intro to soil mechanics -...
TRANSCRIPT
![Page 1: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/1.jpg)
Intro to Soil Mechanics:the what, why & how
José E. Andrade, Caltech
Thursday, June 23, 2011
![Page 2: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/2.jpg)
The What?
Thursday, June 23, 2011
![Page 3: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/3.jpg)
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
![Page 4: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/4.jpg)
sands & gravels clays & silts
Thursday, June 23, 2011
![Page 5: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/5.jpg)
The Why?
Thursday, June 23, 2011
![Page 6: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/6.jpg)
Sandcastles what holds them up?
Thursday, June 23, 2011
![Page 7: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/7.jpg)
Palacio de Bellas ArtesMexico, DF
uniform settlement
Thursday, June 23, 2011
![Page 8: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/8.jpg)
The leaning tower of Pisa
differential settlement
Thursday, June 23, 2011
![Page 9: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/9.jpg)
!
Teton dam dam failure
Thursday, June 23, 2011
![Page 10: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/10.jpg)
Niigata earthquake liquefaction
Thursday, June 23, 2011
![Page 11: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/11.jpg)
KatrinaNew Orleans
levee failure
Thursday, June 23, 2011
![Page 12: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/12.jpg)
MER: Big Opportunity xTerramechanics
Thursday, June 23, 2011
![Page 13: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/13.jpg)
MER: Big Opportunity xTerramechanics
Thursday, June 23, 2011
![Page 14: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/14.jpg)
The How?
Thursday, June 23, 2011
![Page 15: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/15.jpg)
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
![Page 16: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/16.jpg)
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)
gas (air)
Each phase has volume and mass
Mechanical behavior governed by phase interactionThursday, June 23, 2011
![Page 17: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/17.jpg)
Index & gradation
Key volumetric ratios
e =Vv
Vs
void ratio [0.4,1] sand
[0.3,1.5] clays
η =Vv
Vt
porosity[0,1]
S =Vw
Vv
saturation[0,1]
Key mass ratio
w =Mw
Ms
water content<1 for most soils
>5 for marine, organic
Key link mass & volumeρ = M/V
moist, solid, water, dry, etc.
ratios used in practice to characterize soils & properties
solid
water+air=voids
Thursday, June 23, 2011
![Page 18: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/18.jpg)
Gradation & classification
sands &gravels
Grain size is main classification feature
• can see grains• mechanics~texture• d>0.05 mm
clays &silts
• cannot see grains• mechanics~water• d<0.05 mm
Soils are currently classified using USCS (Casagrande)Thursday, June 23, 2011
![Page 19: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/19.jpg)
Fabric in coarsely-grained soils
“loose packing”, high
“dense packing”, low
e
erelative
ID = emax−eemax−emin
emax greatest possible, loosest packingemin lowest possible, densest packing
relative density
strongly affects engineering behavior of soils
e =Vv
Vs
Thursday, June 23, 2011
![Page 20: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/20.jpg)
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?)
• Need: stresses & matl behavior
SAND FILL
PISA CLAY
ROCK (UNDERFORMABLE, IMPERMEABLE)
Thursday, June 23, 2011
![Page 21: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/21.jpg)
Modeling tools
Thursday, June 23, 2011
![Page 22: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/22.jpg)
!s
"#
"
x1
x2
X
x
!f
"#
f
Theoretical framework
• continuum mechanics
• constitutive theory
• computational inelasticity
• nonlinear finite elements
Thursday, June 23, 2011
![Page 23: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/23.jpg)
Theoretical framework
• continuum mechanics
• constitutive theory
• computational inelasticity
• nonlinear finite elements
!p
Kf+ ! · v = "! · q
! · ! + " = 0
balance of mass
balance of momentum
Thursday, June 23, 2011
![Page 24: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/24.jpg)
Theoretical framework
• continuum mechanics
• constitutive theory
• computational inelasticity
• nonlinear finite elements
q = k ·!h
!!= cep
: "
darcy
hooke
k permeability tensor
cep
mechanical stiffness
controls fluid flow
controls deformation
Thursday, June 23, 2011
![Page 25: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/25.jpg)
!"
!#
!$
!n
!n+1
!n+1
tr
Fn+1
Fn
Theoretical framework
• continuum mechanics
• constitutive theory
• computational inelasticity
• nonlinear finite elements
Thursday, June 23, 2011
![Page 26: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/26.jpg)
Pressure nodeDisplacement node
Theoretical framework
• continuum mechanics
• constitutive theory
• computational inelasticity
• nonlinear finite elements
Thursday, June 23, 2011
![Page 27: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/27.jpg)
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
!!"!#$
%!"!#$
&
&
' '() '(* '(+ '(, '(- '(. '(/ '(0 '(1 )'
'()
'(*
'(+
'(,
'(-
'(.
'(/
'(0
'(1
)
'
'()
'(*
'(+
'(,
'(-
'(.
'(/
'(0
'(1
)
Thursday, June 23, 2011
![Page 28: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/28.jpg)
FEM recipe
Strong from
Weak form
Galerkin form
Matrix form
Thursday, June 23, 2011
![Page 29: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/29.jpg)
Multi-D deformation with FEM
!g
!h
!
! · ! + f = 0 in !
u = g on "g
! · n = h on "h
equilibriume.g., clampe.g., confinement
Constitutive relation:
ugiven get !
e.g., elasticity, plasticity
Thursday, June 23, 2011
![Page 30: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/30.jpg)
1. Set geometry2.Discretize domaiin3. Set matl parameters4. Set B.C.’s5. Solve
4.417
B
H
Modeling Ingredients
Thursday, June 23, 2011
![Page 31: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/31.jpg)
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
Modeling Ingredients
Thursday, June 23, 2011
![Page 32: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/32.jpg)
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
Modeling Ingredients
Thursday, June 23, 2011
![Page 33: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/33.jpg)
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
!a
!r
Modeling Ingredients
Thursday, June 23, 2011
![Page 34: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/34.jpg)
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
1. Set geometry2. Discretize domain3. Set matl parameters4. Set B.C.’s5. Solve
!a
!r
Modeling Ingredients
Thursday, June 23, 2011
![Page 35: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/35.jpg)
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
CONTINUE
CONTINUE
CONTINUE
T = T + !T
FEM Program
constitutivemodel
Thursday, June 23, 2011
![Page 36: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/36.jpg)
Material behavior: shear strength
• Void ratio or relative density
• Particle shape & size
• Grain size distribution
• Particle surface roughness
• Water
• Intermediate principal stress
• Overconsolidation or pre-stress
Engineers have developed models to account for most
of these variables
Elasto-plasticity framework of choice
Thursday, June 23, 2011
![Page 37: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/37.jpg)
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 homogeneousCons: complex
Thursday, June 23, 2011
![Page 38: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/38.jpg)
log - ’p
v
!vp
A
B
C
Material models for sands should capture
• Nonlinearity and irrecoverable deformations
• Pressure dependence
• Difference tensile and compressive strength
• Relative density dependence
• Nonassociative plastic flow
Thursday, June 23, 2011
![Page 39: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/39.jpg)
0 100 200 300
100
200
-p’ (kPa)q
(kPa)
!’vc
191 kPa
241
310380
Material models for sands should capture
• Nonlinearity and irrecoverable deformations
• Pressure dependence
• Difference tensile and compressive strength
• Relative density dependence
• Nonassociative plastic flow
Thursday, June 23, 2011
![Page 40: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/40.jpg)
!’"
Von Mises
Mohr Coulomb
!’#
!’$
Loose sand Dense sand
Material models for sands should capture
• Nonlinearity and irrecoverable deformations
• Pressure dependence
• Difference tensile and compressive strength
• Relative density dependence
• Nonassociative plastic flow
Thursday, June 23, 2011
![Page 41: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/41.jpg)
0 2 4 6 8 10
200
0
400
600
800
1000
1
2
3
4
0
-1
!r
!a
"!#$
!a
r(k
Pa)
"
%a (%)
% v(%
)Dense SandLoose Sand
Material models for sands should capture
• Nonlinearity and irrecoverable deformations
• Pressure dependence
• Difference tensile and compressive strength
• Relative density dependence
• Nonassociative plastic flow
Thursday, June 23, 2011
![Page 42: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/42.jpg)
-p’
q
Yield Function
Plastic Potential
Flow vector
Material models for sands should capture
• Nonlinearity and irrecoverable deformations
• Pressure dependence
• Difference tensile and compressive strength
• Relative density dependence
• Nonassociative plastic flow
Thursday, June 23, 2011
![Page 43: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/43.jpg)
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 = λg, g := ∂G/∂σ
K-T optimality λF = 0 λH = −∂F/∂α · α
Elastoplastic constitutive tangent
cep = ce − 1χ
ce : g ⊗ f : ce, χ = H − g : ce : f
Thursday, June 23, 2011
![Page 44: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/44.jpg)
Examples
Thursday, June 23, 2011
![Page 45: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/45.jpg)
F = F (σ�, πi)G = G(σ�, πi)H = H(p�
, πi, ψ)
v
v1
vc
v2
-!i
-p’
ln-p’
"~
#
#i
CSL
Figure 2: Geometric representation of the state parameter ψ.
where the stored energy is an isotropic function of the elastic strain tensor �e. The total stress
tensor σ is decomposed according to Terzaghi’s well known expression for fully saturated soils
i.e., σ = σ�−pδ, where p is the pore fluid pressure and is negative under compression, following
continuum mechanics convention. The tensor δ is the second-order identity.
Let us define three independent invariants for the effective stress tensor σ�,
p� =13
trσ�, q =�
23�s��, 1√
6cos 3θ =
tr s�3
χ3(2.2)
where s� = σ�−p�δ is the deviatoric component of the effective stress tensor, and χ =√
tr s�2.
The invariant p� is the mean normal effective stress and is assumed to be negative throughout.
Further, θ is the so-called Lode’s angle whose values range from 0 ≤ θ ≤ π/3; it defines an
angle emanating from a tension corner on a deviatoric plane (see Figure 3).
The elastic region in effective stress space is contained by the yield surface which is a
function of the three stress invariants introduced above,
F�σ�,πi
�= F
�p�, q, θ, πi
�= ζ (θ) q + p�η
�p�,πi
�(2.3)
6
Example of elasto-plastic model
q
p'
(a) (b)
M
!3
!2!
1
"=1
"=7/9
# i
N=0 N=0.5
CSL
$
Figure 3: Three invariant yield surface on (a) deviatoric plane at different values of � and (b)meridian plane at different values of N .
with
η =
M [1 + ln (πi/p�)] if N = 0
M/N�1− (1−N) (p�/πi)N/(1−N)
�if N > 0.
(2.4)
The function ζ controls the cross-sectional shape of the yield function on a deviatoric plane
as a function of the Lode’s angle. We adopt the shape function proposed by Gudehus and
Argyris [18, 19] because of its mathematical simplicity i.e.,
ζ (θ) =(1 + �) + (1− �) cos 3θ
2�(2.5)
where as shown in Figure 3, the ellipticity constant � controls the form of the cross-section
going from perfectly circular � = 1 to convex triangular for � = 7/9. The shape function
with � < 1 reflects a classical feature in geomaterials, which exhibit higher strength in triaxial
compression. Figure 3 also reflects the geometrical interpretation for parameters M and N
which govern the slope of the CSL and the curvature of the yield surface on a meridian plane,
respectively.
Central to the formulation is the additive decomposition of the strain rate tensor into
7
Thursday, June 23, 2011
![Page 46: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/46.jpg)
2.3. CALIBRATIONS AND PREDICTIONS 23
Figure 2.4: Triaxial compression calibrations, (a) and (c), and plane strain compressionpredictions, (b) and (d), versus experimental data for loose and dense Brasted Sandspecimens.
model validation: drained txc and ps
Thursday, June 23, 2011
![Page 47: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/47.jpg)
undrained txc loose sands
Thursday, June 23, 2011
![Page 48: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/48.jpg)
true triaxial b=constant
Thursday, June 23, 2011
![Page 49: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/49.jpg)
H − HL
Plane-strain liquefaction numerical simulationThursday, June 23, 2011
![Page 50: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/50.jpg)
H − HL
Plane-strain liquefaction numerical simulationThursday, June 23, 2011
![Page 51: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/51.jpg)
50
100
150
200
250
.0000
.0005
.0010
.0015
.0020
.0025
0
200000
400000
600000
(b) Deviatoric Strain
(a) Pore Pressure (in kPa)
ELASTIC
H −HL
Field scale predictionLevee failure
(recall Katrina)
Thursday, June 23, 2011
![Page 52: Intro to Soil Mechanics - kiss.caltech.edukiss.caltech.edu/workshops/xTerra/presentations1/andrade_sc.pdf · What is Soil Mechanics? erdbaumechanik The application of the laws of](https://reader033.vdocument.in/reader033/viewer/2022051406/5abb44607f8b9af27d8c92de/html5/thumbnails/52.jpg)
References
Thursday, June 23, 2011