> 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
Multi-physics approach to model concrete as porous material
Model Concepts for Fluid- Fluid and Fluid- Solid InteractionsFreudenstadt- Lauterbad, March 20- 22, 2006
> 2 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 3 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Critical temperature & Critical point Scanning Electron Microscopy
10 000 x
Physical ModelPhases of moisture & inner structure of concrete poro sity
> 4 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 5 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 6 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 7 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 8 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 9 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 10 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Energy balance equation
→ taking into account phase changes & related source terms
conduction (Fourier’s law)
convection
Entropy inequality
limitations for constitutive relationships
Theoretical ModelMacroscopic (space averaged) balance equations
> 11 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
Theoretical ModelMacroscopic (space averaged) balance equations
> 12 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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)
> 13 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 14 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 15 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 16 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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 & &
Mathematical modelMacroscopic balance equations
Accumulation terms
Flux terms
Source terms
> 17 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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:
Mathematical modelMacroscopic balance equations
Inertial terms
Accumulation terms Flux term
Source term
> 18 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 19 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 20 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 21 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
( )
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
> 22 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 23 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 24 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
( ) ( )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:
> 25 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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 [-
]
Chemo - hygrothermal interactionsEvolution of the hydration process
> 26 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Evolution of the hydration degree[Ulm & Coussy, 1996]
Chemo - hygrothermal interactionsEvolution of the hydration process
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
> 27 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Hydration rate as a function of chemicalaffinity
[Cervera, Olivier, Prato, 1999]
Chemo - hygrothermal interactionsEvolution of the hydration process
( ) ( ) ( )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]
> 28 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Evolution of the hydration degree[Gawin, Pesavento, Schrefler, IJNME 2006 part 1 and part 2]
Chemo - hygrothermal interactionsEvolution of the hydration process
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
> 29 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Evolution of the hydration degree[Gawin, Pesavento, Schrefler, IJNME 2006 part 1 and part 2]
Chemo - hygrothermal interactionsEvolution of the hydration process
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
> 30 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
RELATIVE HUMIDITY [-]
CO
EF
FIC
IEN
T ββ ββ
φφ φφ [-
]
Hygral - chemical interactionsEvolution of the hydration process
( )141 a aϕβ ϕ
− = − −
[Bazant, 1988]
> 31 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Influence of the hydration rate on the heat- & mass- sources
Chemo - hygrothermal interactionsEvolution of the hydration process
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
> 32 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
-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]
> 33 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 34 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Capillary pressure & disjoining pressure Forces:
in water
Capillary pressure
Disjoining pressure
in skeleton
Capillary pressure
Disjoining pressure
Hygro-mechanical interactionsShrinkage of concrete
> 35 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
Hygro-mechanical interactionsShrinkage of concrete
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
> 36 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Hygro-mechanical interactionsShrinkage of concrete
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]
> 37 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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]
> 38 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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:
Hygro-structural - chemical interactionsEvolution of the material properties
> 39 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Evolution of concrete strength properties:[De Schutter, 2002]
Hygro-structural - chemical interactionsEvolution of the material properties
> 40 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 41 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
( ) ( )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
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
Details: [Bazant Z.P. Prasannan S.,Solidification theory for concrete creep I: formulation, J. Eng. Mech., 115(8), 1989]
Mathematical modelConstitutive law for creep strain
where:
> 42 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
( ) ( )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
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:
Mathematical modelConstitutive law for creep strain
( ) 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
> 43 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
Details: [Bazant Z.P. Prasannan S.,Solidification theory for concrete creep I: formulation, J. Eng. Mech., 115(8), 1989]
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 = −
Mathematical modelConstitutive law for creep strain
> 44 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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+ =
Mathematical modelConstitutive law for creep strain
> 45 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
Mathematical modelConstitutive law for creep strain
> 46 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 47 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 48 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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)
> 49 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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)
Numerical exampleMaturing of a cylindrical specimen (Bentz-Laplante te st)
> 50 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleMaturing of a cylindrical specimen (Bentz-Laplante te st)
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
> 51 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleMaturing of a cylindrical specimen (Bentz-Laplante te st)
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
> 52 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleMaturing of a cylindrical specimen (Bentz-Laplante te st)
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
> 53 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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 exampleAutogenous shrinkage in High Performance Cement Pas te
(Lura, Jensen, van Breugel test)
> 54 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 55 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleAutogenous shrinkage in High Performance Cement Pas te
(Lura, Jensen, van Breugel test)
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 [
-]
> 56 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleAutogenous shrinkage in High Performance Cement Pas te
(Lura, Jensen, van Breugel test)
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
> 57 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleAutogenous shrinkage in High Performance Cement Pas te
(Lura, Jensen, van Breugel test)
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
> 58 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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;
Boundary conditions:
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
> 59 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleTests 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]
ST
RA
IN [-
]
exp. Sealed - load sealed - loadexp. shrinkageshrinkageexp. total creepdrying & load
Pickett’s effect (8 days)
> 60 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleTests of Bryant and Vadhanavikkit (1987)
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;
Boundary conditions:
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
> 61 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Numerical exampleTests of Bryant and Vadhanavikkit (1987)
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
> 62 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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;
Boundary conditions:
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
> 63 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 64 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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.
> 65 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 66 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 67 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 68 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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% %
> 69 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 70 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 71 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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.
> 72 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
]
> 73 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
( )( 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
> 74 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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= ⋅
> 75 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
moisture clog
High pressure of vapour
Cement matrix
Aggregate
Physical phenomena in heated concreteCauses of thermal spalling phenomenon
> 76 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
moisture clog
High pressure of vapour
stresses: aggregate - cem. matrix
Cement matrix
Aggregate
thermaldegradation
+
Physical phenomena in heated concreteCauses of thermal spalling phenomenon
> 77 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
moisture clog
High pressure of vapour
stresses: aggregate - cem. matrix
Stress – strain curve
thermaldegradation
+
Physical phenomena in heated concreteCauses of thermal spalling phenomenon
> 78 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
moisture clog
High pressure of vapour
stresses: aggregate - cem. matrix
thermal degradation
+
Cement matrix Aggregate
chemical degradation+
water from dehydration
+v vsp p ϕ= ⋅
Physical phenomena in heated concreteCauses of thermal spalling phenomenon
> 79 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
moisture clog
High pressure of vapour
chemical degradation
+water from
dehydration
+thermal
degradation
+
stresses: aggregate - cem. matrix
Physical phenomena in heated concreteCauses of thermal spalling phenomenon
> 80 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
moisture clog
High pressure of vapour
chemical degradation
+water from
dehydration
+thermal
degradation
+
stresses: aggregate - cem. matrix
Mechanical stresses
Constraints for thermal dilatation
Physical phenomena in heated concreteCauses of thermal spalling phenomenon
> 81 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
> 82 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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
Numerical simulationsSquare column subjected to ISO-Fire
> 83 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
C-60 concrete
Temperature & relative humidity after 20 min. of fi re
Numerical simulationsSquare column subjected to ISO-Fire
> 84 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
C-60 concrete
Gas pressure & total damage after 20 min. of fire
Numerical simulationsSquare column subjected to ISO-Fire
> 85 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
Spalling of the C-60 unloaded column
Numerical simulationsSquare column subjected to ISO-Fire during the test
> 86 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006
Multi-physics approach to model concrete as porous material
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.