These slides have a special sentimental value to me: It...

These slides have a special sentimental value to me: It presents a joint work with Olivier Coussy, one of my dearest friends.

Olivier Coussy

Achievements •  Plenary lecture, Geoproc 2008 •  Invited scientist, Princeton University, July 2007 •  Keynote Lecture, Thermo-mechanical modelling for solids, Ecole polytechnique, 2007 •  Invited Professor, Universitat Polytècnica de Cataluyna, May 2007 •  Professor, Ecole des Ponts, in charge of the Lafarge research and master program, 2006 •  Senior scientist of outstanding rank, 2006 •  Berkeley-France funds award, 2005 •  Visiting Research Scholar, University of Princeton, Spring semester 2005 •  Visiting Miller Research Professor, University of California Berkeley, Fall semester 2004 •  Plenary Lecture, Congrès Français de Thermique, 2004 •  Biot Medal lecture, 2003, 16th ASCE Engineering Mechanics Conference, Seattle, 2003 •  ASCE Biot Medal Award, 2003 •  T.C. Powers Lecture, Concreep VI, 2001 •  Knight of the National Order of Merit, 2000 •  Plumey Award, 1999 (Award from the French Academy of Sciences in the field of

Mechanical Sciences) •  Associate Professor, 1985-1997, Ecole Polytechnique •  MTS Visiting professor, University of Minnesota, Spring semester 1995 •  Associate Professor, 1992-1998, Université de Marne-la-Vallée •  Plenary Lecture, Symposium Rock at Great Depths, 1989 •  Jean Mandel Award, 1985

Our joint papers •  P.J.M. Monteiro, Olivier Coussy, Denise A. Silva, Effect of cryo-suction

and air void transition layer on the hydraulic pressure developed during freezing of concrete, ACI Materials Journal, V. 103, No.2, 136-140, 2006.

•  Olivier Coussy_and Paulo J.M. Monteiro, Unsaturated poroelasticity for crystallization in pores, Computers and Geotechnics 34 (2007) 279–290.

•  O. Coussy_and P. J.M. Monteiro, Poroelastic model for concrete exposed to freezing temperatures, Cement and Concrete Research, Volume: 38, 40-48, 2008.

•  A. Fabbri, O. Coussy, T. Fen-Chong and P.J.M. Monteiro, Are Deicing Salts Necessary to Promote Scaling in Concrete? Journal of Engineering Mechanics – ASCE, Volume: 134 Issue: 7 Pages: 589-598, JUL 2008.

•  O. Coussy and P.J.M. Monteiro, Errata to “Poroelastic model for concrete exposed to freezing temperatures”[Cement and Concrete Research 38 (2008) 40–48], Cement and Concrete Research 39 (2009) 371–372.

If you’re interested in poromechanics

NAVIER INSTITUTE

Institut Navier Olivier Coussy

Unsaturated poroelasticity of freezing water-infiltrated materials with air voids

Miller Institute for Basic Research in Science, UC Berkeley

Paulo Monteiro Berkeley University

3 rd Biot Conference on Poromechanics, May 24-27 2005, Norman, Oklahoma

Bulk damage induced by frost action on a limestone

Outlines

1.  Unsaturated poromechanics 2.  Liquid saturation curve and

interfacial energy 3.  Hydraulic pressure

4.  Cryosuction 5.  Air voids

Beaudoin, Mc Innis, 1974

motivations

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

definitions

Solid matrix

unsaturated porous material

(as a whole)

interfaces

crystal

liquid

interfaces

+ =

porous solid =

solid matrix +

porous volume

bulk crystal and liquid

constituents

+

Skeleton Scale at which porosity, saturation degree, pore radius distribution are

relevant

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

ρ φC C

ρ φL L

i jσ

i jε

0σ ε φ φ+ + − =i j i j C C L L skd p d p d dF

partial porosity

Skeleton

( ) ( ), , ,sk ij C L LF W U Sε ϕ ϕ φ φ= +

Interfacial energy Porous solid elastic free energy

Skeleton energy including the

interfacial energy

2 2 03 3

σ ε ϕ ϕ⎛ ⎞ ⎛ ⎞+ − + − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

ij ij C C L Ld p U d p U d dW ( )C LL

dUp pdS

− − = −

0C C CSφ φ ϕ= +

Crystal saturation degree

Change due to deformation

Porous solid constitutive equations

Upscaling methods: K, G, bJ, NIJ are known as functions of ks, gs, SL, φ0

2 2 03 3

σ ε ϕ ϕ⎛ ⎞ ⎛ ⎞+ − + − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

ij ij C C L Ld p U d p U d dW

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

Solid matrix elastic modulis

0

0

1/ 1/ 2 / 31/ 1/ 2 / 3

ij ijkl C L ij

C C C C CC CL C

L L L L CL LL L

b bS b N N p US b N N p U

σ ε

ϕ φ φ

ϕ φ φ

Λ − −⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟= − = − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟= − − −⎝ ⎠ ⎝ ⎠⎝ ⎠

metastable

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

C LL

dUp pdS

− = −

( )− = Σ −C L f fp p T T

(Thomson-Gibbs)

( )L LS Tσ=

from the capillary curve or by direct experimentation

-45°C 0°C 100°C

fT state function

*

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

−= f

L

T TS f

T

2γ γγ∗ ∗

∗

= =Σ Σ

CL CL

f LG f

NT RT

12C L

CL

p pRγ

−=

12G L

LG

p pRγ

−=

0

100

200

300

0 20 40 60 80 100 0

25

50

75

100

SL (%)

p G -

pL (

MPa

)

h r (%

)

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

−= G L

Lp pS fN

Capillary curve

-5 -4 -3 -2 -1 0 1 2 3

0.5

1

1.5

2

2.5

0.9

0.8

0.5 0.7

0.6

0.9 0.8

0.5 0.4

0.6 0.7

R *   dS L / dR

S L

−( T f  −T ) /   T* R / R *

( )1 11 1

1 / 1 1m m

m m

fL C L

T T RS S T TT R

σ− −

− −

∗∗

∗

⎡ ⎤ ⎡ ⎤−⎛ ⎞ ⎛ ⎞⎢ ⎥= − = = + = +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

U/S

fT *

( T f  −T ) /   T*

0.9 0.8 0.5 0.7 0.6

Interfacial energy Liquid saturation curves and pore radius distributions

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

*

*

164.26nm

T KR=

=

cement paste

CRYOSTAT

Data acquisition 4 cm

By direct measurement: capacitive test

0

0,2

0,4

0,6

0,8

1

1,2

1,4

-3 -2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2

E / C = 0,5.   T *= 7° C   et   R *= 9,1   nm

E / C = 0,4.   T *= 9° C   et   R *= 7,3   nm

-­‐( T f   -­‐T ) / T * R / R *

R * X dS L / dRS L

propagation Scherer, 1993

fT

fThomogeneous nucleation

2 ?CLf

f

T T RRSγ

= −

1 fT T<

2 1T T<

2R

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

1 fT T<2R

2 1T T<

2R

hydraulic pressure (undeformable, Powers, 1949-1953)

liquid

air void

ice

Lp Z

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

Undeformable

0

01 CL

L

dTpdt

ρρ

⎛ ⎞∝ − ×⎜ ⎟⎝ ⎠

Deformable: ε ?Unsaturated

poromechanics

liquid

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

( )− = Σ −C L f fp p T T

pore radius distribution and saturation cryo-swelling

0 0.4 0.8 1.2 1.6 20

0.1

0.2

0.3

( T f  −T ) /   T*

0.9

0.8

0.5 0.7 0.6 0.4

Sf

10 3

-5 -4 -3 -2 -1 0 1 2 3

0.5

1

1.5

2

2.5

0.9

0.8

0.5 0.7

0.6

0.9 0.8

0.5 0.4

0.6 0.7

R *   dS L / dR

S L

−( T f  −T ) /   T* R / R *

ice

T

T

air void ( )0

= −Σ −

=

L f f

C

p T T

p

unsaturated poromechanics liquid saturation curve hydraulic pressure cryosuction air voids

( )− = Σ −C L f fp p T T

T stops

P. Monteiro

air void

( )0

= −Σ −

=

L f f

C

p T T

p

overall shrinkage

T stops

0.5 1 1.5 - 2

- 1

0

1

2

3

t / τ * = ( T f - T) / T *

O ×

10 3

m = 0.7 m = 0.5

All effects accounted for, including thermal deformation !

no air voids

0 1 2 3 R/R*

120oC h−×

13oC h−×

with air voids Equilibrium

Damage due to salt crystallization

( )− = Σ −C L f fp p T T0

lnυ

− =C LC

nRT xp px

supersaturation

Extension to salt crystallization?

From Rijniers et al.

Conclusions

1.  unsaturated poromechanics can be uselful

2.  Liquid saturation degree as a thermodynamic state function of the temperature

3.  Theory and experiments include:

1.  pore radius distribution 2.  surface effects 3.  hydraulic pressure 4.  cryosuction 5.  air voids

… presence of chemicals, deicer salt, salt crystallization

Acknowledgments

Miller Institute

A. Fabbri (PhD, UMLV) T. Fen-Chong (Institut Navier)

G. Scherer (Princeton)