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