B.A. Schrefler, F. Pesavento

Transcription

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. Pesavento Department of Structural and Transportation Engineering University of Padova ITALY D. Gawin Department of Building Physics and Building Materials Technical University of Lodz POLAND > 1 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

2 Layout 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 > 2 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

3 Physical Model Phases of moisture & inner structure of concrete porosity Critical temperature & Critical point Scanning Electron Microscopy x > 3 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

4 Theoretical Model Fundamental hypotheses 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) > 4 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

5 Theoretical Model Fundamental hypotheses Various mechanisms of moisture- and energy- transport characteristic 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. > 5 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

6 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: Theoretical Model Transport Mechanisms advective flow (gas pressure gradient) diffusive flow (dry air concentration gradient) > 6 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

7 Theoretical Model Chemical reactions & Phase Changes Dehydration: Hydration: solid matrix + energy bound water water chemically bound water solid matrix + energy Evaporation: Condensation: Desorption: Adsorption: capillary water + energy water vapour water vapour capillary water + energy phys. adsorbed water + energy water vapour water vapour phys. adsorbed water + energy > 7 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

8 Balance equations: Theoretical Model Micro- macro- description local formulation (micro- scale) d v Volume Averaging Theory by Hassanizadeh & Gray, 1979,1980 macroscopic formulation (macro- scale) δ V Representative Elementary Volume 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 > 8 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

9 Theoretical Model Macroscopic (space averaged) balance equations 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 Solid phase Liquid phase Gas phase > 9 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

10 Theoretical Model Macroscopic (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 > 10 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

11 Theoretical Model Macroscopic (space averaged) balance equations Linear momentum balance equations multiphase system capillary water adsorbed water water vapour dry air (equilibrium equation) (Darcy s law) (Fick s law) (Darcy s and Fick s law) (Darcy s and Fick s law) Angular momentum balanc equation symmetry of stress tensor > 11 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

12 Theoretical Model State variables & internal variables Gas pressure p g Capillary pressure p c Temperature T; Displacement vector [u x, u y, u z ] Hydration/Dehydration degree Γ hydr Mechanical damage degree d Themo-chemical damage degree V 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 Gawin, Pesavento, Schrefler, IJNME 2006 (part 1, part2) > 12 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

13 Theoretical Model State variable for moisture transport Energetic interpretation of capillary pressure Chemical equilibrium b gw µ = µ f b = f gw f bs = f gws Above critical point of water Adsorbed water potential gw p H ads = RT ln bs f Ψ = H / M ads w gw RT p Ψ = ln gws M w f Below critical point of water c gw p RT p = ln w gws ρ M w p c w p = Ψρ > 13 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

14 Theoretical Model Macroscopic balance equations & evolution equations 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 > 14 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

15 Mathematical model Macroscopic balance equations Solid skeleton mass balance equations ( 1 ) s s s n D ρ D n + s ( 1 n ) div v = s ρ Dt Dt m& dehydr s ρ Dry air mass balance equations s s s ga D S D T S n D ρ 1 1 n βs n Sg + Sgdiv + + div ga ga g + div n S ga gρ Dt Dt ρ Dt ρ ρ ( 1 ) g ( 1 ) v J ( v ) w s ga ga gs n S D m& = s ρ Accumulation terms Flux terms Source terms g s ρ sγdehydr dehydr Γdehydr Dt s ρ S g > 15 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

16 Mathematical model Macroscopic balance equations Water species (liquid+vapor) mass balance equations Flux terms s s s gw w gw D S w gw s * D T D ρ gw w g v swg g J g w ( ) ( ) n ρ ρ + ρ S + ρ S αdiv β + S n + div Dt Dt Dt w ws gw gs w gw ( ) ( ) ( ) ( 1 n v v ) + div n S ρ + div n S ρ ρ S + ρ S w g w g s m& = ρ + ρ ρ ρ ( Sw Sg ) hydr w gw s s Source terms where: Energy balance equation (for the whole system) w w g g ( ) ( ) ( ) ρ Γ > 16 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006 ρ s hydr Accumulation terms D ( )( ) s Γ Dt hydr * gw w w swg s 1 n Sg Sw n w Sw β = β ρ + ρ + β ρ T ρ C + ρ C v + ρ C v grad T div χ grad T = m& H m& H t p eff w p g p eff vap vap hydr hydr

17 Mathematical model Macroscopic balance equations Linear momentum balance equation (for the multiphase system) Accumulation terms Flux term s w ws ws w g gs gs g ρa n Swρ a + v grad v n Sgρ a + v grad v + div σ+ ρ g = 0 Inertial terms Source term where: ( 1 ) s w g w g ρ = n ρ + n S ρ + n S ρ > 17 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

18 Numerical solution Discretization and linearization the model equations 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) > 18 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

19 Numerical solution Discretization and linearization the model equations Discretized form (nonlinear equations) the Newton - Raphson method Solution of the final, linear equation set the frontal method Computer code (COMES family) > 19 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

20 Numerical solution Matrix form of the FEM-discretised governing equations g c p p T u g c gg + gc + gt + gu + gg + gc + gt = g C C C C K p K p K T f t t t t c p T u g c cc + ct + cu + cg + cc + ct = c C C C K p K p K T f t t t c p T u g c tc + tt + tu + tg + tc + tt = t C C C K p K p K T f t t t g c p p T u C + C + C + C = f t t t t ug uc ut uu u where K ij - related to the primary variables C ij - related to the time derivative of the primary variables f i - related to the other terms, eg. BCs (i,j=g,c,t,u) > 20 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

21 Numerical solution Time-discretisation Finite Difference Method ( n ) [ θ t ] [ ( θ ) t ] Ψ X = C + K X n+ θ n+ 1 C 1 K X t F = 0 n+ θ n n+ θ where g c t u ( X ) ( X ), ( X ), ( X ), ( X ) Ψ = Ψ Ψ Ψ Ψ n+ 1 n+ 1 n+ 1 n+ 1 n+ 1 T X Xn+ 1 X = t t n+θ ( 1 ) X = θ X + θx n+θ n n+ 1 n 0 < θ 1 θ =1 θ =0.5 θ =0 fully implicit scheme (Euler backward); Crank-Nicholson scheme; fully explicit scheme (Euler forward); > 21 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

22 Numerical solution Solution of the equations set i i ( l ) l n+ + X Ψ X Ψ 1 = Xn X 1, l n+ 1 where l ( ) l ( ) l g c l l n+ 1,, n 1 n 1 n 1, n+ 1 X = p p T u T Monolithic approach Frontal solver (with preconditioning & pivoting, but without disc storage) (or partitioned approach) (or domain decomposition & parallel multi-frontal solver) > 22 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

23 Introduction Modelling of concrete at early ages and beyond Creep in concrete Bazant, Wittmann (eds) Bazant et al Harmathy Bazant, Chern Hansen Bazant, Prasannan De Schutter, Taerwe Sercombe, Hellmich, Ulm, Mang Hydration of cement Jensen van Breugel De Schutter, Taerwe Singh et al Ulm, Coussy Bentz et al. 1998, 1999 Sha et al > 23 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

24 Chemo - hygrothermal interactions Evolution of the hydration process Maturity-type model: Γ ( ), hydr = Γhydr t eq where: t β β eq ϕ T ( ) = t t β ( τ) βϕ( τ) dτ, ( τ) o T { } 4 1, ( τ) = 1+ α 4 [ 1+ ϕ( τ) ] U = exp R hydr 1 T o 1 T ( τ), Γ hydr - cement hydration degree; t eq - equivalent period of hydration; β T - coefficient describing effect of the concrete temperature on the hydration rate; β ϕ - coefficient describing effect of the concrete relative humidity on the hydration rate; i ( ) ( ) Details: T [Bazant Z.P. (ed.), Mathematical Modeling of Creep and Shrinkage of Concrete, Wiley, 1988] i g c i i i X n 1 =,, 1 1 n 1, + n+ n+ + n + 1 p p T u + X 1 = X + X i i i n+ 1 n+ 1 n+ 1 > 24 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

25 Chemo - hygrothermal interactions Evolution of the hydration process Influence of the temperature and relative humidity [Bazant, 1988] 7 1 COEFFICIENT β T [-] U/R=const U/R=f(T) COEFFICIENT β φ [-] TEMPERATURE [K] RELATIVE HUMIDITY [-] > 25 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

26 Chemo - hygrothermal interactions Evolution of the hydration process Evolution of the hydration degree [Ulm & Coussy, 1996] Free water Chemically bound water dγhydr Ea = A% Γ ( Γ hydr )exp dt RT where χ Γ = = hydr χ m m hydr hydr A% Γ ( Γ hydr ) - hydration degree-related, normalized affinity, χ - hydration extent, E a hydration activation energy, R universal gas constant, t time. > 26 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

27 Chemo - hygrothermal interactions Evolution of the hydration process Hydration rate [1/h] Hydration rate as a function of chemical affinity [Cervera, Olivier, Prato, 1999] Hydration degree [-] A A% A 1 κ Γ 1 exp η κ 2 ( Γ ) = + Γ ( Γ ) ( Γ ) hydr hydr hydr hydr > 27 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

28 Chemo - hygrothermal interactions Evolution of the hydration process Evolution of the hydration degree [Gawin, Pesavento, Schrefler, IJNME 2006 part 1 and part 2] x (unhydrated cement) + w H 2 O z (hydrated cement) A Γ x µ unhydr + w µ water - z µ hydr where A Γ ( Γ hydr ) dnunhydr dnwater dn dχ = = = x w z hydr - hydration degree-related chemical affinity, χ - hydration extend, µ chemical potential, N mole number, x, w, z - stoichiometric coefficients. > 28 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

29 Chemo - hygrothermal interactions Evolution of the hydration process Evolution of the hydration degree [Gawin, Pesavento, Schrefler, IJNME 2006 part 1 and part 2] dγhydr Ea = A% Γ ( Γhydr ) β ( Γhydr, ) exp ϕ ϕ dt RT Effect of relative humidity where χ Γ hydr = = χ m m hydr hydr A% Γ ( Γ hydr ) A% Γ - hydration degree-related, normalized affinity, χ - hydration extent, E a hydration activation energy, R universal gas constant, t time. A 1 1 exp κ 2 ( Γ ) = A + κ Γ ( Γ ) ( ηγ ) hydr hydr hydr hydr from: [Cervera, Olivier, Prato, 1999] > 29 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

30 Hygral - chemical interactions Evolution of the hydration process Influence of relative humidity on the hydration rate COEFFICIENT β φ [-] βϕ ( ϕ ) = 1 a a [Bazant, 1988] RELATIVE HUMIDITY [-] > 30 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

31 Chemo - hygrothermal interactions Evolution of the hydration process Influence of the hydration rate on the heat- & mass- sources Hydration rate [1/s] 5,E-06 4,E-06 3,E-06 2,E-06 1,E-06 0,E ,2 0,4 0,6 0,8 1 Hydration degree [-] Qhydr Γhydr mhydr = Qhydr = H t t t m& hydr mhydr = = t Γ t hydr m Hydration rate hydr where: m hydr mass of chemically bound water, Q hydr heat of hydration hydr > 31 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

32 Chemo- mechanical interactions Strain components STRAINS [-] 1.E-04 0.E+00-1.E-04-2.E-04-3.E-04-4.E-04-5.E-04-6.E-04-7.E-04-8.E-04-9.E-04-1.E TIME [hour] total creep chemical mech. & shrinkage thermal dε ( ) = β Γ dγ chem chem hydr hydr ε chem - chemical strain (irreversible) details: [Gawin, Pesavento, Schrefler, IJNME part1 and part2] Caused by: mechanical load and shrinkage > 32 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

33 Hygro-mechanical interactions Shrinkage of concrete Phenomenological modelling of shrinkage strain dε sh = β dϕ sh 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] > 33 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

34 Hygro-mechanical interactions Shrinkage of concrete Capillary pressure & disjoining pressure Forces: in water Capillary pressure Disjoining pressure in skeleton Capillary pressure Disjoining pressure > 34 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

35 Hygro-mechanical interactions Shrinkage of concrete Effective stress principle: s s s e = + α p σ σ I 3,5E-01 3,0E-01 2,5E-01 2,0E-01 1,5E-01 Coefficient x s g ws c = χs p p p [Gray & Schrefler, 2001] 1,0E-01 5,0E-02 0,0E+00 0,0 0,2 0,4 0,6 0,8 1,0 Saturation [-] where ws x s is the solid surface fraction in contact with the wetting film, I- unit, second order tensor, α- Biot s coefficient, p s - pressure in the solid phase > 35 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

36 Hygro-mechanical interactions Shrinkage of concrete Effective stress principle: Shrinkage strain [m/m] experimental results theory by Coussy theory by Schrefler & Gray 0 0% 20% 40% 60% 80% 100% Relative humidity [% RH] [Gray & Schrefler, 2001] s e σ = σ + α I p s g ws c = χs p p p [Coussy, 1995] dσ dσ Idp s " = + α s g c = w dp dp S dp s Experimental data from: [Baroghel- Bouny, et al., Cem. Concr. Res. 29, 1999] > 36 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

37 Hygro-structural - chemical interactions Evolution of the material properties Evolution of concrete porosity & permeability: 6,0E-01 1,0E-14 POROSITY [-] 5,0E-01 4,0E-01 3,0E-01 2,0E-01 1,0E-01 PERMEABILITY [m 2 ] 1,0E-15 1,0E-16 1,0E-17 1,0E-18 1,0E-19 1,0E-20 k = f (hydr) k = f (porosity) 0,0E+00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 HYDRATION DEGREE [-] 1,0E-21 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 HYDRATION DEGREE [-] ( Γ ) = + ( Γ 1) n n A hydr n hydr ( ) ( ) 10 Akn n n = k n k A ( Γ ) = 10 k hydr k k ΓΓ hydr [Halamickova, Detwiler, Bentz, Garboczi, 1995] [Halamickova et al., 1995, Kaviany, 1999] > 37 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

38 Hygro-structural - chemical interactions Evolution of the material properties Evolution of concrete porosity & permeability: POROSITY [%] W/C= 0.3 W/C= 0.4 W/C= 0.5 W/C= 0.6 W/C= HYDRATION DEGREE [%] Experimental results from [Cook & Hover, 1999] > 38 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

39 Hygro-structural - chemical interactions Evolution of the material properties Evolution of concrete strength properties: [De Schutter, 2002] > 39 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

40 Mathematical model Constitutive law for creep strain s F σ e ( t) where & ε v ( t) = & γ ( t) & ε f ( t) Γ ( t) Hydration degree & εc & εv & ε f γ η hydr s ( ) σ e ( ) ( t) s F σ e t t η - creep strains (as sum of viscoelastic and viscous (flow) term); - viscoelastic term; - flow term; Solidification theory for basic creep ( ) ( ε ε ε ε ) d σ = D d ε d ε d ε d ε s e c th ch + dd c th ch - viscoelastic microstrain; - apparent macroscopic viscosity Details: [Bazant Z.P. Prasannan S.,Solidification theory for concrete creep I: formulation, J. Eng. Mech., 115(8), 1989] > 40 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006 = dε = dε + dε c v f Effective stress

41 Mathematical model Constitutive law for creep strain Non-linear dependance of creep on the stress: s 1+ s σ e = Ω = = 1 Ω f 2 s 10 F σ e ( t) ; s ; s ( t) c where: F s σ s e Ω f c - function of stresses ; - normalized stress; - effective stress; - damage at high stress; - compressive strenght; i ( ) ( ) Details: i i i [Bazant Z.P. Prasannan S.,Solidification theory for concrete creep I: formulation, n+ 1 J. n Eng. + 1 Mech., n (8), 1989] i g c i i i X n 1 =,, 1 1 n 1, + n+ n+ + n + 1 p p T u + X 1 = X + X T > 41 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

42 Mathematical model Constitutive law for creep strain Log-Power law: 1 1 = v t Γ 1 p 1 η ( ) t ( t) = m λ0 = + cps hydr α where Microprestress theory ξ Φ( t t ') = q2 ln 1 + λ0 ( t t ') - compliance function Details: T [Bazant Z.P. et al. Microprestressi solidification theory for concrete creep i : aging i and drying i effects, J. ( ) ( ) Eng. Mech., 123(11), 1997] n+ 1 n+ 1 n+ 1 i g c i i i X n 1 =,, 1 1 n 1, + n+ n+ + n + 1 p p T u + X 1 = X + X Φ q > 42 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, S p, c v( t) λ 0 α, m n - parameter from B3 model - microprestress ; - positive constants; - volume fraction of the solified matter; - constant usually equal to 1; - constants of the material;

43 Mathematical model Constitutive law for creep strain Rheologic model: (non-aging Kelvin chain) ( ) ( ξ ) ξ / τ µ Φ t t ' = Φ = A µ 1 e N µ t t E µ with and A µ = 1 E µ ξ = t t - time (age of concrete); - time (age of concrete) at loading time; - elastic modulus of Kelvin unit ' η1 η2 ηn σ Ε1 Ε2 ΕN γ 1 γ 2 γ N γ σg(v,t) Φ(t-t') η v(t) h(t) σ dv(t) dh(t) σ E ε v ε f ε 0 STRAINS elastic (with aging) visco-elastic viscous (flow) +chemical +thermal +cracking Details: [Bazant Z.P. Prasannan S.,Solidification theory for concrete creep I: formulation, J. Eng. Mech., 115(8), 1989] creep > 43 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

44 Mathematical model Constitutive law for creep strain Retardation spectrum technique with τ A µ + 1 = 10τ µ 1 = = L E and where ( τ )( ln10) ( logτ ) µ µ µ µ L ( τ ) µ q2n( n 1) ( 3τ ) µ 1+ 3 n ( τ ) µ n L( τ ) ( logτ µ ) - retardation spectrum; - time interval between two adjacent Kelvin units; Details: [Bazant Z.P. Xi Y.,Continuous retardation spectrum for solidification theory of concrete creep, J. Eng. Mech., 121(2), 1993] > 44 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

45 Mathematical model Constitutive law for creep strain Evaluation of Young s modulus: N 1 1 λµ 1 E = + q 1 + A0 Γhydr µ = 1 Eµ Γ Hydration degree Aging elasticity hydr 1 where the elastic instantaneous term is: q = Γ 1 q / 1 hydr q1 Γ λ µ A hydr 0 - parameter from B3 model; - hydration degree; - coefficient from the exponential algorithm; - term 0 in the Kelvin chain > 45 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

46 Mathematical model Constitutive relationships for thermal properties Thermal capacity Thermal conductivity Thermal capacity [J/m 3 K] 2.45E E E E E E E E E+06 Pc= 0 [Pa] 5.00E E E E E+08 Thermal conductivity [W/(mK)] Pc= 0 [Pa] 1.00E E E E E E Temperature [ o C] Temperature [ o C] ( ) ( 1 ) s w ( 1 ) g gw ( ) ρ C = n ρ C + n S ρ C + S ρ C + ρ C C p ps pw pga pgw pga ( ) ρ C = ρ C 1+ A T T ps pso c o w 4nρ S χ eff = χd ( T ) 1+ ( 1 s n) ρ ( ) χ = χ 1+ Aχ T T d do o > 46 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

47 Mathematical model Constitutive relationships for porous solid Sorption isotherm Saturation degree [%] 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Ordinary Concrete at T=20 C 0% 20% 40% 60% 80% 100% Relative humidity [%] [Baroghel-Bouny, Mainguy, Lassabatere, Coussy, Cement Concrete Res. 29, 1999] > 47 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

48 Numerical example Maturing of a cylindrical specimen (Bentz-Laplante test) Cylinder size - d= 4 cm ; h= 60 cm Initial conditions: T o = K, ϕ o = 99.9% RH, Γ hydr =0.1, for Ordinary Concrete (Bentz test); T o = K, ϕ o = 99.0% RH, Γ hydr =0.1, for High Performance Concrete (Laplante test); Boundary conditions: - convective heat and mass exchange: sealed (adiabatic) - surface mechanical load: unloaded CASE A CASE B CASE C full model; simplified model (no Relative humidity, β ϕ (ϕ)=1) full model adjusted (better agreement with experimental tests) > 48 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

49 Numerical example Maturing of a cylindrical specimen (Bentz-Laplante test) Characteristic properties of different types of concrete (in dry state, after 28 days of maturing) Parameter Symbol Unit OC HPC C-30 Water / cement ratio w/c [-] Aggregate / cement ratio c/a [-] Silica fume / cement ratio s/a [-] Porosity n [%] Intrinsic permeability k [m 2 ] Activation energy E a /R [K] Parameter A 1 A 1 [1/s] ( ) Parameter A 2 A 2 [-] Parameter κ κ [-] Parameter η η [-] (4.7) 5.3 Heat of hydration Q [MJ/m 3 ] hydr Parameter a a [-] Parameter b b [-] Apparent density ρ eff [kg/m 3 ] Specific heat (C p ) eff [J/kgK] Thermal conductivity λ eff [W/mK] Young s modulus E [GPa] Poisson s ratio ν [-] Compressive strength f c [MPa] > 49 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

50 Numerical example Maturing of a cylindrical specimen (Bentz-Laplante test) TEMPERATURE [K] Experiment Numerical TEMPERATURE [K] case A case B case C Experiment TIME [h] TIME [h] Ordinary Concrete High Performance Concrete > 50 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

51 Numerical example Maturing of a cylindrical specimen (Bentz-Laplante test) 1 1 RELATIVE HUMIDITY [-] NSC HPC RELATIVE HUMIDITY [-] case A case B case C TIME [h] TIME [h] OC-HPC comparison (case A) HPC - A-B-C- comparison > 51 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

52 Numerical example Maturing of a cylindrical specimen (Bentz-Laplante test) HYDRATION DEGREE [-] OC HPC HYDRATION DEGREE [-] case A case B case C TIME [h] TIME [h] OC-HPC comparison (case A) HPC - A-B-C- comparison > 52 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

53 Numerical example Autogenous shrinkage in High Performance Cement Paste (Lura, Jensen, van Breugel test) Cubic specimen 50x50x200 mm; Initial conditions: T o = K, ϕ o = 99.0% RH, Γ hydr =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 > 53 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

54 Numerical example Autogenous shrinkage in High Performance Cement Paste (Lura, Jensen, van Breugel test) RELATIVE HUMIDITY [-] RH b=4, fc=10 b=2.8, fc=10 HYDRATION DEGREE [-] Experimental b=4 or 2.8, fc= TIME [h] TIME [h] R.H. development in time Hydr. Degree development in time > 54 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

55 Numerical example Autogenous shrinkage in High Performance Cement Paste (Lura, Jensen, van Breugel test) TEMPERATURE [K] SATURATION [-] TIME [h] TIME [h] Temperature development in time Saturation development in time > 55 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

56 Numerical example Autogenous shrinkage in High Performance Cement Paste (Lura, Jensen, van Breugel test) 0.0E E-05 EFFECTIVE STRESS [Pa] -5.0E E E E E E E+06 TOTAL STRAIN [-] 0.0E E E E E E E-04 total shrinkage chemical creep thermal -4.0E TIME [h] -3.5E RELATIVE HUMIDITY [-] Effective stress development in time Total strains vs R.H. > 56 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

57 Numerical example Autogenous shrinkage in High Performance Cement Paste (Lura, Jensen, van Breugel test) 0.0E+00 TOTAL STRAIN [-] -5.0E E E E E-04 total - experiment shrinkage - experiment total - numerical shrinkage - numerical -3.0E RELATIVE HUMIDITY [-] Effect of shrinkage creep coupling > 57 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

58 Numerical example Tests of Bryant and Vadhanavikkit (1987) Slab thickness=30 cm; concrete: C50 Initial conditions: T o = K, ϕ o = 99.8% RH, Γ hydr =0.3; Boundary conditions: Shrinkage - convective heat and mass exchange: α c =5W/m 2 K; β c =0.002m/s - surface mechanical load: unloaded or load=7 MPa Sealed - convective heat and mass exchange: α c =5W/m 2 K; sealed - surface mechanical load: unloaded or load=7 MPa Age of loading: 8,28,84,182 days > 58 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

59 Numerical example Tests of Bryant and Vadhanavikkit (1987) 0.0E E E E E E-04 TOTAL STRAIN [-] -6.0E E E E E E-03 exp. shrinkage shrinkage exp. 8 days total 8 days exp. 28 days total 28 days exp. 84 days tot. 84 days exp. 182 days tot. 182 days STRAIN [-] -6.0E E E E E E E-03 exp. Sealed - load sealed - load exp. shrinkage shrinkage exp. total creep drying & load -1.8E TIME [days] -2.0E TIME [days] Drying & Load Pickett s effect (8 days) > 59 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

60 Numerical example Tests of Bryant and Vadhanavikkit (1987) Square prism 15x15x60 cm; equiv. cylinder φ=16.3 cm; concrete: C50 Initial conditions: T o = K, ϕ o = 99.8% RH, Γ hydr =0.3; Boundary conditions: Shrinkage - convective heat and mass exchange: α c =5W/m 2 K; β c =0.002m/s; RH amb =60% - surface mechanical load: unloaded or load=7 MPa Sealed - convective heat and mass exchange: α c =5W/m 2 K; sealed - surface mechanical load: unloaded or load=7 MPa Age of loading: 8,28,84,182 days > 60 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

61 Numerical example Tests of Bryant and Vadhanavikkit (1987) 0.0E E+00 TOTAL STRAIN [-] -5.0E E E E-03 exp. shrinkage shrinkage exp. 8 days total 8 days exp. 28 days total 28 days exp. 84 days total 84 days exp. 182 days tot. 182 days STRAIN COMPONENT [-]. -5.0E E E E-03 exp. shrinkage num. shrinkage exp. load num. load exp. load + shrinkage num. load + shrinkage -2.5E TIME [days] -2.5E TIME [days] Drying & Load Pickett s effect (8 days) > 61 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

62 Numerical example Tests of L Hermite (1965) Square prism 7x7x28 cm; equiv. cylinder φ=7.6 cm; concrete: C30 Initial conditions: T o = K, ϕ o = 99.9% RH, Γ hydr =0.3; Boundary conditions: Shrinkage - convective heat and mass exchange: α c =5W/m 2 K; β c =0.002m/s; RH amb =50% - surface mechanical load: unloaded or load=4.9, 9.8 and 12.3 MPa Sealed - convective heat and mass exchange: α c =5W/m 2 K; sealed - surface mechanical load: unloaded or load=4.9, 9.8 and 12.3 MPa Age of loading: 7, 21, 90 and 180 days > 62 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

63 Numerical example Tests of of L Hermite (1965) 0.0E E E E-04 STRAIN COMPONENT [-]. -1.0E E E E-03 exp.shrinkage shrinkage exp. 7days total 7 days exp. 21 days total 21 days exp. 90 days total 90 days exp. 180 days total 180 days STRAIN COMPONENT [-] -4.0E E E E E E E-03 exp.shrinkage shrinkage exp MPa total MPa exp MPa total 9.81MPa exp MPa total MPa -3.0E TIME [days] -1.8E TIME [days] Drying & Load Effect of load (180 days) > 63 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

64 Conclusions The FE model of concrete based on a thermodynamically consistent mechanistic 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 theory for description of concrete cracking during early stages of hydration is in progress. > 64 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

65 Introduction Modelling of concrete at high temperarure Concrete at high temperature Bazant, Thonguthai 1978 Thelandersson 1987 England, Khoylou Bazant, Kaplan 1996 Gerard, Pijaudier-Cabot, Laborderie Gawin, Majorana, Schrefler Bentz 2000 Nechnech, Reynouard, Meftah Thermal spalling Phan 1996 Ulm, Coussy, Bazant 1999 Sullivan 2001 Phan, Lawson, Davis HITECO project NIST workshop 1997 UPTUN project > 65 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

66 Thermo-chemical interactions Concrete at high temperature Dehydration of concrete Dehydration degree [-] 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, Dehydration rate [1/s] 0,035 0,03 0,025 0,02 0,015 0,01 0, ,2 0,4 0,6 0,8 1 Temperature [ o C] Dehydration degree [-] Γ = Γ = ( Γ ) a A exp dehydr ( t ) [ T ( t )] dehydr max Γ t Γ E RT T temperature of concrete A Γ chemical affinity > 66 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

67 Mechanical mat. degradation interactions Mechanical material degradation description Non-local isotropic damage theory [Mazars & Pijaudier-Cabot, 1989] σ F cc d = α d + α d t t c c ε % = 3 ( εi + ) i= 1 2 K o K oc ε max ε ε 1 ( x) = ( x s) ( s) dv V ( x) Ψ ε % r V F ct Ψ( x s) = Ψo exp x s 2l 2 c 2 > 67 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

68 Chemo - mechanical interactions Concrete at high temperature Correlation of dehydration degree & thermo-chemical damage Thermo-chemical damage [-] 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, ,1 0,2 0,3 0,4 V = 1 S σ % = σ = S% Eo( T ) E ( T ) o a σ ( 1 d )( 1 V ) Dehydration degree [-] 0 e 0 σ = (1 d)(1 V ) Λ : ε = (1 D) Λ : ε e E( T ) E( T ) Eo( T ) D = 1 = 1 = 1 1 d 1 V E ( T ) E ( T ) E ( T ) o a o o a ( ) ( ) > 68 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

69 Thermochemical - mechanical interactions Concrete at high temperature Effect of thermo-chemical material degradation on the stress strain curves for C-60 concrete Stress [MPa] 500 C 400 C 300 C 200 C 20 C Strain [-] Experimental results from the EURAM-BRITE HITECO project > 69 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22,

70 Mechanical - hygral interactions Concrete at high temperature Intrinsic permeability damage relationship 1.E-13 Water intr. permeability [m 2 ] 1.E-14 1.E-15 1.E-16 1.E-17 C-60 C-60 SF C-70 C-90 k p f ( T ) p = ko g p o g A A D D 1.E Damage parameter [-] Details: [Gawin, Pesavento & Schrefler, CMAME 2003] Experimental results from the EURAM-BRITE HITECO project > 70 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

71 Chemo - mechanical interactions Concrete at high temperature Load Free Thermal Strain (LFTS) model T hermal dilatation of C-90 Linear strain [m/m] 1,0E-02 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 experimental total therm_dilat shrinkage thermo-chem Temperature [ o C] dε V dv tchem = βtchem ( ) LFTS - irreversible part of thermal strain V thermo-chemical damage parametr LFTS model according to: [Gawin, Pesavento, Schrefler, Mat&Struct 2004] > 71 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

72 Thermo - mechanical interactions Concrete at high temperature Load Induced Thermal Strain (LITS = thermal creep ) model Linear strain [m/m] 1,00E-02 7,50E-03 5,00E-03 2,50E-03 0,00E+00-2,50E-03-5,00E-03-7,50E-03 Transient thermal creep of C-90 0% load 15% load 30% load 60% load 45% load βtr ( V ) dεtr = Q :σ f ( T ) c a 3-D modelling of LITS: [Thelandersson, 1987] s e dt 1 Q = γδ δ γ δ δ + δ δ 2 ( )( ) ijkl ij kl ik jl il jk -1,00E β tr (V) relation according to: Temperature [oc] [Gawin, Pesavento & Schrefler, Mat&Struct 2004] > 72 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

73 Thermo - mechanical interactions Concrete at high temperature Desorption isotherms Porosity of concrete Saturation [-] E E E E E E E E E+10 Porosity [-] silicateconcrete basaltconcrete limestoneconcrete Temperature [ C] Temperature [ C] S ( ) ( 1 1/ b G ) = + b b 1 c E c = (, ) = G G T p p a φ = φ + ( ) A T T 0 φ 0 > 73 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

74 Physical phenomena in heated concrete Causes of thermal spalling phenomenon v vs p = p ϕ 1,6 Saturated vapour pressure (MPa) 1,4 1,2 1 0,8 0,6 0,4 0, ,15 323,15 373,15 423,15 473,15 Temperature (K) moisture clog High pressure of vapour > 74 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

75 Physical phenomena in heated concrete Causes of thermal spalling phenomenon Cement matrix Aggregate moisture clog High pressure of vapour > 75 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

76 Physical phenomena in heated concrete Causes of thermal spalling phenomenon Cement matrix Aggregate stresses: aggregate - cem. matrix + thermal degradation moisture clog High pressure of vapour > 76 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

77 Physical phenomena in heated concrete Causes of thermal spalling phenomenon Stress strain curve stresses: aggregate - cem. matrix + thermal degradation moisture clog High pressure of vapour > 77 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

78 Cement matrix Physical phenomena in heated concrete Causes of thermal spalling phenomenon Aggregate v vs p = p ϕ stresses: aggregate - cem. matrix + thermal degradation + chemical degradation moisture clog High pressure of vapour + water from dehydration > 78 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

79 Physical phenomena in heated concrete Causes of thermal spalling phenomenon stresses: aggregate - cem. matrix + thermal degradation + chemical degradation moisture clog High pressure of vapour + water from dehydration > 79 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

80 Physical phenomena in heated concrete Causes of thermal spalling phenomenon Constraints for thermal dilatation Mechanical stresses stresses: aggregate - cem. matrix + thermal degradation + chemical degradation moisture clog High pressure of vapour + water from dehydration > 80 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

81 ICs: Problem description C- 60 concrete 30x30 cm column ϕ in = 60 % RH, T in = K BCs: Numerical simulations Square column subjected to ISO-Fire convective & radiative for heat exchange: α c =18 W/m 2 K, eσ o = W/(m 2 K 4 ), T e =ISO fire curve, convective for moisture exchange p v = 1000 Pa, β c = m/s > 81 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

82 Numerical simulations Square column subjected to ISO-Fire Material properties C-60 concrete Parameter Symbol Unit Value Porosity n [-] Intrinsic permeability k [m 2 ] Apparent density ρ [kg/m 3 ] 2564 Specific heat s C p [J/kgK] 855 Thermal conductivity λ [W/mK] 1.92 Young s modulus E [GPa] Poisson s ratio ν [-] 0.18 Compressive strength f c [MPa] 60 Tensile strength f t [MPa] 6.0 > 82 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

83 Numerical simulations Square column subjected to ISO-Fire Temperature & relative humidity after 20 min. of fire C-60 concrete > 83 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

84 Numerical simulations Square column subjected to ISO-Fire Gas pressure & total damage after 20 min. of fire C-60 concrete > 84 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

85 Numerical simulations Square column subjected to ISO-Fire during the test Spalling of the C-60 unloaded column > 85 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006

86 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. > 86 Model Concepts for Fluid-Fluid and Fluid-Solid Interactions, Freudenstadt-Lauterbad, March 20-22, 2006