Geotechnical modelling and critical state soil mechanicsNaples, May 2007

David Muir Wood

University of Bristol

13. Designer models: addition of extra features

(GM 2, 3, EM, GeoF)

Designer models: addition of extra features

1.Kinematic yielding2. Cam clay

3. Mohr-Coulomb

shear stress

mean stress

elastic - stiff

plastic – less stiff

shear stress

shear strain

classical elastic-plastic modelling of soil

for example, Cam clay (1963, 1968)

stress

yield?

strain

typical actual response

void ratio

vertical stress(log scale)

preconsolidation pressure

classical identification of yield from stress:strain response

geometrical construction for estimation of preconsolidation pressure

-150

-100

-50

0

50

100

150

200

0 100 200 300 400

Cam clay?

p': kPa

q: kPa

Fig 3: Anisotropic yield locus for one-dimensional stress history (after Al-Tabbaa, 1984)

Cam clay providing inspiration:

search for ‘Cam clay like’ yield loci

eg kaolin (Al-Tabbaa, 1984)

yield loci for natural clays

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

p/svc

q/svc

Rang de Fleuve

Belfast

Winnipeg

St Alban

Lyndhurst

Mastemyr

collected by Graham et al (1988)

typical experimental observation: stiffness falls steadily with monotonic straining:

is there an elastic region?

shear stiffness degradation data for Quiou sand from resonant column and torsional shear tests (after LoPresti et al, 1997)

limit of elastic response??

how do we objectively identify yielding?occurrence of irrecoverable strain?

dissipation of energy in loading/unloading cycles?change in slope of stress:strain response?

stress

strain

a.

stress

strain

b.

-20

-10

0

10

20

30

40

50

60

70

0 10 20 30 40 50 60

Y1 yield locus

Y2 yield locus

Y3 yield locus

q: kPa

p': kPa

yielding of Bothkennar clay:

boundaries deduced from inspection of stress:strain response

Y1 approximately centred on in situ stress state

Y3 reflects natural structure – damaged by any irrecoverable strain - evanescent

after Smith et al (1992)

kaolin revisited: isotropic consolidation histories

-50

0

50

100

150

200

0 50 100 150 200 250 300 350 400 450

mean effective stress p': kPa

deviator stress q: kPa

a.

data from Al-Tabbaa (1987)

kaolin revisited: one-dimensional consolidation histories

data from Al-Tabbaa (1987)

-50

0

50

100

150

200

0 50 100 150 200 250 300 350 400

mean effective stress p': kPa

deviator stress q: kPa b.

q kPa

q kPa

q kPa

q kPa

p' kPa p' kPa

p' kPap' kPa

plastic strain increments: approximate normality to kinematic yield loci

kaolin: Al-Tabbaa, 1987

Deviatoric stress response envelopes

qz:kPa

qx:kPaA

sx

sy

sz

A270

C-A300

C-A330 C-A0 C-A30

C-A60

C-A90

C-A120

C-A150C-A180C-A210

C-A240

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250 A

sz

sy

sx

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250

•distortional stress probe rosettes•constant mean stress•cross anisotropy?•Ev > Eh

εd = 0.05, 0.2, 0.4, 0.6, 0.8, 1, 1.2%

σx σy

σz

σxσy

σz

A: isotropic compression

Hostun sand

Sadek, 2006

ABC30

qz:kPa

qx:kPaA

sx

sy

sz

ABC60

ABC90

ABC120 ABC150

ABC180

ABC210

ABC240

ABC270ABC300

ABC300

B

ABC330

C

ABC360

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250

Stress paths in the octahedral plane for Rosette AB(CCA: p' = 200kPa, Hostun Sand: D r = 65%)

AB30

sz

sy

sx

Aqx:kPa

qz:kPa

AB60AB90

AB120

AB150

AB180

AB210

AB240

AB270

B

AB300AB330

AB360

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250

distortional probing

constant mean stress

non-monotonic stress paths

stress probe rosettes

ABC … probe

AB … probe

σz

σx

σx

σy

σy

σz

Sadek, 2006

A

sz

sy

qx: kPa

sx

B

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250A

sz

sy

sx

qx: kPa

BC

-250

-150

-50

50

150

250

-250 -150 -50 50 150 250

distortional strain

0.05%: history recalled

1%: history ‘forgotten’

radial shearing ABtwo corners

ABC

Stress response envelopes: Hostun sand: small-medium strain

εd = 0.05, 0.2, 0.4, 0.6, 0.8, 1, 1.2%

Sadek, 2006

sz

sx

sy

qx: kPaA

BC

a

bc

-150

-50

50

150

-150 -50 50 150

comparison of 0.05% strain response envelopes for histories A, AB, ABC

stress response envelopes

small/medium strain stiffness

kinematic hardening

centre as indicator of current fabric

but strain too large

Sadek, 2006

Designer models: addition of extra features 1. Kinematic yielding

2. Cam clay3. Mohr-Coulomb

Cam clay

elastic-hardening plastic model

volumetric hardening

associated flow – normality

Cam clay

response in drained triaxial compression tests with constant p'

asymptotic approach to critical state

effect of overconsolidation ratio

sharp division between elastic and plastic response

compare response of soil on nonmonotonic loading

with capability of single yield surface model

extension to simple models using kinematic hardeningand bounding surface plasticity

compare response of soil on nonmonotonic loading with capability of single yield surface model

elastic-hardening plastic model expects elastic behaviour on reversal, sudden drop in stiffness at yield

soils typically show hysteretic behaviour on unload-reload cycles, steady change in incremental stiffness

kinematic hardening extension

yield locus carried around with stress state – 'bubble' – strongly influenced by recent history

stiffness falls as yield 'bubble' approaches bounding surface – controlled by distance b

when loading with 'bubble' in contact with bounding surface model is identical to Cam clay

assume relative size R of 'bubble'

assume rule for translation of 'bubble'

assume interpolation rule linking plastic stiffness with b

…otherwise identical to Cam clay

kaolin

constant p' cycles

hysteresis

build up of volumetric strainexperiment simulation

volumetric strain

distortional strain

η

η

η

η

migration of 'bubble' during constant p' unloading after one-dimensional normal compression

hardening of 'bubble' and bounding surface

q kPa

p' kPa

experiment simulation

constant q cycles after one-dimensional normal compression

add further effects in a similarly hierarchical way

cementation and structure: extension to 'bubble' model

natural soils often contain structure: bonding between particles: destroyed with mechanical or chemical damage

design model in which yield surface has increased size as a result of the bonding

with plastic straining (or chemical weathering) the yield surface gradually shrinks to the yield surface, for remoulded, structureless material

extension of 'bubble' kinematic extension of Cam clay

all features of 'bubble' model retained

ratio of sizes of structure surface and reference surface gives indication of current degree of structure

add measure of structure or bonding: single scalar parameter r

'bounding' surface now called 'structure' surface: size r times larger than a reference surface

structure lost whenever plastic strains occur

damage law:

damage plastic strain increment δεdp combines plastic volumetric

and plastic distortional strain increments:

– additional parameter to control their relative importance

structure progressively disappears:

r 1 as plastic deformation increases

pd1r

kr

logical: structureless soil is one which has been so mechanically pummelled that it has no remaining bonds between particles

particular forms of laboratory testing (triaxial testing, for example) may not be able to provide sufficient damage

evolution law and definition of damage strain may need to include some more subtle reference to the nature of the strain path

shearing with rotation of principal axes is likely to be especially damaging

feasible to introduce other evolution laws which relate change (increase or decrease) of scalar measure of structure r to chemical environment or time or temperature effects

Cam clay can be regained by setting r = 1, R = 1

hierarchical extension of 'bubble' model to include effects of structure

other evolution laws: relate change (increase or decrease) of scalar measure of structure r to chemical environment or time or temperature effects

Cam clay can be regained by setting r = 1, R = 1

Norrköping clay – calibration tests

Rouainia & Muir Wood (2000)

Norrköping clay – parametric variation

Rouainia & Muir Wood (2000)

Norrköping clay – undrained – isotropic consolidation

Rouainia & Muir Wood (2000)

Norrköping clay – undrained – anisotropic consolidation

Rouainia & Muir Wood (2000)

Norrköping clay – undrained – isotropic overconsolidation

Rouainia & Muir Wood (2000)

simulation experimentBothkennar clay

results normalised by Hvorslev equivalent consolidation pressure p'e for structureless soil

Gajo & Muir Wood, 2001

Hierarchical extensions of Cam clay

•it is relatively straightforward to add extra features to a soil model

•advantage in using well known model as basis – check implementation – acceptability

•extra features imply additional soil parameters and additional calibration tests

•seek adequate complexity in modelling – match complexity of model to availability of data and needs of application

Designer models: addition of extra features 1. Kinematic yielding

2. Cam clay

3. Mohr-Coulomb

standard elastic-perfectly plastic Mohr-Coulomb model

non-associated plastic flow

simplicity

sharp stiffness changes

tangent stiffness either elastic or zero

continuing volume change

standard elastic-perfectly plastic Mohr-Coulomb model

available in all numerical analysis programs

subjectivity in selecting values of soil parameters – stiffness, strength, dilatancy

elastic-hardening plastic Mohr-Coulomb model

non-associated flow steady fall in stiffness continuing volume change

is post-peak softening important?

design a model to include softening

Mohr-Coulomb family

post-peak softening to critical state

three regimes of response

adaptation of hardening Mohr-Coulomb model

for η < ηp response is elastic: η < ηp, ηy = ηp δεqp = 0

after peak, linear fall in yield stress ratio with strain

0 < εqp < b: (distortional 'hardening' law)

eventual perfectly plastic critical state: εqp b, ηy = M

non-associated flow rule as before

bM

pq

p

yp

conventional triaxial drained compression test

elastic

post-peak softening

critical state

triaxial undrained compression test

p

ppp

p

i

M12

M211

bK

'p'p

effective stress path

elastic (isotropic: δp' = 0)

post-peak softening

critical state

limited model

concentration on single aspect of response

Mohr-Coulomb model with strength dependent on state variable

Severn-Trent sand

influence of density

softening

dilatancy

simplicity

build on Mohr-Coulomb model

describe journey from initial elastic response to ultimate critical state

include nonlinearity, peak strength and softening

simplicity?

adequate complexity?

Severn-Trent sand

Been & Jefferies

state parameter ψ = volume distance from critical state line

function of density and stress level

more useful than void ratio alone – indicating effect of density and stress 'dense'

'loose'

ψ

critical state line

mean stress

specific volume

Severn-Trent sand: strength

what is peak strength?

data confirm link between strength and state parameter ψ

Mohr-Coulomb model with current strength dependent on current state parameter

Been & Jefferies

peak strength

Severn-Trent sand: strength

'dense'

'loose'

ψ

mean stress

critical state line

specific volume

state parameter ψ

what is peak strength?

•property of the soil which changes with stress level, density

data confirm link between strength and state parameter ψ

Mohr-Coulomb model with current strength dependent on current state parameter

Been & Jefferies

peak strength

Severn-Trent sand: strength

'dense'

'loose'

ψ

mean stress

specific volume

state parameter ψ

Severn-Trent sand: dilatancy

Benahmed

dilatancy: volume change during shearing

'dense' sand expands

'loose' sand contracts

dilatancy depends on density

dilatancy varies during test

what do we mean by 'dense' and 'loose'?

volume strain

shear strain

data confirm link between dilatancy and state parameter ψ

if soil is not at critical state when it is being sheared (ψ 0):

then volume changes occur towards the critical state: dilatancy

'loose': ψ > 0: contraction

'dense': ψ < 0: dilation

'loose''dense'

state parameter ψ

Been & Jefferies

dilatancy

'dense'

'loose'

ψ

critical state line

mean stress

specific volume

Severn-Trent sand: dilatancy

mobilised strength 'mob

shear strain

currently mobilised strengthcurrently available strength

monotonic relationship

1ratio

available strength ':varies with state parameter

distortional hardening

monotonic increase of ratio of mobilised to available strength (η/ηp) with distortional strain εq

p

hyperbolic hardening law: simple

Mohr-Coulomb model with strength dependent on state variable

volume change accompanies shearing

hence change in state variable

hence change in available strength

model automatically homes in on critical state

softening emerges without being described mathematically

peak strength is moving target reached at infinite distortional strain – then identical with critical state strength

conventional drained triaxial compression tests

different initial density (state variable)

current peak strength

characterisation of variation of tangent stiffness

soil response perfectly plastic model

nonlinearity and reversed plasticity observed when direction of loading is reversed

elastic-hardening plastic model: behaviour purely elastic for stress ratios lower than the previous maximum stress ratio

Severn-Trent sand

add kinematic hardening:

elastic region of high stiffness carried round with recent stress history

boundary of elastic region is the yield surface

use bounding surface plasticity:

plastic hardening stiffness depends on separation of the yield surface and bounding surface

kinematic hardening Mohr-Coulomb: strength dependent on state variable: hierarchical development

Severn-Trent sand

calibrated against triaxial test data for Hostun sand

effect of different density/stress level automatically described

ignore practical problem of maintaining homogeneity within softening sample

Gajo & Muir Wood, 1999

use model to simulate cyclic undrained loading leading to eventual liquefaction

model fails after 25 cycles

actual soil (Hostun sand) fails after 89 cycles

number of cycles to liquefaction is not a particularly reliable parameter to use for model calibration

obvious significant difference between samples which liquefy in one or two cycles and those which survive for many cycles

character of cyclic pore pressure build-up reproduced in model

messages:

•possible to develop elegant models which reproduce desirable mechanical characteristics

•especially effects of density and stress level

•mathematical complexity not essential

•build up from well known model – Mohr-Coulomb