Intro to Soil Mechanics: the what, why & how. José E. Andrade, Caltech

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1 Intro to Soil Mechanics: the what, why & how José E. Andrade, Caltech

2 The What?

3 What is Soil Mechanics? erdbaumechanik The application of the laws of mechanics (physics) to soils as engineering materials Karl von Terzaghi is credited as the father of erdbaumechanik

4 sands & gravels clays & silts

5 The Why?

6 Sandcastles what holds them up?

7 Palacio de Bellas Artes Mexico, DF uniform settlement

8 The leaning tower of Pisa differential settlement

9 ! Teton dam dam failure

10 Niigata earthquake liquefaction

11 Katrina New Orleans levee failure

12 MER: Big Opportunity xterramechanics

13 MER: Big Opportunity xterramechanics

14 The How?

15 Topics in classic Soil Mechanics Index & gradation Soil classification Compaction Permeability, seepage, and effective stresses Consolidation and rate of consolidation Strength of soils: sands and clays

16 Index & gradation Definition: soil mass is a collection of particles and voids in between (voids can be filled w/ fluids or air) solid particle fluid (water) Each phase has volume and mass gas (air) Mechanical behavior governed by phase interaction

17 Index & gradation solid water+air=voids Key volumetric ratios Key mass ratio e = V v V s void ratio [0.4,1] sand [0.3,1.5] clays w = M w M s water content <1 for most soils >5 for marine, organic η = V v V t S = V w V v porosity [0,1] saturation [0,1] Key link mass & volume ρ = M/V moist, solid, water, dry, etc. ratios used in practice to characterize soils & properties

18 Gradation & classification Grain size is main classification feature sands & gravels clays & silts can see grains mechanics~texture d>0.05 mm cannot see grains mechanics~water d<0.05 mm Soils are currently classified using USCS (Casagrande)

19 Fabric in coarsely-grained soils loose packing, high dense packing, low e e relative e = V v V s e max e min greatest possible, loosest packing lowest possible, densest packing I D = e max e e max e min relative density strongly affects engineering behavior of soils

20 Typical problem(s)xb in Soil Mechanics Compact sand fill Calculate consolidation of clay Calculate rate of consolidation Determine strength of sand Calculate F.S. on sand (failure?) SAND FILL Need: stresses & matl behavior PISA CLAY ROCK (UNDERFORMABLE, IMPERMEABLE)

21 Modeling tools

22 Theoretical framework continuum mechanics constitutive theory computational inelasticity s x nonlinear finite elements X f x 2 f x 1

23 Theoretical framework continuum mechanics constitutive theory computational inelasticity nonlinear finite elements balance of mass φ ṗ K f + v = q σ + γ = 0 balance of momentum

24 Theoretical framework continuum mechanics constitutive theory computational inelasticity q = k h σ = c ep : ɛ k permeability tensor darcy hooke nonlinear finite elements controls fluid flow c ep mechanical stiffness controls deformation

25 Theoretical framework continuum mechanics constitutive theory computational inelasticity F n F n+1 tr n+1 n+1 n nonlinear finite elements

26 Theoretical framework continuum mechanics constitutive theory computational inelasticity nonlinear finite elements Displacement node Pressure node

27 ) '(1 '(0 '(/ '(. '(, '(+ '(* '() ' & ' '() '(* '(+ '(, '(- '(. '(/ '(0 '(1 ) & ) '(1 '(0 '(/ '(. '(- '(- '(, '(+ '(* '() ' Finite Element Method (FEM) Designed to approximately solve PDE s PDE s model physical phenomena Three types of PDE s: Parabolic: fluid flow %!"!#$!!"!#$ Hyperbolic: wave eqn Elliptic: elastostatics

28 FEM recipe Strong from Weak form Galerkin form Matrix form

29 Multi-D deformation with FEM σ + f = 0 in Ω u = g on Γ g σ n = h on Γ h equilibrium e.g., clamp e.g., confinement Γ g Ω Constitutive relation: given u get σ Γ h e.g., elasticity, plasticity

30 Modeling Ingredients 1. Set geometry 2.Discretize domaiin 3. Set matl parameters H Set B.C. s 5. Solve B

31 Modeling Ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve

32 Modeling Ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve

33 Modeling Ingredients! a 1. Set geometry 2. Discretize domain! r 3. Set matl parameters 4. Set B.C. s 5. Solve

34 Modeling Ingredients! a 1. Set geometry 2. Discretize domain! r 3. Set matl parameters 4. Set B.C. s 5. Solve

35 FEM Program TIME STEP LOOP ITERATION LOOP ASSEMBLE FORCE VECTOR AND STIFFNESS MATRIX ELEMENT LOOP: N=1, NUMEL GAUSS INTEGRATION LOOP: L=1, NINT CALL MATERIAL SUBROUTINE constitutive model CONTINUE CONTINUE CONTINUE T = T +!T

36 Material behavior: shear strength Void ratio or relative density Particle shape & size Grain size distribution Engineers have developed models to account for most of these variables Particle surface roughness Water Intermediate principal stress Elasto-plasticity framework of choice Overconsolidation or pre-stress

drained, forced failure, non-homogeneous Pros: control

37 A word on current characterization methods Direct Shear Triaxial Pros: cheap, simple, fast, good for sands Cons: drained, forced failure, non-homogeneous Pros: control drainage & stress path, principal dir. cnst., more homogeneous Cons: complex

38 Material models for sands should capture Nonlinearity and irrecoverable deformations v Pressure dependence A Difference tensile and compressive strength v p Relative density dependence Nonassociative plastic flow C B log - p

39 Material models for sands should capture Nonlinearity and irrecoverable deformations Pressure dependence Difference tensile and compressive strength q (kpa) vc 191 kpa Relative density dependence Nonassociative plastic flow p (kpa)

40 Material models for sands should capture Nonlinearity and irrecoverable deformations Pressure dependence Mohr Coulomb Von Mises Difference tensile and compressive strength Relative density dependence Nonassociative plastic flow Loose sand Dense sand

41 Material models for sands should capture Nonlinearity and irrecoverable deformations Pressure dependence Difference tensile and compressive strength a r (kpa) a r a (%) Relative density dependence 4 3 Nonassociative plastic flow v (%) Dense Sand Loose Sand

42 Material models for sands should capture Nonlinearity and irrecoverable deformations Pressure dependence q Yield Function Plastic Potential Flow vector Difference tensile and compressive strength Relative density dependence Nonassociative plastic flow -p

43 Elasto-plasticity in one slide Hooke s law σ = c ep : Additive decomposition of strain = e + p Convex elastic region F (σ, α) =0 Non-associative flow p = λg, g := G/ σ K-T optimality λf =0 λh = F/ α α Elastoplastic constitutive tangent c ep = c e 1 χ ce : g f : c e, χ = H g : c e : f

44 Examples

45 Example of elasto-plastic model! 3 "=1 "=7/9 q N=0 N=0.5 $ # i p'! 1! 2 M v CSL CSL v 1 i ~ F = F (σ, π i ) v c v 2 G = G(σ, π i ) - i -p ln-p H = H(p, π i, ψ)

46 model validation: drained txc and ps

47 undrained txc loose sands

48 true triaxial b=constant

49 H H L Plane-strain liquefaction numerical simulation

50 H H L Plane-strain liquefaction numerical simulation

(a) Pore Pressure (in kpa) (b) Deviatoric Strain H H L 250 200 150 100 50.0025.0020.0015.0010.

51 (a) Pore Pressure (in kpa) (b) Deviatoric Strain H H L ELASTIC Field scale prediction Levee failure (recall Katrina)

52 References

MODELING GEOMATERIALS ACROSS SCALES JOSÉ E. ANDRADE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING EPS SEMINAR SERIES MARCH 2008

MODELING GEOMATERIALS ACROSS SCALES JOSÉ E. ANDRADE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING EPS SEMINAR SERIES MARCH 2008 COLLABORATORS: DR XUXIN TU AND MR KIRK ELLISON THE ROADMAP MOTIVATION