Lecture 8 Critical State Soil Mechanics.pdf

Geo-E2010 Advanced Soil Mechanics L Wojciech Sołowski 14 March 2018

Plasticity refresh

Theory of Plasticity - Summary 1.

Elastic Strain

2.

Yield surface

3.

Plastic Potential

4.

Flow rule

dε ep  1 K ′ 0   dp ′   e =    dq ′  ′ d ε 0 1 3 G    q 

f ( p′, q, p0′ ) = 0 g ( p′, q, ζ ) = 0 ∂g ∂g = d ε pp d= ; d ε qp d λ λ ∂p′ ∂q

Theory of Plasticity - Summary 5.

Hardening law

p0′ = p0′ (ε pp , ε qp ) dp0′ =

6.

∂p0′ ∂p0′ p + ε d dε qp p p p ∂ε p ∂ε q

Plastic deformations  ∂f ∂g  ∂p′ ∂p′ dε pp  −1  ∂f ∂g  p = d ε     ′ ′ ∂f  ∂p0 ∂g ∂p0 ∂g    q   ∂ε p ∂p′ + ∂ε p ∂q   ∂p′ ∂q ′ p ∂  0  p q 

7.

Total deformations

dε p  =  d ε q  

∂f ∂g  ∂q ∂p′   dp′ ∂f ∂g   dq   ∂q ∂q 

 d ε ep   d ε pp   e+ p  d ε q   d ε q 

Critical State Soil Mechanics

Critical state soil mechanics To learn / refresh : - what is critical state - how soil behaves when over-consolidated / dense and when normally consolidated / loose - volumetric behaviour of soil - oedometer, drained triaxial tests - semi logarithmic scale… - how we can approximate the elastic behaviour of soil in the semi-logarithmic plot (p’ – specific volume) - how we can approximate the elasto-plastic behaviour of soil in the semi-logarithmic plot (p’ – specific volume)

Department of Civil Engineering Advanced Soil Mechanics. W. Sołowski 6

Plastic Behaviour of Soils Mohr-Coulomb in Principal Stress Space σ’3

σ’1=σ’2= σ’3

σ’2

σ’1

. / Mohr – Coulomb failure surface is a irregular hexagon in the principal stress space

Salient Features and Drawbacks of MC Model . / It is assumed that the material behaves elastically until the failure surface is reached. . / In reality, plastic deformation begins well before the failure conditions are met. p' H

+

E B p0 A

C

G

D

F

εv

Oedometer / consolidation

ν = ν1 − κ ⋅ ln p′

(c) Muir Wood

Drucker et al. (1957) model

. / Drucker et al. (1957) proposed the introduction of a yield surface bounded by a “cap” which is allowed to expand and increase in size with increasing stress level. . / The behaviour of the soil is purely elastic only inside the failure surface σ3 and the cap.

. / If the stress increase is directed outwards the cap, plastic deformations are generated and the cap expands.

σ2 σ1

Cam clay models In Cambridge University, several researchers tried to improve the Drucker model to describe the behaviour of reconstituted Kaolin clay, on which a large library of axisymmetric tests was available for normally consolidated and overconsolidated specimens. In particular, Schofield and Wroth proposed the model which goes by the name (Original) Cam Clay model A real revolution in the soil modelling field. Soon afterwards Burland (1967) & Roscoe & Burland

(1968) presented the: Modified Cam Clay model which we will discuss in detail

Formulation of Modified Cam Clay model I n t he stress space, a yield function f behaviour is elastic when f < 0.

is defined so that the soil

The surface described by f is the so-called yield locus and is formulated in function of the stress tensor and scalar values called hardening parameters. These scalar values are related to the size of the yield surface in the stress space, as it will be clarified. Moreover, inside the yield locus the soil behaviour is described by the exponential law:

ν = ν1 − κ ⋅ ln p′ where ν

is the specific volume, expressed in terms of void ratio by: κ is the slope of the recompression ν ≡ 1+ e line in ν-lnp' plane

Formulation of Modified Cam Clay model: elasticity

ν = ν1 − κ ⋅ ln p′

(c) Muir Wood

Formulation of Modified Cam Clay model: elasticity ν ≡ 1+ e

ν = ν1 − κ ⋅ ln p′

(c) Muir Wood

Void ratio and volumetric strains are correlated by:

dε v = −

de dν =− 1+ e ν

κ is the slope of the unloading – reloading (url) line in ν-lnp' plane Other elastic parameter : Poisson ratio or shear modulus G

Clay behaviour

Mitchell & Soga 2005

Department of Civil Engineering Advanced Soil Mechanics . W. Sołowski 15

Clay behaviour

Glacial till

Mitchell & Soga 2005

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Modified Cam Clay – volumetric behaviour

Starting point of Cam-Clay theory: Logarithmic isotropic compression and unloading ln p’

pc

Primary loading:

e − e0 = λ ln p pc

1

λ

v=1+e Unloading / reloading:

e − e0 = κ ln p pc

κ

iso-NCL 1

Behaviour of clays under triaxial states Isotropic Loading

Modified Cam Clay

For other than isotropic stress paths from the origin there is another (parallel) NCL

q

failure

ln p’ 1

K0

λ

iso p’

iso-NCL K0-NCL failure-NCL

Clay behaviour

Glacial till

Mitchell & Soga 2005

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Behaviour of clays under triaxial states Shearing (drained) Volumetric deformations during shearing (dilatancy):

• Initial elastic contraction

∆v

• Followed by an elastoplastic behaviour: NC soils or slightly OC (OCR ≈ 1 to 4), a volumetric contraction is observed OC soils (OCR > 4), a volumetric expansion is observed

ε1

Clay behaviour

Glacial till

Mitchell & Soga 2005

Department of Civil Engineering Advanced Soil Mechanics . W. Sołowski 22

Behaviour of clays under triaxial states Shearing (undrained)

Pore pressure during shearing : • Initial elastic (positive) change of pore water pressure

• Followed by elastoplastic changes:

pw

NC or slightly SC

NC soils or slightly SC (OCR ≈ 1 to 4), positive changes of the pore water pressure are observed

OC soils (OCR > 4), negative changes of the pore water pressure are observed

ε1 hightly OC

Clay behaviour

Glacial till

Mitchell & Soga 2005

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Behaviour of clays under triaxial states Stage II: Shearing v

The final values of the void ratio, at constant volume, show a unique line parallel to the virgin compression line:

v = Γ − λ ln p′

ln p’

Behaviour of clays under triaxial states Shearing (drained) q

Linear relationship between q and p’ on the critical states:

q = Mp′

M

and M is constant p’

Mechanical behaviour of clays under triaxial states Shearing (drained)

Behaviour of clays under triaxial states

Stress paths with

σ3′ σ1′ = KC =const.

q

In particular, the oedometric test corresponds to

ε H = 0 → KC = K 0 Notation:

K0 = p’

σ 3′ σ1′

when lateral deformation is zero

σ3′ σ1′ = KC =const.

Stress paths with q

The relationship of (virgin) volumetric compression:

K0

dv = −λ dp′ p′ p, p’ And, hence, NC lines are v

Isotropic virgin compression line

parallel to isotropic virgin compression line and critical state line Yield surface expands as a function of plastic deformations

K0

ln p’

Modified Cam Clay - plasticity The shape of the yield function is an ellipse, symmetrical with respect to the p’ axis. The equation of the yield function is:

f = q 2 − M 2 p ′( p 0′ − p ′)

q

M – slope of the critical state line (CSL)

p0 /2

p0

p'

Thank you