Intro to Soil Mechanics: the what, why & how José E. Andrade, Caltech
Thursday, June 23, 2011 The What?
Thursday, June 23, 2011 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
Thursday, June 23, 2011 sands & gravels clays & silts
Thursday, June 23, 2011 The Why?
Thursday, June 23, 2011 Sandcastles what holds them up?
Thursday, June 23, 2011 Palacio de Bellas Artes Mexico, DF uniform settlement
Thursday, June 23, 2011 The leaning tower of Pisa
differential settlement
Thursday, June 23, 2011 !
Teton dam dam failure
Thursday, June 23, 2011 Niigata earthquake liquefaction
Thursday, June 23, 2011 Katrina New Orleans levee failure
Thursday, June 23, 2011 MER: Big Opportunity xTerramechanics
Thursday, June 23, 2011 MER: Big Opportunity xTerramechanics
Thursday, June 23, 2011 The How?
Thursday, June 23, 2011 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
Thursday, June 23, 2011 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
Thursday, June 23, 2011 solid Index & gradation water+air=voids
Key volumetric ratios Key mass ratio void ratio water content Vv Mw e = [0.4,1] sand w = <1 for most soils V M s [0.3,1.5] clays s >5 for marine, organic Key link mass & volume Vv porosity η = Vt [0,1] ρ = M/V moist, solid, water, dry, etc. Vw saturation S = Vv [0,1] ratios used in practice to characterize soils & properties
Thursday, June 23, 2011 Gradation & classification
Grain size is main classification feature
sands & clays & gravels silts
• can see grains • cannot see grains • mechanics~texture • mechanics~water • d>0.05 mm • d<0.05 mm Soils are currently classified using USCS (Casagrande)
Thursday, June 23, 2011 Fabric in coarsely-grained soils
“loose packing”, high e Vv relative e = “dense packing”, low e Vs
emax greatest possible, loosest packing emin lowest possible, densest packing
emax e ID = e e− relative density max− min strongly affects engineering behavior of soils
Thursday, June 23, 2011 Typical problem(s)xb in Soil Mechanics
• Compact sand fill
• Calculate consolidation of clay
• Calculate rate of consolidation
• Determine strength of sand
SAND FILL • Calculate F.S. on sand (failure?)
• Need: stresses & matl behavior PISA CLAY
ROCK (UNDERFORMABLE, IMPERMEABLE)
Thursday, June 23, 2011 Modeling tools
Thursday, June 23, 2011 Theoretical framework
• continuum mechanics
• constitutive theory js W W • computational inelasticity 0 x
X • nonlinear finite elements jf
f
x2 W0
x1
Thursday, June 23, 2011 Theoretical framework
• continuum mechanics
• constitutive theory balance of mass
• computational inelasticity p˙ φ + ∇·v = −∇ · q Kf • nonlinear finite elements ∇·σ + γ = 0
balance of momentum
Thursday, June 23, 2011 Theoretical framework
• continuum mechanics = h • constitutive theory q k ·∇ darcy ! ep σ˙ = c : "˙ hooke • computational inelasticity k permeability tensor • nonlinear finite elements controls fluid flow ep c mechanical stiffness controls deformation
Thursday, June 23, 2011 Theoretical framework
• continuum mechanics
t3 • constitutive theory tr tn+1 Fn+1 tn+1 F • computational inelasticity n tn
• nonlinear finite elements
t1 t2
Thursday, June 23, 2011 Theoretical framework
• continuum mechanics
• constitutive theory
• computational inelasticity
• nonlinear finite elements
Displacement node Pressure node
Thursday, June 23, 2011 Finite Element Method (FEM)
• Designed to approximately solve PDE’s
• PDE’s model physical phenomena
• Three types of PDE’s:
) & )
'(1 '(1
'(0 '(0
'(/ '(/
'(. '(.
"!#$ '(- '(- ! • Parabolic: fluid flow % '(, '(,
'(+ '(+
'(* '(*
'() '()
' & ' ' '() '(* '(+ '(, '(- '(. '(/ '(0 '(1 ) !!"!#$
• Hyperbolic: wave eqn
• Elliptic: elastostatics
Thursday, June 23, 2011 FEM recipe
Strong from
Weak form Galerkin form Matrix form
Thursday, June 23, 2011 Multi-D deformation with FEM
∇·σ + f = 0 in Ω equilibrium u = g on Γg e.g., clamp σ · n = h on Γh e.g., confinement
Γg
Constitutive relation: Ω given u get σ Γ h e.g., elasticity, plasticity
Thursday, June 23, 2011 Modeling Ingredients
1. Set geometry 2.Discretize domaiin
H 3. Set matl parameters 4.417 4. Set B.C.’s 5. Solve
B
Thursday, June 23, 2011 Modeling Ingredients
1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve
Thursday, June 23, 2011 Modeling Ingredients
1. Set geometry 2. Discretize domain 3. Set matl parameters 4. Set B.C.’s 5. Solve
Thursday, June 23, 2011 Modeling Ingredients !a
1. Set geometry 2. Discretize domain !r 3. Set matl parameters 4. Set B.C.’s 5. Solve
Thursday, June 23, 2011 Modeling Ingredients !a
1. Set geometry 2. Discretize domain !r 3. Set matl parameters 4. Set B.C.’s 5. Solve
Thursday, June 23, 2011 FEM Program
TIME STEP LOOP
ITERATION LOOP
ASSEMBLE FORCE VECTOR AND STIFFNESS MATRIX
ELEMENT LOOP: N=1, NUMEL
GAUSS INTEGRATION LOOP: L=1, NINT constitutive CALL MATERIAL SUBROUTINE model
CONTINUE
CONTINUE
CONTINUE
T = T + !T
Thursday, June 23, 2011 Material behavior: shear strength
• Void ratio or relative density
• Particle shape & size Engineers have developed models to account for most • Grain size distribution of these variables • Particle surface roughness
• Water Elasto-plasticity framework of choice • Intermediate principal stress
• Overconsolidation or pre-stress
Thursday, June 23, 2011 A word on current characterization methods
Direct Shear Triaxial
Pros: cheap, simple, fast, Pros: control drainage & stress good for sands path, principal dir. cnst., Cons: drained, forced failure, more homogeneous non-homogeneous Cons: complex
Thursday, June 23, 2011 Material models for sands should capture
• Nonlinearity and irrecoverable deformations v
• Pressure dependence A
• Difference tensile and p compressive strength Dv
C • Relative density dependence B
• Nonassociative plastic flow
log -p ’
Thursday, June 23, 2011 Material models for sands should capture
• Nonlinearity and irrecoverable deformations
s’ • Pressure dependence 200 vc 191 kPa 241 310 • Difference tensile and 380 compressive strength (kPa) 100
q
• Relative density dependence 0 100 200 300 • Nonassociative plastic flow -p’ (kPa)
Thursday, June 23, 2011 Material models for sands should capture
• Nonlinearity and irrecoverable deformations t’3 Von Mises • Pressure dependence Mohr Coulomb
• Difference tensile and compressive strength
• Relative density dependence t’ t’ • Nonassociative plastic flow 1 2
Loose sand Dense sand
Thursday, June 23, 2011 Material models for sands should capture
• Nonlinearity and irrecoverable deformations 1000
(kPa) 800
| • Pressure dependence 600 sa
ar 400
|s -s • Difference tensile and 200 s compressive strength r 0 0 2 4 6810 ea (%) • Relative density dependence 4 3 • Nonassociative plastic flow 2 Dense Sand 1 Loose Sand
(%) 0
v
e -1
Thursday, June 23, 2011 Material models for sands should capture
• Nonlinearity and irrecoverable deformations Yield Function q Plastic Potential • Pressure dependence Flow vector
• Difference tensile and compressive strength
• Relative density dependence
• Nonassociative plastic flow -p’
Thursday, June 23, 2011 Elasto-plasticity in one slide
Hooke’s law σ˙ = cep : ˙ Additive decomposition of strain ˙ = ˙ e + ˙ p Convex elastic region F (σ, α)=0 Non-associative flow ˙ p = λ˙ g, g := ∂G/∂σ K-T optimality λ˙ F =0 λ˙ H = ∂F/∂α α˙ − · Elastoplastic constitutive tangent 1 cep = ce ce : g f : ce, χ = H g : ce : f − χ ⊗ −
Thursday, June 23, 2011 Examples
Thursday, June 23, 2011 Example of elasto-plastic model
!3 "=1 q "=7/9
N=0 N=0.5 $ # i p' M ! ! 1 2
v CSL CSL (a) (b) v1 Figure 3: Th~ree invariant yield surface on (a) deviatorF ic=planeFat d(iffeσrentv,aluπes of) and (b) l i yi meridian plane at different values of N.
vc y G = G(σ, π¯ ) v2 with i
M [1 + ln (πi/p)] if N = 0 η = (2.4) ln-p’ N/(1 N) M/N 1 (1 N) (p/πHi) − = Hif N(>p0., πi, ψ) -p’ − − -pi The function ζ controls the cross-sectional shape of the yield function on a deviatoric plane Thursday, June 23, 2011 Figure 2: Geometricasrepra fuesenctionntationofof ththe stateLode’sparameangle.ter ψW. e adopt the shape function proposed by Gudehus and where the stored energy is an isotropiArgyc furisnction[18,of19]thebelasticecausstraie ofnitstensormatheme. Thateictotalal simplistresscity i.e., tensor σ is decomposed according to Terzaghi’s well known expression for fully saturated soils (1 + ) + (1 ) cos 3θ i.e., σ = σ pδ, where p is the pore fluid pressure and is negative underζcom(θ)pres= sion, following − (2.5) − 2 continuum mechanics convention. The tensor δ is the second-order identity.
Let us define three independentwhereinvarianastsshoforwnthe ineffeFiguctiverestr3,essthetensorellipσticit, y constant controls the form of the cross-section
going from perfectly circular 3= 1 to convex triangular for = 7/9. The shape function 1 2 1 tr s p = tr σ, q = s , cos 3θ = 3 (2.2) 3 with <31 reflects√a6 classical featurχ e in geomaterials, which exhibit higher strength in triaxial where s = σ p δ is the deviatoriccomcomppresonsion.ent of thFigue effreectiv3 ealsostressreteflenctssor,theand geχ =ome√trtrsical2. interpretation for parameters M and N − The invariant p is the mean normalwhiceffectivh goe vstresernstheandslopis asseuofmedthtoe CbeSLnegatiandveththre coughurvaturout. e of the yield surface on a meridian plane, Further, θ is the so-called Lode’s angle whose values range from 0 θ π/3; it defines an respectively. ≤ ≤ angle emanating from a tension corner on a deviatoric plane (see Figure 3). Central to the formulation is the additive decomposition of the strain rate tensor into The elastic region in effective stress space is contained by the yield surface which is a function of the three stress invariants introduced above, 7
F σ, πi = F p, q, θ, πi = ζ (θ) q + pη p, πi (2.3)
6 2.3. CALIBRATIONS AND PREDICTIONS 23
model validation:Figure 2.4 drained: Triax itxcal c andomp psression calibrations, (a) and (c), and plane strain compression predictions, (b) and (d), versus experimental data for loose and dense Brasted Sand
Thursday, June 23,s p2011ecimens. undrained txc loose sands
Thursday, June 23, 2011 true triaxial b=constant
Thursday, June 23, 2011 H H¯ − L
Plane-strain liquefaction numerical simulation
Thursday, June 23, 2011 H H¯ − L
Plane-strain liquefaction numerical simulation
Thursday, June 23, 2011
250 200 150 100
50 (a) Pore Pressure (in kPa) .0025 .0020 .0015 .0010 .0005 Field scale prediction .0000 (b) Deviatoric Strain ELASTIC Levee failure
600000 (recall Katrina) 400000
200000
0 H H − L
Thursday, June 23, 2011 References
Thursday, June 23, 2011