NUMERICAL MODELING OF INSTABILITIES IN SAND

Transcription

1 NUMERICAL MODELING OF INSTABILITIES IN SAND KIRK ELLISON March 14, 2008 Advisor: Jose Andrade Masters Defense

2 Outline of Presentation Randomized porosity in FEM simulations Liquefaction in FEM simulations

3 INTRODUCTION

4 OUTLINE INTRODUCTION What is an Instability? Simplest example: BUCKLING of a column The column may bow outward in any direction At this point, an instability has developed because the constitutive relation has bifurcated into multiple feasible solutions Eventually, the column will collapse due to the resulting eccentricity

5 INTRODUCTION Types of Instabilities GEOMETRIC instabilities develop when bifurcation arises due to the shape of the material (e.g. buckling) MATERIAL instabilities develop when bifurcation arises due to the properties of the material itself (e.g. deformation banding)

6 INTRODUCTION Types of Instabilities GEOMETRIC instabilities develop when bifurcation arises due to the shape of the material (e.g. buckling) MATERIAL instabilities develop when bifurcation arises due to the properties of the material itself (e.g. deformation banding) Geometric instabilities are extremely rare in geomaterials; therefore, we will only consider material instabilities in this presentation

7 INTRODUCTION Material Instabilities Hill 1962 proposes that a material is unstable if a small perturbation to the body grows and stable if it does not. LOCALIZED instabilities are associated with the growth of a localized region of deformation Example in sands: SHEAR BANDING DIFFUSE instabilities are NOT necessarily associated with a localized region of deformation Example in sands: LIQUEFACTION

8 INTRODUCTION Instabilities in Sand SHEAR BANDING LIQUEFACTION n after Alshibli et al., 2003 after Kramer, 1996 Results in a loss of shear Results in a localized region strength as rapid loading occurs of excess deformations faster than water can drain away

9 INTRODUCTION Modeling Instabilities Across Scales after Andrade et al., 2008

10 THE CONSTITUTIVE MODEL

11 THE CONSTITUTIVE MODEL Overview of the Constitutive Framework The constitutive model was originally developed by Andrade and Borja, 2006 based upon the classic Nor-Sand model Important features of the original Nor-Sand model include: Critical state plasticity The state parameter (i.e. ) ψ = ν ν c New features of the Andrade and Borja, 2006 model include: Hyperelasticity A three invariant yield surface

12 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Three-invariant yield surface! 3 "=1 "=7/9 q $ N=0 N=0.5 # i p'! 1! 2 M CSL Deviatoric Plane Meridian Plane Three-dimensional View

13 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Three-invariant yield surface - invariants! 3 Deviatoric Stress Lode Angle "=1 "=7/9 q $ N=0 N=0.5 # i p'! 1 Deviatoric Plane! 2 CSL M Meridian Plane Mean Normal Effective Stress

14 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Three-invariant yield surface - shape parameters! 3 "=1 "=7/9 q $ N=0 N=0.5 # i p'! 1! 2 M Shape Parameters CSL Deviatoric Plane Meridian Plane

15 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Three-invariant yield surface - image pressure! 3 "=1 "=7/9 Image Pressure q $ N=0 N=0.5 # i p'! 1! 2 M CSL Deviatoric Plane Meridian Plane

16 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model State parameter v CSL v 1 i ~ v c v 2 - i -p ln-p

17 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model State parameter v CSL Critical State Line Image State Parameter v 1 v c i ~ Compression Gradient State Parameter v 2 - i -p ln-p

18 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model State parameter - significance The state parameter influences the rate of change of the size of the yield surface because: D = ( ɛ p v ) = αψ i ɛ p s Solve for the limiting image pressure π i And the hardening law is: π i = h(π i π i ) ɛ p s

19 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model State parameter - significance The state parameter influences the rate of change of the size of the yield surface because: ( ɛ p v Rates of Change of Volumetric and Shear Plastic Strain Components D = Limiting Dilatancy ɛ p s And the hardening law is: ) = αψ i 3.5 (Jefferies, 1993) Solve for the limiting image pressure π i π i = h(π i π i ) ɛ p s Hardening Parameter

20 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Input parameters Symbol Parameter Theory µ 0 Shear modulus Elasticity v c0 Reverence specific volume CSSM ˆκ Gradient of swelling CSSM ˆλ Compression gradient CSSM M CSL slope on meridian plane CSSM N Curvature of yield surface Plasticity N Curvature of plastic potential Plasticity ρ Ellipticity of yield surface Plasticity ρ Ellipticity of plastic potential Plasticity p c Preconsolidation pressure Plasticity α 0 Coupling coefficient Plasticity h Hardening constant Plasticity

21 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Input parameters - assuming associativity and decoupling Symbol Parameter Theory µ 0 Shear modulus Elasticity v c0 Reverence specific volume CSSM ˆκ Gradient of swelling CSSM ˆλ Compression gradient CSSM M CSL slope on meridian plane CSSM N Curvature of yield surface Plasticity N Curvature of plastic potential Plasticity ρ Ellipticity of yield surface Plasticity ρ Ellipticity of plastic potential Plasticity p c Preconsolidation pressure Plasticity α 0 Coupling coefficient Plasticity h Hardening constant Plasticity

22 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Input parameters - assuming associativity and decoupling Easily found from laboratory experiments Symbol Parameter Theory µ 0 Shear modulus Elasticity v c0 Reverence specific volume CSSM ˆκ Gradient of swelling CSSM ˆλ Compression gradient CSSM M CSL slope on meridian plane CSSM N Curvature of yield surface Plasticity N Curvature of plastic potential Plasticity ρ Ellipticity of yield surface Plasticity ρ Ellipticity of plastic potential Plasticity p c Preconsolidation pressure Plasticity α 0 Coupling coefficient Plasticity h Hardening constant Plasticity

23 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Input parameters - assuming associativity and decoupling Can easily be determined from correlations with the friction angle Symbol Parameter Theory µ 0 Shear modulus Elasticity v c0 Reverence specific volume CSSM ˆκ Gradient of swelling CSSM ˆλ Compression gradient CSSM M CSL slope on meridian plane CSSM N Curvature of yield surface Plasticity N Curvature of plastic potential Plasticity ρ Ellipticity of yield surface Plasticity ρ Ellipticity of plastic potential Plasticity p c Preconsolidation pressure Plasticity α 0 Coupling coefficient Plasticity h Hardening constant Plasticity

24 THE CONSTITUTIVE MODEL Main Ingredients of the Constitutive Model Input parameters - assuming associativity and decoupling Must be calibrated to capture the soil the response Symbol Parameter Theory µ 0 Shear modulus Elasticity v c0 Reverence specific volume CSSM ˆκ Gradient of swelling CSSM ˆλ Compression gradient CSSM M CSL slope on meridian plane CSSM N Curvature of yield surface Plasticity N Curvature of plastic potential Plasticity ρ Ellipticity of yield surface Plasticity ρ Ellipticity of plastic potential Plasticity p c Preconsolidation pressure Plasticity α 0 Coupling coefficient Plasticity h Hardening constant Plasticity

25 THE CONSTITUTIVE MODEL Evaluation of the Constitutive Model Three simulations of laboratory tests were performed to evaluate the constitutive model under homogeneous conditions 1. Calibration of true triaxial tests on dense sand PURPOSE: to evaluate the robustness of the model 2. Calibration/prediction of TXC/PS tests on loose and dense sand PURPOSE: to evaluate the predictiveness of the model 3. Calibration/prediction of static liquefaction for undrained TXC PURPOSE: to demonstrate the robustness and predictiveness of the the model for undrained conditions

26 THE CONSTITUTIVE MODEL Evaluation of the Constitutive Model True triaxial testing of dense Monterey No. 0 Sand Results generally show close agreement between the laboratory results and the model calibrations BUT The model has a little trouble capturing both TXC and TXE simultaneously

27 THE CONSTITUTIVE MODEL Evaluation of the Constitutive Model True triaxial testing of loose and dense Brasted Sand Only the hardening parameter, h, was allowed to vary between the loose and dense tests The calibration under TXC fits the data very well and the predictions under PS are also reasonable

28 THE CONSTITUTIVE MODEL Evaluation of the Constitutive Model Undrained TXC testing of very loose Hostun Sand Thus, the model is also able to replicate STATIC LIQUEFACTION

29 RANDOMIZED POROSITY IN FEM

30 RANDOMIZED POROSITY IN FEM Motivation X-ray computed tomography image of porosity distribution Meso-scale porosity distribution in an FEM simulation

31 RANDOMIZED POROSITY IN FEM Overview of the FEM Testing Program 150 drained, plane strain compression simulations were performed on 5x10 cm specimens using 20 x 40 quadrilateral isoparametric elements Simulations account for combinations of: Anisotropy Ratios- 1, 10, 100 Anisotropy Orientations- 0, 30, 45, 60, 90

32 RANDOMIZED POROSITY IN FEM Overview of the FEM Testing Program 150 drained, plane strain compression simulations were performed on 5x10 cm specimens using 20 x 40 quadrilateral isoparametric elements Simulations account for combinations of: Anisotropy Ratios- 1, 10, 100 Anisotropy Orientations- 0, 30, 45, 60, 90 PURPOSE: 1. to assess the sensitivity of the PS compression test to variations in void ratio heterogeneities 2. to provide a general framework coupling elastoplasticity and geostatistical tools to assess the effects of potential heterogeneities

33 RANDOMIZED POROSITY IN FEM Distribution of the Porosity Values Semivariogram for an isotropic simulation Histogram of porosity distribution for a single simulation

34 OUTLINE RANDOMIZED POROSITY IN FEM Shear Banding with Randomized Porosity Example: 60 anisotropy angle, anisotropy ratio=10 Shear Band Initial specific volume distribution Final deviatoric strain distribution

35 RANDOMIZED POROSITY IN FEM Range of Results Failure is defined as the onset of localization (i.e. det A = 0 )

36 LIQUEFACTION MODELING IN FEM

37 LIQUEFACTION MODELING IN FEM Formulation of the Liquefaction Criterion Hill s Instability Criterion [[ σ]] : [[ ɛ]] = 0

38 LIQUEFACTION MODELING IN FEM Formulation of the Liquefaction Criterion Hill s Instability Criterion [[ σ]] : [[ ɛ]] = 0 + Terzaghi s Effective Stress Equation σ = σ + p1

39 LIQUEFACTION MODELING IN FEM Formulation of the Liquefaction Criterion Hill s Instability Criterion [[ σ]] : [[ ɛ]] = 0 + Terzaghi s Effective Stress Equation σ = σ + p1 = [[ σ]] : [[ ɛ]] = [[ ɛ]] : c ep : [[ ɛ]] [[ṗ]]1 : [[ ɛ]] = 0

40 LIQUEFACTION MODELING IN FEM Formulation of the Liquefaction Criterion Hill s Instability Criterion [[ σ]] : [[ ɛ]] = 0 + Terzaghi s Effective Stress Equation σ = σ + p1 = [[ σ]] : [[ ɛ]] = [[ ɛ]] : c ep : [[ ɛ]] [[ṗ]]1 : [[ ɛ]] = 0 + Assumptions Incompressible fluids and solids Small strains Two invariant plasticity

41 LIQUEFACTION MODELING IN FEM Hill s Instability Criterion [[ σ]] : [[ ɛ]] = 0 Formulation of the Liquefaction Criterion + Terzaghi s Effective Stress Equation σ = σ + p1 = [[ σ]] : [[ ɛ]] = [[ ɛ]] : c ep : [[ ɛ]] [[ṗ]]1 : [[ ɛ]] = 0 + =... Assumptions Incompressible fluids and solids Small strains Two invariant plasticity

42 LIQUEFACTION MODELING IN FEM Formulation of the Liquefaction Criterion LIQUEFACTION CRITERION H H crit = 0 where H crit = K F Q p p H = 1 λ F π i π i Hardening Modulus 0 x 104!2!4 Onset of Liquefaction!6!8!10 Hardening Modulus Critical Hardening Modulus! Time Step Example of the evolution of H and Hcrit at a Gauss point

43 LIQUEFACTION MODELING IN FEM Formulation of the Liquefaction Criterion LIQUEFACTION CRITERION H H crit = 0 where H crit = K F Q p p H = 1 λ Hardening Modulus F π i π i Plastic Strain Rate Critical Hardening Modulus Hardening Modulus!2!4!6!8 0 x 104 Onset of Liquefaction!10 Hardening Modulus Critical Hardening Modulus! Time Step Example of the evolution of H and Hcrit at a Gauss point

44 LIQUEFACTION MODELING IN FEM Important Caveat For the simulations that will follow, one of the assumptions used in the above formulation is not valid! The constitutive model has THREE INVARIANTS (not two)

45 LIQUEFACTION MODELING IN FEM Important Caveat For the simulations that will follow, one of the assumptions used in the above formulation is not valid! The constitutive model has THREE INVARIANTS (not two) However, comparisons with a more general criterion developed by Borja, 2006 show that both methods are nearly identical: c ep c ep c ep det L = 0 where L = c ep 21 c ep 22 c ep 23 1 c ep c ep c ep }{{}

46 LIQUEFACTION MODELING IN FEM Evolution of the Liquefaction Criterion Example: Undrained plane strain compression test H-Hcrit

47 LIQUEFACTION MODELING IN FEM Evolution of the Liquefaction Criterion Example: Undrained plane strain compression test H-Hcrit

48 LIQUEFACTION MODELING IN FEM Undrained Plane Strain Compression Simulation σ LIQUEFACTION CRITERION Deviatoric Strains Pore Pressures

49 LIQUEFACTION MODELING IN FEM Undrained Plane Strain Compression Simulation Liquefaction Criterion DEVIATORIC STRAINS Pore Pressures

50 LIQUEFACTION MODELING IN FEM Undrained Plane Strain Compression Simulation Liquefaction Criterion Deviatoric Strains PORE PRESSURES

51 LIQUEFACTION MODELING IN FEM Increasing Pore Pressure Gradient Simulation

52 LIQUEFACTION MODELING IN FEM Submarine Slope Simulation Dimensions and loading conditions Simulation results at the onset of liquefaction

53 LIQUEFACTION MODELING IN FEM Loading Rate Effects Effect of loading rate on drained PS compression tests The loading rate can be normalized in terms of displacement rate and hydraulic conductivity: Z = u y k But, Z only applies when the loading is unidirectional

54 LIQUEFACTION MODELING IN FEM Mesh Size Effects Effect of mesh size on undrained PS compression tests

55 CONCLUSIONS

56 Liquefaction Modeling in FEM CONCLUSIONS Main Conclusions The Andrade and Borja, 2006 constitutive model was shown to make reasonably accurate and robust predictions of sand behavior under monotonic loading conditions Minute variations in porosity at the meso-scale can account for a relatively wide range of compressive strengths due to its heavy influence on shear band formation The onset and location of liquefaction can be tracked in finite element analyses using recently developed criteria A unified constitutive framework can distinguish between different modes of instabilities

57 ACKNOWLEDGMENTS...