Mohr-Coulomb model and soil stiffness

Mohr-Coulomb model

CG1 - Santiago, Chile - Octubre de 2012 1

The Mohr-Coulomb model

Dennis Waterman

Plaxis bv

Mohr-Coulomb model and soil stiffness

Objectives:

• To indicate features of soil behaviour

• To formulate Hooke’s law of isotropic linear elasticity

• To formulate the Mohr-Coulomb criterion in a plasticity

framework

• To identify the parameters in the LEPP Mohr-Coulomb model

• To indicate the possibilities and limitations of the MC model

Mohr-Coulomb model


Features of soil behaviour

• Elasticity (reversible deformation; limited) > stiffness

• Plasticity (irreversible deformation) > stiffness, strength

• Failure (ultimate limit state or critical state) > strength

• Presence and role of pore water

• Undrained behaviour and consolidation

• Stress dependency of stiffness

• Strain dependency stiffness

• Time dependent behaviour (creep, relaxation)

• Compaction en dilatancy

• Memory of pre-consolidation pressure

• Anisotropy (directional strength and/or stiffness)

Concepts of soil modelling

• Relationship between stresses (stress rates) and strains (strain rates)

• Elasticity (reversible deformations) dσσσσ=f (dεεεε)

– Example: Hooke’s law

• Plasticity (irreversible deformations) dσσσσ=f (dεεεε,σσσσ,h)

– Perfect plasticity, strain hardening, strain softening

– Yielding, yield function, plastic potential, hardening/softening rule

– Example: Mohr-Coulomb yielding

• Time dependent behaviour (time dependent deformations)

– Biot’s (coupled) consolidation dσσσσ=f (dεεεε,σσσσ,t)

– Creep, stress relaxation

– Visco elasticity, visco plasticity

σ yy

σ yz

σ yxσxy

σxxσxzσzxσzz

σzy

Mohr-Coulomb model


Types of stress-strain behaviour

Linear-elastic Non-linear elastic Elastoplastic

Lin. elast. perfectly-plast. EP strain-hardening EP strain-softening

σ

ε

σ

ε

σ

ε

σ

ε

σ

ε

σ

ε

Stress definitions

• In general, soil cannot sustain tension, only compression

• PLAXIS adopts the general mechanics definition of stress and strain: Tension/extension is positive; Pressure/compression is negative

• In general, soil deformation is based on stress changes in the

grain skeleton (effective stresses)

• According to Terzaghi’s principle: σ’ = σ - pw

σyy

σyy

σxx σxx

σyy

σyy

σxx σxx

Mohr-Coulomb model


Elasticity: Hooke’s law

σσσσσσ

ν ν

ν ν νν ν νν ν ν

νν

ν

εεεγγγ

xx

yy

zz

xy

yz

zx

xx

yy

zz

xy

yz

zx

E

=+ −

−−

−−

−−

( )( )1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

12

12

12

εεεγγγ

ν νν νν ν

νν

ν

σσσσσσ

xx

yy

zz

xy

yz

zx

xx

yy

zz

xy

yz

zx

E

=

− −− −− −

++

+

1

1 0 0 0

1 0 0 0

1 0 0 0

0 0 0 2 2 0 0

0 0 0 0 2 2 0

0 0 0 0 0 2 2

Inverse:

1 1

2 2

3 3

1

1(1 )(1 2 )

1

Eσ ν ν ν εσ ν ν ν ε

ν νσ ν ν ν ε

− = − + − −

Elasticity: Hooke’s law

In principal stress / strain components:

0

0 3

v

s

p K

q G

εε

=

In isotropic and deviatoric stress / strain components:

( )11 2 33

p σ σ σ= + +2 2 2

1 2 2 3 3 1

1( ) ( ) ( )

2q σ σ σ σ σ σ= − + − + −

Mohr-Coulomb model


Two parameters:

- Young’s modulus E- Poisson’s ratio ν

Meaning (axial compr.):

Ed

d= σ

ε1

1

ν εε

= − d

d

3

1

⇓

⇐

- dε1

dε3

- ε1

- σ1

ε3

E

1

1ν

Model parameters in Hooke’s law:dσ1

Shear modulus:

( )G

d

d

Exy

xy

= =+

σγ ν2 1

⇒

dσxy

dγxy

( )( )( )

Ed

d

Eoed = =

−+ −

σε

νν ν

1

1

1

1 1 2

Oedometer modulus:

( )K

dp

d

E

v

= =−ε ν3 1 2

Bulk modulus:

⇓ - dε1

- dσ1

dp

dεv

Alternative parameters in Hooke’s law:

Mohr-Coulomb model


The modeling of non-linear soil behaviour requires a relationshipbetween effective stress rates (dσ’ ) and strain rates (dε)

Symbolic: ( ) 1

' 'e e

d D d d D dσ ε ε σ−

= ⇔ =

12

12

12

1 ' ' ' 0 0 0'

' 1 ' ' 0 0 0'

' ' 1 ' 0 0 0' '

0 0 0 ' 0 0' (1 ')(1 2 ')

0 0 0 0 ' 0'

0 0 0 0 0 ''

xx xx

yy yy

zz zz

xy xy

yz yz

zx zx

d d

d d

d dE

d d

d d

d d

ν ν νσ εν ν νσ εν ν νσ ε

νσ γν ννσ γ

νσ γ

− − −

= −+ − −

−

Hooke’s law for effective stress rates

Basic principle of elasto-plasticity:p

ij

e

ijij εεε += (total strains)p

ij

e

ijij ddd εεε += (strain rates)

Plasticity

Elastic strains: Hooke’s lawPlastic strains: 3 questions

1. Does plasticity occur? -> yield function2. If so, in what direction? -> potential function3. How much plasticity? -> magnitude dλ

Mohr-Coulomb model


Plasticity – does plasticity occur?

• If f < 0 Pure elastic behaviour

• If f = 0 and df < 0 Unloading from plastic state (= elastic behaviour)

• If f = 0 and df = 0 Elastoplastic behaviour

Determination based on yield function f = f (σ’,ε)

Yield function f is (a.o.) a function of the stress state → f=0 can be represented as a border in the

stress space (yield contour)

Within the yield contour: f < 0On the yield contour: f = 0Outside the yield contour: f > 0 (impossible stress state)

Condition: Yield contour must be convex

f=0

f<0f>0

Plasticity – does plasticity occur?

Mohr-Coulomb model


Plasticity – in what direction?

Determination based on potential function g = g (σ’,ε)

The direction of plastic strain is determined by the vector Perpendicular to the plastic potential function

Metals (a.o): f = g (associated flow)Soils: f ≠≠≠≠ g (non-associated flow)

q,εs

p,εv

f=g

q,εs

p,εv

f

g

Plasticity – how much?

Determination based on magnitude scalar dλλλλ

The magnitude of plastic strain can be found with the so-calledconsistency condition, stating that for plasticity the stress state should remain on the yield surface:

0f f

df d dσ εσ ε

∂ ∂= + =

∂ ∂

Mohr-Coulomb model


Basic principle of elasto-plasticity:p

ij

e

ijij εεε += (total strains)p

ij

e

ijij ddd εεε += (strain rates)

Elastic strain rates:

( ) klijkl

ee

ij dDd '1 σε −

=

Plasticity

Plastic strain rates:

ij

p

ij

gdd

'σλε

∂∂=

dλ = scalar; magnitude of plastic strainsdg/dσ = vector; direction of plastic strains

g = plastic potential function

Origin: F

T

σ’n

τ

Coulomb: T ≤ A + F tanϕ τ ≤ c’ - σ’n tanϕ’

A

ϕT

F

c’

ϕ’τ

σ’n

The Mohr-Coulomb failure criterion

Mohr-Coulomb model


In general:

σ’n

τ

θ

σ’3

σ’1

The condition τ ≤ c’ - σ’n tanϕ’ must hold for arbitrary angle θ


cϕ

c cosϕ

-s* sinϕ

τ

-σn

-s*

t*

-σ1-σ3

MC criterion:

t*≤ c cosϕ - s* sinϕ


Mohr-Coulomb model


MC criterion: t*≤c’ cosϕ’ - s* sinϕ’

t* = ½(σ’3 - σ’1)

s* = ½(σ’3+σ’1)

( ) ( ) 'sin'''cos''' 1321

1321 ϕσσϕσσ +−≤− c

31 ''sin1

'sin1

'sin1

'cos'2' σ

ϕϕ

ϕϕσ

−+

−−

≤−c

Note: Compression is negative and σ’1≤ σ’2≤ σ’3


c’

ϕ’τ

σ’n

a

-σ’3

-σ’1

1

b 'sin1

'cos'2

ϕϕ

−=

ca

'sin1

'sin1

ϕϕ

−+

=b

Visualisation of the M-C failure criterion

Mohr-Coulomb model


( ) ( ) 'sin'''cos''' 2321

2321 ϕσσϕσσ +−≤− c

( ) ( ) 'sin'''cos''' 3221

3221 ϕσσϕσσ +−≤− c

( ) ( ) 'sin'''cos''' 1321

1321 ϕσσϕσσ +−≤− c

( ) ( ) 'sin'''cos''' 3121

3121 ϕσσϕσσ +−≤− c

( ) ( ) 'sin'''cos''' 1221

1221 ϕσσϕσσ +−≤− c

( ) ( ) 'sin'''cos''' 2121

2121 ϕσσϕσσ +−≤− c

σ1

σ3σ2

Full Mohr-Coulomb criterion

( ) ( ) 'sin'''cos''' 1321

1321 ϕσσϕσσ +−≤− c

Reformulation into yield functions

( ) ( ) 'cos''sin'''' 1321

1321

2 ϕϕσσσσ cf b −++−=

Mohr-Coulomb model


( ) ( ) 'cos''sin'''' 2321

2321

1 ϕϕσσσσ cf a −++−=( ) ( ) 'cos''sin'''' 322

1322


( ) ( ) 'cos''sin'''' 3121

3121

2 ϕϕσσσσ cf a −++−=

( ) ( ) 'cos''sin'''' 1321

1321


( ) ( ) 'cos''sin'''' 1221

1221

3 ϕϕσσσσ cf a −++−=

( ) ( ) 'cos''sin'''' 2121

2121


σ1

σ3σ2

Reformulation into yield functions

Parameters: Effective cohesion (c’) and effective friction angle (ϕ’)

( ) ( ) ψψσσσσ cos'sin'''' 2321

2321

1 cg a −++−=( ) ( ) ψψσσσσ cos'sin'''' 322

1322

11 cg b −++−=

( ) ( ) ψψσσσσ cos'sin'''' 3121

3121

2 cg a −++−=

( ) ( ) ψψσσσσ cos'sin'''' 1321

1321

2 cg b −++−=

( ) ( ) ψψσσσσ cos'sin'''' 2121

2121

3 cg b −++−=

Dilatancy angle ψ instead of friction angle ϕ

Motivation based on simple shear test

( ) ( ) ψψσσσσ cos'sin'''' 1221

1221

3 cg a −++−=

Plastic potentials of the M-C model

Mohr-Coulomb model


ψγε

γε

tan==p

xy

p

yy

xy

yy

d

d

d

dσxy

γxy

εyyψ

γxy

Failure in a simple shear test:

dilatancy

Linear-elastic perfectly-plastic stress-strain relationship

- Elasticity: Hooke’s law- Plasticity: Mohr-Coulomb failure criterion

For this model: Plasticity = Failure

This does NOT apply to all models!!!

The LEPP Mohr-Coulomb model

The LEPP model with Mohr-Coulomb failure contour is in PLAXIS called the Mohr-Coulomb model

Mohr-Coulomb model


Model parameters:

- Young’s modulus (stiffness) E- Poisson’s ratio ν- Cohesion c- Friction angle ϕ- Dilatancy angle ψ

Model parameters must be determined such that real soil behaviour is approximated in the best possible way

The LEPP Mohr-Coulomb model

Parameter determination

Parameter determination from:

• Laboratory tests (triaxial test (CD, CU), oedometer test or CRS,

simple shear test, …)

• Field tests (SPT, CPT, pressure meter (Menard, CPM, SBP),

dilatometer, …)

• Correlations with qc , PI , RD and other index parameters

• Rules-of-thumb, norms, charts, tables

• Engineering judgement

Mohr-Coulomb model


E ’50

1-2ν’

σ1-σ3

-ε1

-ε1

εv

σ’3 = confining pressure

MC approximation of a CD triax. test

32 'cos ' 2 ' sin '

1 sin '

c φ σ φφ

−−

ψψ

sin1

sin2

−

-σ1

-ε1

MC approximation of a compression test

Eoed

(1 )(1 2 )

(1 )oedE E

ν νν

+ −=

−

Mohr-Coulomb model


Possibilities and limitations of the LEPP Mohr-Coulomb model

Possibilities and advantages

– Simple and clear model

– First order approach of soil behaviour in general

– Suitable for many practical applications

– Limited number and clear parameters

– Good representation of failure behaviour (drained)

– Dilatancy can be included

σ1

σ3σ2

Limitations and disadvantages

– Isotropic and homogeneous behaviour

– Until failure linear elastic behaviour

– No stress/stress-path/strain-dependent stiffness

– No distinction between primary loading and unloading or reloading

– Dilatancy continues for ever (no critical state)

– Be careful with undrained behaviour

– No time-dependency (creep)

σ1

σ3σ2

Possibilities and limitations of the LEPP Mohr-Coulomb model

Documents

Mohr-Coulomb model and soil stiffness