Micromechanics of three-phase granular materialspeople.3sr-grenoble.fr/users/bchareyre/ozschool/Chareyre.pdf · Micromechanics of three-phase granular materials Bruno Chareyre Lab

Powders and Grains 2009 – Golden CO, USA, july 13-17

Powders & Grains ConferenceGolden, CO, USA, July 13-17, 2009

Micromechanics of three-phase granular materials

Bruno Chareyre

Lab. 3S-R – Grenoble INP

ALERT O.Z. Course, June 2011, Grenoble

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Powders and Grains 2009 – Golden CO, USA, july 13-17Powders & Grains Conference

Golden, CO, USA, July 13-17, 2009

Plan

1. Wet granulars 2. DEM modeling 3. Macroscopic results and validation 4. Upscaling 5. Generalized effective stress 6. Discussion

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Air

Solid GrainLiquid(water)

1

Capillarity in Unsaturated Granular Materials

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Air

Solid GrainLiquid(water)

At low water content levelsinterfacial phenomena lead to

intergranular water menisci

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5




[Mitchell: Fundaments of soil behavior, Wiley Inter Science,1993]

In granular soils (silts and sands),capillary effects are of primary significance in

the unsaturated induced strength increase

3

6



Capillary Theory (Laplace):

F capillary=2 πσy0+πΔuy02

V meniscus=π∫ y2 x .dx

Δu=u a−uw=σCC=f [ y x ] ; y(x) is the interface profile

2


7



Local Hysteretis

Δu=u a−uc=cst

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Evolution of the capillary force at constant suction :

Un

F cap ,V m


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Local Hysteretis

Δu=u a−uc=cst

D

F cap ,V m

6

Evolution of the capillary force at constant suction :


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pendularregime

funicularregime

capillaryregime

Hydric Domains:

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DEM simulations and results : suction variation

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Golden, CO, USA, July 13-17, 2009 9

DEM simulations and results : suction variation

The range of simulated saturation degree

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using YADE – open DEM (http://yade-dem.org)(based upon the pioneering work of Cundall and Strack, 1979)

Law of Motion (Newton's 2nd Law)

applied to each particleForce-Displacement Lawapplied to each interaction

Positions and contacts update(finite difference scheme)

Resultant forces and moments

kn

ktφcont

5

DEM modelling

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using YADE – open DEM (http://yade-dem.org)(based upon the pioneering work of Cundall and Strack, 1979)

Law of Motion (Newton's 2nd Law)

applied to each particleForce-Displacement Lawapplied to each interaction

Positions and contacts update(finite difference scheme)

Resultant forces and moments

knktφco

nt

5

Fcap

Capillary force

DEM modelling

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10 000 spherical particles randomly positionned into a cubic box A unique value of succion in the sample (thermodynamic equilibrium) compacted through radius expansion to ensure the isotropy of the packing rigid frictionless boundary walls guarantee the homogeneity of the loading

DEM Sample :

σ 2 σ 2

1ε

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DEM simulations and results

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Triaxial loading : dry sample

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For several capillary pressure in the pendular regime (0 < Sr < 12%)u a−uw

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Triaxial loading : wet sample

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Yield surfaces in the (q,p) plane

Cohesion vs Saturation degree

No significant changes in the internal friction angle

Compares well with experiments and simulations done in Montpellier

(Richefeu and al.)

for <D> = 0.045 mmcDEM = 5 kPa

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Lu et al., Tensile strength of unsaturated sands, J. of Geotech. and Geoenv. Eng. (2007)

Quantative validation :

12

σ t DEM σ t Sand =

D Sand D DEM =

0 . 450 . 045


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So what?

loi d'interraction

A DEM model, so what?

expérience numérique

Réponse en contraintes/déformations

Rôle dans le développement de modèles micromécaniques d'homogénéisation

En particulier : modèle micro-directionel de F. Nicot (Sholtès et al. 2009a), modèle micro-structurel de Chang et Hicher (Sholtès et al. 2009b)

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Effective stress tensor in saturated granular materials (Terzaghi 1936 ):

« All measurable effects of a change of stress of the soil, that is, compression, distorsion, and change of shearing resistance, are exclusively due to changes in the effective stress. »

σ ij=σ' ij+uw . δ ij

Effective stress

Generalization for porous elastic materials Biot (1955) :

Standard sand : Ey = 100 Mpa, ν = 0.2 – 0.4Silice : Ey = 100 Gpa, ν = 0.16Biot's alpha coefficient : 0.999 ~ 1

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σ ij=σ' ij+uw . δ ij

Effective stress

P

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σ ij=σ' ij+u . δ ij

Generalized effective stress

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σ ij '=σ ij−u a+χ u a−uw δ ij

Partial saturation :Bishop and Blight, Géotechnique (1963)

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Generalised effective stress

?

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Generalised effective stressχ=S rA common assumption :

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Yield surfaces in the (p',q') plane :

Khalili and Khabbaz, Géotechnique (1998)A unique relationship for χ...

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1 > Sr > 0.59

After Nuth and Laloui (2007)

The advantages of an effective stress is pointed out.

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χ=Sr

Yield surfaces in the (p',q') plane for simulated low saturations :

None of the common definitions willresult in a unique yield surface...

Khalili and Khabbaz (1998)

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χ=Sr

Interpretation of the suction term as an additional confinement :

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Khalili and Khabbaz, Géotechnique (1998)A unique relationship for χ...

Cohesion vs Saturation degree

10 kPa < s < 104 kPa !!

σ ij '=σ ij−u a+χ u a−uw δ ij

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σ ij=1V ∑

c=1

N c ont ac t s

F icont l j

1V ∑

m=1

N me ni sc i

F icap l j

σ=σ contact+σ capillary

F=F contF cap

on each particle n of the assembly :

[Love, 1927]Cauchy stress tensor by homogenisation :

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Generalised effective stress : micromechanical definition

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σ ij= σ ijc +σ ij

cap

Generalised effective stress : micromechanical definition

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Possible definition of the effective stress : σ ijcontact= 1

V ∑c=1

N c ontac t s

F icont l j

Yield surfaces in the (pcont, qcont) plane :

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Generalised effective stress : numerical results

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σ t=3

4πsκΘzmD grains

Provides an explanation of the plateau in c vs. Sr curves. As suggested in e.g. Richefeu et al. (2006), the magnitude of capillary effects scales like :

σ ijcap= 1

V ∑m=1

N me ni sc i

F icap l j


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σ ijcontact= 1

V ∑c=1

N c ontac t s

F icont l j

22

σ t=3

4πsκΘzmD grains

Provides an explanation of the plateau in c vs. Sr curves. As suggested in e.g. Richefeu et al. (2006), the magnitude of capillary effects scales like :


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The shear strength is larger in the drying phase than in the wetting one

No significant changes in the volumetric strain

(nor in the internal friction angle)

Common residual stress state

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« Wetting » vs. « drying » initial states :


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« Wetting » vs. « drying » initial states :

The difference in the shear strength is linked to thenumber of liquid bridges

inside the sample

The liquid bridges tendto the same distribution

with the deformations

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σ ijcontact=σ ij−σ ij

capillary

The usual formalism fail to describe the anisotropy of the fluid contribution :

σ ij '=σ ij+χ u a−uw δ ij

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capillary

Ψ =2σ 1

cap−σ 2cap

σ 1cap +σ 2

cap



Ψ =0 if σ cap =χ u a−u w

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qpeak

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capillary



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Khalili and Khabbaz (1998)

?

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Generalised effective stress : discussion

After the thermodynamical approach of Gray and Schrefler (2006) :

with :

- xsws : solid-liquid surface ratio (not volume fractions);

- last term reflects direct effects of surface tension- still, the effect of the fluids are isotropic

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Conclusions In the same way as frictional phenomena, water effects at low saturation degrees are adequately modelised at the grains scale. Capillary forces generate an apparent cohesion, which compares well with measured values. The mechanical behaviour of the sample is almost constant on the range of saturation degree [2%,10%]. The contribution of the liquid in the effective stress is anisotropic. Capillary forces homogeneized using Love-Weber stress provides a relevant quantity to describe the effect of capillary forces.

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However...

σ ijcontact= 1

V ∑c=1

N c ontac t s

F icont l j = σ ij−σ ij

cap

Elastic response to isotropic compression ( ) vs Wetting ( )Δσ Δσ cap

Δσ

Δσ cap

?

21

Generalised effective stress : micromechanical results

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σ ijcontact= 1

V ∑c=1

N c ontac t s

F icont l j = σ ij−σ ij

cap

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Generalised effective stress : micromechanical results

Local kinematics are different :P.D.Fs. of normal displacement at contacts

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In the same as frictional phenomena, water effectsat low saturation degrees are adequately modelised atthe grains scale as a result of capillary menisci

A multi-scale approach to analyse water induced phenomena then appears as apertinent tool for critical examination of constitutive models.


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Challenges:

- range of water content: shape of the liquid bridges between 3, 4, N particles....

http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.html

- kinetics: interfaces, transfers, variable wetting angle.

- constitutive macro-modeling


4

http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.html

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Sr=∑m=1

N menisc us

V meniscus

V sample

pendularregime

funicularregime

capillaryregime

Suction Variation under isotropic loading : Wetting

Hydric Domains:

Simulation Results :Water retention

9

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σ t=3

4πsκΘzmD grains

Quantitative validation :

Tensile strength :

Richefeu et al., Physical Review E (2006) Shear strength properties of wet granular materials,

σ t=c

tan φ1

D grains∞

σ t DEM σ t Sand =

D Sand D DEM =

0 . 450 . 045

Simulation Results :Unsaturated triaxial paths

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F=F contF cap

on each particle n of the assembly :

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Powders and Grains 2009 – Golden CO, USA, july 13-17

Powders & Grains ConferenceGolden, CO, USA, July 13-17, 2009

Scholtès, L., Chareyre, B., Nicot, F., Darve, F. (2009), Micromechanicsof granular materials with capillary effects. International Journal of Engineering Science (47), pages 64–75. DOI 10.1016/j.ijengsci.2008.07.002

Scholtès, L., Hicher, P.-Y., Chareyre, B., Nicot, F., Darve, F. (2009), On the capillary stress tensor in wet granular materials. International Journal for Numerical and Analytical Methods in Geomechanics (33), pages1289–1313. DOI 10.1002/nag.767

Scholtès, L., Chareyre, B., Nicot, F., Darve, F. (2009), Discrete modelling of capillary mechanisms in multi-phase granular media.Computer Modeling in Engineering and Sciences (52), pages 297–318.

Nuth M, Laloui L. E ective stress concept in unsaturated soils: clarification and validation of a unified framework. Int. Journal forffNumerical and Analytical Methods in Geomechanics 2008; 32:771-801.

K HALILI N., K HABBAZ M., “A unique relationship for χ for the determination of the shear strength of unsaturated soils”, Géotechnique,vol. 48, no. 5, p. 681–687, 1998.

L U N., W U B., TAN C., “Tensile strength characteristics of unsaturated sands”, Journal of Geotechnical and Geoenvironmental Engineering, vol. 133, no.2, p. 144–154, 2007.

TERZAGHI K., “The Shearing Resistance of Saturated Soils and the Angle between the Plane of Shear”, p. 54–56, Harvard University Press, Cambridge, 1936.

BISHOP A., B LIGHT G., “Some aspects of effective stress in saturated and unsaturated soils”, Géotechnique, vol. 13, no. 3, p. 177–197, 1963.

Gray WG, Schrefler BA. Analysis of the solid phase stress tensor in multiphase porous media. Int J Numer Anal Methods Geomech, vol 31-4, 2007.

References

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Micromechanics of three-phase granular materialspeople.3sr-grenoble.fr/users/bchareyre/ozschool/Chareyre.pdf · Micromechanics of three-phase granular materials Bruno Chareyre Lab