Multiaxial Constitutive and Numerical Modeling in Geo-mechanics within Critical State Theory

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. THE UNIVERSITY OF BRITISH COLUMBIA Multiaxial Constitutive and Numerical Modeling in Geo-mechanics

Dafalias 2 Associate Professor, Department of Civil Engineering, The University of British Columbia,

University of California, Davis, CA, USA, and Department of Mechanics, National Technical University of

1 . THE UNIVERSITY OF BRITISH COLUMBIA Multiaxial Constitutive and Numerical Modeling in Geo-mechanics within Critical State Theory Mahdi Taiebat and Yannis F. Dafalias 2 Associate Professor, Department of Civil Engineering, The University of British Columbia, Vancouver, BC, Canada 2 Distinguished Professor, Department of Civil and Environmental Engineering, University of California, Davis, CA, USA, and Department of Mechanics, National Technical University of Athens, Zographou, Greece SCEC Workshop on 3D Site Effects in Physics-Based Ground Motion Simulations Los Angeles, CA. May 5, 25 ACKNOWLEDGEMENT: National Science and Engineering Research Council of Canada, and National Science Foundation of United States Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles / 9

2 Outline Constitutive modeling for clays SANICLAY class Model Performance 2 Constitutive modeling for sands SANISAND class Model Performance 3 Application in numerical modeling Model implementations Nonlinear effective stress seismic site response analysis 4 Discussion related to the SCEC workshop Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles / 9

3 Constitutive modeling for clays SANICLAY class SANICLAY: Simple ANIsotropic CLAY plasticity model SANICLAY (Dafalias et al., 26) M c N q f/ σ g/ σ σ α g = N M e p α p f = p (c) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 2 / 9

4 Constitutive modeling for clays SANICLAY class SANICLAY: Simple ANIsotropic CLAY plasticity model SANICLAY (Dafalias et al., 26) M c N q f/ σ g/ σ σ α SANICLAY-D (Taiebat et al., 2a) M c N q f/ σ g/ σ σ α g = N M e p α p f = p (c) p d p α p g = f = N M e p (d) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 2 / 9

5 Constitutive modeling for clays SANICLAY class SANICLAY: Simple ANIsotropic CLAY plasticity model SANICLAY (Dafalias et al., 26) M c N q f/ σ g/ σ σ α SANICLAY-D (Taiebat et al., 2a) M c N q f/ σ g/ σ σ α SANICLAY-B (Seidalinov and Taiebat, 24) M c N q F/ σ σ g/ σ f = σ σ α g = N M e p α p f = p (c) p d p α p g = f = N M e p (d) σ c p α p p F = g = N M e (e) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 2 / 9

6 Constitutive modeling for clays SANICLAY class SANICLAY: Simple ANIsotropic CLAY plasticity model SANICLAY (Dafalias et al., 26) M c N q f/ σ g/ σ σ α SANICLAY-D (Taiebat et al., 2a) M c N q f/ σ g/ σ σ α SANICLAY-B (Seidalinov and Taiebat, 24) M c N q F/ σ σ g/ σ f = σ σ α p α p p p d p α p p g = f = g = f = N M e (c) Systematic tensorial extension to multiaxial stress space N M e (d) σ c p α p p F = g = N M e (e) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 2 / 9

7 Constitutive modeling for clays SANICLAY class SANICLAY: Simple ANIsotropic CLAY plasticity model SANICLAY (Dafalias et al., 26) M c N q f/ σ g/ σ σ α SANICLAY-D (Taiebat et al., 2a) M c N q f/ σ g/ σ σ α SANICLAY-B (Seidalinov and Taiebat, 24) M c N q F/ σ σ g/ σ f = σ σ α p α p p g = f = N M e (c) p d p α p g = f = Systematic tensorial extension to multiaxial stress space N M e Relatively straightforward calibration process MCC SANICLAY { { SANICLAY-D{ SANICLAY-B { Model constant category Designation Georgia kaolin Cloverdale Ariake Elasticity κ ν Critical state λ M c, M e.29,.27.87,.86.68,.65 Yield surface N.8.68 Rotational hardening C x Destructuration k i 2 Bounding surface h a d p (d) σ c p α p p F = g = N M e (e) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 2 / 9

8 Constitutive modeling for clays Model Performance Monotonic loading (iso. comp., oedometer, undrained triaxial comp./ext.) Lower Cromer Till Bothkennar clay 2.7 Georgia kaolin clay experiment simulation.9 void ratio.4. e destructured sample, ICC structured Sherbrooke sample structured Laval sample σ a (kpa) A D B C (a) CIUC/E on NC clay p (kpa) Compression σ c (kpa): Extension σ c (kpa): p (kpa) Data: after Gens (982) p (kpa) Data: after Smith et al. (992) Data: after Sheu (984) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 3 / 9

9 Constitutive modeling for clays Undrained cyclic triaxial tests Georgia kaolin clay (reconstituted) 2 2 experiment (5 cycles) q cy =36 kpa simulation (5 cycles) Model Performance q cy =36 kpa Cloverdale clay (structured).4 experiment (3 cycles).2 (σ a + σ r ) /2 σ c Data: Sheu (984) ε a (%) 2 ε a (%) 2 experiment ( 7, q cy =4.7 kpa simulation q cy =4.7 kpa 3 5 cycles).2 Data: Zergoun and Vaid (994) ε (%) a simulation (3 cycles) τ /Su =.75 cy c (σ a σ r ) /2 σ e ε a (%) 2 ε a (%) 2 q =65.5 kpa cy simulation q cy =65.5 kpa experiment (3 cycles) ε a (%) Simulations: Seidalinov and Taiebat (24) ε a (%) ε (%) a Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 4 / 9

10 Peak ε a (%) Peak ε a (%) Constitutive modeling for clays Undrained cyclic triaxial tests Cloverdale clay (structured) Experiment.57 compression extension τ cy /Su c = Number of cycles τ /Su =.27 cy c compession extension SANICLAY B Number of cycles Model Performance Peak ε a (%) Cyclic stress ratio, q cy /2 σ c Ariake clay (reconstituted) experiment simulation.3 q cy /2σ c = Number of cycles experiment simulation Fitted curves Number of cycles Data: Yasuhara et al. (992) Data: Zergoun and Vaid (994) Simulations: Seidalinov and Taiebat (24) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 5 / 9

11 Constitutive modeling for sands SANISAND class SANISAND: Simple ANIsotropic SAND plasticity model Dafalias, Manzari, Li, Papadimitriou, Taiebat (997-22) Formulation in triaxial stress space f = η α m = η = q p ε p q = η hb b = (M b η) ε p v = A d d ε p q d = (M d η) Deviatoric stress, q q p M b M M d Mean effective stress, p Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 6 / 9

12 Constitutive modeling for sands SANISAND class SANISAND: Simple ANIsotropic SAND plasticity model Dafalias, Manzari, Li, Papadimitriou, Taiebat (997-22) Formulation in triaxial stress space f = η α m = η = q p ε p q = η hb b = (M b η) ε p v = A d d ε p q d = (M d η) Deviatoric stress, q q p M b M M d Mean effective stress, p Dependence on state parameter M b = M exp ( n b ψ) M d = M exp (n d ψ) Void ratio, e e c = e e c e CSL p Mean effective stress, p Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 6 / 9

Constitutive modeling for sands Generalization and model constants NISAND Systematic cont d tensorial extension to stematic multiaxial tensorial extension stress space to multiaxial to stress stress

13 Constitutive modeling for sands Generalization and model constants NISAND Systematic cont d tensorial extension to stematic multiaxial tensorial extension stress space to multiaxial to stress stress space space SANISAND class Relatively straightforward calibration process Taiebat et al. (2b) M d M d M M M b M b M d M d M M b M b onstitutive ingredients to account to for for te state parameter dependent dilatancy stress-ratio (Manzari (Manzari and and Dafalias, Dafalias, 997; 997; Li and Li and Dafalias 2) 2) Soil-specific set of constants for different densities & confining pressures. evolving fabric fabric anisotropy (Dafalias & Manzari, & 24) 24) inherent fabric fabric anisotropy (Dafalias (Dafalias et al., et al., 24) 24) Constitutive ingredients to account for plastic strains strains under under const. const. stress-ratio & particle & particle crushing crushing (Taiebat (Taiebat & Dafalias, & Dafalias, 28) 28) anisotropic critical critical state state (Li and (Li and Dafalias, Dafalias, 22) 22) ψ-dependent dilatancy stress-ratio (Manzari and Dafalias, 997; Li and Dafalias, 2) i at Taiebat (UBC) (UBC) evolving fabric anisotropy (Dafalias and Los Angeles, Los Manzari, Angeles, CA. May CA. 25 May 24) inherent fabric anisotropy (Dafalias et al., 24) plastic strains under const. stress-ratio & particle crushing (Taiebat and Dafalias, 28) anisotropic critical state (Li and Dafalias, 22) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 7 / 9

14 Constitutive modeling for sands Model Performance 6 Triaxial loading and unloading on Toyoura sand Drained triaxial tests ARTICLE IN PRESS Void ratio, e Undrained triaxial tests Void ratio, e Fig. 8. Simulations vs. experiments in drained triaxial compression tests on isotropically consolidated samples of Toyoura san Experiment Simulation M. Taiebat et al. / Soil Dynamics and Earthquake Engineering 3 (2) Experiment Simulation 2 Experiment 2 Simulation 4 Experiment 4 Simulation 9 6 p in=5 kpa p in = kpa 9 6 p in=5 kpa p in = kpa 3 2 e=.735, D r=63.7% e=.833, D r=37.9% e=.97, D r =8.5% 3 2 e=.735, D r=63.7% e=.833, D r=37.9% e=.97, D r=8.5% Axial strain (%) Axial strain (%) Axial strain (%) Axial strain (%) 2 9 p in=5 kpa p in= kpa 2 9 p in=5 kpa p in= kpa 4 3 e=.735, D r=63.7% e=.833, D r=37.9% e=.97, D r=8.5% 4 3 e=.735, D r=63.7% e=.833, D r=37.9% e=.97, D r=8.5% Void ratio, e Void ratio, e p (kpa) 2 3 p (kpa). Simulations vs. experiments in drained triaxial compression tests on isotropically consolidated samples of Toyoura Fig. 9. sand Simulations [3]. vs. experiments in undrained triaxial compression tests on isotropically consolidated samples of Toyoura s Data: Verdugo and Ishihara (996); Simulations: Taiebat et al. (2b) Experiment Simulation Taiebat 4 (UBC) & Dafalias (UCD, NTUA) 4 constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 8 / 9

15 Constitutive modeling for sands Model Performance Constant-p cyclic triaxial on Toyoura sand Loose sample Experiment ARTICLE IN PRESS Simulation M. Taiebat et al. / Soil Dynamics and Earthquake Engineering 3 (2) Dense sample Experiment ARTICLE IN PRESS Simulation M. Taiebat et al. / Soil Dynamics and Earthquake Engineering 3 (2) Stress ratio, q/p 2 p= kpa (const.) e in=.845 Experiment Stress ratio, q/p 2 p= kpa (const.) e in=.845 Simulation Stress ratio, q/p 2 p= kpa (const.) e in=.653 Experiment Stress ratio, q/p 2 p= kpa (const.) e in=.653 Simulation Shear strain, (%) Shear strain, γ (%) γ Shear strain, γ (%) Shear strain, γ (%) Volumetric strain, ε v (%) Volumetric strain, ε v (%) Volumetric strain, ε v (%).3.3 Volumetric strain, ε v (%) Shear strain, (%) Shear strain, γ (%) γ Shear strain, (%) Shear strain, γ (%) γ Volumetric strain, ε v (%) Stress ratio, q/p Volumetric strain, ε v (%) Stress ratio, q/p Fig.. Simulations vs. experiments in constant-p cyclic triaxial tests on a relatively loose sample of Toyoura sand [32]. strain Taiebat tensor (by(ubc) neglecting & the Dafalias quadratic(ucd, portion NTUA) density constitutive first case ðe ¼ :8Þ. & numerical A schematic modeling illustrationwithin of CS theory SCEC workshop. Los Angeles 9 / 9 as the Volumetric strain, ε v (%) Stress ratio, q/p Data: Pradhan et al. (989); Simulations: Taiebat et al. (2b) Volumetric strain, ε v (%) Stress ratio, q/p Fig.. Simulations vs. experiments in constant-p cyclic triaxial tests on a relatively dense sample of Toyoura sand [32].

16 Application in numerical modeling Model implementations Finite Element platform and model implementation OpenSees: The Open System for Earthquake Engineering Simulation Fully coupled nonlinear dynamic finite element program Open-source: Variety of relevant element types for continuum modeling of soil medium 2D (quad) and 3D (brick) single phase (solid, u) and double phase (solid and pore fluid, u p) Variety of analysis types, integration schemes, and solvers Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles / 9

17 Application in numerical modeling Model implementations Finite Element platform and model implementation OpenSees: The Open System for Earthquake Engineering Simulation Fully coupled nonlinear dynamic finite element program Open-source: Variety of relevant element types for continuum modeling of soil medium 2D (quad) and 3D (brick) single phase (solid, u) and double phase (solid and pore fluid, u p) Variety of analysis types, integration schemes, and solvers SANICLAY and SANISAND implementation: SANICLAY: Refined explicit integration scheme with automatic sub-stepping and error control (Seidalinov and Taiebat, 24) SANISAND: Various explicit and implicit integration schemes (Ghofrani and Arduino, 24) All implementations are in full tensorial forms of stresses and strains (3D) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles / 9

Application in numerical modeling Application of SANICLAY models Nonlinear effective stress seismic site response analysis Modeling of infinite slope subjected to earthquake excitation: 2 m deep (2%

18 Application in numerical modeling Application of SANICLAY models Nonlinear effective stress seismic site response analysis Modeling of infinite slope subjected to earthquake excitation: 2 m deep (2% grade) deposit of NC clay SANICLAY & SANICLAY-B models m water table, permeability: 8 m/s Periodic BCs to emulate D analysis Modeling 9-node quad u p element (Biot s theory) Periodic BCs to emulate D analysis Base dashpot to account for the finite rigidity of the underlying elastic medium Velocity time history u(t) and high V s,base Imperial Valley record scaled to PGA=.35g Acceleration (g) time (s) 2 m! m! m! u-dof! (solid displ)! p-dof! (pore water pressure)! g y! g! g x! g x =g.sin(incl)! g y =g.cos(incl)! incl = atan(grade)! Lysmer-Kuhlemeyer! (969) dashpot! Bedrock! Adopted from McGann and Arduino (23) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles / 9

19 Application in numerical modeling Nonlinear effective stress seismic site response analysis Results: shear stress (τ) & shear strain (γ) at depth of 5.5 m SANICLAY τ (kpa) SANICLAY-B τ (kpa) γ (%) time (s) γ (%) time (s) time (s) time (s) τ (kpa) τ (kpa) γ (%) γ (%) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 2 / 9

20 Application in numerical modeling Nonlinear effective stress seismic site response analysis Results: Displacement profile and spectral accelerations Horizontal displacement profiles at the end of shaking 5 SANICLAY SANICLAY B depth (m) Residual horizontal displacement (m) Spectral acceleration at the base and top of soil columns.4.2 base motion ground surface (SANICLAY) ground surface (SANICLAY B) S a (g) Period, T (sec) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 3 / 9

21 Application in numerical modeling Application of SANISAND model in PRENOLIN Nonlinear effective stress seismic site response analysis Modeling of free field soil column subjected to earthquake excitation in Sendai: 8 m deep deposit of sand 7 m: SANISAND, and 7 8 m elastic.5 m water table, permeability: 5 m/s Periodic BCs to emulate D analysis Modeling SSPquadUP element (Biot s theory) Periodic BCs to emulate D analysis Base dashpot to account for the finite rigidity of the underlying elastic medium Velocity time history u(t) and high V s,base Several motions from downhole arrays at Sendai site [Collaborative study with UW] Adopted from McGann and Arduino (23) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 4 / 9

22 Soil properties Application in numerical modeling Nonlinear effective stress seismic site response analysis Calibration based on data of drained monotonic triaxial tests at three different confining pressures undrained cyclic triaxial tests on two frozen samples at depths of 3.5 and 5.5 m, resulting in plots of G/G max and ξ vs. (ε a) SA G/G max G/G max and Damping for the 5th Cycles sendai ξ (%) Stiffness adjusted based on profile of V s Half amplitude of axial strain, (ϵ a ) SA (%).2 G/G max and Damping for the 5th Cycles 3 sendai G/G max.6 5 ξ (%) Half amplitude of axial strain, (ϵ a ) SA (%) Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 5 / 9

23 6 Results Application in numerical modeling Nonlinear effective stress seismic site response analysis Analysis results for one of the ground motions East-West Motion Results 8 East-West 6 Motion Acceleration Acceleration (m/s 2 ) (m/s 2 ) S a (m/s 2 ) S a (m/s 2 ) Surface Recorded Time (s) 4 5 Figure 2: 5 TS Time2 History Time (s) Figure 2: TS Time History Surface Recorded Surface Input Recorded Surface Input Recorded 5 2 Period, T (s) 2 Figure 3: TS Acceleration Period, T (s) PWP (kpa) PWP (kpa) PWP/σ v PWP/σ v Time (sec) 5 Figure 5: TS Pore Water Pressure Time (sec) Figure 3: TS Acceleration Figure 6: TS Excessive Pore Water Pressure Ratio Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 6 / Figure 5: TS Pore Water Pressure z7 z6 z5 z4 z7 z3 z6 z2 z5 z z4 z z3 z2 z z z7 z6 z5 z4 z7 z3 z6 z2 z5 z z4 z z3 z2 z z Time (sec) Figure 6: TS Excessive Pore Water Pressure Ratio Time (sec) 8

24 Discussion related to the SCEC workshop Challenges for 3D seismic site response?! Moving from D to 3D in regional-scale simulations: Our models have always been 3D 3D is the same in any scale and our scale is that of continuum Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 7 / 9

25 Discussion related to the SCEC workshop Challenges for 3D seismic site response?! Moving from D to 3D in regional-scale simulations: Our models have always been 3D 3D is the same in any scale and our scale is that of continuum he Japanese Geotechnical Society Further works in constitutive modeling Fabric-related strongly anisotropic response 28 (next slide) Constitutive modeling of intermediate soils... Validating the models for multiaxial loading... Kammerer (22) Yamada and Ishihara (983) Figure 4.: Schematic illustration of idealized multi-directional loading conditions imposed in testing program Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 7 / 9

26 Discussion related to the SCEC workshop Challenges for 3D seismic site response?! Moving from D to 3D in regional-scale simulations: Our models have always been 3D 3D is the same in any scale and our scale is that of continuum he Japanese Geotechnical Society Further works in constitutive modeling Fabric-related strongly anisotropic response 28 (next slide) Constitutive modeling of intermediate soils... Validating the models for multiaxial loading... Kammerer (22) Yamada and Ishihara (983) Calibration and simulation Figure State 4.: Schematic parameters illustration of including idealized multi-directional internal loading variables from in-situ testing results?! conditions imposed in testing program Statistical methods to deal with scarce and sparse input parameters?! Professional programming and use of HPC techniques?! Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 7 / 9

Discussion related to the SCEC workshop Fabric-related strongly anisotropic response σ2 σ3 b = σ σ 3 NII-Electronic Lib imental results and model simulations of triaxial compression Yoshimine ( =,

27 Discussion related to the SCEC workshop Fabric-related strongly anisotropic response σ2 σ3 b = σ σ 3 NII-Electronic Lib imental results and model simulations of triaxial compression Yoshimine ( =, b=) et al. and (998) Fig. 7. Experimental extension results ( =9, and b=) model Datasimulations after Yoshimine for different et values of 326 / JOURNAL OF ENGINEERING MECHANICS ASCE / NOVEMBER 2 Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 8 / 9

28 THANK YOU! Acknowledgments: Collaborators: Prof. Pedro Arduino (UW) Students: Mr. Gaziz Seidalinov (UBC) Mr. Graeme McAllister (UBC) Mr. Alborz Ghofrani (UW) Mr. Long Chen (UW)

29 Bibliography I References Dafalias, Y. F. and Manzari, M. T. (24), Simple plasticity sand model accounting for fabric change effects, ASCE Journal of Engineering Mechanics 3(6), Dafalias, Y. F., Manzari, M. T. and Papadimitriou, A. G. (26), SANICLAY: simple anisotropic clay plasticity model, Int l Journal for Numerical and Analytical Methods in Geomechanics 3(2), Dafalias, Y. F., Papadimitriou, A. G. and Li, X. S. (24), Sand plasticity model accounting for inherent fabric anisotropy, Journal of Engineering Mechanics 3(), Gens, A. (982), Stress strain and strength of a low plasticity clay, Ph.D. thesis, Imperial College, London University. 856 pages. Kammerer, A. M. (22), Undrained Response of Monterey /3 Sand Under Multidirectional Cyclic Simple Shear Loading Conditions, PhD thesis, University of California, Berkeley. Li, X. S. and Dafalias, Y. F. (2), Dilatancy for cohesionless soils, Géotechnique 54(4), Li, X. S. and Dafalias, Y. F. (22), Anisotropic critical state theory: role of fabric, Journal of Engineering Mechanics 38(3), Manzari, M. T. and Dafalias, Y. F. (997), A critical state two surface plasticity model for sands, Géotechnique 47(2), McGann, C. and Arduino, P. (23), Effective stress site response analysis of a layered soil column. Layered_Soil_Column. Pradhan, T. B., Tatsuoka, F. and Sato, Y. (989), Experimental stress dilatancy relations of sand subjected to cyclic loading, Soils and Foundations 29(), Seidalinov, G. and Taiebat, M. (24), Bounding surface SANICLAY plasticity model for cyclic clay behavior, International Journal for Numerical and Analytical Methods in Geomechanics 38(7), Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 8 / 9

30 Bibliography II References Sheu, W. (984), Modeling of stress-strain-strength behavior of a clay under cyclic loading, Ph.D. dissertation, Univeristy of Colorado, Boulder, Colorado, USA. Smith, P. R., Jardine, R. J. and Hight, D. W. (992), Yielding of bothkennar clay, Géotechnique 42(2), Taiebat, M. and Dafalias, Y. F. (28), SANISAND: simple anisotropic sand plasticity model, International Journal for Numerical and Analytical Methods in Geomechanics 32(8), Taiebat, M., Dafalias, Y. F. and Peek, R. (2a), A destructuration theory and its application to SANICLAY model, International Journal for Numerical and Analytical Methods in Geomechanics 34(), 9 4. Taiebat, M., Jeremić, B., Dafalias, Y. F., Kaynia, A. M. and Cheng, Z. (2b), Propagation of seismic waves through liquefied soils, Soil Dynamics and Earthquake Engineering 3(4), Verdugo, R. and Ishihara, K. (996), The steady state of sandy soils, Soils and Foundations 36(2), 8 9. Vucetic, M. and Dobry, R. (99), Effect of soil plasticity on cyclic response, Journal of Geotechnical Engineering 7(), Wood, D. M. (974), Some Aspects of the Mechanical Behaviour of Kaolin under Truly Triaxial Conditions of Stress and Strain., PhD thesis, University of Cambridge. Yamada, Y. and Ishihara, K. (983), Undrained deformation characteristics of sand in multi-directional shear, Soils and Foundations 23(), Yasuhara, K., Hirao, K. and Hyde, A. (992), Effects of cyclic loading on undrained strength and compressibility of clay, Soils and Foundations 32(), 6. Yoshimine, M., Ishihara, K. and Vargas, W. (998), Effects of principal stress direction and intermediate principal stress on drained shear behavior of sand, Soils and Foundations 38(3), Zergoun, M. and Vaid, Y. (994), Effective stress response of clay to undrained cyclic loading, Canadian Geotechnical Journal 3(5), Taiebat (UBC) & Dafalias (UCD, NTUA) constitutive & numerical modeling within CS theory SCEC workshop. Los Angeles 9 / 9

Propagation of Seismic Waves through Liquefied Soils

Propagation of Seismic Waves through Liquefied Soils Mahdi Taiebat a,b,, Boris Jeremić b, Yannis F. Dafalias b,c, Amir M. Kaynia a, Zhao Cheng d a Norwegian Geotechnical Institute, P.O. Box 393 Ullevaal