Probabilistic ElastoPlasticity and Its Application in Finite ElementSimulations of Stochastic ElasticPlastic Boundary Value Problems

By

KALLOL SETTBachelor of Civil Engineering (Jadavpur University, Calcutta, India) 1997

Master of Science in Civil Engineering (University of Houston, Houston, Texas) 2003

DISSERTATION

Submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in

Civil Engineering

in the

OFFICE OF GRADUATE STUDIES

of the

UNIVERSITY OF CALIFORNIA

DAVIS

Approved:

Committee in Charge

2007

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Abstract

Probabilistic ElastoPlasticity and Its Application in Finite Element Simulations of

Stochastic ElasticPlastic Boundary Value Problems

by

Kallol Sett

Doctor of Philosophy in Civil Engineering

University of California, Davis

Professor Boris Jeremic, Chair

A computational framework has been developed for simulations of the behaviors of solids and

structures made of stochastic elasticplastic materials. Particular emphasis has been given

to soil, a highly nonlinear (elasticplastic) and highly uncertain material, and geotechnical

engineering applications.

Uncertain elasticplastic material properties are modeled as random fields, which,

in the governing partial differential equation of mechanics, appear as the coefficient term. A

spectral stochastic elasticplastic finite element method with FokkerPlanckKolmogorov

equation approach based probabilistic constitutive integrator is proposed for solution of this

nonlinear (elasticplastic) partial differential equation with stochastic coefficient. To this

end, the random field material properties are discretized, in both spatial and stochastic

dimension, into finite numbers of independent basic random variables, using Karhunen

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Loeve expansion. Those random variables are then propagated through the elasticplastic

constitutive rate equation using FokkerPlanck-Kolmogorov equation approach, to obtain

the evolutionary material properties, as the material plastifies. The unknown displacement

(solution) random field is then assembled, as a function of known basic random variables

with unknown deterministic coefficients, using polynomial chaos expansion. The unknown

deterministic coefficients of polynomial chaos expansion are obtained, by minimizing the

error of finite representation, by Galerkin technique.

The applicability of the developed methodology is demonstrated in obtaining the

probabilistic solutions of 1D static pushover test and response of 1D structure due to

sinusoidal base displacement. In addition, pure constitutive level simulations of von Mises,

Drucker-Prager, and Cam Clay material models are also shown. The results are verified

with analytical solution, where available, or Monte Carlo solution and good agreements

are obtained. Finally, the complete solution process, based on the developed computa-

tional framework, of a geotechnical engineering problem - seismic wave propagation through

stochastic elasticplastic soil - is illustrated using real-life soil data and with real earthquake

motion.

Professor Boris JeremicDissertation Committee Chair

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To my wife

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Contents

List of Figures ix

List of Tables xvi

Acknowledgements xvii

I Motivation and Theoretical Background 1

1 Introduction 21.1 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Scope of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Summary of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Original Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Probability Theory Background 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Properties of Single Random Variable . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Probability Distribution Function . . . . . . . . . . . . . . . . . . . . 132.3.2 Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . 152.3.3 Moments of Probability Distributions . . . . . . . . . . . . . . . . . 152.3.4 Characteristic Function and its relation with Moments and Cumulants 182.3.5 Relation between Moments and Cumulants . . . . . . . . . . . . . . 20

2.4 Properties of Two or More Random Variables . . . . . . . . . . . . . . . . . 212.4.1 Joint and Marginal Probability Distribution . . . . . . . . . . . . . . 212.4.2 Joint Cumulative Probability Distribution . . . . . . . . . . . . . . . 262.4.3 Conditional Probability Distribution . . . . . . . . . . . . . . . . . . 272.4.4 Dependency and Correlation between Random Variables . . . . . . . 282.4.5 Joint Characteristic Function . . . . . . . . . . . . . . . . . . . . . . 302.4.6 Some useful Properties of Two Random Variables . . . . . . . . . . . 30

2.5 Random Processes and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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2.5.1 Classification of Random Processes/Fields . . . . . . . . . . . . . . . 352.5.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.6.1 Mean Square Convergence of a Random Function . . . . . . . . . . . 432.6.2 Mean Square Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 442.6.3 Riemanian Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . 452.6.4 RiemannStieltjes Stochastic Integral . . . . . . . . . . . . . . . . . 472.6.5 Ito Stochastic Differential Equation . . . . . . . . . . . . . . . . . . 472.6.6 FokkerPlanckKolmogorov Equation . . . . . . . . . . . . . . . . . 49

II Uncertain and Spatially Uncertain Material Properties 50

3 Characterization & Quantification of Uncertainties in Material Proper-ties 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Classification of Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Uncertainties in Soil Properties . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Uncertain Spatial Variability . . . . . . . . . . . . . . . . . . . . . . . . . . 583.5 Probabilistic Geotechnical Site Characterization . . . . . . . . . . . . . . . . 61

4 Random Field Modeling of Uncertain Material Properties 644.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Finite Scale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.2 Fractal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Example Estimation of Statistical Parameters . . . . . . . . . . . . . . . . . 72

III Material (Constitutive) Level Stochastic Simulation: ProbabilisticElastoPlasticity 82

5 Probabilistic ElastoPlasticity: Theory 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 One Dimensional Development . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.1 Specialization of General Formulation to Particular Constitutive Laws 955.2.2 Solution Method of Probabilistic ElastoPlastic Equation . . . . . . 107

5.3 Three Dimensional Development . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Probabilistic ElastoPlasticity: Numerical Examples and Verifications 1166.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Linear Elastic Shear Constitutive Behavior . . . . . . . . . . . . . . . . . . 1196.3 ElasticPlastic Shear Constitutive Behavior with Mean Yield Criteria . . . 125

6.3.1 Drucker-Prager Associative Model . . . . . . . . . . . . . . . . . . . 1266.3.2 Cam Clay Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

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6.4 ElasticPlastic Shear Constitutive Behavior with Probabilistic Yield Criteria 1476.4.1 von Mises Associative Model . . . . . . . . . . . . . . . . . . . . . . 1506.4.2 Drucker-Prager Associative Model . . . . . . . . . . . . . . . . . . . 155

IV Stochastic Simulations of Solids and Structures with UncertainMaterial Properties: Stochastic Finite Elements 159

7 Stochastic Finite Elements: Theory 1607.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.2 Discretization of Governing Stochastic Partial Differential Equation . . . . . 164

7.2.1 Stochastic Discretization of Input Random Field . . . . . . . . . . . 1657.2.2 Stochastic Discretization of Unknown Solution Random Field . . . . 1747.2.3 Spatial Discretization of the Differential Operator . . . . . . . . . . 178

7.3 Stochastic Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . 1797.3.1 Non-Linear (ElasticPlastic) Formulation . . . . . . . . . . . . . . . 1827.3.2 Dynamic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.4 Post-Processor: Estimation of Response Statistics . . . . . . . . . . . . . . . 1897.4.1 Mean and Autocovariance . . . . . . . . . . . . . . . . . . . . . . . . 1907.4.2 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . 191

8 Stochastic Finite Elements: Numerical Examples and Verifications 1948.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.2 One Dimensional Static Problem . . . . . . . . . . . . . . . . . . . . . . . . 195

8.2.1 Problem Statement and Formulation . . . . . . . . . . . . . . . . . . 1958.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.3 One Dimensional Dynamic Problem . . . . . . . . . . . . . . . . . . . . . . 2118.3.1 Problem Statement and Formulation . . . . . . . . . . . . . . . . . . 2118.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.4 Seismic Wave Propagation through Stochastic ElasticPlastic Soil . . . . . 2178.4.1 Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2198.4.2 Simulation Results and Discussions . . . . . . . . . . . . . . . . . . . 221

Conclusions and Future Research Directions 229Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Bibliography 241

Appendices 255

A 256A.1 Derivation of Akl (Eq. (5.26)) . . . . . . . . . . . . . . . . . . . . . . . . . . 256A.2 Derivation of B (Eq. (5.27)) . . . . . . . . . . . . . . . . . . . . . . . . . . . 259A.3 Derivation of KP (Eq. (5.28)) . . . . . . . . . . . . . . . . . . . . . . . . . . 261

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A.4 Derivation of Eq. (5.11): Ensemble average form of stochastic continuityequation (Eq. (5.8)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

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List of Figures

1.1 Interpreted Effective Stress Strength Parameters at Opelika NGES (afterMayne et al. [70]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Soil: Inside Failure of Uniform MGM Specimen (after Swanson et al. [94]) 4

2.1 Gaussian Probability Density Function of Friction Coefficient Random Vari-able (A) with mean = 0.3 and Standard Deviation = 0.05 . . . . . . . . . . 14

2.2 Gaussian Cumulative Density Function of Friction Coefficient Random Vari-able (A) with mean = 0.3 and Standard Deviation = 0.05 . . . . . . . . . . 16

2.3 Gaussian Joint Probability Density Function of Shear Modulus (G) and fric-tion coefficient (A) with G = 2.5 MPa, (SD)G = 1.0 MPa, A = 0.3,(SD)A = 0.05, and GA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Marginal PDF of friction coefficient (A) from joint PDF of shear modulus,G, and friction coefficient, A (Fig. 2.3) . . . . . . . . . . . . . . . . . . . . . 24

2.5 Gaussian Joint Cumulative Density Function of Shear Modulus (G) and fric-tion coefficient (A) with G = 2.5 MPa, (SD)G = 1.0 MPa, A = 0.3,(SD)A = 0.05, and GA = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 A realization of Shear Modulus Random Field . . . . . . . . . . . . . . . . . 33

3.1 Graphical depiction of the deterministic, epistemic and aleatory uncertaintyrelated to geotechnical simulations. This is, in a sense, macro scale interpre-tation of Heisenberg uncertainty principle. . . . . . . . . . . . . . . . . . . 53

3.2 Measured values of mechanical properties of soil from Mexico city, typicalsoft spot (after Baecher and Christian [6]) . . . . . . . . . . . . . . . . . . . 61

4.1 Shear modulus random field: Trend and residual around trend . . . . . . . 664.2 CPT Sounding locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 East-West soil profile interpreted from CPT soundings . . . . . . . . . . . . 744.4 Typical qT data: Borehole 1 sounding . . . . . . . . . . . . . . . . . . . . . 754.5 Maximum likelihood estimated Gauss-Markov autocovariance function along

with method of moment estimate (for borehole 1 sounding) . . . . . . . . . 754.6 Maximum likelihood estimated Gauss-Markov autocorrelation function along

with method of moment estimate (for borehole 1 sounding) . . . . . . . . . 76

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4.7 (a) Measured (at borehole 2) and (b) Simulated (finite scale Gauss-Markovmodel) realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.8 Deterministic Trend as obtained through global regression over 16 CPTsoundings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.9 Periodogram of borehole 1 sounding . . . . . . . . . . . . . . . . . . . . . . 804.10 Maximum likelihood estimated fractal (1/f -type noise with lower cut-off fre-

quency) power spectral density function corresponding to borehole 1 sounding 804.11 Fractal (1/f -type noise with lower cut-off frequency) autocovariance function

for borehole 1 sounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1 Anticipated Influence of Material Fluctuations on Stress-Strain Behavior . . 845.2 Movements of Cloud of Initial Points, described by density (, 0), in the

-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Spatial Discretization of the FokkerPlanckKolmogorov PDE . . . . . . . . 110

6.1 Approximation of Dirac delta initial condition (Eq. (5.52)) with a Gaussianfunction of a zero mean and a standard deviation of 0.00001 MPa . . . . . . 119

6.2 Time (or strain) evolution of probability density function of shear stress forelastic constitutive rate equation with random shear modulus obtained usingFPKE approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.3 Time (or strain) evolution of probability density function of shear stress forelastic constitutive rate equation with random shear modulus obtained usingtransformation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.4 Comparison of contours of time (or strain) evolution of probability densityfunction for shear stress for elastic constitutive rate equation with randomshear modulus for FPKE solution and variable transformation method solution124

6.5 Comparison of evolution of mean and standard deviation of shear stress withtime (or shear strain) for elastic constitutive rate equation with random shearmodulus for FPKE solution and variable transformation method solution . 124

6.6 Effect of approximating function of Dirac delta initial condition: PDF ofstress at yield for different approximations of initial condition with actual(variable transformation method) solution . . . . . . . . . . . . . . . . . . . 125

6.7 Initial condition for Fokker-Planck-Kolmorogov equation for probabilisticsimulation of post-yield region . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.8 Time (or strain) evolution of probability density function of shear stress forDrucker-Prager elastic-plastic constitutive rate equation with random shearmodulus (Problem-I) (View obtained when one looks perpendicular to thetime/strain axis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.9 Time (or strain) evolution of probability density function of shear stressfor Drucker-Prager elastic-plastic constitutive rate equation with randomshear modulus (Problem-I) (View obtained when one looks parallel to thetime/strain axis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.10 Contour of time (or strain) evolution of probability density function for shearstress for Drucker-Prager elastic-plastic constitutive rate equation with ran-dom shear modulus (Problem-I) . . . . . . . . . . . . . . . . . . . . . . . . . 133

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6.11 Comparison of evolution of PDF of shear stress for Drucker-Prager elastic-plastic linear hardening material model and extended linear elastic modelwith random shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.12 Comparison of evolution of mean and standard deviation of shear stress forDrucker-Prager elastic-plastic constitutive rate equation with random shearmodulus (Problem-I), obtained from FPKE solution and Monte Carlo solution134

6.13 Initial condition for Fokker-Planck-Kolmorogov equation for probabilisticsimulation of Drucker-Prager post-yield region with random friction coef-ficient (Problem-II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.14 Time (or strain) evolution of PDF of shear stress for Drucker-Prager elastic-plastic constitutive rate equation with random friction coefficient (Problem-II) (only post-yield region is shown, note that the pre-yield region is deter-ministic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.15 Contours of time (or strain) evolution of PDF for shear stress, along withevolutions of mean and standard deviation of shear stress, for Drucker-Prager elastic-plastic constitutive rate equation with random friction coef-ficient (Problem-II) (only post-yield region is shown, note that the pre-yieldregion is deterministic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.16 Comparison of evolutions of mean and standard deviation of shear stress forDrucker-Prager elastic-plastic constitutive rate equation with random fric-tion coefficient (Problem-II), obtained from FPKE solution and Monte Carlosolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.17 Low OCR Cam Clay response with random normally distributed shear mod-ulus (G): (a) Evolution of PDF and (b) Evolution of contours of PDF, mean,mode, standard deviations, and deterministic solution of shear stress (12)with time (t)/shear strain (12) . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.18 Comparison of FPK approach and Monte Carlo approach in obtaining lowOCR Cam Clay response with random normally distributed shear modulus(G) in terms of evolution of mean and standard deviation of shear stress (12)with time (t)/shear strain (12) . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.19 Low OCR Cam Clay response with random normally distributed shear mod-ulus (G) and random normally distributed slope of critical state line (M): (a)Evolution of PDF and (b) Evolution of contours of PDF, mean, mode, stan-dard deviations, and deterministic solution of shear stress (12) with time (t)/shear strain (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.20 Low OCR Cam Clay response with random normally distributed shear mod-ulus (G), random normally distributed slope of critical state line (M), andrandom normally distributed overconsolidation pressure (p0): (a) Evolutionof PDF and (b) Evolution of contours of PDF, mean, mode, standard devi-ations, and deterministic solution of shear stress (12) with time (t)/shearstrain (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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6.21 Low OCR Cam Clay response with random normally distributed shear mod-ulus (G) and random normally distributed overconsolidation pressure (p0):(a) Evolution of PDF and (b) Evolution of contours of PDF, mean, mode,standard deviations, and deterministic solution of shear stress (12) with time(t)/shear strain (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.22 Comparison of shear stresses at 1.62% shear strain obtained from low OCRCam Clay model with different degrees of randomnesses . . . . . . . . . . . 146

6.23 High OCR Cam Clay response with random normally distributed shear mod-ulus (G) and random normally distributed slope of critical state line (M): (a)Evolution of PDF and (b) Evolution of contours of PDF, mean, mode, stan-dard deviations, and deterministic solution of shear stress (12) with time (t)/shear strain (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.24 Comparison of FPK Approach and MonteCarlo Approach for High OCRCam Clay Response with Random Normally Distributed Shear Modulus (G)and Random Normally Distributed Slope of Critical State Line (M) in termsof Evolution of Mean and Standard Deviation of Shear Stress (12) with Time(t)/Shear Strain (12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.25 CDF of shear strength for von Mises model: (a) very uncertain case, (b)fairly certain case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.26 von Mises associative plasticity model with uncertain shear modulus andshear strength (yield parameter): (a) Evolution of PDF of stress with strain(PDF=10000 was used as a cutoff for surface plot) and (b) Contours ofevolution of stress PDF with strain. . . . . . . . . . . . . . . . . . . . . . . 152

6.27 von Mises elasticplastic model: (a) Shear modulus: very uncertain; shearstrength: fairly certain, (b) Shear modulus: fairly certain; shear strength:very uncertain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.28 CDF of yield stresses for Drucker-Prager model: (a) very uncertain and (b)fairly certain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.29 Drucker-Prager associative elasticplastic model with uncertain shear mod-ulus and frictional coefficient: (a) Evolution of probability density function(PDF) of stress with strain (PDF=10000 was used as a cutoff for surfaceplot) and (b) Contours of evolution of PDF with strain . . . . . . . . . . . . 156

6.30 Drucker-Prager elasticplastic model: (a) Shear modulus: very uncertain;frictional coefficient: fairly certain, (b) Shear modulus: fairly certain; fric-tional coefficient: very uncertain. . . . . . . . . . . . . . . . . . . . . . . . . 158

7.1 KL eigenvalues of exponential covariance kernel having variance = 1000 kPa2

and correlation length = 1 m . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.2 KL eigenvectors of exponential covariance kernel having variance = 1000

kPa2 and correlation length = 1 m . . . . . . . . . . . . . . . . . . . . . . . 1717.3 Exact exponential covariance kernel having variance = 1000 kPa2 and cor-

relation length = 1 m (C(x1, x2) = 2 e|x1x2|/r, 2 = 1000 kPa2 and r =

1.0 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.4 KL approximations (with estimated errors) of exponential covariance kernel

having variance = 1000 kPa2 and correlation length = 1 m . . . . . . . . . 173

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7.5 Exact exponential covariance kernel with variance = 1000 kPa2 and correla-tion length = 0.05 m (C(x1, x2) =

2 e|x1x2|/r, 2 = 1000 kPa2 and r =0.05 m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.6 KL approximations (with estimated errors) of exponential covariance kernelhaving variance = 1000 kPa2 and correlation length = 0.05 m . . . . . . . . 175

8.1 Schematic of static 1D soil column (shear beam) example . . . . . . . . . . 1968.2 Mean and standard deviations of displacement at the top node of the soil

column, with linear elastic material model (with KL-dimension = 2, order ofPC = 2). Monte Carlo simulation is also shown . . . . . . . . . . . . . . . . 204

8.3 Mean and standard deviations of displacement at the top node of the soilcolumn, with von Mises elastic-plastic material model (with KL-dimension= 2, order of PC = 2). Monte Carlo simulation is also shown . . . . . . . . 204

8.4 Comparison of PDF of top node displacements of the soil column: Elasticversus von Mises elastic-plastic material model . . . . . . . . . . . . . . . . 205

8.5 Correlation length and KL dimension: Mean displacement along depth ofthe 1D soil column with linear elastic material model, having very smallvariance (COV = 1%) of shear modulus and very large ratio of correlationlength of shear modulus to domain length (= 100) . . . . . . . . . . . . . . 206

8.6 Correlation length and KL dimension: Standard deviation of displacementalong depth of the 1D soil column with linear elastic material model, havingvery small variance (COV = 1%) of shear modulus and very large ratio ofcorrelation length of shear modulus to domain length (= 100) . . . . . . . . 207

8.7 Correlation length and KL dimension: Mean displacement along depth ofthe 1D soil column with linear elastic material model, having very smallvariance (COV = 1%) of shear modulus and very small ratio of correlationlength of shear modulus to domain length (= 0.0001) . . . . . . . . . . . . . 207

8.8 Correlation length and KL dimension: Standard deviation of displacementalong depth of the 1D soil column with linear elastic material model, havingvery small variance (COV = 1%) of shear modulus and very small ratio ofcorrelation length of shear modulus to domain length (= 0.0001) . . . . . . 208

8.9 Variance and order of PC: Mean displacement along depth of the 1D soilcolumn with linear elastic material model, having large variance (COV =20%) of shear modulus and ratio of correlation length of shear modulus todomain length = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.10 Variance and order of PC: Standard deviation of displacement along depth ofthe 1D soil column with linear elastic material model, having large variance(COV = 20%) of shear modulus and ratio of correlation length of shearmodulus to domain length = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . 209

8.11 Variance and order of PC: Mean displacement along depth of the 1D soilcolumn with linear elastic material model, having very small variance (COV= 1%) of shear modulus and ratio of correlation length of shear modulus todomain length = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

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8.12 Variance and order of PC: Standard deviation of displacement along depthof the 1D soil column with linear elastic material model, having very smallvariance (COV = 1%) of shear modulus and ratio of correlation length ofshear modulus to domain length = 0.1 . . . . . . . . . . . . . . . . . . . . . 210

8.13 Schematic of dynamic 1D soil column example . . . . . . . . . . . . . . . . 2128.14 Base displacement applied to the bottom node of the 1D soil column . . . 2128.15 Visualization of 1D soil column with base displacement, as shown in Fig. 8.13,

as a soil column-stiff spring system . . . . . . . . . . . . . . . . . . . . . . . 2138.16 Time evolution of mean of displacement at the top node of the 1D soil col-

umn, with linear elastic material model, due to sinusoidal base displacementshown in Fig. 8.14 (with KL-dimension = 2, order of PC = 2) . . . . . . . . 215

8.17 Time evolution of mean of displacement at the top node of the 1D soilcolumn, with von Mises elasticplastic material model, due to sinusoidal basedisplacement shown in Fig. 8.14 (with KL-dimension = 2, order of PC = 2) 216

8.18 Time evolution of standard deviation of displacement at the top node of the1D soil column, with linear elastic material model, due to sinusoidal basedisplacement shown in Fig. 8.14 (with KL-dimension = 2, order of PC = 2) 216

8.19 Time evolution of standard deviation of displacement at the top node ofthe 1D soil column, with von Mises elasticplastic material model, due tosinusoidal base displacement shown in Fig. 8.14 (with KL-dimension = 2,order of PC = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8.20 Time evolution of mean standard deviation of displacement at the top nodeof the 1D soil column, with linear elastic material model, due to sinusoidalbase displacement shown in Fig. 8.14 (with KL-dimension = 2, order of PC= 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.21 Time evolution of mean standard deviation of displacement at the topnode of the 1D soil column, with von Mises elasticplastic material model,due to sinusoidal base displacement shown in Fig. 8.14 (with KL-dimension= 2, order of PC = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.22 Base displacement applied to the bottom node of the 1D soil column: Mod-ified 1938 Imperial Valley motion . . . . . . . . . . . . . . . . . . . . . . . . 221

8.23 Time evolution of mean of displacement at the top node of the 1D soilcolumn, with linear elastic material model, due to modified 1938 ImperialValley base displacement as shown in Fig. 8.22 (with KL-dimension = 2,order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.24 Time evolution of mean of displacement at the top node of the 1D soilcolumn, with von Mises elasticplastic material model, due to modified 1938Imperial Valley base displacement as shown in Fig. 8.22 (with KL-dimension= 2, order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.25 Time evolution of standard deviation of displacement at the top node of the1D soil column, with linear elastic material model, due to modified 1938Imperial Valley base displacement as shown in Fig. 8.22 (with KL-dimension= 2, order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

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8.26 Time evolution of standard deviation of displacement at the top node ofthe 1D soil column, with von Mises elasticplastic material model, due tomodified Imperial Valley base displacement as shown in Fig. 8.22 (with KL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . . . 223

8.27 Time evolution of mean standard deviation of displacement at the top nodeof the 1D soil column, with linear elastic material model, due to modified1938 Imperial Valley base displacement as shown in Fig. 8.22 (with KL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . . . 224

8.28 Time evolution of mean standard deviation of displacement at the topnode of the 1D soil column, with von Mises elasticplastic material model,due to modified 1938 Imperial Valley base displacement as shown in Fig. 8.14(with KL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . . . . . . 224

8.29 Time evolution of coefficient of variation (COV) of displacement at the topnode of the 1D soil column, with linear elastic material model, due to mod-ified 1938 Imperial Valley base displacement as shown in Fig. 8.22 (withKL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . 225

8.30 Time evolution of coefficient of variation (COV) of displacement at the topnode of the 1D soil column, with von Mises elasticplastic material model,due to modified 1938 Imperial Valley base displacement as shown in Fig. 8.14(with KL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . . . . . . 225

8.31 Time evolution of probability density function (PDF) of displacement at thetop node of the 1D soil column, with linear elastic material model, due tomodified 1938 Imperial Valley base displacement as shown in Fig. 8.22 (withKL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . . . . . . . . . . 226

8.32 Time evolution of probability density function (PDF) of displacement at thetop node of the 1D soil column, with von Mises elasticplastic materialmodel, due to modified 1938 Imperial Valley base displacement as shown inFig. 8.22 (with KL-dimension = 2, order of PC = 1) . . . . . . . . . . . . . 227

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List of Tables

2.1 Classification of random processes based on state and parameter space . . . 35

3.1 Representative values of variabilities in consolidation parameters (after Baecherand Christian [6]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Representative values of variabilities in laboratory measured effective frictionangle (after Baecher and Christian [6]) . . . . . . . . . . . . . . . . . . . . . 54

3.3 Representative values of variabilities in some common in-situ soil properties(after Phoon and Kulhawy [79]) . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Best fitting probability density functions (PDFs) for various soil properties(after Lacasse and Nadim [56]) . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Representative values of testing errors of soil index properties (after Hammitt[36], reproduced from the book by Baecher and Christian [6]) . . . . . . . . 57

3.6 Representative testing error of some laboratory tests that evaluates strengthproperties (after Phoon and Kulhawy [79]) . . . . . . . . . . . . . . . . . . . 57

3.7 Representative testing error of some field tests (after Phoon and Kulhawy [79]) 583.8 Transformation uncertainties of some common strength property correla-

tions(after Phoon and Kulhawy [80]) . . . . . . . . . . . . . . . . . . . . . . 583.9 Representative scale of fluctuation of some common soil properties (after

Phoon and Kulhawy [79]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Maximum likelihood estimated constant mean Gauss-Markov model parameters 764.2 Maximum likelihood estimated constant mean fractal (1/f-type noise with

lower cut-off frequency) model parameters obtained using periodogram ap-proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.1 Comparison of results (at top node) of FPKE-based spectral stochastic finiteelement with direct spectral stochastic finite element, for 1D soil columnexample, with linear elastic material . . . . . . . . . . . . . . . . . . . . . . 211

-xvi-

Acknowledgments

I would like to express my sincere gratitude to my advisor, Professor Boris Jeremic

for his guidance, advice, encouragement, and kindness all throughout my doctoral study. I

have been very fortunate to have an advisor like him. He gave me freedom to explore on

my own. However, with careful guidance and constructive criticism, he helped me to stay

focused. I am indebted to him for helping me to develop my critical thinking and research

skills.

My special gratitude goes to Professor M. Levent Kavvas. He has not only intro-

duced me to stochastic methods, on which this dissertation is based, but also helped me to

develop my concept of the topic. Long discussions with him helped me to sort out technical

details of my work.

I would also like to extend my thanks to Professors Ross Boulanger and Niels

Grnbech Jensen for serving on my dissertation committee and taking time to critique my

work.

My work has benefited from conversations with the past and present members of

Computational Geomechanics Group. For their generous help, I would like to acknowledge

Jim Putnam, Guanzhou Jie, Zhao Cheng, Matthias Preisig, Mahdi Taiebat, and Alisa

Neeman.

Financial supports provided by the Department of Civil and Environmental En-

gineering (through Block Grant Living Allowance Fellowship) and the National Science

Foundation (through award # CMMI 0600766) are also gratefully acknowledged.

-xvii-

1Part I

Motivation and Theoretical

Background

2Chapter 1

Introduction

1.1 Hypothesis

Failure of geomaterial is generally preceded by formation of shear band and subse-

quent bifurcation of response. Shear bands form due to strain localization, which, according

to some recent studies (Carmeliet and De Borst [9]; Gutierrez and De Borst [34]), stems

from uncertain material non-uniformity 1 and speaking of natural geomaterials, they are

inherently uncertain and very non-uniform. Figure 1.1 shows variation of a typical soil

property, the undrained shear strength, measured with different field and laboratory tests,

with depth. Note large variability as a function of depth (inherent variability), as well

as with different test methods (testing uncertainty). Fig. 1.2 shows that even in a care-

fully prepared uniform laboratory specimen the failure is due to strain localization and

subsequent formation of shear bands.

1though the usual practice in numerical simulation of shear band is to deterministically specifying theimperfection in the numerical specimen and thus help in formation of shear band

3Figure 1.1: Interpreted Effective Stress Strength Parameters at Opelika NGES (after Mayneet al. [70])

The variabilities of in-situ soil properties are generally quite large. The usual

procedure in this case is to choose either a constant variation of properties or fit a smooth

curve (usually a straight line) to the properties of interest, for example undrained shear

strength as in the example above, with depth. This approach completely neglects variations,

which might have a large effect on response of this soil profile to, for example, seismic

excitation. The effects of neglecting the natural variability are not known a priori and the

usual remedy is to increase factors of safety in dealing with such (uncertain) soils. Book

by Lambe and Whitman [57] gives many more interesting, nonuniform profiles with quite

uncertain and nonuniform properties from various geotechnical sites.

In recent years, civil engineering practice, and in particular the geotechnical engi-

4Figure 1.2: Soil: Inside Failure of Uniform MGM Specimen (after Swanson et al. [94])

5neering practice, has seen an increasing emphasis on reliability. In geotechnical earthquake

engineering, for example, the earthquake ground motions are usually given with certain

probability of occurrence. However, whats missing is the actual quantification of this prob-

ability that come from uncertain and nonuniform soil layers. In other words, it is important

to discern how much of that uncertainty in ground motion is due to the uncertain and vari-

able motion coming from the hypocenter and how much is due to the uncertain and variable

soil properties in top layers.

Hence the question that arises is how to account for this non-uniformity and un-

certainty of soil parameters in analytical and numerical simulations (statics and dynamics)

of behaviors of dams, levees, shallow and deep foundations and other solids and structures

made of geomaterials. The best way to account for uncertainty is to quantify them and this

quest for quantifying uncertainties in response behavior lead to the development of proba-

bilistic simulation tools. In addition, there exist several other practical reasons to pursue

this research project:

Modern building codes (regulations) are increasingly being based on reliability methods,

however the analyses are still largely (exclusively) deterministic. Other industries,

for example nuclear and offshore, are already using probabilistic analysis to a large

extent.

Financial decision making by object owners tends to lead towards the use of probabilistic

theories. For example, decisions on the best course of action for developing new objects

or upgrading and repairing existing objects are highly probabilistic. Crucial decisions

on the extent of work, financing and scheduling are also made using probability theory.

6In contrast, the actual performance assessments (simulations of behavior) used in

designing new or in upgrading and repairing existing objects are still almost exclusively

deterministic.

Consistent development of a probabilistic framework for geotechnical simulations will pro-

vide a rational way to address our confidence (or lack thereof) in simulated behavior.

For example, probabilistic simulation tools will empower engineers to demonstrate

the need for more, uniform data on material properties, to develop novel site charac-

terization techniques, and to design the geotechnical systems that (probably) achieve

best performance.

Further to the above practical factors related to civil engineering industry (with its fields

of geotechnical, structural, construction engineering) there are some wider concerns as well:

Societal needs: Society is demanding greater safety and reliability of infrastructure sys-

tems (structural, geotechnical). An important aspect of meeting this goal is devel-

opment of performance assessment capabilities (numerical simulations on models).

Such probabilistic performance estimates of infrastructure systems (such as buildings,

bridges, dams, levees and lifeline networks) will greatly benefit the society in allowing

the public and decisions makers to appropriately assess the critical needs of economy,

safety, and usability for new infrastructure or upgradation of existing infrastructure

objects.

Complexity of Geotechnical media: The behavior of the soils and structures involves

complex physics of highly nonlinear, heterogeneous materials with uncertain consti-

tutive properties. In particular, soils are known to have coefficients of variance that

7are most of the time greater than 20 % (Phoon and Kulhawy [79, 80]). This large

uncertainty in soil properties renders any deterministic simulations almost useless,

unless large factors of safety are applied. However, use of large factors of safety is be-

coming unacceptable as it leads to design solutions that are not economical and many

times not even safe (e.g., Duncan [20]). Development of probabilistic simulation tools

will greatly improve our ability to consistently and efficiently perform simulations

of inelastic behavior of geomaterial solids and structures with non-homogeneous and

uncertain material parameters. In addition to that, it will also help in better under-

standing of the scale effects in nonhomogeneity of soils and, consequently will help

in performing better site characterizations.

Impracticality of Monte Carlo approach for large-scale probabilistic simulations:

While there are some recent works (e.g., Griffiths et al. [33], Paice et al. [78], Fenton

and Griffiths [25, 27]) on probabilistic simulations of stochastic soils, mostly on spatial

nonuniformity of material properties, it is computationally very expensive and in fact

impossible in any meaningful time to conduct necessary number of Monte Carlo runs

to obtain the complete probabilistic behavior in terms of probability density function.

2 This is especially true when the problem under consideration has more than one

material properties as random. Usually the number of random material properties in

any geotechnical problem is larger than one. For example, even in 1D both elastic

and inelastic 3 material properties are probabilistic. For a 3-D elasto-plastic model

2The capability of any approach to obtain the complete probabilistic description in terms of probabilitydensity function (and not, only the first few statistical moments e.g. mean and standard deviation) is veryimportant as geomaterials often fails at low probabilities and hence the tails of probability density functionare equally important as the first few statistical moments

3the number of inelastic material properties depends on the type of elasticplastic model. The more

8with 3 random variable material parameters (e.g., Youngs modulus E, friction angle

and cohesion c for simplest DruckerPrager model with no hardening) one needs

4 at least 10, 0003 = 1012 runs of computationally expensive deterministic model.

This is even more the case for elasto-plastic models with larger number of random

material parameters. In addition to this huge effort, there is an added burden of

post-processing the high volume of data. Reductions of number of statistically appro-

priate realizations are possible, but at the expense of increasing the error in Monte

Carlo simulations. For example, if one assumes that, the variables are correlated and

that statistically only 1, 000 realizations are appropriate for each of three variables,

the finite element model for given foundation or levee or lifeline network needs to be

solved a billion times (1, 0003 = 109) in order to obtain statistics and probabilities

of response. Due to this high computational cost in analysis and data processing,

the Monte Carlo approach is impractical for large-scale probabilistic simulation and

is used only for verification of other approaches.

1.2 Scope of Study

The main objective of this research is to develop a computational framework for

simulations of behaviors of solids and structures made of stochastic elasticplastic materials,

with particular emphasis on soil. More specifically the objectives are:

Probabilistic characterization and rational quantification of uncertainties in material

properties, which will act as input to the stochastic framework.

advanced the model, the more the number of (random) parameters4assuming statistically appropriate 10,000 realizations per random variable

9 Development of probabilistic elastoplasticity for constitutive (material) level simula-

tion.

Development of a stochastic elasticplastic finite element method for solving boundary

value problems in simulating the behaviors of solids and structures made of stochastic

elasticplastic materials.

Examples to illustrate the applicability of the developed methodologies in solving

real-life geotechnical engineering problems.

1.3 Summary of Contents

This dissertation is divided into four parts. Part - I describes the motivation

and discusses the features of probability theory that have been used in the subsequent

chapters. Part - II deals with characterization and quantification of uncertainties in material

properties, with particular emphasis on soils. It also discusses the random field modeling

of spatially uncertain material properties and techniques to estimate the model parameters

from measured soil properties. Part - III develops a technique for probabilistic constitutive

simulation of elasticplastic materials with example simulations of various material models.

Part - IV builds upon the developments in Part - III in formulating stochastic elasticplastic

finite element method for stochastic solution of boundary value problems with example

simulations. In addition, Part - IV also illustrates the complete solution process, based on

the developed computational framework, of a geotechnical engineering problem with real-life

data.

10

1.4 Original Features

The intellectual merit of this research project is in the merging of state-of-the-art

probability theory with theory of elastoplasticity and subsequent development of a stochastic

elasticplastic finite element method. This is, to our knowledge, a unique endeavor with

minimal prior work in probabilistic elastoplasticity to rely on. Of particular importance

is the application of developed methodology to soils, which exhibit a high degree of non-

linearity and where material properties are highly uncertain.

The impact of the proposed research is expected to be much wider than just in the

area of geotechnical engineering. The phenomena of spatial variability and uncertainty in

material properties is present in all materials. The appropriate formulation and implemen-

tation that incorporate above phenomena into advanced numerical simulations will impact

mechanical, biomedical, materials, aerospace as well as other areas of civil engineering.

11

Chapter 2

Probability Theory Background

2.1 Introduction

The intention of this chapter is to provide some background on probability theory.

Effort here has been made to outline only the mathematical tools that are used in the

subsequent chapters. This chapter develops mainly following the classnotes on Applied

Stochastic Methods in engineering by Professor M. Levant Kavvas [46], with appropriate

examples from mechanics. For a thorough outline of mathematical theory of probability, the

readers are encouraged to refer to standard mathematical texts on probability theory like

by Montgomery and Runger [72] or for advanced topics like stochastic calculus by Gardiner

[30].

2.2 Basic Definitions

The definitions are mainly following Montgomery and Runger [72]

12

Random Experiment: An experiment that can result in different outcomes, even though it

is repeated in the same manner every time, is called a random experiment.

Sample Space: The set of all possible outcomes of a random experiment is called the sample

space.

Probability of an Outcome: There are two schools of thoughts in interpreting probability of

an outcome - degree of belief interpretation and relative frequency interpretation. Degree of

belief interpretation is subjective in the sense that there is always a possibility that different

person will assign different probabilities to the same outcome. On the other hand relative

frequentists interpret probability based on the conceptual model of repeated replications of

any random experiment. According to them probability of an outcome is the limiting value

of the proportions of times any outcome occurs in n repetitions of a random experiment

as n increases beyond all bounds. For example, if the probability that the shear modulus

of a soil is equal to 2.5 MPa is 20%, the relative frequentists will interpret this as follows:

if we do many identical tests (using the same testing devices with the same method of

registering test data etc.) on the same soil, 20% of the tests will result in shear modulus of

the soil equal to 2.5 MPa. In mechanics, both interpretations of probability of an outcome

is important since for any material (and to a larger extent for soils) we have both natural

variability and knowledge uncertainty (discussed in details in next chapter (Chapter [3])).

Probability of an event A: Probability of an event A, denoted as P [A], is equal to sum of

the probabilities of the outcomes in A.

Random Variable: A function that assigns a real number to each outcome in the sample

13

space of a random experiment is called a Random Variable.

Realizations or Sample Values: The various values a random variable can take during a

random experiment are called Realizations or Sample Values.

2.3 Properties of Single Random Variable

The objective of this section is to describe the mathematical tools available to

analyze the complete probabilistic description of a single random variable.

2.3.1 Probability Distribution Function

For many distributions of probability over a sample space, there exists a function

fX(x) from which the probability P [E] of any event, E can be obtained by a summation of

the form:

P [E] =E

fX(x) (2.1)

For discrete sample spaces, the above function is called Probability Mass Function (PMF)

and for continuous sample spaces, is called a Probability Density Function (PDF).

By far the most common probability density function is the Gaussian or normal

probability density function, which, for a single random variable, say friction coefficient 1,

A is written as:

fA() =1

2(SD)2A

e

1

2(SD)2A

( A)2(2.2)

1defined as A = 2sin/3(3 sin), where is the friction angle of a material

14

where A and (SD)A are the mean and standard deviation of the friction coefficient random

variable (A). Fig. 2.1 shows the probability density function of friction coefficient, assuming

Gaussian distribution with mean and standard deviation of 0.3 and 0.05 respectively. From

the probability density function, the probability of an outcome, for example, the probability

of A being 0.25 in the above example 2 can be interpreted as:

P [A = 0.25] = P [0.245 < A < 0.255] =

0.2550.245

fA()d = 0.0483

0.2 0.3 0.4 0.5

2

4

6

8

10P@AD

Figure 2.1: Gaussian Probability Density Function of Friction Coefficient Random Variable(A) with mean = 0.3 and Standard Deviation = 0.05

2This is somewhat counter-intuitive, because for a continuous random variable X and any outcome x,P (X = x) = 0, as every point has zero width. However,in practice when a particular outcome is observed,such as 0.25 in this example, the result can be interpreted as the rounded value of a particular outcomethat is actually in a range such as 0.245 < A < 0.255 in this example. Therefore, the probability that therounded value 0.25 is observed as the value for A is the probability that A assumes a value in the interval[0.245, 0.255], which is not zero.

15

2.3.2 Cumulative Distribution Function

Cumulative distribution function is an alternate way of describing the probability

distribution of any random variable. For a continuous random variable, it is defined as:

F (x) = P (X x) = x

f(u)du for - < x

16

0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1F@AD = P@A D

Figure 2.2: Gaussian Cumulative Density Function of Friction Coefficient Random Variable(A) with mean = 0.3 and Standard Deviation = 0.05

transforms to the arithmetic average, which is called the mean or expected value and it is

mathematically represented as:

X = E(X) = +

xfX(x)dx for < x < (2.5)

The expected value of a function h(X) of a continuous random variable is defined similarly

as:

h(X) = E[h(X)] = +

h(x)fX(x)dx for < x < (2.6)

For example, if the elastic shear modulus (G) of a material is a normally distributed random

variable with a mean of 2.5 MPa and a standard deviation of 1 MPa, then the mean of

1D, elasticplastic shear modulus obeying von Mises associative plasticity (defined 3 as

3derived in Subsection 5.2.1

17

GG2/(G+ cu/3)), where cu = 0.3 (deterministic)) is the rate of evolution of internal

variable cu (unconfined compressive strength) with plastic strain) can be obtained as:

G G

2

G +130.3

=

G

G2

G+130.3

{0.398942 e0.5(2.5+G)

2}dG

= 0.161806 MPa

The nth moment about the mean is called the nth central moment and can be

represented mathematically as:

E[X E(X)]n = +

[x E(X)]nfX(x)dx for < x < (2.7)

The most common central moment is the second central moment. It is called the variance

and is defined as:

V ar[X] = E[X E(X)]2

=

+

[x E(X)]2fX(x)dx for < x < (2.8)

=

+

x2fX(x)dx (E[X])2 for < x < (2.9)

The square root of variance is called the standard deviation and the dimensionless ratio of

mean over standard deviation is called the coefficient of variation (COV).

18

Having defined variance, from Eqs. (2.6) and (2.9), one can obtain the variance of

a function of random variable. Thus, the variance of elasticplastic shear modulus obeying

von Mises associative plasticity can be obtained as:

V ar

G G2

G +130.3

=

G

G2

G+130.3

2 {0.398942 e0.5(2.5+G)

2}dG

G G

2

G+130.3

2

= 0.00427604 MPa2

and the COV of von Mises elasticplastic shear modulus as:

COV

G G2

G +130.3

=

V arG G2

G +130.3

G G

2

G +130.3

= 0.004276040.161806

= 0.404

2.3.4 Characteristic Function and its relation with Moments and Cumu-

lants

As with probability distribution function and cumulative distribution function,

characteristic function also completely characterizes the probabilistic behavior of a random

variable. The characteristic function of a random variable X is defined as:

X(u) =eiuX

=

DeiuXfX(x)dx (2.10)

19

where X is continuous on some domain D on the real line. In case of continuous random

variable, taking the inverse Fourier transform, one can obtain the pdf, fX(x) of X:

fX(x) =1

2

+

X(u)eiuXdu (2.11)

By expanding the characteristic function, x in terms of MacLaurin series:

X(u) = X(u)|u=0 + udX(u)du

|u=0 + u2

2

d2X(u)

du2|u=0 + (2.12)

and evaluating the various derivatives using Eq. (2.10):

X(u)|u=0 = +

fX(x)dx = 1 (2.13)

dX(u)

du|u=0 = i

+

xfX(x)dx = i X (2.14)

djX(u)

duj|u=0 = ij

+

xjfX(x)dx = ijXj

(2.15)

one can express the jth moment of X in terms of characteristic function of X as:

Xj=

1

ijdjX(u)

duj|u=0 (2.16)

The cumulants or semi-invariants of a random variable X are defined by the rela-

20

tionship:

X(u) = e

P

j=1

(iu)j

j!

Xj

(2.17)

Rearranging the above equation (Eq. (2.17)), the cumulants of any random variable (X)

can be written in terms of its characteristic function (X) as:

X = dX(u)du

|u=0 1i

(2.18)

X2

=d2X(u)

du2|u=0 1

i2(2.19)

Xn = dnX(u)

dun|u=0 1

in(2.20)

2.3.5 Relation between Moments and Cumulants

Comparing between Eqs. (2.12) and (2.17), one can write:

1 +(iu)2

2

X2+ + (iu)

n

n!Xn = e

P

n=1

(iu)n

n!Xn

= 1 +n=1

(iu)n

n!Xn+

{n=1

(iu)n

n!Xn

}22

+

{n=1

(iu)n

n!Xn

}n6

+

(2.21)

Equating the coefficients of equal powers in Eq. (2.21), one can obtain the moments in terms

of cumulants as:

21

X = X (2.22)X2=X2

+X22

(2.23)

X3=X3

+ 3 X X2+ X3 (2.24)It is interesting to note here that higher order moments are larger in magnitude

than the lower order moments. On the other hand, higher order cumulants are smaller in

magnitude than the lower order cumulants. Hence while dealing with series approximation,

it is always advantageous to go for the cumulant expansion method rather than the moment

expansion method.

2.4 Properties of Two or More Random Variables

In this section, the focus is mainly on complete probabilistic characterization of two

random variables (commonly known as bi-variate analysis). However, the concept described

here is applicable and can be easily extended to analysis of n number of random variables

(multi-variate analysis).

2.4.1 Joint and Marginal Probability Distribution

If X and Y are two random variables, the probability distribution that defines their

simultaneous behavior is called a Joint Probability Distribution. It is specified by providing

a method for calculating the probability that X and Y assume a value in any given region

of a 2D space. The most common joint probability distribution is Gaussian, which for two

continuous random variables, say shear modulus G (with mean G and standard deviation

22

(SD)G) and friction coefficient A (with mean A and standard deviation (SD)A) is defined

as:

fGA(G) =1

2(SD)G(SD)A

1 2

GA

e

8 1, the infinite variance contributions come from

low frequencies (so the field becomes non-homogeneous).

In rendering the fractal model useful for practical applications Fenton [23] sug-

gested use of either upper cut-off frequency (when 0 < 1) or lower cut-off frequency

(when > 1) or both (when = 1). In addition to making the random field variance finite,

the cut-off frequencies make the field homogeneous. Among different available methods

Fenton [23] suggested periodogram approach in estimating the statistical parameters of the

fractal model, which is defined in power spectral density space as:

P () =P0

0 < (4.16)

The log-likelihood of seeing the periodogram estimates Pj = P (j), j = 1, 2, ..., k, where

k = (n 1)/2, and j = 2j/D, D being the domain length and n being the number of

measured locations is:

L(P |) = k lnP0 + k

j=1

ln j 1P0

kj=1

j Pj (4.17)

72

where = [P0, ] is the unknown parameter vector, which is estimated by maximizing the

log-likelihood equation (Eq. (4.17) as follows:

P0 =1

k

kj=1

Pjj (4.18)

and the estimate for by solving the following equation:

kj=1 Pj

j ln j

1

k

kj=1

Pjj

k

j=1

ln j = 0 (4.19)

4.3 Example Estimation of Statistical Parameters

In this section, the CPT data [99], collected by the USGS Western Earthquake

Hazards Team for miscellaneous field and project investigations in Alameda County, CA,

is analyzed for vertical spatial variability to a depth of 13.5m (with top 1m removed, so

effective soil depth is 12.5m). 16 CPT soundings was considered over an area of approxi-

mately 7 KM2 as shown in Fig. 4.2. The site is sloping from east to west. However, for

illustration purpose the site is considered horizontal. The subsoil to a depth of 13.5m is

mostly composed of soft clay with lenses of stiff clay and sand. The soil classifications at

the sounding locations are arranged from east to west and is shown in Fig. 4.3.

The vertical spatial variability of CPT tip resistance (qT ) has been modeled as 1-

D homogeneous random field, using both finite scale (Gauss-Markov) and fractal (1/f-type

noise) model as described in Section 4.2. A typical sounding (measurement at borehole 1)

of qT is shown in Fig. 4.4.

73

250 750 20001250

500

1000

1500

2000

2500

3000

3500

WE Coordinate (m)

SN Coordinate (m)1

2

34

5 6

78

910

11

1213

14

151617

18

Figure 4.2: CPT Sounding locations

The method of moment estimated and finite scale Gauss-Markov maximum like-

lihood autocovariances for Borehole 1 sounding (refer to Fig. 4.4) are shown in Fig. 4.5.

The autocorrelation, which is the normalized form of autocovariance, is shown in Fig. 4.6

as estimated using both the methods from borehole 1 sounding.

Maximum likelihood estimated constant mean Gauss-Markov model statistical pa-

rameters for all the 16 soundings are tabulated in Table 4.1, along with the mean and

standard deviations of the estimates. A simulated realization of the resulting random field

(using the mean values of the estimates) is shown in Fig. 4.7(b). It was simulated using the

74

Figure 4.3: East-West soil profile interpreted from CPT soundings

75

0 15 30

qT

12

10

8

6

4

2

Dep

th (m

)

(MPa)

Figure 4.4: Typical qT data: Borehole 1 sounding

2 4 6 8 10 12

20

10

10

20

Aut

ocov

aria

nce

(MPa

^2)

ML estimate

Method of momemt estimate

Lag Distance (m)

Figure 4.5: Maximum likelihood estimated Gauss-Markov autocovariance function alongwith method of moment estimate (for borehole 1 sounding)

76

2 4 6 8 10 12

0.75

0.5

0.25

0.25

0.5

0.75

1

Aut

ocor

rela

tion

ML estimate Method of moment estimate

Lag Distance (m)

Figure 4.6: Maximum likelihood estimated Gauss-Markov autocorrelation function alongwith method of moment estimate (for borehole 1 sounding)

Table 4.1: Maximum likelihood estimated constant mean Gauss-Markov model parameters

Estimated Variance Interpreted VarianceBorehole Method of Maximum True Testing Correlation

No. Moment Likelihood Spatial Error Mean Length

1 25.078 24.94 24.94 0.138 6.41 0.532 6.87 7.2 7.2 0 4.8 0.63 3.76 4.42 4.42 0 4.42 0.214 15.22 14.74 14.74 0.48 3.97 0.725 11.72 9.56 9.56 2.16 3.19 0.396 27.52 29.54 29.54 0 6.39 0.337 11.71 4.25 4.25 7.46 2.31 0.419 6.41 5.93 5.93 0.48 3.9 0.3810 20.06 23.18 23.18 0 4.3 0.2611 48.19 44.92 44.92 3.27 4.15 1.5112 10.69 3.53 3.53 7.16 2.2 0.3913 66.13 57.43 57.43 8.7 7.4 0.9214 44.92 42.16 42.16 2.76 7.04 0.5115 31.6 31.82 31.82 0 6.3 0.5916 23.44 25.09 25.09 0 6.01 0.4218 93.97 82.08 82.08 11.89 7.06 1.64

Mean - - 25.67 2.78 4.99 0.61SD - - 22.22 3.86 1.69 0.41

77

Cholesky decomposition of covariance matrix as follows (cf. Mardia and Marshall [65]):

qT = qT + (L)qT Z (4.20)

where qT is the mean vector (assumed constant with depth in this example), (K2)qT =

(L)qT (L)qT is the Cholesky decomposition of the covariance matrix ((K2)qT ), and Z is

the zero mean unit variance normally distributed random matrix (Z N(0, I), I being

the identity matrix). Fig. 4.7(a) shows one of the measured realizations. Note that it is

not expected to be identical, except having same statistical nature, as they are merely two

possible realizations of the same random field.

20 40

12

10

8

6

4

2

20 40

12

10

8

6

4

2

q (MPa) q (MPa)q (MPa)

Dep

th (m

)

Dep

th (m

)T T

(a) (b)

Figure 4.7: (a) Measured (at borehole 2) and (b) Simulated (finite scale Gauss-Markovmodel) realizations

For comparison, the same set of data (16 CPT soundings over approximately

78

7KM2) was analyzed using fractal method as described in Subsection 4.2.2. It was assumed

that the random field mean and variance was calculated by other methods e.g. a constant

(or any polynomially varying with depth) mean deterministic trend could be extracted

using global regression analysis over 16 soundings as shown in Fig. 4.8 along with measured

data. And the field variance could be estimated by method of moment using Eq. (4.12) as

tabulated in Table 4.1.

The periodogram for the borehole 1 sounding is shown in Fig. 4.9 and correspond-

ing the 1/f-type noise model with lower cut-off frequency is shown in Fig. 4.10. The

statistical parameters (0, , and P0) required to define the 1/f-type noise model with

lower cut-off frequency given as:

P () =

P0/0 if 0 < 0

P0/ if > 0

(4.21)

was estimated using maximum likelihood technique (as discussed in Subsection 4.2.2) and

is tabulated in Table 4.2 for all the 16 CPT soundings, along with the mean and stan-

dard deviation of the estimates. The equivalent correlation lengths (computed as r0 =

P0/20 , refer to Fenton [24]) are also tabulated. Having defined the fractal model with

cut-off frequency(ies), the autocovariance function can be easily computed using the Weiner-

Khintchine relationship as (cf. Fenton [24]):

C(r) =

0

P ()cos(r)d

=

00

P00

cos(r)d +

0

P0

cos(r)d (4.22)

The autocovariance function as computed using Eq. (4.22) and borehole 1 estimates is

plotted in Fig. 4.11.

79

20 40

q T

12

10

8

6

4

2

0D

epth

(m)

(MPa)

Figure 4.8: Deterministic Trend as obtained through global regression over 16 CPTsoundings

80

0.1 1 10 100

0.01

0.1

1

10

Spec

tral P

ower

Frequency (rad/sec)

Figure 4.9: Periodogram of borehole 1 sounding

0.1 1 10 100

0.01

0.1

1

10

Spec

tral P

ower

Frequency (rad/sec)Figure 4.10: Maximum likelihood estimated fractal (1/f -type noise with lower cut-off fre-quency) power spectral density function corresponding to borehole 1 sounding

81

Table 4.2: Maximum likelihood estimated constant mean fractal (1/f-type noise with lowercut-off frequency) model parameters obtained using periodogram approach

BH 0 (rad/sec) P0 Equivalent Correlation Length, r0 (m)

1 2.08 1.78 19.42 0.662 3.13 2.09 12.51 0.523 9.36 1.27 1.49 0.074 1.33 1.76 8.18 1.015 1.66 1.64 6.37 0.736 2.04 1.39 10.42 0.437 0.69 1.77 3.85 1.989 2.99 1.86 7.7 0.4810 4.39 1.52 15.19 0.2411 1.14 2.06 28.62 1.4112 1.65 2.14 10.14 1.0113 1.93 2.13 74.46 0.8614 1.42 1.64 22.22 0.8615 1.91 1.8 23.85 0.7316 2.02 1.59 13.36 0.5818 1.09 2.02 52.29 1.44

Mean 2.43 1.78 19.38 0.81SD 2.06 0.26 19.13 0.48

2 4 6 8 10 12

5

10

15

20

25

Lag Distance (m)

Aut

ocov

aria

nce

(MPa

^2)

Figure 4.11: Fractal (1/f -type noise with lower cut-off frequency) autocovariance functionfor borehole 1 sounding

82

Part III

Material (Constitutive) Level

Stochastic Simulation:

Probabilistic ElastoPlasticity

83

Chapter 5

Probabilistic ElastoPlasticity:

Theory

5.1 Introduction

Advanced elastoplasticity based constitutive models, when properly calibrated,

are very accurate in capturing important aspects of material behavior within continuum.

But all materials, in particular geomaterials (soil, rock, concrete, powder, bone etc.) be-

haviors are uncertain due to inherent spatial uncertainties and testing and transformation

uncertainties (as discussed in details in Part II of this dissertation). These uncertainties in

material properties, needed for calibrating constitutive models, could outweigh the advan-

tages gained by using advanced constitutive models. For example, Fig. 5.1 shows a schematic

of anticipated influence of material uncertainties on a bilinear elastic-plastic stress-strain

behavior. Depending on uncertainties in material property(ies) and interaction between

84

them, the behavior of the same material could be very different. This could be even more

complicated for non-linear materials.

Figure 5.1: Anticipated Influence of Material Fluctuations on Stress-Strain Behavior

The uncertainties in material properties are inevitable in real life and the best way

to deal them with is to account for them in our modeling and simulation. In traditional

deterministic constitutive modeling, material models are calibrated against set of experi-

mental data. Though the experimental data generally exhibit some statistical distribution,

the models are usually calibrated against the mean of the data and the uncertainties with

respect to the mean are neglected. Hence, when these constitutive models are used for

further modeling (e.g., for modeling the behavior of solids and structures made with those

materials), the uncertainties in material properties are lost from the simulation results.

The strategy for propagating uncertainties through governing differential equa-

85

tions can be broadly classified into two categories - stochastic differential equation (SDE)

with random forcing and SDE with random coefficient. For SDE with random forcing, when

the governing equation is of Ito type (refer to Subsection 2.6.5, for details refer to Gardiner

[30]), highly developed mathematical theory exists - the solution is a Markov process and the

probability density of the solution obeys a Fokker-Planck-Kolmogorov (FPK) partial differ-

ential equation (refer to Subsection 2.6.6, for details refer to Gardiner [30]). The advantage

of the FPK equation is that it transforms the original nonlinear SDE in real space into a

linear deterministic PDE in probability density space. On the other hand, for SDEs with

random coefficients, which is of immediate interest of this study, approximate numerical

methods (e.g., perturbation method (cf. Klieber and Hien [51])) are very popular especially

for nonlinear problems. Monte Carlo method, which is based on law of large numbers, is

also very popular. It is carried out sequentially by generating randomized parameters and

using them as input into a set of deterministic models. This set of models is then used in

a multitude of simulations to determine the value of desired response function. Finally the

statistics of the response variable are quantified. The advantage of Monte Carlo method is

that accurate solutions can be obtained for any problem (either linear or nonlinear) whose

deterministic solution (either analytical or numerical) is known. The major drawback of

Monte Carlo method is that it is computationally very expensive. This is even more the

case for problems where no closed-form solution exists for solving the deterministic prob-

lem. On the other hand, perturbation method, applicable to both linear and non-linear

stochastic problems, uses Taylor series expansion with respect to the mean and considers

first few terms of the expansion. Inherent to the Taylor series expansion, regular pertur-

86

bation methods often exhibit closure problems (cf. Kavvas [47]), where information on

higher-order moments are necessary to solve for lower-order moments. Also, the regular

perturbation approach is applicable only to small fluctuations in the state variable since

the linearization approximation fails when the input parameters exhibit large coefficient of

variations (COVs) (cf. Matthies et al. [69]).

First attempt to propagate uncertainties through elasticplastic constitutive equa-

tions considering random Youngs modulus was published only recently, e.g., Anders and

Hori [1]. They took perturbation expansion at the stochastic mean behavior and considered

only the first term of the expansion. In computing the mean behavior they took advantage

of bounding media approximation. Though this method doesnt suffer from computational

difficulty associated with Monte Carlo method for problems having no closed-form solution,

it inherits closure problem and the small COV requirements for the material parame-

ters. Furthermore, with bounding media approximation, difficulty arises in computing the

mean behavior when one considers uncertainties in internal variable(s) and/or direction(s)

of evolution of internal variable(s).

Recently, Kavvas [47] obtained a generic EulerianLagrangian form of FPK equa-

tion, exact to second-order, corresponding to any nonlinear ordinary differential equation

with random coefficients and random forcings. The FPK equation approach doesnt suffer

from the drawbacks of Monte Carlo method and perturbation technique. In this chapter,

the development by Kavvas [47] is applied in obtaining probabilistic solution for a gen-

eral, incremental elasticplastic constitutive equation with random coefficient. The solution

methodology is designed with several applications in mind, namely to

87

obtain probabilistic stressstrain behavior from spatial average form (upscaled form)

of constitutive equation, when input uncertain material properties to the constitutive

equation are random fields; and

obtain probabilistic stress-strain behavior from point-location scale constitutive equa-

tion, when input uncertain material properties to the constitutive equation are random

variables.

5.2 One Dimensional Development

In this section, one-dimensional general formulation of probabilistic elastoplasticity

is shown first, followed by its specialization in obtaining particular (obeying particular

elastoplasticity model) point-location scale constitutive behaviors and solution methodol-

ogy of the resulting equation. Governing equations for probabilistic solutions of von Mises,

Drucker-Prager, and Cam Clay models have been derived. In addition, the governing equa-

tion for probabilistic solution of linear elastic constitutive equation has also been derived

as a special case of general nonlinear derivation.

The most general, incremental (rate) form of spatial-average elastic-plastic consti-

tutive equation can be written as

dij(xt, t)

dt= Dijkl(xt, t)

dkl(xt, t)

dt(5.1)

where the stiffness tensor Dijkl(xt, t) can be either elastic or elastic-plastic:

88

Dijkl =

Delijkl when f < 0 (f = 0 df < 0)

Delijkl Delijmn

U

mn

f

pqDelpqkl

f

rsDelrstu

U

tu fq

rwhen f = 0 df = 0

(5.2)

where Delijkl is the elastic stiffness tensor and Depijkl is the elasticplastic stiffness tensor, f

is the yield function, which is a function of stress (ij) and internal variables (q), U is the

potential function (also a function of stress and internal variables). The internal variables

(q) could be scalar (for perfectly-plastic and isotropic hardening models), second-order

tensor (for translational and rotational kinematic hardening) or fourth-order tensor (for

distortional hardening) or combinations of the above. The same classification applies to

the direction of evolution of internal variables (r). Therefore, the most general form of

constitutive rate equation in terms of its parameters can be written as

dij(xt, t)

dt= ijkl(ij , D

elijkl, q, r;xt, t)

dkl(xt, t)

dt(5.3)

Due to randomnesses in elastic constants (Delijkl) and/or internal variables (q)

and/or rate of evolution of internal variables (r) the material stiffness operator ijkl in

Eq. (5.3) becomes stochastic and hence Eq. (5.2) becomes a set of linear/non-linear ordi-

nary differential equations with stochastic coefficients. On the other hand, the randomness

in forcing term (kl) (e.g., seismic loading), Eq. (5.3) becomes a set of linear/non-linear

ordinary differential equations with stochastic forcing. In general, randomnesses in mate-

rial properties and forcing function, Eq. (5.3) becomes a set of linear/non-linear ordinary

89

differential equation with stochastic coefficients and stochastic forcing.

In order to gain better understanding of the effects of random material parameters

and forcing on response, focus is shifted from a general 3-D case to a 1-D case. Focusing

on 1-D behavior, the Eq. (5.3) can be written as

d(xt, t)

dt= (,Del, q, r;xt, t)

d(xt, t)

dt(5.4)

which is a non-linear ordinary differential equation with stochastic coefficient and stochastic

forcing. The right hand side of Eq. (5.4) is replaced with the function as

(,Del, q, r, ;x, t) = (,Del, q, r;xt, t)d(xt, t)

dt(5.5)

so that now Eq. (5.4) can be written as

(xt, t)

t= (,Del, q, r, ;x, t) (5.6)

with initial condition,

(x, 0) = 0 (5.7)

In the above equation (Eq. (5.6)), can be considered to represent a point in the

-space and hence it can be said that Eq. (5.6) determines the velocity for the point in the

-space. This may be visualized, from the initial point, and given initial condition 0, as

a trajectory that describes the corresponding solution of the non-linear stochastic ordinary

differential equation (ODE) (Eq. (5.6)). Considering a cloud of initial points (refer to

Fig. 5.2), described by a density (, 0) in the -space and movement of the points dictated

90

by Eq. (5.6), the phase density of (x, t) varies in time according to a continuity equation

which expresses the conservation of all these points in the -space.

Figure 5.2: Movements of Cloud of Initial Points, described by density (, 0), in the -space

This continuity equation can be expressed in mathematical terms, using Kubos

stochastic Liouville equation (cf. Kubo [55]):

((x, t), t)

t=

[(x, t), Del(x), q(x), r(x), (x, t)].[(x, t), t] (5.8)

with an initial condition,

(, 0) = ( 0) (5.9)

where is the Dirac delta function and Eq. (5.9) is the probabilistic restatement of the

original deterministic initial condition (Eq. (5.7)). Then by using Van Kampens Lemma

91

(cf. Van Kampen [95]), one can write

(, t) = P (, t) (5.10)

where, the symbol denotes the expectation operation, and P (, t) denotes evolutionary

probability density of the state variable of the constitutive rate equation (Eq. (5.4)).

In order to obtain the deterministic probability density (P (, t)) of the state vari-

able, , it is necessary to obtain the deterministic partial differential equation (PDE) of

the -space mean phase density (, t) from the linear stochastic PDE system (Eqs. (5.8)

and (5.9)). This necessitates the derivation of the ensemble average form of Eq. (5.8) for

(, t). This ensemble average can be derived as (for detailed derivation refer to Appendix

(Section A.4)):

((xt, t), t)t

=

{[((xt, t), D

el(xt), q(xt), r(xt), (xt, t))

t0dCov0

[((xt, t), D

el(xt), q(xt), r(xt), (xt, t));

((xt , t ), Del(xt ), q(xt ), r(xt )(xt , t )

]]((xt, t), t)

}

+

{[ t0dCov0

[((xt, t), D

el(xt), q(xt), r(xt), (xt, t));

((xt , t ), Del(xt ), q(xt ), r(xt ), (xt , t ))] ] ((xt, t), t)

}

(5.11)

to exact second order (to the order of the covariance time of ). In Eq. (5.11), Cov0[] is

92

the time ordered covariance function defined by

Cov0 [(x, t1), (x, t2)] = (x, t1)(x, t2) (x, t1) (x, t2) (5.12)

By combining Eqs. (5.11) and (5.10) and rearranging the terms the following

EulerianLagrangian form of Fokker-Planck-Kolmogorov equation (FPKE) (cf. Kavvas [47])

yields:

P ((xt, t), t)

t=

[{((xt, t), D

el(xt), q(xt), r(xt)(xt, t))

+

t0dCov0

[((xt, t), D

el(xt), q(xt), r(xt)(xt, t))

;

((xt , t ), Del(xt ), q(xt ), r(xt ), (xt , t )]}

P ((xt, t), t)

]

+2

2

[{ t0dCov0

[((xt, t), D

el(xt, t), q(xt, t), r(xt, t), (xt, t));

1((xt , t ), Del(xt ), q(xt ), r(xt ), (xt , t ))]}

P ((xt, t), t)

]

(5.13)

to exact second order. This is the most general relation for probabilistic behavior of inelastic

(nonlinear) 1-D stochastic constitutive rate equation. The solution of this deterministic

linear FPKE (Eq. (5.13)) in terms of the probability density P (, t) under appropriate

initial and boundary conditions will yield the PDF of the state variable of the original

1-D non-linear stochastic constitutive rate equation (Eq. (5.4)). It is important to note that

while the original equation (Eq. (5.4)) is non-linear, the FPKE (Eq. (5.13)) is linear in terms

of its unknown, the probability density P (, t) of the state variable . This deterministic

93

linearity, in turn, provides significant advantages in the solution of the probabilistic behavior

of the constitutive rate equation (Eq. (5.4)).

One should also note that Eq. (5.13) is a mixed Eulerian-Lagrangian equation

since, while the real space location xt at time t is known, the location xt is an unknown.

If one assumes small strain theory, one can relate the unknown location xt from the

known location xt by using the strain rate, (=d/dt) as,

d = =xt xt

xt(5.14)

or, rearranging

xt = (1 )xt (5.15)

Once the evolutionary probability density function P (, t) is obtained it can be

used to obtain the evolutionary statistical moments of state variable () by usual expectation

operation (refer to Subsection 2.3.3) e.g., the evolutionary mean by:

(t) =

(t)P ((t))d(t) (5.16)

Another interesting aspect of this development, but could be possibly of math-

ematical interest, is to obtain the equivalent Ito stochastic differential equation (refer to

Subsections 2.6.5, for more details refer to Gardiner [30]) corresponding to the general

FPKE (Eq. (5.13)). Using the equivalency between Ito stochastic differential equation and

FPKE (cf. Gardiner [30]) one can obtain the equivalent Ito form:

94

d(x, t) =

{((xt, t), D

el(xt), q(xt), r(xt), (xt, t))

+

t0dCov0

[((xt, t), D

el(xt), q(xt), r(xt), (xt, t))

;

((xt , t ), Del(xt ), q(xt ), r(xt ), (xt , t ))]}

dt

+ b(, t)dW (t) (5.17)

where,

b2(, t) = 2

t0dCov0

[((xt, t), D

el(xt), q(xt), r(xt), (xt, t));

((xt , t ), Del(xt ), q(xt ), r(xt ), (xt , t ))]

(5.18)

and, dW (t) is an increment of Wiener process W(t) (refer to Subsections 2.6.4 and 2.6.5, for

more details refer to Gardiner [30]) with dW (t) = 0. It is also interesting to note that all

the stochasticities of the original equation (Eq. (5.4)) are lumped together in the last term

(Wiener increment term) of the right-hand-side of Eq. (5.17). But the problem in solving for

the statistical moments of the state variable () using the Ito form e.g., say the evolution

of the mean of the state variable (), which can be written mathematically as (taking

advantage of the independent increment property of the Wiener process (dW (t) = 0), for

details refer to Kavvas [47]):

95

d(x, t)dt

=

((xt, t), D

el(xt), q(xt), r(xt), (xt, t))

+

t0dCov0

[((xt, t), D

el(xt), q(xt), r(xt), (xt, t))

;

((xt , t ), Del(xt ), q(xt ), r(xt ), (xt , t ))]

(5.19)

is the non-linear stochasticity (note that state variable also appears within () on the right-

hand-side of Eq. (5.19) and is random) in the resulting equation. No analytical treatment

is available for dealing with this type of problem. There exist approximate numerical

method e.g., perturbation approach (cf. Anders and Hori [2]) but the closure problem

will appear. Also, due to linearization approximation using Taylor series expansion, the

error in perturbation approximation is a function of COV, which for soil is usually quite

large.

5.2.1 Specialization of General Formulation to Particular Constitutive

Laws

Having obtained the relation for probabilistic behavior of 1-D inelastic (nonlinear)

constitutive rate equation with stochastic coefficient and stochastic forcing in most general

form, in this subsection the general relation will be specialized to four particular types of

point-location scale shear constitutive behavior: a) 1-D linear elastic, b) 1-D Drucker-Prager

associative elasticplastic linear hardening c) 1-D von Mises associative elasticplastic linear

hardening, and d) 1-D Cam Clay elasticplastic.

96

Probab