a multiphase concrete model with application to …sscs 2012 - aix-en-provence, france , may 29-june...

SSCS 2012 - Aix-en-Provence, France , May 29-June 1, 2012

A multiphase concrete model with application to high temperature, structural repair, leaching and Alkali-Silica reaction

> 1

F. Pesavento, B.A. Schrefler, G. Sciumè

Dept. of Civil, Environmental and Architectural EngineeringUniversity of Padova – ITALY

D. Gawin

Department of Building Physics and Building MaterialsTechnical University of Lodz – POLAND

A multiphase concrete model with application

to high temperature, structural repair,

leaching and Alkali-Silica reaction



> 3

Layout

� Introduction

� Mathematical model of concrete as multiphase porous material

� (Numerical model)

� Applications of the model and numerical examples

� Modelling of concrete at early ages and beyond

� Modelling of concrete leaching

� Modelling of ASR in concrete structures

� Concrete at high temperature

� (Simplification of the model and its application to concrete structures repair)

� Conclusions & final remarks



> 4

� Concrete treated as a deformable, multiphase porous material

� Phase changes and chemical reactions (hydration, leaching, etc) are

taken into account

� Full coupling:

hygro-thermo-mechanical (stress – strain) ⇔ chemical reaction(cement hydration, leaching, ASR, etc)

� Transport mechanisms of moisture- and energy- typical for the specific phases of concrete are considered.

� Evolution in time (aging) of material properties, e.g. porosity, permeability,strength properties according to the hydration degree.

� Non-linearity of material properties due to temperature, gas pressure,moisture content and material degradation.

Theoretical ModelFundamental hypotheses



> 5

Capillary water (free water):� advective flow (water pressure gradient)

Physically adsorbed water:� diffusive flow (water concentration gradient)

Chemically bound water:� no transport

Water vapour:� advective flow (gas pressure gradient)� diffusive flow (water vapour concentration gradient)

Dry air:� advective flow (gas pressure gradient)� diffusive flow (dry air concentration gradient)

Ions (Ca++, Na+):� advective flow(water pressure gradient)� diffusive flow (calcium ions concentration gradient)� electrical flow (electrical potential gradient)

Theoretical ModelTransport Mechanisms

E=0



> 6

� Dehydration: solid matrix + energy ⇒ bound water

� Hydration: chemically bound water ⇒ solid matrix + energy

� Evaporation: capillary water + energy ⇒ water vapour

� Condensation: water vapour ⇒ capillary water + energy

� Desorption: phys. adsorbed water + energy ⇒ water vapour

� Adsorption: water vapour ⇒ phys. adsorbed water + energy

� Leaching: calcium in solid skeleton ⇒ calcium in liquid solution

� ASR: silica+alkali+water ⇒ hydrophilic gel+absortpion

Theoretical ModelChemical reactions & Phase Changes



> 7

Balance equations:

local formulation (micro- scale)

upscaling

Volume Averaging Theory

by Hassanizadeh & Gray, 1979,1980

macroscopic formulation

(macro- scale)

Rational Thermodynamics

Theoretical ModelMicro- ⇒ macro- description

Model development:

� Lewis & Schrefler, 1998; Schrefler, 2002;

�Gray & Schrefler, 2007; Gray, Pesavento, Schrefler, 2 009;



> 8

� The dry air and skeleton mass balance

� The water species and skeleton mass balance

� The multiphase medium enthalpy balance

� The calcium mass conservation equation

� Electrical charge balance equation

� The multiphase medium momentum balance (mechanical equilibrium)

� Evolution equation for hydration/dehydration

� Evolution equation for mechanical damage

� Evolution equation for thermo-chemical damage (leaching, etc.)

� Evolution equation for ASR process

Theoretical ModelMacroscopic balance equations & evolution equations

Electrical field neglected



> 9

( )

( )

ρβρ ρ

ρρρ

− − − + + + +

= −ɺ

11

1

s gas sg gasw

s g g gga ga

ga gs disg gga s

S nD S D T Dn n S S div divDt Dt Dt

mdiv nS S

v J

v

Dry air (and skeleton) mass balance equation

( ) ( )1 1

s s shydr disss

s s s

mn D D n mn div

Dt Dt− ρ − + − = −ρ ρ ρ

vɺ ɺ

Solid skeleton mass balance equation

Mathematical modelMacroscopic balance equations



> 10

Water species (liquid+vapor) (and skeleton) mass ba lance equation


( ) ( )

( ) ( ) ( ) ( )

( )

ρρ ρ ρ ρ α β

ρρ ρ ρ ρρ

ρ ρρ

− + + − + + +

− Γ+ − + = Γ

− +ɺ

*

1

s gws sgw gw gww w sw

w g swg g g

ss sgw gs gww ls w leach

w g w g sleach

gw wdiss

g ws

D S D T Dn S S div S n div

Dt Dt Dt

n DDdiv nS div nS S S

D Dt

mS S

v J

v v

Energy balance equation (for the whole system)

( ) ( ) ( )ρ ρ ρ χ∂ + + ⋅ − = − ∆ + ∆∂

ɺ ɺg gw w

p w p g p eff vap vap dis diseff

TC C C gradT div gradT m H m H

tv v



> 11


( )

( )

ρ αρ ρ

ρρ ρ ρ

Γ− + + + + +

Γ

−ɺ ɺ

1 11

1=

s s ss ss Caleach w Ca

Ca w Ca w Ca w ds wleach

w ls dis disCa w Ca ww w s

D D S D CDn C S nC nS C S div div

D Dt Dt Dt

m mdiv C nS C S

v J

v

Calcium mass balance equation

( ) ρ =ɺ ɺ+ 0totaldiv t g

Linear momentum balance equation (for the multipha se system)



> 12

� Gas pressure – pg

� Capillary pressure – pc

� Temperature – T;

� Calcium concentration – cCa

� Displacement vector – [ux, uy , uz]

� Hydration/Dehydration degree – Γhydr

� Mechanical damage degree – d

� Themo-chemical damage degree – V

� ASR reaction extent – ΓASR

� Leaching degree – Γleach

Theoretical ModelState variables & internal variables

Theoretical fundamentals and model development:

� Gawin, Pesavento, Schrefler , CMAME 2003, Mat.&Struct. 2004, Comp.&Conc. 2005

�Gawin, Pesavento, Schrefler , IJNME 2006 (part 1, part2)

�Gawin, Pesavento, Schrefler , IJSS 2008 (part 1, part2), CMAME 2009

�Pesavento et al. CMAME 2012



> 14

Modelling of concrete at early ages and beyond

Creep in concrete

� Bazant, Wittmann (eds)- 1982

� Bazant et al. - 1972-2002

� Harmathy - 1969

� Bazant, Chern – 1978 - 1987

� Hansen - 1987

� Bazant, Prasannan - 1989

� De Schutter, Taerwe - 1997

� Sercombe, Hellmich, Ulm, Mang - 2000

Hydration of cement

� Jensen - 1995

� van Breugel - 1995

� De Schutter, Taerwe - 1995

� Singh et al. – 1995

� Ulm, Coussy -1996

� Bentz et al. – 1998, 1999

� Sha et al. - 1999



> 16

hydrhydr

hydr

m

m∞ ∞Γ = =χ

χ

( ) ( ), exphydr ahydr hydr

d EA

dt RTΓΓ = Γ Γ −

ɶ ϕβ ϕ

( ) ( ) ( )21 1 exphydr hydr hydr hydr

AA AΓ ∞

∞

Γ = + κ Γ − Γ −ηΓ κ ɶ

where

- hydration degree-related, normalized affinity, χ - hydration extent,

Ea – hydration activation energy, R – universal gas constant, t – time.

( )hydrAΓ Γɶ

� from: [Cervera, Olivier, Prato, 1999]

Effect of relative humidity

Chemo - hygrothermal interactionsEvolution of the hydration process

Evolution of the hydration degree ΓΓΓΓhydr[Gawin, Pesavento, Schrefler, IJNME 2006 part 1 and part 2]



> 17

Chemo- mechanical interactionsStrain components

ch ch hydrd d= β Γε I

In general, a total strain of maturing concrete, εεεεtot, can be split into the following components:

1. free thermal strain2. thermo-chemical strain3. creep strain4. mechanical strain (caused by mechanical load and shrinkage)

Shrinkage strain Thermo-chemical strain

Free thermal strain strain

t sd dTβ=ε I

( )3

ws c ws csh

Td d p dp

Kα χ χ= − +ε I

mech tot c th chd d d d d= −ε ε ε − ε − εε ε ε − ε − εε ε ε − ε − εε ε ε − ε − εStrain decomposition



> 18

Capillary pressure & disjoining pressure Forces:

in waterCapillary pressure

Disjoining pressure

in skeletonCapillary pressure

Disjoining pressure

Hygro-mechanical interactionsShrinkage of concrete



> 19

Effective stress principle:

where is the solid surface fraction in contact with the wetting film,

I - unit, second order tensor, α - Biot’s coefficient,

ps - pressure in the solid phase and pc is given by

with - disjoining pressure

Hygro-mechanical interactionsShrinkage of concrete

wssx

Coefficient x

0,0E+00

5,0E-02

1,0E-01

1,5E-01

2,0E-01

2,5E-01

3,0E-01

3,5E-01

0,0 0,2 0,4 0,6 0,8 1,0

Saturation [-]�[Gray & Schrefler, 2001]�[Gray & Schrefler 2006]�[Gray, Pesavento, Schrefler 2009]

c f wg wwgp s J= Π −

fΠ

0,0E+00

1,0E-04

2,0E-04

3,0E-04

4,0E-04

5,0E-04

6,0E-04

7,0E-04

8,0E-04

0% 20% 40% 60% 80% 100%

RELATIVE HUMIDITY [%]

SH

RIN

KA

GE

ST

RA

IN [m

/m] OC - experiment

OC - theory

HPC - experiment

HPC - theory



> 20

Evolution of concrete porosity & permeability:

Hygro-structural - chemical interactionsEvolution of the material properties

0,0E+00

1,0E-01

2,0E-01

3,0E-01

4,0E-01

5,0E-01

6,0E-01

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

HYDRATION DEGREE [-]

PO

RO

SIT

Y [

-]

1,0E-21

1,0E-20

1,0E-19

1,0E-18

1,0E-17

1,0E-16

1,0E-15

1,0E-14

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

HYDRATION DEGREE [-]

PE

RM

EA

BIL

ITY

[m

2 ]

k = f (hydr)

k = f (porosity)

( ) 10 k hydrAhydrk k ΓΓ

∞Γ = ⋅( ) ( )10 knA n nk n k ∞−∞= ⋅( ) ( )1hydr n hydrn n A∞Γ = + Γ −

[Halamickova, Detwiler, Bentz, Garboczi, 1995] [Halamickova et al., 1995, Kaviany, 1999]



> 21

Evolution of concrete strength properties:[De Schutter, 2002]

Hygro-structural - chemical interactionsEvolution of the material properties



> 22

Mathematical modelConstitutive law for creep strain

Details:� [Bazant Z.P. Prasannan S.,Solidification theory for concrete creep I: formulation, J. Eng. Mech., 115(8), 1989]

where ( ) ( )( )v

hydr

tt

t

γε =

Γɺ

ɺ ( ) ( )( )

se

f

tt

t

σε

η=ɺ

- creep strains (as sum of viscoelastic and viscous (flow) term);

- viscoelastic term;

- flow term;

- viscoelastic microstrain;

- apparent macroscopic viscosity

c

v

f

εεεγη

ɺ

ɺ

ɺ

Solidification theory for basic creep

Effective stress

Hydration degree

c v fd d d= +ε ε ε( )

( )

se c th ch

c th ch

d d d d d

d

= −

+ −

D

D

σ ε ε − ε − εσ ε ε − ε − εσ ε ε − ε − εσ ε ε − ε − ε

ε ε − ε − εε ε − ε − εε ε − ε − εε ε − ε − ε



> 23

� Cubic specimen – 50x50x200 mm;

� Initial conditions:To= 293.15 K, ϕo= 99.0% RH, Γhydr=0.1;

� Boundary conditions:

- convective heat and mass exchange: sealed (adiabatic)- surface mechanical load: unloaded

� Properties of the cement paste:- Elastic modulus measured prior the test (1, 3 and 7 days), curing temperature 20°C

Numerical example Autogenous shrinkage in High Performance Cement Paste

(Lura, Jensen, van Breugel test)



> 24



0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 24 48 72 96 120 144 168

TIME [h]

RE

LAT

IVE

HU

MID

ITY

[-]

RHb=4, fc=10b=2.8, fc=10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 12 24 36 48 60 72

TIME [h]H

YD

RA

TIO

N D

EG

RE

E [-

]

Experimental

b=4 or 2.8, fc=10

R.H. development in time Hydr. Degree development in time



> 25



Effect of shrinkage – creep coupling

-3.0E-04

-2.5E-04

-2.0E-04

-1.5E-04

-1.0E-04

-5.0E-05

0.0E+00

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

RELATIVE HUMIDITY [-]

TO

TA

L S

TR

AIN

[-]

total - experiment

shrinkage - experiment

total - numerical

shrinkage - numerical



> 26

Numerical example Tests of Bryant and Vadhanavikkit (1987)

� Slab – thickness=30 cm; concrete: C50

� Material: concrete C50� - final porosity: 0.122, density: ρ= 1900 kg/m3,

� - intrinsic permeability: ko= 5·10-19 m2,

� - Young modulus: E= 49.2 GPa,

� - water/cement ratio w/c=0.45.

� Initial conditions:To= 293.15 K, ϕo= 99.8% RH, Γhydr=0.3;


Shrinkage (from day 7)

- convective heat and mass exchange: αc=5W/m2K; Tamb=293.15K; βc=0.002m/s; RHamb=60%

- surface mechanical load: unloaded or load=7 MPa at 8,28,84,182 days

Sealed (for the first 7 days and basic creep)

- convective heat and mass exchange: αc=5W/m2K; sealed - surface mechanical load: unloaded or load=7 MPa at 8,28,84,182 days



> 27

Numerical example Tests of Bryant and Vadhanavikkit (1987)

Drying & Load

-1.8E-03

-1.6E-03

-1.4E-03

-1.2E-03

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

1 10 100 1000

TIME [days]

TO

TA

L S

TR

AIN

[-]

exp. shrinkageshrinkageexp. 8 daystotal 8 daysexp. 28 daystotal 28 daysexp. 84 daystot. 84 daysexp. 182 daystot. 182 days

-2.0E-03

-1.8E-03

-1.6E-03

-1.4E-03

-1.2E-03

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

1 10 100 1000

TIME [days]S

TR

AIN

[-]

exp. Sealed - load sealed - loadexp. shrinkageshrinkageexp. total creepdrying & load

Pickett’s effect (8 days)



> 28

Modelling of concrete leachingD

em

ine

rali

ze

d

wa

ter

Picture of a leached cement paste sample, obtained by means of microscopic analysis in a fluorescent light.

Equilibrium based models: Gérard (1996), Kuhl et al. (2004), Kuhl and Meschke (2007)

Process kinetics based models:Ulm, Torrenti, Adenot – J. Engineering. Mechanics, 1999

Gawin, Pesavento, Schrefler - Part 1 & Part 2 , Solids and Structures, 2008Gawin, Pesavento, Schrefler, CMAME 2009.



> 30

Concrete leaching process kinetics Curve describing equilibrium between solid and liquid calcium

Calcium conc. in the pore solution [mol/m3]

Ca

lciu

m c

on

cte

nt

in t

he

so

lid

sk

ele

ton

[k

mo

l/m

3]

Calcium leaching



> 31

Concrete leaching process kinetics Curve describing equilibrium between solid and liquid calcium

Calcium conc. in the pore solution [mol/m3]

Ca

lciu

m c

on

cte

nt

in t

he

so

lid

sk

ele

ton

[k

mo

l/m

3]

Calcium leaching

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16 18 20 22

Ca ions concentr. [mol/m 3]

Sol

id C

a co

ncen

tr. [

kmol

/m3 ]

0.00E+00

5.00E-07

1.00E-06

1.50E-06

2.00E-06

2.50E-06

dSC

a/dc

Ca

[mol

/m3 s]

SCa-Ca,T=25°C

source,T= 25°C



> 32

Kinetics of leaching processThermodynamic description of the process

η- parameter dependent on micro-diffusion of Ca2+,

µs – chemical potential of liquid calcium,

Γ – chemical potential of solid calcium.

[Ulm, Torrenti, Adenot – J. Engng. Mech. 1999]

dA

s= RT

dcCa

cCa

− B : dεεεε − dεεεε p( )+ kdχ −κ dsCa

Effect of Caconcentration

Effect of the material elastic properties

Effect of micro-cracking

Effect of Ca content in skeleton

A

s= µ s − Γ = η

dsCa

dt



> 33


η- parameter dependent on micro-diffusion of Ca2+,

µs – chemical potential of liquid calcium,

Γ – chemical potential of solid calcium.

[Ulm, Torrenti, Adenot – J. Engng. Mech. 1999]

dA

s= RT

dcCa

cCa

− B : dεεεε − dεεεε p( )+ kdχ −κ dsCa

Effect of Caconcentration

Effect of the material elastic properties

Effect of micro-cracking

Effect of Ca content in skeleton

A

s= µ s − Γ = η

dsCa

dt



> 34

τ

s= η

RT Characteristic time of the process

R – universal constant of gas,

κ – process equilibrium constant,

η - parameter dependent on micro-

diffusion of Ca2+


∂sCa

∂t= 1

ηκ s ,T( )ds − κ s ,T( )ds

sCa0

sCa

∫sCa0

sCa*

∫



> 35

cCa

ef T( )= cCa × exp −Eleach

R

1

T− 1

Tref

dAs = RT

dcCa

cCa

− κ dsCa = 0

The values of the equilibrium constant

can be found from the thermodynamic equilibrium condition at temperature T, which can be written in the incremental form as

κ sCa ,T( )

κ sCa ,T( )=

RT

cCa

dsCa

dcCa T

−1

in this way we get

and taking into account the expression

one can write

κ sCa ,T( )=κ sCa ,Tref( ) T

Tref

This allows writing the general relationship describing the thermodynamic of the process as:

∂sCa

∂t= 1

τ leach T( )1

RTref

κ s ,Tref( )ds − κ s ,Tref( )dssCa

0

sCa

∫sCa0

sCa*

∫

i.e. the dependence on the temperature is concentrated in the function τ leach T( )

Kinetics of leaching processExtension to the non-isothermal conditions



> 37

Evolution of a material porosity & permeability:

Chemo - hygral interactionsEvolution of the material properties

[Kuhl et al. - 2004] [Saito and Deguchi - 2000]

( )0 10 leachAk k n ΓΓ= ⋅

( )0 ,ch leachn n n= + ∆ Γ

0dissch Ca leachdiss

Mn s

ρ∆ = ⋅ Γ

** 10 nA nk B= ⋅

k = k n

0( )⋅10An∆nch*

Effect of mechanical and chemical damage

[Gawin et al. - 2009, CMAME accepted]

AΓ = An

∆nch

Vpaste

=A

n

Vpaste

Mdiss

ρdisss

Ca0

with



> 38

Hygral - chemical interactionsCalcium transport

0CaD2,0E-10

3,0E-10

4,0E-10

5,0E-10

6,0E-10

7,0E-10

8,0E-10

9,0E-10

0 2 4 6 8 10 12 14 16 18 20 22 24

Ca++ IONS CONCENTRATION [mol/m 3]

Ca+

+ IO

NS

DIF

FU

SIV

ITY

[m

2 /s]

T= 298.15 K

T= 283.15 K

T= 313.15 K

0,0E+00

1,0E-10

2,0E-10

3,0E-10

4,0E-10

5,0E-10

0 2 4 6 8 10 12 14 16 18 20 22

Ca++ IONS CONCENTRATION [mol/m 3]C

a++

ION

S T

HE

RM

OD

IFF

. [m

2 /s]

T= 298.15 K

T= 283.15 K

T= 313.15 K

Ions thermo-diffusionIons diffusion

( )0 expCa Ca Ca CaT T w T T refD

cn S D T T

Tτ α = = ⋅ − D I I

[Samson et al. - 2007]

D

dCa = D

dCaI = n S

wτ D

d 0Ca ⋅exp A

4T − T

ref( )

I

[Samson et al. - 2007, Kuhl and Meschke - 2003 ]



> 39

Numerical simulation resultsModeling of a wall subject to reaction-diffusion-advection process

Boundary Conditions



> 40

Numerical simulation resultsReaction-diffusion-advection process: 25°C and 60°C-25°C cases (2 and 3)

Ca ions concentration Calcium content Solid skeleton



> 41



Chemical Damage



> 42





> 43



Chemical Damage



> 44

Numerical simulation resultsModeling of a wall subject to reaction-diffusion-advection process

Leaching front evolution

14

2

3



> 45

Modelling of Silica Alkali ReactionIntroduction

Difficulties related to ASR process simulation

� The reaction between alkali and the aggregates takes place at microscopic level and consists in the formation of a hydrophilic gel from the chemical combination of the silica contained in the aggregates, the alkali in the cement klinker (ions K+ and Na+ mainly) and the saline water solution, which fills partly or fully the pores of the material.

� While the first stage of the ASR (i.e. the dissolution of the silica) is known and can be simulated by means of a first order kinetic law, the second part of the process is less clear. In particular the mechanism of the formation of the amorphous gel, its combination with water and the related swelling, which leads to the macroscopically observable expansion and deterioration of the concrete structures, is under discussion.



> 46

Existing models and model proposed

� The existing models can be classified essentially into two classes: models which attribute the process progression to the water imbibitions (temperature independent) [1, 2] and models which consider temperature as the main parameter, i.e. ASR is considered as a pure thermally activated process [3, 4].

� In the present model the ASR is modelled as a two-stage process, involving chemical reactions causing first silica dissolution and then gel formation, [1]. The effect of moisture content on the kinetics of the first process is considered similarly as in [1, 2] and effect of temperature following approach proposed in [3, 4]. It is assumed that the gel formation process causes expansion of the material skeleton and the maximal chemical strain is dependent on the moisture content and to a much lesser extent on the temperature value. The gel is assumed to be in equilibrium with moisture in its pores, hence any variation of relative humidity causes an immediate change of chemical strains, also during decrease of water content.

① A. Steffens et al., Aging Approach to Water Effect on Alkali–Silica Reaction Degradation of Structures, Journal of Engineering Mechanics, 129, 50-59, (2003).

② F. Bangert et al., Chemo-hygro-mechanical modelling and numerical simulation of concrete deterioration caused by alkali-silica reaction, Int. J. Numer. Anal. Meth. Geomech., 28(78), 689-714, (2004).

③ Larive, C. Apports combinés de l'expérimentation et la modélisation à la compréhension de l'alcali-réaction et de ses effets mécaniques, Monograph LPC, 0A28, Laboratoire Central des Ponts et Chaussées, Paris (1998).

④ F-J Ulm et al., Thermo-chemo-mechanics of ASR expansion in concrete structures, Journal of Engineering Mechanics, 126(3), 233-242, (2000).

Modelling of Silica Alkali ReactionIntroduction



> 47

First stage of the process: evolution of the reaction extent ΓΓΓΓASR

tr = kr/A0 is the reaction time (A 0 is the initial chemical affinity and kr is the kinetic coefficient)

Mathematical modelModel of Silica Alkali Reaction

where (at constant T and Sw):

The chemical affinity is:

The reaction time is: where

with

Influence of saturation level

Latency time

Characteristic time

� Ref.: Larive et al , Journal of Engineering Mechanics 2000

τ L0 ,τ r 0 , AL , BL are material parameters



> 48

Mathematical modelStrain components

ch ch hydrd d= β Γε I

In general, a total strain of concrete, εtot, can be split into the following components:

1. free thermal strain2. ASR strain3. thermo-chemical strain4. creep strain5. mechanical strain (caused by mechanical load and shrinkage)

Shrinkage strain Thermo-chemical strain


t sd dTβ=ε I

( )3

ws c ws csh

Td d p dp

Kα χ χ= − +ε I

Strain decomposition



> 49


� Ref.: Steffens et al , Journal of Engineering Mechanics 2003

Second stage of the process: evolution of ASR strain εεεεASR

The water-gel combination rate is now:

with

Due to peculiar physico-chemical processes a loss of swelling potential is experimentally observed; this loss can be considered as a sort of “material aging”:

Water Aging

in which is the water combination coefficient at saturation level SwM ASR Sw( )

� Ref.: Pesavento et al , Comp. Methods in Appl. Mech. and Eng. 2012



> 50


Hence, the ASR strain evolution law is now:

After some mathematical transformations Steffens showed that:

Second stage of the process: evolution of ASR strain εεεεASR

M ASR Sw( ) = M ASR0ɶAASR ⋅ Sw + ɶBARS( ) linear formulation

M ASR Sw( ) = M ASR0 ⋅ exp ɶCASR ⋅ Sw + ɶDASR( ) power formulation

� Ref.: Steffens et al , Journal of Engineering Mechanics 2003

�Ref.: Bangert et al , IJNME 2004

M ASR0 = M ASR Sw = 1( )in which and and are material parametersɶAASR, ɶBARS, ɶCASRɶDASR

� Ref.: Pesavento et al , Comp. Methods in Appl. Mech. and Eng. 2012



> 51

� Density of the solid phase:

Mathematical modelModel of Silica Alkali Reaction: constitutive relationships

Const. relationships for the solid density

� In general the constitutive relationship for the solid skeleton has the form :

� Rate of the first invariant of the effective stress tensor:

Variation of the pore volume



> 52

Mathematical modelModel of Silica Alkali Reaction: constitutive relationships

Const. relationships for the intrinsic permeability

k = k mASR,d( ) = ko ⋅ 10Ak mASR/mASR∞( )10Add

Const. relationships for the porosity

n = constant

Sorption isotherms

� Suction curve for “unreacted” and for “fully reacted” material (Van Genuchten type):

� Evolution of the sorption isotherm during ASR process:



> 53

Mathematical modelModel of Silica Alkali Reaction: additional shrinkage

Evolution of the solid surface fraction due to ASR

with

Solid pressure

New microstructure due to the gel formation ⇒ modification of the solid pressure formulation by means a new form of the solid surface fraction

This allows of taking into account an additional shrinkage due to water loss from the ASR gel during the drying phases (the gel occupies less volume)



> 56

Modelling Silica Alkali ReactionExperimental-Numerical results comparison

� Material properties as in:

S. Poyet, Etude de la dégradation des ouvrages en béton atteints de la réaction alcali–silice: approcheexpérimentale et modélisation numérique multi-échelle des dégradations dans un environnement

hydrochemo–mécanique variable, PhD thesis, Université de Marne la Vallée, France (2003).

� Size of the specimens:

Cylindrical specimens – radius=1cm, height=16cm


- convective heat and mass exchange:

� RH∞=82%, and then 59%, 76%, 82%, 96% or 100% kept constant in time

with βc=0.002 m/s (drying cases) and βc=0.002 m/s (swelling cases)� T=60°C with αc=5 W/Km2

- surface mechanical load: unloaded

Poyet’s experimental tests at constant relative humidity



> 57


Poyet’s experimental tests at constant relative humidity

Mass variation

ASR strain evolution



> 58


� Material properties as in:

S. Poyet, Etude de la dégradation des ouvrages en béton atteints de la réaction alcali–silice: approcheexpérimentale et modélisation numérique multi-échelle des dégradations dans un environnement

hydrochemo–mécanique variable, PhD thesis, Université de Marne la Vallée, France (2003).

� Size of the specimens:

Cylindrical specimens – radius=1cm, height=16cm


- convective heat and mass exchange:

� RH∞=59-96% variable in time with

two different cycles: short (14 days) and long (28 days)βc=0.002 m/s (drying phases) and βc=0.002 m/s (swelling phases)

� T=60°C with αc=5 W/Km2

Poyet’s experimental tests at variable relative humidity




Poyet’s experimental tests at variable relative humidity (long cycle)

Mass variation

ASR strain and react. extent evolution




Poyet’s experimental tests at variable relative humidity (short cycle)

Mass variation

ASR strain and react. extent evolution



> 67

Modelling of concrete at high temperature

Concrete at high temperature

� Bazant, Thonguthai – 1978

� Thelandersson – 1987

� England, Khoylou - 1995

� Bazant, Kaplan – 1996

� Gerard, Pijaudier-Cabot, Laborderie - 1998

� Gawin, Majorana, Schrefler - 1999

� Bentz – 2000

� Nechnech, Reynouard, Meftah

- 2001

Thermal spalling

� Phan – 1996

� Ulm, Coussy, Bazant – 1999

� Sullivan – 2001

� Phan, Lawson, Davis - 2001

� HITECO project – 1995-2000

� NIST workshop – 1997

� UPTUN project – 2002-2006



> 68

Dehydration of concrete

Thermo-chemical interactionsConcrete at high temperature

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 200 400 600 800 1000

Temperature [ oC]

Deh

ydra

tion

degr

ee [

-]

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

0 0,2 0,4 0,6 0,8 1

Dehydration degree [-]

Deh

ydra

tion

rate

[1/

s]( ) ( )[ ]maxdehydr dehydrt T tΓ = Γ ( )

−⋅Γ=∂Γ∂

Γ RT

EexpA

ta

AΓ – chemical affinityT –temperature of concrete



> 69

Non-local isotropic damage theory

[Mazars & Pijaudier-Cabot, 1989]

Mechanical – mat. degradation interactionsMechanical material degradation description

2

2( ) exp

2oc

x sx s

l

−Ψ − = Ψ −

1( ) ( ) ( )

( )r V

x x s s dvV x

ε = Ψ − ε∫ ɶ

( )3 2

1i

i+

=

ε = ε∑ɶ

Ko

Koc

Fct

Fcc

εmax

σ−

ε−

t t c cd d d= +α α



> 70

Chemo - mechanical interactionsConcrete at high temperature

Correlation of dehydration degree & thermo-chemical damage

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 0,1 0,2 0,3 0,4

Dehydration degree [-]

The

rmo-

chem

ical

dam

age

[-]

( )1

( )o

o a

E TV

E T= −

( ) ( )( ) ( ) ( )1 1 1 1 1

( ) ( ) ( )o

o a o o a

E T E T E TD d V

E T E T E T= − = − = − − ⋅ −

0 0(1 )(1 ) : (1 ) :e ed V D= − − = −σ Λ ε Λ εσ Λ ε Λ εσ Λ ε Λ εσ Λ ε Λ ε

( ) ( )1 1S

d VS= =

− −σσσσσ σσ σσ σσ σɶ

ɶ



> 71

Effect of thermo- chemical material degradation on the stress – strain curves for C-60 concrete

Thermochemical - mechanical interactionsConcrete at high temperature

-70

-60

-50

-40

-30

-20

-10

0

-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0

Strain [-]

Str

ess

[MP

a]

20°C200°C

300°C

400°C

500°C

Experimental results from the EURAM-BRITE “HITECO” project



> 72

Intrinsic permeability – damage relationship

1.E-18

1.E-17

1.E-16

1.E-15

1.E-14

1.E-13

0 0.2 0.4 0.6 0.8 1

Damage parameter [-]

Wat

er in

tr.

perm

eabi

lity

[m2 ]

C-60

C-60 SF

C-70

C-90

Mechanical - hygral interactionsConcrete at high temperature

( )10 10p

D

Agf T A D

o go

pk k

p

= ⋅ ⋅ ⋅

Details:[Gawin, Pesavento & Schrefler,CMAME 2003]

Experimental results from the EURAM-BRITE “HITECO” project



> 73

Stress and strains of concreteStrain decomposition

Mechanical strain

mech tot th tchem trd d d d d= − − −ε ε ε ε ε

� mechanical strain (contains also shrinkage strain)

� total strain

� thermal strain

� thermo-chemical strain

� thermal transient strain

mech

tot

th

tchem

tr

ε

ε

ε

ε

ε

Where:



> 74

Shrinkage strain

LFTS model according to: [Gawin, Pesavento, Schrefler, Materials and Structures, 2004]

Stress and strains of concreteLoad Free Thermal Strain (LFTS) model

( )ws c ws csh

Td d p dp

Kα χ χ= +ε I tchem tot th shd d d d= − −ε ε ε ε

By using Load Free Thermal Strain (LFTS, i.e. irreversible part of thermal strain) model we get:

1. free thermal strain2. shrinkage strain3. thermo-chemical strain

( )tchem tchemd V dVβ=εConstitutive relationship for εεεεtchem

Thermo-chemical strain assessment


( )th sd T dTβ=ε



> 76

3-D modelling of LITS:

[Thelandersson, 1987]

βtr(V) relation according to:

[Gawin, Pesavento & Schrefler, Mat&Struct 2004]

c

( )f ( )

tr str e

a

Vd dT

Tβ=ε Q : σ

( ) ( )11

2ijkl ij kl ik jl il jkQ = − + + +γδ δ γ δ δ δ δ

Transient thermal creep of C-90

15% load

30% load45% load60% load

0% load

-1,00E-02

-7,50E-03

-5,00E-03

-2,50E-03

0,00E+00

2,50E-03

5,00E-03

7,50E-03

1,00E-02

0 100 200 300 400 500 600 700 800

Temperature [oC]

Line

ar s

trai

n [m

/m]

Stress and strains of concreteLoad Induced Thermal Strain (LITS) model



AIM: qualitative analysis of the effect of cooling processes on the hygro-thermo-chemo-mechanical state of a square column, manufactured with HSC, during thedevelopment of a natural fire in a high-rise building.

Numerical simulationsAnalysis of cooling processes in HSC square columns

> 79



Cooling Rates of the Air:�“Slow cooling”: – 0.20 ºC/s� “Fast cooling”: – 20.0 ºC/s

C60 (HSC)

HEATING PARAMETRIC CURVE:- Slow fire curve-Eurocode 1, Part 1-2 is used (O = 0,013 m1/2)

(Θg [ºC] , t [h])

0,0077

0,06545 0,7315

20 1.325 (1 0,324

0,204 0,472 )

tg

t t

e

e e

− ⋅

− ⋅ − ⋅

Θ = + ⋅ − ⋅ −

− ⋅ − ⋅

+2 TYPES OF ENVIRONMENTAL COOLING:

> 80



Par. 82 min. from fire start – at the beginning of cooling 122 minutes from fire start – at the end of slow cooling

Analysis of cooling processesSLOW (40 min) ENVIRONMENTAL COOLING

> 82



Par. 82 min. from fire start – at the beginning of cooling 122 minutes from fire start – at the end of slow cooling


> 83



ZONES WITH AN INCREASE OF DAMAGE DUE TO COOLING (ESPECIALLY CLOSE TO

THE SURFACE)

Param. 82 min. fromfire start – at the beginningof slow cooling 122 minutesfromfire start – at the end of slow cooling


> 84



DURING COOLING PHASE IN THE ZONES

CLOSE TO THE EXTERNAL SURFACES WE

OBSERVE AN INCREASE OF DAMAGE D

Analysis of cooling processesFAST (20 sec) ENVIRONMENTAL COOLING

> 85



Param. 82 min. fromfire start – at the beginningof fast cooling≈100 mins from fire start –at the end of the phase at constant ambient temperature

TO

TAL

DA

MA

GE

[-]

Zones of the squares column with Total Damage higher than 0.75, at the beginning of the fast environmental cooling and at the end of the phase at constant ambient temperature

Analysis of cooling processesFAST (20 sec) ENVIRONMENTAL COOLING

> 86



� A general model based on the mechanics of multiphase porous media for the analysis of thermo-hygral-chemical and mechanical behaviour of concrete has been presented.

� Some relevant applications of this model have been shown:

� Concrete maturing/hardening process (i.e. hydration) and long term behaviourof concrete structures (i.e. creep development);

� Concrete structures under extreme conditions in terms of pressure and temperature (high temperature);

� Concrete exposed to chemical degradation by pure water (i.e. leaching) in both isothermal and non-isothermal conditions;

� Concrete subject to Alkali-Silica Reaction;

� The leaching model considers not only the diffusive calcium transport, as other existing models, but additionally also the advective calcium flow.

� Calcium leaching process is modeled by considering thermodynamic imbalance of the calcium in solid and liquid phases, what allows for the description of process kinetics;

� The ASR model accounts for both water content (i.e. saturation) and temperature influence on the reaction evolution and the strain development;

� A simplified version of the model suitable for practical engineering application (e.g. concrete structures repair) has been formulated.

Conclusions

a multiphase concrete model with application to …sscs 2012 - aix-en-provence, france , may 29-june...

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