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> 1 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

Multi-physics approach to model concrete as porous material

B.A. Schrefler , F. PesaventoDepartment of Structural and Transportation Engineering

University of Padova – ITALY

D. GawinDepartment of Building Physics and Building MaterialsTechnical University of Lodz – POLAND


Model Concepts for Fluid- Fluid and Fluid- Solid InteractionsFreudenstadt- Lauterbad, March 20- 22, 2006



Introduction

Mathematical model of concrete as multiphase porous material

Modelling of concrete at early ages and beyond

Thermodynamic model of the kinetics of hydration process

Components of concrete strains & description of concrete shrinkage

Hygro- thermo- chemo- mechanical couplings in concrete at early ages

Numerical solution & examples of its application

Space- & temporal discretisation

Solution of the nonlinear equation set

Validation of the model: numerical examples

Concrete at high temperature

Physics of concrete exposed to high temperature

Spalling phenomena

Conclusions & final remarks

Layout



Critical temperature & Critical point Scanning Electron Microscopy

10 000 x

Physical ModelPhases of moisture & inner structure of concrete poro sity



Thermodynamical equilibrium state locally (slow phenomena)

Concrete treated as a deformable, multiphase porous material

Phase changes and chemical reactions (hydration) taken into account

Full coupling:

hygro-thermo-mechanical (stress – strain) ⇔ chemical reaction

(cement hydration)

Theoretical ModelFundamental hypotheses



Various mechanisms of moisture- and energy- transportcharacteristic 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



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)

Theoretical ModelTransport Mechanisms



Dehydration: solid matrix + energy ⇒ bound water 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

Theoretical ModelChemical reactions & Phase Changes



Balance equations:

local formulation (micro- scale)

Volume Averaging Theory

by Hassanizadeh & Gray, 1979,1980

macroscopic formulation

(macro- scale)

δV

dv

Theoretical ModelMicro- ⇒⇒⇒⇒ macro- description

Model development:

Lewis & Schrefler: “The FEM in the Static and Dynamic ...”, Wiley, 1998

Schrefler: ”Mechanics and thermodynamics of saturated/unsaturated…”, Appl. Mech. Rev., 55(4), 2002

Representative

Elementary

Volume



Mass balance equations

→ taking into account phase changes & related source terms

solid matrix

chemically bound water

capillary water (free water)

physically adsorbed water

water vapour

dry air

Liquid phase

Gas phase

Solid phase

Theoretical ModelMacroscopic (space averaged) balance equations



Energy balance equation

→ taking into account phase changes & related source terms

conduction (Fourier’s law)

convection

Entropy inequality

limitations for constitutive relationships




Linear momentum balance equations

multiphase system (equilibrium equation)

capillary water (Darcy’s law)

adsorbed water (Fick’s law)

water vapour (Darcy’s and Fick’s law)

dry air (Darcy’s and Fick’s law)

Angular momentum balance equation

symmetry of stress tensor




Gas pressure – pg

Capillary pressure – pc

Temperature – T;

Displacement vector – [ux, uy , uz]

Hydration/Dehydration degree – Γhydr

Mechanical damage degree – d

Themo-chemical damage degree – V

Theoretical ModelState variables & internal variables

Theoretical fundamentals and model development:

Lewis & Schrefler : “The FEM in the Static and Dynamic ...”, Wiley, 1998

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

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



Energetic interpretation of capillary pressure

Theoretical ModelState variable for moisture transport

Chemical equilibrium

c wp = −Ψρ

lngw

ads bs

pH RT

f

∆ = −

/ads wH MΨ = −∆

lngw

gwsw

RT p

M f

Ψ =

b gwµ = µ

lnc gw

w gwsw

p RT p

M p

− = ρ

b gwf f=

Above critical point of water

bs gwsf f=

Adsorbed water potential

Below critical point of water



The dry air and skeleton mass balance

The water species and skeleton mass balance

The multiphase medium enthalpy balance

The multiphase medium momentum balance (mechanical equilibrium)

Evolution equation for hydration/dehydration

Evolution equation for material damage

Evolution equation for thermo-chemical damage

Theoretical ModelMacroscopic balance equations & evolution equations



Solid skeleton mass balance equations

( ) ( )1 1

s s sdehydrs

s s

mn D D nn div

Dt Dt− ρ − + − =ρ ρ

v&

Dry air mass balance equations

( ) ( )( )

1 11

1

s s s gagw s ga ga gs

s g g g gga ga ga

ssg dehydr dehydr

gs sdehydr

S nD S D T Dn n S S div div div n S

Dt Dt Dt

n S D mS

Dt

ρ− − β − + + + + ρρ ρ ρ

− Γ∂ρ− =∂Γρ ρ

v J v

&

Mathematical modelMacroscopic balance equations

Accumulation terms Flux terms Source terms



Water species (liquid+vapor) mass balance equations

Energy balance equation (for the whole system)

( ) ( )

( ) ( ) ( ) ( )

( )

*

1

s s s gwww gw w gw s gw

w g swg g g

sshydrw ws gw gs w gw

w g w g shydr

hydr w gw sw gs

D S D T Dn S S div S n div

Dt Dt Dt

Dndiv n S div n S S S

Dt

mS S

ρρ − ρ + ρ + ρ α − β + +

Γ− ∂ρ+ ρ + ρ − ρ + ρ∂Γρ

= − ρ + ρ − ρρ

v J

v v

&

( ) ( )* 1 gw w wswg s g w w wn S S n Sβ = β − ρ + ρ + β ρwhere:

( ) ( ) ( )w w g gp w p g p eff vap vap hydr hydreff

TC C C grad T div grad T m H m H

t∂ρ + ρ + ρ ⋅ − χ = − ∆ − ∆∂

v v & &


Accumulation terms

Flux terms

Source terms



Linear momentum balance equation (for the multipha se system)

+ 0s w ws ws w g gs gs gw gn S grad n S grad div−ρ − ρ + ⋅ − ρ + ⋅ + ρ = a a v v a v v gσσσσ

( )1 s w gw gn n S n Sρ = − ρ + ρ + ρwhere:


Inertial terms

Accumulation terms Flux term

Source term



Governing equations of the model

Galerkin’s (weighted residuum)

method

Variational (weak) formulation

FEM (in space)

FDM (in the time domain)

Discretized form (non-linear set of equations)

Numerical solutionDiscretization and linearization the model equation s



Discretized form (nonlinear equations)

the Newton - Raphson method

Solution of the final, linear equation set

the frontal method

Computer code (COMES family)

Numerical solutionDiscretization and linearization the model equation s



tg

g cg c

gg gc gt gu gg gc gt g

cg c

cc ct cu cg cc ct c

cg c

tc tt tu tc tt t

g c

ug uc ut uu u

t t t t

t t t

t t t

t t t t

∂ ∂ ∂ ∂+ + + + + + =∂ ∂ ∂ ∂

∂ ∂ ∂+ + + + + =∂ ∂ ∂∂ ∂ ∂+ + + + + =∂ ∂ ∂

∂ ∂ ∂ ∂+ + + =∂ ∂ ∂ ∂

p p T uC C C C K p K p K T f

p T uC C C K p K p K T f

p T uC C C K p K p K T f

p p T uC C C C f

where Kij - related to the primary variablesCij - related to the time derivative of the primary variablesf i - related to the other terms, eg. BCs (i,j=g,c,t,u)

Numerical solutionMatrix form of the FEM-discretised governing equations



( )

1

1

0 1

1

n n

n

n n n

t t+

+θ

+θ +

∂ − = ∂ ∆

< θ ≤= − θ + θ

X X X

X X X

θ =1 – fully implicit scheme (Euler backward);θ =0.5 – Crank-Nicholson scheme;θ =0 – fully explicit scheme (Euler forward);

( ) [ ]( )[ ]1 1

1 0

n nn

n nn

t

t tθ

θθ

θθ

+ ++

++

Ψ = + ∆ +− − − ∆ − ∆ =

X C K X

C K X F

( ) ( ) ( ) ( ) ( )1 1 1 1 1, , ,Tg c t u

n n n n n+ + + + +Ψ = Ψ Ψ Ψ Ψ X X X X Xwhere

Numerical solutionTime-discretisation – Finite Difference Method



where

Monolithic approach (or partitioned approach)

Frontal solver (or domain decomposition &

(with preconditioning & pivoting, parallel multi-frontal solver)

but without disc storage)

( )1

1 1,ln

ii l l

n n

+

+ +∂

= − ∆∂

X

ΨΨ X X

X

( ) ( ) Tl ll l l

n 1 n 1 n 1n 1 n 1, , ,g c

+ + ++ + ∆ = ∆ ∆ ∆ ∆

X p p T u

Numerical solutionSolution of the equations set



IntroductionModelling 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



( ) ( )1 1 11 1, , ,

Ti ii g c i in n nn n+ + ++ +

∆ = ∆ ∆ ∆ ∆ X p p T u 1

1 1 1i i in n n++ + += + ∆X X X

where:

( ) ,eqhydrhydr tΓ=Γ

( ) ( ) ( ) ,ττβτβ= ∫ ϕ dttt

o Teq

( ) ( ) ,

τ−=τβ

T1

T1

R

Uexp

o

hydrT

( ) ( )[ ] ,144 11

−

ϕ τϕ+α+=τβ

Details: [Bazant Z.P. (ed.), Mathematical Modeling of Creep and Shrinkage of Concrete, Wiley, 1988]

Γhydr - cement hydration degree;

teq - equivalent period of hydration;

βT - coefficient describing effect of the concrete temperature on the hydration rate;

βϕ - coefficient describing effect of the concrete relative humidity on the hydration rate;

Chemo - hygrothermal interactionsEvolution of the hydration process

Maturity-type model:



Influence of the temperature and relative humidity[Bazant, 1988]

0

1

2

3

4

5

6

7

273 298 323 348 373

TEMPERATURE [K]

CO

EF

FIC

IEN

T ββ ββ

T [-

] U/R=const

U/R=f(T)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

RELATIVE HUMIDITY [-]

CO

EF

FIC

IEN

T ββ ββ

φφ φφ [-

]




Evolution of the hydration degree[Ulm & Coussy, 1996]


hydrhydr

hydr

mm

χχ∞ ∞

Γ = =

( )exphydr ahydr

d EA

dt RTΓΓ = Γ −

%

where

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

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

( )hydrAΓ Γ%

Free water → Chemically bound water



Hydration rate as a function of chemicalaffinity

[Cervera, Olivier, Prato, 1999]


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

AA A κ ηκΓ ∞

∞

Γ = + Γ − Γ − Γ

%

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.2 0.4 0.6

Hydration degree [-]

Hyd

ratio

n ra

te [

1/h]



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


x · (unhydrated cement) + w · H2O → z · (hydrated cement)

AΓ ≡ x · µ unhydr + w · µ water - z · µ hydr

( )hydrAΓ Γ

where

- hydration degree-related chemical affinity, χ - hydration extend,

µ – chemical potential, N – mole number, x, w, z - stoichiometric coefficients.

unhydr hydrwaterdN dNdNd

x w z= = =

− −χ



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


hydrhydr

hydr

mm∞ ∞

Γ = =χχ

( ) ( ), exphydr ahydr hydr

d EA

dt RTΓΓ = Γ Γ −

% ϕβ ϕ

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

AA AΓ ∞

∞

Γ = + κ Γ − Γ −ηΓ κ %

where

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

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

( )hydrAΓ Γ%

from: [Cervera, Olivier, Prato, 1999]

Effect of relative humidity



Influence of relative humidity on the hydration rate

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1


CO

EF

FIC

IEN

T ββ ββ

φφ φφ [-

]

Hygral - chemical interactionsEvolution of the hydration process

( )141 a aϕβ ϕ

− = − −

[Bazant, 1988]



Influence of the hydration rate on the heat- & mass- sources


hydr hydrhydr hydr

mm m

t t ∞∂ ∂Γ

= =∂ ∂

&

hydr hydr hydrhydr hydr

Q mQ H

t t t∞∂ ∂Γ ∂

= = ∆∂ ∂ ∂

0,E+00

1,E-06

2,E-06

3,E-06

4,E-06

5,E-06

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

Hydration degree [-]

Hyd

ratio

n ra

te [

1/s]

Hydration rate

where:mhydr – mass of chemically bound water,

Qhydr – heat of hydration



-1.E-03

-9.E-04

-8.E-04

-7.E-04

-6.E-04

-5.E-04

-4.E-04

-3.E-04

-2.E-04

-1.E-04

0.E+00

1.E-04

0 72 144 216 288 360 432 504 576 648 720

TIME [hour]

ST

RA

INS

[-]

total creep chemical mech. & shrinkage thermal

Chemo- mechanical interactionsStrain components

Caused by:mechanical load and shrinkage

( )chem chem hydr hydrd d= β Γ Γε

εchem - chemical strain

(irreversible)

details:

[Gawin, Pesavento, Schrefler,

IJNME part1 and part2]



ϕβ dd shsh ⋅=ε

No physical mechanisms are taken into account

Based directly on experimental data

e.g:

[Bazant (ed.), Mathematical Modelling of Creep and Shrinkage of Concrete, Wiley, 1988]

Phenomenological modelling of shrinkage strain

Hygro-mechanical interactionsShrinkage of concrete



Capillary pressure & disjoining pressure Forces:

in water

Capillary pressure

Disjoining pressure

in skeleton

Capillary pressure

Disjoining pressure




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


s g ws csp p p= − χ

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]

s s se p= + ασ σ I




Experimental data from: [Baroghel- Bouny, et al., Cem. Concr. Res. 29, 1999]

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

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

Relative humidity [% RH]

Shr

inka

ge s

trai

n [m

/m]

experimental results

theory by Coussy

theory by Schrefler & Gray

s g cwdp dp S dp= −

[Coussy, 1995]

" sd d dp= + ασ σ I

Effective stress principle:

s g ws csp p p= − χ

s se p= + ασ σ I

[Gray & Schrefler, 2001]



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]



Experimental results from [Cook & Hover, 1999]

15

20

25

30

35

40

45

50

55

30 40 50 60 70 80 90 100

HYDRATION DEGREE [%]

PO

RO

SIT

Y [

%]

W/C= 0.3

W/C= 0.4

W/C= 0.5

W/C= 0.6

W/C= 0.7

Evolution of concrete porosity & permeability:




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




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 ( ) ( )( ) ( )

se

vhydr

F tt t

t

σε γ

=Γ

& & ( ) ( ) ( )( )

s se e

f

F t tt

t

σ σε

η =&

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

- viscoelastic term;

- flow term;

- viscoelastic microstrain;

- apparent macroscopic viscosity

c

v

f

εεεγη

&

&

&

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



( ) ( )1 1 11 1, , ,


∆ = ∆ ∆ ∆ ∆ X p p T u 1

1 1 1i i in n n++ + += + ∆X X X

Non-linear dependance of creep on the stress:

- function of stresses ;

- normalized stress;

- effective stress;

- damage at high stress;

- compressive strenght;

se

c

F

s

f

σΩ

′

( ) ( )2101

; ;1

ses

ec

tsF t s s

f

σσ + = Ω = = ′− Ω



where:



( ) ( )1 1 11 1, , ,


∆ = ∆ ∆ ∆ ∆ X p p T u 1

1 1 1i i in n n++ + += + ∆X X X

Details:[Bazant Z.P. et al. Microprestress- solidification theory for concrete creep : aging and drying effects, J. Eng. Mech., 123(11), 1997]

Log-Power law:


( ) 20

' ln 1n

t t qξλ

Φ − = +

11 pcpSη

−=

( )01

m

v t t

λ α = +

Microprestress theory

- compliance function

- parameter from B3 model

- microprestress ;

- positive constants;

- volume fraction of the solified matter;

- constant usually equal to 1;

- constants of the material;

( )2

0

'

,

( )

,

t t

q

S

p c

v t

m

λα

Φ −where( ) ( )1 1

hydrv t t=

Γ



η1 Ε1

η2 Ε2

ηN ΕN

γ 1

γ 2

γ N

γ

Φ(t-t')

σg(v,t)

σ

v(t) dv(t)

σ

η

h(t) dh(t)

ε0

εf

εv

σE

STRAINS

elastic (with aging)

visco-elastic

viscous (flow)

+chemical+thermal+cracking

creep


with

Rheologic model: (non-aging Kelvin chain)

( ) ( ) /' 1N

t t A e µξ τµ

µξ − Φ − = Φ = − ∑

1A

Eµµ

=

and

t

t

Eµ

′- time (age of concrete);

- time (age of concrete) at loading time;

- elastic modulus of Kelvin unit

't tξ = −




Details: [Bazant Z.P. Xi Y.,Continuous retardation spectrum for solidification theory of concrete creep,

J. Eng. Mech., 121(2), 1993]

with

Retardation spectrum technique

and where

( )( )

log

L

µ

ττ∆

- retardation spectrum;

- time interval between two adjacent Kelvin units;

( )( ) ( )1ln10 logA L

Eµ µ µµ

τ τ= = ∆

( ) ( ) ( )( )2

31

1 3

n

nL q n n

µµ

µ

ττ

τ≈ −

+1 10µ µτ τ+ =




where the elastic instantaneous term is:

Evaluation of Young’s modulus:

1

0

hydr

q

Aµλ

Γ-parameter from B3 model;

-hydration degree;

- coefficient from the exponential algorithm;

- term 0 in the Kelvin chain

1

1 01

11 1N

hydr hydr

E q AE

µ

µ µ

λ−

=

−′′ ′= + + Γ Γ

∑

1 1 / hydrq q′ = Γ

Aging elasticityHydration degree




Thermal conductivity

( )4

( ) 1+1

w

eff d s

n ST

n

ρχ = χ − ρ

( )1d do oA T Tχ χ = χ + −

1.1

1.2

1.3

1.4

1.5

1.6

1.7

20 70 120 170 220 270 320 370

Temperature [ oC]

The

rmal

con

duct

ivity

[W

/(mK

)]

Pc= 0 [Pa]

1.00E+08

2.00E+08

4.00E+08

6.00E+08

5.00E+07

Thermal capacity

2.00E+06

2.05E+06

2.10E+06

2.15E+06

2.20E+06

2.25E+06

2.30E+06

2.35E+06

2.40E+06

2.45E+06

20 70 120 170 220 270 320 370

Temperature [ oC]

The

rmal

cap

acity

[J/m

3 K]

Pc= 0 [Pa]

5.00E+07

1.00E+08

2.00E+08

4.00E+08

6.00E+08

( ) ( ) ( )( )1 1s w g gwp ps pw pga pgw pgaC n C n S C S C C C ρ = − ρ + ρ + − ρ + ρ −

( )1ps pso c oC C A T Tρ = ρ + −

Mathematical modelConstitutive relationships for thermal properties



Mathematical modelConstitutive relationships for porous solid

[Baroghel-Bouny, Mainguy, Lassabatere, Coussy, Cement Concrete Res. 29, 1999]

Sorption isotherm

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

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

Relative humidity [%]

Sat

urat

ion

degr

ee [

%] Ordinary Concrete

at T=20°C



Cylinder size - d= 4 cm ; h= 60 cm

Initial conditions:

To= 293.15 K, ϕo= 99.9% RH, Γhydr=0.1, for Ordinary Concrete (Bentz test);To= 293.15 K, ϕo= 99.0% RH, Γhydr=0.1, for High Performance Concrete (Laplante test);

Boundary conditions:

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

CASE A ⇒ full model;

CASE B ⇒ simplified model (no Relative humidity, β ϕ(ϕ)=1)CASE C ⇒ full model adjusted (better agreement with experimental tests)

Numerical exampleMaturing of a cylindrical specimen (Bentz-Laplante te st)



Parameter Symbol Unit OC HPC C-30 Water / cement ratio w/c [-] 0.45 0.35 0.35 Aggregate / cement ratio c/a [-] 4.0 4.55 3.0 Silica fume / cement ratio s/a [-] 0.00 0.09 0.00 Porosity n [%] 12.2 8.2 11.8 Intrinsic permeability k [m2] 3⋅10-18 1⋅10-18 3⋅10-18 Activation energy Ea/R [K] 5000 4000 5000 Parameter A1 A1 [1/s] 7.78⋅104 1.11⋅103

(8.88⋅102) 7.78⋅104

Parameter A2 A2 [-] 0.5⋅10-5 1⋅10-4 0.5⋅10-5 Parameter ∞κ ∞κ [-] 0.72 0.58 0.66

Parameter η η [-] 5.3 6.0 (4.7) 5.3

Heat of hydration hydrQ ∞ [MJ/m3] 202 172 103

Parameter a a [-] 18.6237 46.9364 18.6237 Parameter b b [-] 2.2748 2.0601 2.2748 Apparent density ρ eff [kg/m3] 2285 2373 2295 Specific heat (Cp)eff

[J/kgK] 1020 1020 1020 Thermal conductivity λ eff [W/mK] 1.5 1.78 1.5 Young’s modulus E [GPa] 24.11 39.61 29.52 Poisson’s ratio ν [-] 0.20 0.20 0.18 Compressive strength fc [MPa] 26 70 30

Characteristic properties of different types of concrete (in dry state, after 28 days of maturing)





293.15

303.15

313.15

323.15

333.15

343.15

353.15

363.15

0 24 48 72 96 120 144 168

TIME [h]

TE

MP

ER

AT

UR

E [

K] Experiment

Numerical

293.15

303.15

313.15

323.15

333.15

343.15

353.15

0 4 8 12 16 20 24

TIME [h]

TE

MP

ER

AT

UR

E [

K]

case Acase Bcase CExperiment

Ordinary Concrete High Performance Concrete




OC-HPC comparison (case A) HPC - A-B-C- comparison

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 24 48 72 96 120 144 168

TIME [h]

RE

LAT

IVE

HU

MID

ITY

[-]

NSCHPC

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 12 24 36 48 60 72

TIME [h]

RE

LAT

IVE

HU

MID

ITY

[-]

case Acase Bcase C




OC-HPC comparison (case A) HPC - A-B-C- comparison

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 84 96

TIME [h]

HY

DR

AT

ION

DE

GR

EE

[-]

OCHPC

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]

HY

DR

AT

ION

DE

GR

EE

[-]

case Acase Bcase C



Cubic specimen – 50x50x200 mm;

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


- 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 exampleAutogenous shrinkage in High Performance Cement Pas te

(Lura, Jensen, van Breugel test)





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]

HY

DR

AT

ION

DE

GR

EE

[-]

Experimental

b=4 or 2.8, fc=10

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





Temperature development in time Saturation development in time

293.15

294.15

295.15

296.15

297.15

298.15

299.15

300.15

301.15

302.15

303.15

0 24 48 72 96 120 144 168

TIME [h]

TE

MP

ER

AT

UR

E [K

]

0.9

0.91

0.92

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]

SA

TU

RA

TIO

N [

-]





Effective stress development in time Total strains vs R.H.

-4.0E+06

-3.5E+06

-3.0E+06

-2.5E+06

-2.0E+06

-1.5E+06

-1.0E+06

-5.0E+05

0.0E+00

0 24 48 72 96 120 144 168

TIME [h]

EF

FE

CT

IVE

ST

RE

SS

[P

a]

-3.5E-04

-3.0E-04

-2.5E-04

-2.0E-04

-1.5E-04

-1.0E-04

-5.0E-05

0.0E+00

5.0E-05

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

RELATIVE HUMIDITY [-]T

OTA

L S

TR

AIN

[-]

total

shrinkage

chemical

creep

thermal





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


TO

TA

L S

TR

AIN

[-]

total - experiment

shrinkage - experiment

total - numerical

shrinkage - numerical



Numerical exampleTests of Bryant and Vadhanavikkit (1987)

Slab – thickness=30 cm; concrete: C50

Initial conditions:

To= 293.15 K, ϕo= 99.8% RH, Γhydr=0.3;


Shrinkage- convective heat and mass exchange: αc=5W/m2K; βc=0.002m/s

- surface mechanical load: unloaded or load=7 MPa

Sealed

- convective heat and mass exchange: αc=5W/m2K; sealed - surface mechanical load: unloaded or load=7 MPa

Age of loading: 8,28,84,182 days




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]

ST

RA

IN [-

]

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

Pickett’s effect (8 days)




Square prism – 15x15x60 cm; equiv. cylinder φ=16.3 cm; concrete: C50

Initial conditions:

To= 293.15 K, ϕo= 99.8% RH, Γhydr=0.3;


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

- surface mechanical load: unloaded or load=7 MPa

Sealed

- convective heat and mass exchange: αc=5W/m2K; sealed - surface mechanical load: unloaded or load=7 MPa

Age of loading: 8,28,84,182 days




Drying & Load Pickett’s effect (8 days)

-2.5E-03

-2.0E-03

-1.5E-03

-1.0E-03

-5.0E-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 daystotal 84 daysexp. 182 daystot. 182 days

-2.5E-03

-2.0E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

1 10 100 1000

TIME [days]

ST

RA

IN C

OM

PO

NE

NT

[-].

exp. shrinkagenum. shrinkageexp. loadnum. loadexp. load + shrinkagenum. load + shrinkage



Numerical exampleTests of L’Hermite (1965)

Square prism – 7x7x28 cm; equiv. cylinder φ=7.6 cm; concrete: C30

Initial conditions:

To= 293.15 K, ϕo= 99.9% RH, Γhydr=0.3;


Shrinkage- convective heat and mass exchange: αc=5W/m2K; βc=0.002m/s; RHamb=50%

- surface mechanical load: unloaded or load=4.9, 9.8 and 12.3 MPa

Sealed

- convective heat and mass exchange: αc=5W/m2K; sealed - surface mechanical load: unloaded or load=4.9, 9.8 and 12.3 MPa

Age of loading: 7, 21, 90 and 180 days



Numerical exampleTests of of L’Hermite (1965)

Drying & Load Effect of load (180 days)

-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]

ST

RA

IN C

OM

PO

NE

NT

[-]

exp.shrinkageshrinkageexp. 4.905 MPatotal 4.905 MPaexp. 9.81 MPatotal 9.81MPaexp.12.2625 MPatotal 12.2625 MPa

-3.0E-03

-2.5E-03

-2.0E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

1 10 100 1000

TIME [days]

ST

RA

IN C

OM

PO

NE

NT

[-].

exp.shrinkageshrinkageexp. 7daystotal 7 daysexp. 21 daystotal 21 daysexp. 90 daystotal 90 daysexp. 180 daystotal 180 days



Conclusions

The FE model of concrete based on a thermodynamically consistentmechanistic theory and its application for concrete at early ages and beyond has been presented.

Concrete is considered as multiphase, porous visco-elastic material.

Phase changes, chemical reactions (hydration), different fluid flows, material non-linearities (with respect to temperature, moisture content & evolution of cement hydration) and coupling between these processes are taken into account.

Futher research on introducing into the model a plastic damage theoryfor description of concrete cracking during early stages of hydration is in progress.



IntroductionModelling of concrete at high temperarure

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



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



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∫ %

( )3 2

1i

i+

=

ε = ε∑%

Ko

Koc

Fct

Fcc

εmax

σ−

ε−

t t c cd d d= +α α



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 σ% %



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



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



Chemo - mechanical interactionsConcrete at high temperature

Load Free Thermal Strain (LFTS) model

LFTS - irreversible part of thermal strain

V – thermo-chemical damage parametr

LFTS model according to:

[Gawin, Pesavento, Schrefler, Mat&Struct2004]

( )tchem tchemd V dVβ=ε

Thermal dilatation of C-90

-8,0E-03

-6,0E-03

-4,0E-03

-2,0E-03

0,0E+00

2,0E-03

4,0E-03

6,0E-03

8,0E-03

1,0E-02

0 100 200 300 400 500 600 700 800

Temperature [ oC]

Line

ar s

trai

n [m

/m]

experimental

total

therm_dilat

shrinkage

thermo-chem.



Thermo - mechanical interactionsConcrete at high temperature

Load Induced Thermal Strain (LITS = „thermal creep”) model

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

]



( )( 1/ )1

bS G

−= + ( ) 1,

b

bc cE

G G T p pa

− = =

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 100 200 300 400

Temperature [°C]

Sat

urat

ion

[-]

1.00E+051.00E+065.00E+061.00E+072.00E+075.00E+071.00E+081.00E+091.00E+10

Porosity of concrete

( )0 0A T Tφφ = φ + −

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 100 200 300 400 500 600

Temperature [°C]P

oros

ity [

-]

silicate-concrete

basalt-concretelimestone-concrete

Desorption isotherms

Thermo - mechanical interactionsConcrete at high temperature



Physical phenomena in heated concreteCauses of thermal spalling phenomenon

moisture clog

High pressure of vapour0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

273,15 323,15 373,15 423,15 473,15Temperature (K)

Sat

urat

ed v

apou

r pr

essu

re

(MP

a)

v vsp p ϕ= ⋅



moisture clog

High pressure of vapour

Cement matrix

Aggregate




moisture clog


stresses: aggregate - cem. matrix

Cement matrix

Aggregate

thermaldegradation

+




moisture clog



Stress – strain curve

thermaldegradation

+




moisture clog



thermal degradation

+

Cement matrix Aggregate

chemical degradation+

water from dehydration

+v vsp p ϕ= ⋅




moisture clog


chemical degradation

+water from

dehydration

+thermal

degradation

+





moisture clog


chemical degradation

+water from

dehydration

+thermal

degradation

+


Mechanical stresses

Constraints for thermal dilatation




Problem description

C- 60 concrete 30x30 cm column

ICs:

ϕin= 60 % RH, Tin= 298.15 K

BCs:

convective & radiative for heat exchange:

αc=18 W/m2K, eσo=5.1⋅10-8 W/(m2K4),

Te=ISO fire curve,

convective for moisture exchange

pv= 1000 Pa, βc= 0.018 m/s

Numerical simulationsSquare column subjected to ISO-Fire



Parameter Symbol Unit Value Porosity n [-] 0.082 Intrinsic permeability k [m2] 2×10-18 Apparent density ρ [kg/m3] 2564 Specific heat Cp

s [J/kgK] 855 Thermal conductivity λ [W/mK] 1.92 Young’s modulus E [GPa] 34.52 Poisson’s ratio ν [-] 0.18 Compressive strength fc [MPa] 60 Tensile strength ft [MPa] 6.0

C-60 concrete

Material properties




C-60 concrete

Temperature & relative humidity after 20 min. of fi re




C-60 concrete

Gas pressure & total damage after 20 min. of fire




Spalling of the C-60 unloaded column

Numerical simulationsSquare column subjected to ISO-Fire during the test



Conclusions

Numerical model of hygro-thermo-chemo-mechanical performance of concrete at high temperature has been developed.

Energy and mass transport mechanisms typical for different phases of concrete, as well as phase changes and chemical reactions are taken into account.

Full coupling between hygral-, thermal-, chemical- and mechanical processes is considered.

The model is validated by means of averaging theories and thermodynamics.

The model is verified by means of the investigation of its numerical properties.

b.a. schrefler , f. pesavento - uni-stuttgart.de · > 1 model concepts for fluid-fluid and...

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