LECTURE 2 EXTANT THEORIES OF CLAYS AND SANDSprusv/ncmm/workshops/wog/... · 2007. 10. 3. · de...
Transcript of LECTURE 2 EXTANT THEORIES OF CLAYS AND SANDSprusv/ncmm/workshops/wog/... · 2007. 10. 3. · de...
LECTURE 2
EXTANT THEORIES OF CLAYS AND SANDSEXTANT THEORIES OF CLAYS AND SANDS
Ian F CollinsDepartment of Engineering Science, University of Auckland,
New Zealand.
AUCKLAND BY NIGHT
““COULOMB” MODELSCOULOMB” MODELS
TWO-DIMENSIONAL COULOMB MODEL
τ
σ
φc
dilation angle
WHAT IS COHESION?WHAT IS COHESION?
ONE SOURCE of COHESION
r/T2)uu(u ae =−−≡
Gauge water suction at which air enters
For sand r=1mm suction is 1.4kPa
For clays r=1 micron, suction is 140kPa
CLAY CLIFFS
RICE PADDIES
0
20
40
60
80
100
20 40 60 80 100
normal effective stress
shearstress
Quartz
Kaolinite
Illite
Montmorillonite
(psi)
(psi)
Effective failure envelopes for clays (Olson (1974))
PROBLEMS WITH COULOMB• WHAT IS THE COHESION?
• WHAT IS THE FRICTION ANGLE?
• THESE ARE NOT MATERIAL • CONSTANTS!
• SLIPLINE AND LIMIT ANALYSIS PROCEDURES REQUIRE NORMAL FLOW RULE-TOO MUCH DILATION!
τ
σc φ+π 2/
φ
FAILURE IN PLANE STRAIN-1
1σ
2/4/ φ−π
FAILURE CONDITION ACHIEVED ON THESE LINES
φσ+=τ tanc
STRESS CHARACTERISTICSARE NOT ORTHOGONAL
SOKOLOVSI “SOLUTION” FOR FOOTING
FAILURE IN PLANE STRAIN-2
1σ
STRESS CHARACTERISTICS
φσ+=τ tanc
STRESS CHARACTERISTICSARE NOT ORTHOGONAL.
IF MATERIAL IS INCOMPRESSIBLE,VELOCITY CHARACTERISTICS ARE ORTHOGONAL
VELOCITYCHARACTERISTICS
DOUBLE SHEARING MODELSDOUBLE SHEARING MODELS
MANDEL, SPENCER, DE JOSSELIN MANDEL, SPENCER, DE JOSSELIN DE JONG, MEHRABADI, COWIN, DE JONG, MEHRABADI, COWIN,
HARRISHARRIS
ATTEMPTS TO OVERCOME THE NON- COINCIDENCE OFSTRESS AND VELOCITY CHARACTERISTICS PROBLEM
SPENCER’S MODEL
(a) Material is isotropic, perfectly rigid plastic(b) In plane strain, deformation consists of
superposition of two shears on stress characteristics.
(c) Principal axes of stress and plastic strain rate no longer coincide.
(d) Constitutive law is relation between plastic strain rate,stress and stress rate-”hypoplastic”.
NON-ASSOCIATED NON-ASSOCIATED PLASTICITYPLASTICITY
BASIC EQUATIONS-1
Ph
)ˆ:Q(Dand,F
P:eF
h:ModulusHardening
,ˆ:)PQh1:C(DDD:equationsRate
PD:RuleFlow
F
F
Q......G
G
P;0)e,(GPotentialPlastic
0D:eFˆ:F0)e,(F:ConditionYield:Plasticity
D:Mˆ,ˆ:CD:Elasticity
PP
PE
P
P
PP
P
EE
σ=
σ∂∂
∂∂−
=
σ+=+=
λ=σ∂
∂σ∂
∂
≡
σ∂∂
σ∂∂
≡=σ
=∂∂+σ
σ∂∂⇒=σ
=σσ=
BASIC EQUATIONS - 2
"ulusmodsofteningcritical"isP:E:Qhwhere
hhP:E)D:E:Q(D:Eˆ:EquationsRateInverse
0
0
=
+−=σ
“QUASI-THERMODYNAMIC” RESTRICTIONS-1
0)P:ˆ(0h/)P:ˆ)(ˆ:Q(
0D:ˆ:"POSTULATE"S'DRUCKER
signsamehavehand)ˆ:Q(
0P:POTENTIALSHAPEDSTAR
0h/)P:)(ˆ:Q(D::"NDISSIPATIO"
p
p
≥σ⇒≥σσ⇒
≥σ
σ⇒
>σ⇒
≥σσ≡σ⇒
“QUASI-THERMODYNAMIC” RESTRICTIONS-2
NO+VE-VE
OK-VE-VE
NO-VE+VE
OK+VE+VE
Q:σ̂ P:σ̂ DRUCKER
P
Q
1
23
4
DRUCKER’S POSTULATE HAS NO RELEVANCE TO NON-ASSOCIATED PLASTICITY
“HILL’S STABILITY” POSTULATE
"DEFINITEPOSITIVEisM........."....................
:UNIQUENESSFOR
CONDITIONSUFFICIENTWORKVIRTUALAPPLY
.solutionspossibletwobetweendifferenceiswhere
,0D:M:Dˆ:DD:MˆALSO
0D:M:D:POSTULATES'HILL
hh)E:Q)(P:E(EMwhere,D:Mˆ:EQUATIONRATE
0D:ˆ:POSTULATES'HILL
O
⇒
∆
≥∆∆=σ∆∆⇒∆=σ∆⇒
≥⇒
+−≡=σ
≥σ
THE RANIECKI FORMULA SUFFICIENT FORMULA FOR UNIQUENESS:
0hQP
)}Q:E:P()Q:E:Q()P:E:P{(21hh
C
21
21
C
=⇒=
−=>
strain
stress
P=Q
P ≠ Q
LOCALIZATION
)}P:E:n()n:E:n)(n:E:Q{(hh 10
−=+
strain
stress
Loss of Uniqueness
n
0)nMndet()Tdet( =≡ (RICE)
Possible Localization
CRITICAL STATE MODELSCRITICAL STATE MODELS
A BASIC CRITICAL STATE MODEL
VOIDS RATIO-PRESSURE PLOT
e
Ln (p)
O
DENSE
LOOSE
CSL
NB A “DENSE’ SPECIMEN CAN BECOME “LOOSE” IF p IS INCREASED
ISOTROPIC COMPRESSION
e
Ln(p)
Elastic loading
Elastic unloading
Plastic loading
q
pO
CSLEg CONSTANT PRESSURE DRAINED PATHS
LOOSE-COMPACTS
DENSE-DILATES
BOTH LOOSE AND DENSE SPECIMENS END UP ON CSL
CHARACTERISTICS REVISITED
Normal flow rule CSL
A
AT POINT “A” THERE ARE TWO CONSTRAINTS, AND HENCE TWO FAMILIES OF STRESS CHARACTERISTICS
p
q
O
FAILURE MECHANISMS
q
p
CSL
Failure by plastic compression
“Peak” failure or Shear Band formation
Tension cut off
OHvorslev Surface
STATE BOUNDARY SURFACE
THREE DIMENSIONAL MODELSLADE
THREE DIMENSIONAL MODELS
ttanconsII
SURFACEYIELDDUNCANLADE3
31 =−
ttanconsIII
SURFACEYIELDNAKAIMATSUOKA3
21 =−
EXTENSIONS OF CRITICAL EXTENSIONS OF CRITICAL STATE MODELSSTATE MODELS
STATE PARAMETER MODELSBEEN&JEFFERIES, DAFALIAS,..
eeCSL −=ψ
CSL
ln(p)
e
dense
loose
)Iln(
ppI
p
cslp =
MULTI-SURFACE MODELSHASHIGUCHI, DAFALIAS,WOOD,….
p
q“BUBBLE”
“BOUNDING SURFACE”
O
ACHIEVEMENTS OF CSSM• Coulomb condition is
a failure condition, not a yield condition.
• Provides an explanation of non-coincident characteristics.
• Models both loose and dense behaviour.
• Provides a relatively simple elastic/plastic model.
• Provides design criteria
• Is in good agreement with experiments on “lightly” over-consolidated clays.
SHORTCOMINGS OF CSSM• Does not well model
behaviour of sands and heavily over-consolidated clays.
• Assumes isotropy.• Is limited to normal
flow rules.• Assumes a critical
state actually exists
• Makes no allowance for the “internal structure” of soil.
• The underlying thermomechanics is wrong!!
EXTANT THERMOMECHANICAL ARGUMENT(Schofield and Wroth)
• “Plastic work equation”
• Rewrite as a “Dilatancy relation”
• ODE for potential function
• Integrate to give potential and yield locus
assumes a normal flow rule.
pppv depMqdepde γγ =+
Mdede
pq
p
pv =+γ
Mdpdq
pq =−
)p/pln(Mpq c=
(ORIGINAL CAM CLAY)
{Frictional dissipation}
Two examples of extant procedure
⇒+=Φ γ
2p22pv eMepˆ
q
p
q
pO
q
p
q
pO
⇒=Φ γpeMpˆ
Schofield and Wroth“Original Cam Clay”
Roscoe and Burland“Modified Cam Clay”
CRITIQUE OF EXTANT PROCEDURE-1
• Original Cam Clay violates the Second law of Thermodynamics.
• Confuses plastic work with dissipation.• Fails to recognize that the dissipation function
automatically gives the yield condition.• Has no memory of prior consolidation.• No attempt to model granular nature of
material.
pppv depMqdepde γγ =+
MODERN THERMOMECHANICSMODERN THERMOMECHANICS
In next lecture I will show that the In next lecture I will show that the modern procedures of the theory of modern procedures of the theory of thermomechanics of dissipative materials thermomechanics of dissipative materials overcomes many of these problems.overcomes many of these problems.
CSSM will be re-evaluated and aspects of CSSM will be re-evaluated and aspects of the microstructure such as micro-level the microstructure such as micro-level inhomogeneity, and induced dilatancy and inhomogeneity, and induced dilatancy and anisotropy can be modelledanisotropy can be modelled
ANY QUESTIONS?
NEW ZEALAND – A GOOD PLACE TO STUDY SAND BEHAVIOUR