Intro to Soil Mechanics: the what, why & how José E. Andrade, Caltech
The What?
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
sands & gravels clays & silts
The Why?
Sandcastles what holds them up?
Palacio de Bellas Artes Mexico, DF uniform settlement
The leaning tower of Pisa differential settlement
! Teton dam dam failure
Niigata earthquake liquefaction
Katrina New Orleans levee failure
MER: Big Opportunity xterramechanics
MER: Big Opportunity xterramechanics
The How?
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
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
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
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)
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
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)
Modeling tools
Theoretical framework continuum mechanics constitutive theory computational inelasticity s x nonlinear finite elements X f x 2 f x 1
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
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
Theoretical framework continuum mechanics constitutive theory computational inelasticity F n F n+1 tr n+1 n+1 n nonlinear finite elements
Theoretical framework continuum mechanics constitutive theory computational inelasticity nonlinear finite elements Displacement node Pressure node
) '(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
FEM recipe Strong from Weak form Galerkin form Matrix form
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
Modeling Ingredients 1. Set geometry 2.Discretize domaiin 3. Set matl parameters H 4.417 4. Set B.C. s 5. Solve B
Modeling Ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve
Modeling Ingredients 1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C. s 5. Solve
Modeling Ingredients! a 1. Set geometry 2. Discretize domain! r 3. Set matl parameters 4. Set B.C. s 5. Solve
Modeling Ingredients! a 1. Set geometry 2. Discretize domain! r 3. Set matl parameters 4. Set B.C. s 5. Solve
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
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
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
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
Material models for sands should capture Nonlinearity and irrecoverable deformations Pressure dependence Difference tensile and compressive strength q (kpa) 200 100 vc 191 kpa 241 310 380 Relative density dependence Nonassociative plastic flow 0 100 200 300 -p (kpa)
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
Material models for sands should capture Nonlinearity and irrecoverable deformations Pressure dependence Difference tensile and compressive strength a r (kpa) 1000 800 600 400 200 a r 0 0 2 4 6 8 10 a (%) Relative density dependence 4 3 Nonassociative plastic flow v (%) 2 1 0-1 Dense Sand Loose Sand
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
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
Examples
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)
model validation: drained txc and ps
undrained txc loose sands
true triaxial b=constant
H H L Plane-strain liquefaction numerical simulation
H H L Plane-strain liquefaction numerical simulation
(a) Pore Pressure (in kpa) (b) Deviatoric Strain H H L 250 200 150 100 50.0025.0020.0015.0010.0005.0000 ELASTIC 600000 400000 200000 0 Field scale prediction Levee failure (recall Katrina)
References