Lecture 4 - Cam Clay

7/23/2019 Lecture 4 - Cam Clay

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Critical-State Soil Models

Department of Civil and Envir. Eng., Rensselaer Polytechnic Institute, Troy, NY

April, 2014


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Critical-State Soil Mechanics

Developed by Roscoeand others at Cambridge inthe fifties

Some Useful Literature:

Schofield, A., & Wroth, P. (1968). Critical state soil mechanics.

Wood, D. (1990). Soil behaviour and critical state soil mechanics.Cambridge University Press.

Helwany, S. (2007).Applied soil mechanics with ABAQUS applications.John Wiley & Sons.

Is an effective stress framework for describing soilresponse


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Critical-State Soil Mechanics

Critical State models tie together:

1. Behavior of confining pressure (p)2. Behavior of deviatoric stress (q)

3. Behavior of volume changes (e)

Introduce concept of critical state of soils:

State at which shear strains occur without furtherchanges in confining stress, deviatoric stress orvolume.


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Critical-State Models Some Critical State Soil Models:

Granta Gravel

Cam Clay Modified Cam Clay


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Prevosts Multi-surface Modified Cam-Clay

Prevosts Multi-surface Modified Cam-Clay

Kinematic hardening Isotropic hardening

Multiple yield surfaces Yield surface and Critical Statesurface

Yield surfaces not capped Capped yield surfaces

Yield surface size independent of e Dependency on e

Ignores OCR Accounts for OCR effect


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Kinematic Hardening

xz

xyxy

xz

xy


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Isotropic Hardening

b

a


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Isotropic Hardening


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Normal Consolidation Line and

UnloadingReloading Lines


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Critical-State

Critical state: State at

which shear stress

application leads tono volume change as

the soil is continually

sheared.


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Critical-State

p (kPa)

CSL M


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Critical-State Line

Elastic Yielding


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Critical-State Line in e-p-q space

3D Figure


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Parameters for formulation

or


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Parameters for formulation


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Inside the field surface (Elastic state)

Elastic but

NOT linear


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Soil Plasticity

Flow Rule: Describes the direction of the

vector of plastic strain increment

Hardening Law: Relates the magnitude of a

plastic strain to the magnitude of an

increment of stress as the stress traverses the

yield surface and the material hardens


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Yield Surface

Yield Surface: Separates states of stress which cause only elastic

strains from states of stress that cause plastic (and elastic) strains

Preconsolidation Pressure


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Plastic Strains

Definitions:

des:Deviatoric Elastic strain increment

dp

s:Deviatoric Plastic strain incrementds= d

es+ dps

dev:Volumetric Elastic strain incrementdpv: Volumetric Plastic strain increment

dv= dev+ d

pv

0


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Volumetric Strains

Derived from isotropicconsolidation test:

also:

Differentiating:

Elastic Volumetric

Strains Plastic VolumetricStrains

Total Volumetric Strains


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Deviatoric Plastic Strains Cam Clay Model assumes an associatedflow rule

controlled by the shape of the yield surface:

Stress increment has same direction as strain increment

These increments are perpendicular to the yield surface

(normality rule)


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Deviatoric Plastic Strains

Yield surface:

Using:

Differentiating (dq/dp), we find normalof the yield surface (-dp/dq):

2 2

2

M


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Deviatoric Plastic Strains

Normal of theyield surface

2 2

2PS Vd d

M

/P P

S S Vd d d dp dq


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Plastic Strains (General)

d


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Summary

Volumetric Strains:

Deviatoric Strains:


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Critical-State Soil Models

Department of Civil and Envir. Eng., Rensselaer Polytechnic Institute, Troy, NY

April, 2014


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Summary

p

(kPa)

CSL

M


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Summary


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Yield Surface

Yield Surface: Separates states of stress which cause only elastic

strains from states of stress that cause plastic (and elastic) strains

Preconsolidation Pressure


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Summary

Volumetric Strains:

Deviatoric Strains:

G l F l i


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General FormulationBorja, R. I., & Lee, S. R. (1990). Cam-clay plasticity, part I:

implicit integration of elasto-plastic constitutive relations.

Computer Methods in Applied Mechanics and

Engineering, 78(1), 49-72

G l F l i


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General Formulation


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Hardening Behavior


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Hardening Behaviorq

P

e

Critical State

Line

State Boundary

Surface

NormalConsolidation

Line


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Hardening Behaviorq

P

e

Critical State

Line

State Boundary

Surface

NormalConsolidation

Line


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Softening Behavior

S f i B h i


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Softening Behavior

q

P

e

Critical State

Line

State Boundary

Surface

NormalConsolidation

Line


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Softening Behaviorq

P

e

Critical State

Line

State Boundary

Surface

NormalConsolidation

Line

NCL

CSL

e

P

d d l


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Extended Cam Clay

d d C Cl


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Extended Cam Clay


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Effect of shape of Yield Function

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Lecture 4 - Cam Clay