

This section was not intended to be a comprehensive review of the theories and themethods currently used to model the mechanical behavior of soils or more in general ofsolids. The ones briefly mentioned here are a subset of a much broader class of theories incontinuum and discrete mechanics. However, it is the writers belief that some of themodels presented in this section are of a significative interest in the problem of modelingsoil behavior.


2.3 Phenomenological Modeling of Soil Behavior

It is apparent from the previous section that the mechanical behavior of soils may bedescribed with a variety of mathematical idealizations. At the present time none of thetheories that have been cited is able to capture simultaneously all of the features exhibitedby soils, particularly under dynamic excitation and for wide ranges of strain and stress levels.Each proposed model has its own domain of validity, and it may predict reasonable resultsif applied to problems satisfying the assumptions laid at basis of the theory. Often in themechanics of materials, the conditions of applicability of a specific theory are dictated bythe intrinsic spatial and temporal scales of a problem. The spatial scale(s) of a problem maybe defined as a measure of the relationships existing among the size of some of itscharacteristic elements. Each problem has its own spatial (and temporal) scale whichpermits attributing meaning to relative terms such as small and large . The temporalscale of a problem provides a quantitative description of the relationships existing amongthe duration of some of its characteristic temporal events. By specifying the temporal scaleassociated to a given problem it becomes possible to attribute a relative meaning to termssuch as fast and slow. All of the problems associated with the mechanical behavior of soilsand other materials are characterized by intrinsic spatial and temporal scales. This statementcan ultimately be justified by the experimental evidence that all natural events are neithercontinuous nor instantaneous.

An appropriate assessment of the spatial scales of a problem may show for instance,that even if there are profound differences between the theories of continuum and discretemechanics, the discrepancies between their predictions may be irrelevant for practicalpurposes. In seismology, where most of the seismic energy propagates within the frequencyrange of about 0.001-100 Hz (Aki and Richards, 1980), the discrete nature of the mediumhas no role to play when compared with the lengths of the propagating seismic waves.Sometimes however, multi-scale phenomena may complicate the analysis of a problem. Forinstance in a composite medium characterized by the presence of randomly distributed localinhomogeneities (scatterers) whose size is comparable with the wavelength of the seismicwaves, a continuous model may be inadequate to represent the scattered wave field.

The time dependent deformation processes exhibited by many rheological materialsincluding soils are examples of phenomena involving intrinsic temporal scales. For instance,the process of energy dissipation occurring when a seismic wave propagates within a soildeposit involves the superposition of several dissipation mechanisms, each characterized byits own temporal scale (Liu et al., 1976). It is the frequency of excitation that will dictate therelative importance of the mechanisms activated during the overall process of energydissipation (a measure of the amount of unrecoverable energy produced during thedeformation of inelastic materials is the internal entropy density).

Throughout this study the mechanical behavior of soils was modeled using thephenomenological approach of classical one-constituent continuum mechanics. Theconstitutive model used to simulate soil response to dynamic excitations at very small strain


levels was linear-isotropic viscoelasticity. One of the purposes of the next section is tojustify this choice by illustrating some experimental results.

2.4 Experimental Observations

2.4.1 Overview

During the last 25-30 years, a considerable amount of research has been performed inan effort to better understand the mechanical response of soils to dynamic excitations.These studies were performed using a variety of laboratory techniques (e.g., resonantcolumn tests, cyclic torsional shear tests, cyclic direct simple shear tests, and cyclic triaxialtests), which allowed researchers to investigate the influence of variables including strainamplitude and frequency of excitation on soil behavior. The results obtained from this workhave helped in identifying the most important variables and factors affecting the dynamicbehavior of soils.

These variables and factors can be broadly divided into two categories according to theirorigin: external variables and intrinsic variables. The external variables correspond toexternally applied actions and include the stress/strain path, stress/strain magnitude,stress/strain rate, and stress/strain duration depending on the nature of the applied action(i.e., stress-controlled versus strain-controlled tests). The intrinsic variables correspond tothe inherent characteristics of soil deposits and include the soil type, the size of soilparticles, and the state parameters. The latter include the geostatic effective stress tensor(which is a measure of the current state of in-situ effective stress), some measure of thearrangement of soil particles (e.g., the fabric tensor or at least the void ratio, which howeveris scale-dependent), and some measure of the stress-strain history (e.g., the yield surface orat least the preconsolidation pressure). Figure 2.1 summarizes the relationships betweencauses and effects in the response of soils to dynamic excitations.

As described in the previous section, soil behavior may be studied using either aphenomenological or a micromechanical approach. In the phenomenological approach themain concern is understanding the relationship between causes and effects from amacroscopic point of view, without attempting an explanation of the observed phenomenaat a microscopic level. This microscopic interpretation is the objective of themicromechanical approach, which can be implemented by using the framework of eithercontinuum or discrete mechanics. As already mentioned the approach used in this work tomodel the dynamic behavior of soils is phenomenological, and coincides with that ofclassical, one-constituent continuum mechanics.

2.4.2 Threshold Strains

Experimental evidence shows that among the external variables affecting soil responseto dynamic excitations, the one that plays the most important role is the magnitude of theapplied stress or strain, or more precisely, the magnitude of the deviatoric strain tensor in


strain controlled tests. This quantity is a measure of the level of shear strains that wereinduced in the soil mass during the dynamic excitation. Based on these findings, it was thenpossible to define a shear strain spectrum for simple shear conditions where four distincttypes of soil behaviors were identified (EPRI, 1991; Vucetic, 1994).

The very small strain region is defined for values of shear strain in the range 0< g g tl

where g tl is the so-called linear cyclic threshold shear strain (Vucetic, 1994). Within this region

soil response to cyclic excitation is linear, but not elastic since energy dissipation occurseven at these very small strain levels (Lo Presti et al., 1997; Kramer, 1996). Although nostiffness reduction is observed in the soil response for g g t

l (linear behavior), thehysteretic loop in the stress-strain plane is characterized by a non-null area. Thephenomenon of energy dissipation at very small strain levels is caused by the existence of atime-lag between say, a driving cyclic strain and the driven cyclic stress in a strain-controlledtest (the word hysteresis comes from the ancient Greek and means lag or delay). This timelag between cyclic stress and strain is responsible for energy losses over a finite period oftime, which is typical of a viscoelastic behavior. There is little experimental evidence tosupport the existence of appreciable phenomena of instantaneous energy dissipation forg g t

l , which would be typical of an elastoplastic behavior.

Another important feature of soil behavior at very small strain levels is that soilproperties do not degrade as the number of cycles increases, and, as a result, the shape of

ExternalCauses

Stress/Strain Path

Stress/Strain Magnitude

Stress/Strain Rate

Stress/Strain Duration

Soil Type

Size of Soil Particles

Soil Natural State

- Geostatic Stresses

- Void Ratio/Fabric

- Stress/Strain History

IntrinsicCauses

Soil Response

Phenomenological

Linear ViscoelasticNon-Linear Viscoelastic

Non-Linear Elasto-Visco-Plastic

Micromechanical

Response Functions

Threshold Strains Stiffness Degradation Energy Dissipation

Figure 2.1 Cause-Effect Relationships in Soil Response to Dynamic Excitations


the hysteretic loop does not change with the continued loading (EPRI, 1991; Ishihara,1996). The value of g t

l varies considerably with the soil type. For example, g tl for sands is

on the order of 10-3%, whereas for normally consolidated clays with a plasticity index (PI) of50, g t

l is on the order of 10-2% (Lo Presti, 1987; Lo Presti, 1989).

The small strain region corresponds to shear strain levels in the range g g gtl

tv<

where g tv is the so-called volumetric threshold shear strain (Vucetic, 1994). The name for this

threshold strain is suggested from the experimental observation that soil response to cyclicexcitation for values of g exceeding g t

v is characterized by irrecoverable changes involume in drained tests and development of pore-water pressure in undrained tests(Vucetic, 1994). This region of the shear strain spectrum is characterized by a non-linear,inelastic soil response. However, the material properties do not change dramatically withincreasing shear strain, and very little degradation of these properties is observed as thenumber of cycles increases (soil hardening or softening). Values of g t

v , the upper limit forthis region of behavior, are on the order of 5.10-3% for gravels, 10-2% for sands, and 10-1%for normally consolidated, high plasticity clays (Bellotti et al., 1989; Lo Presti, 1989; Vuceticand Dobry, 1991).

Values of g g gtv

tpf< identify an intermediate strain region where g t

pf may be called

pre-failure threshold shear strain since values of g g> tpf characterize soil behavior at large

deformations preceding the conditions at failure. In the intermediate strain region bothinstantaneous energy dissipation and energy losses over a finite period of time take place asthe number of cycles progresses. This is mostly due to the irrecoverable microstructuralchanges that affect soils once the volumetric cyclic threshold shear strain g t

v is exceeded(Vucetic, 1994). In this range of deformation the degradation of soil properties with theshear strain is apparent not only within the hysteretic loop but also with the increase of thenumber of cycles (Ishihara, 1996).

Finally, values of g g gtpf

tf< identify the region of large strains (EPRI, 1991; Vucetic,

1994) where soil response to cyclic excitation is highly non-linear and inelastic. This is thestate of soils preceding the condition of failure, which is postulated to occur at the failurethreshold shear strain g t

f . Table 2.1 shows the shear strain spectrum with the four postulatedtypes of soil response to cyclic excitation.

Among the four threshold shear strains previously defined, namely g tl , g t

v , g tpf , and

g tf , two of them are particularly meaningful: the linear cyclic threshold shear strain g t

l and

the volumetric threshold shear strain g tv . The threshold strain g t

l is important because itseparates the linear (even though inelastic) response from the non-linear response of soilssubjected to cyclic excitations. The volumetric threshold shear strain g t

v instead, is a


threshold strain used to distinguish between different types of irrecoverable deformationsoccurring in soils undergoing harmonic oscillations. For g g t

v it can be postulated that allthe energy losses taking place in a soil specimen are of a viscoelastic nature, i.e. they onlyoccur over a finite period of time. However, for g g> t

v both phenomena of instantaneousand finite-time energy dissipation, which is typical of a visco-plastic soil behavior, areobserved experimentally.

The threshold shear strains g tl and g t

v were defined by considering simple shear strainpaths. Soil response to both static and dynamic loadings is strain/stress-path dependent andhence, different values of g t

l and g tv (and also of g t

pf , and g tf ) would be obtained if

different strain/stress-paths had been used. However the relevance of these concepts andtheir implications in understanding the dynamic behavior of soils would be unchanged.

Another factor that affect the values of the t

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