Simulations of the Thermo-HydroMechanical Behaviour of an Annular Reinforced Concrete Structure Heated Up to 200°C Marcus V. G. de Morais*,**, Benoit Bary*, Guillaume Ranc***, Sabine Durand****, Alexis Courtois***** *CEA, DEN-DANS/DPC/SCCME/LECBA Saclay – Bât. 158, 91191 Gif sur Yvette, France [email protected] [email protected] ** University of Cergy-Pontoise, DGC/L2MGC 5 mail Gay-Lussac Neuville sur Oise, 95031 Cergy-Pontoise Cedex, France *** CEA, DEN/DTEC/S2EC/LCEC Valrho – Bât. 051, 30207 Bagnols sur Cèze, France ****CEA, DEN-DANS/DM2S/SEMT/LM2S Saclay – Bât. 607, 91191 Gif sur Yvette, France *****EDF-SEPTEN Division Génie Civil, Installations et Structures Groupe Dynamique et Séisme 12-14 avenue Dutriévoz, 69628 Villeurbanne, France We perform the thermo-hydro-mechanical simulation of the behaviour of an annular reinforced concrete structure with external diameter of 2.2 m and height of 3 m heating up to 200°C during two months. This structure reproduces a typical concrete structure in interim waste storage conditions. The numerical results are compared with experimental ones in terms of evolution of temperatures, gas pressures, relative humidity and strains in the median plan as a function of time. This comparison shows a reasonable agreement. ABSTRACT:

KEY WORDS: modelling, thermo-hydro-mechanical coupling, numerical simulations, concrete, damage, reinforced concrete.

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THM Behaviour of an Annular Reinforced Concrete Structure Heated Up to 200°C

1. Introduction The behaviour of concrete is a topic of great concern in the context of radioactive waste management and nuclear plant containments undergoing temporary accidental situations. For these structures, concrete is typically subjected to thermal and mechanical loadings inducing moisture transfers and leading possibly to cracking. We focus here on the thermo-hydro-mechanical (THM) behaviour of concrete subjected to temperature increases up to 200°C. To describe this behaviour, a simplified coupled model has been developed on the basis of the mechanics of partially saturated porous media (Bary et al., 2008). The assumption that the gaseous phase is composed uniquely of vapour permits to substantially simplify the model by combining the two initial mass conservation equations of water in liquid and gaseous phase into a single one. The constitutive equation for the skeleton relies classically on the Bishop concept of effective stresses. Cracking is introduced in the model by means of an isotropic micromechanical model where microcracks are represented by penny-shaped ellipsoids. Estimations of the mechanical and transfer parameters are then provided by applying appropriate effective-medium approximation schemes. We carry out the simulations of the behaviour of a hollow reinforced concrete structure with external diameter of 2.2 m and height of 3 m, called MAQBETH mock-up. The numerical results are compared with experimental ones in terms of evolution of temperatures, gas pressures, relative humidity and strains in the median plan of the structure as a function of time. This comparison shows a relatively good agreement.

2. Description of the MAQBETH mock-up The MAQBETH mock-up is a large scale experiment developed at the French Atomic Energy Agency (CEA Saclay) in order to support the conception of nuclear interim waste storage structures (Ranc et al., 2003). The design of this mock-up was guided mainly by specific characteristics and loading conditions of this type of structure (high quality concrete, large thick for radioprotection, slow heating kinetics, and moderate temperature level maintained for long periods). The main objective of this study was to provide a series of temporal data regarding the THM behaviour of a large scale experiment. MAQBETH is a reinforced hollow concrete cylinder with an inner and outer diameter of 1 and 2.2 meters, respectively, and a height of 3 meters (Figure 1a). The structure is heated from the inside surface with the help of hollow aluminium cylinders equipped with resistors, and is subjected to a thermal loading path depicted in Figure 1a. The mock-up is laid on a 60mm thick wood plate and its top is recovered with a 200mm thick insulator layer and a 30mm thick wood layer.

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Figure 1. Description of the concrete hollow cylinder: geometry (a), numerical axisymmetrical model (b), and thermal loading path (c).

3. THM governing equations The principal equations of the simplified THMs model developed in (Bary et al., 2008) are briefly described in the following. In this formulation, it is assumed that the gas phase is constituted only of vapour water, i.e., the dry air is neglected. This hypothesis allows combining both mass conservation equations of water (liquid and vapour) into a single one. 3.1. Mechanical behaviour The mechanical behaviour of reinforced steel is assumed as linear elastic. We suppose a perfect adherence between concrete matrix and reinforced steel. The classical Biot formulation adapted to the unsaturated condition case is applied as concrete’s mechanical behavior (see e.g. (Lewis and Schrefler, 1998; Gawin, Majorana et al., 1999; Coussy, 2004)). The stress tensor σ is expressed as:

σ = 2 µ J : ε + 3kL : ε − bIpl − 3α kθ I

[1]

where ε is the strain tensor, L and J are projection operators such that L = 1/ 3I ⊗ I and J = I ⊗I − L with I the second order identity tensor, µ and k the shear and bulk moduli in isothermal drained conditions, b the Biot coefficient, pl is the liquid pressure, α the coefficient of thermal expansion, and θ = T − T0 with T and T0 the current and initial temperatures, respectively. We model the concrete matrix as a porous material composed of a homogeneous matrix in which are dispersed spherical inclusions representing porosity, and penny-shaped ellipsoids

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THM Behaviour of an Annular Reinforced Concrete Structure Heated Up to 200°C

describing microcracks. We assume to simplify that the ellipsoids are identical and randomly oriented, this latter assumption being necessary to comply with the isotropy condition. The ellipsoid distribution is classically characterized by its 3 density ρ = Na , with N the density number and a the radius of the ellipsoids. Denoting as 2c the crack opening (dimension of the ellipsoid in the revolution axis direction), the volume fraction of microcracks is then φc = 4π / 3 ρ X , with X = c a the aspect ratio of the ellipsoid. Homogenization procedures are then applied to estimate the various hydromechanical and transfer parameters as a function of saturation degree and microcrack density parameter, see (Bary et al., 2008) for further details.

3.2. Mass conservation equation The mass conservation of water can be written as (Bary et al., 2008):

φ   ρlv

  ∂T ∂ρ  ∂S  k ∂ε  ∂ρl + Sv v  + ϖ l +  φd′ + 3α − 3α s (1 − φ )   + b*  Sl T T T k t ∂ ∂ ∂ ∂ ∂t   s   

 ∂S b − φ Sl ∂ρ   ∂p φ  ∂ρl dɺ + + ∇ ⋅ ( wl + wv ) +  −ϖ l + + + S v v  l =  Sl ρlv  ∂pl ρlv k s ∂t ρlv ks ∂pl   ∂t  ∂pc Ξ ∂ρ

[2]

1

in which Sl is the liquid saturation degree, φ is the porosity, ρi is the density of component i , φd is the porosity fraction due to dehydration of the solid phase, Ξ = pl ∂b / ∂ρ −(ε − 3αθ)∂k / ∂ρ + φ′c + ϖk s ∂Sl / ∂ρ , k s is the bulk modulus of the solid phase, b* is the saturated Biot coefficient, ϖ = φ(ρl − ρv ) / ρlv − pl / ks (φ − ∂b / ∂Sl ) , and ρlv = Sl ρl + S v ρv . It is implicitly supposed that Sl varies as a function of pc , T and ρ , ∂k s / ∂T = 0 , and k s ≫ σ s − pl . The classical Darcy’s law is used for expressing the mass flux wi = − ( ρ i K ηi ) k ri ( S l )∇pi , where K is the intrinsic permeability tensor, kri and ηi are the relative permeability and the dynamic viscosity of the phase i , respectively. We denote kei the permeability coefficient such that k ei ( S l , ρ) I = K . The water mass dehydration d is expressed by d = k d (T − 60) , if T ≥ 60°C , and k d = 0.018kg / °C (Ranc, Sercombe et al., 2003). The function S l ( pc , T , ρ) integrates a high level of microstructural information in terms of pore size distribution and connectedness. The Kelvin’s law is further used to link the capillary pressure pc to the relative humidity hr = pv / pvs by pc = − ρ l ( RT M v ) ln hr , where M v is the molar mass of water.

3.3. Heat equation The entropy conservation equation is classically written in the form (see (Bary et al., 2008)):

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c(Sl , d )

∂T ∂t

+ ( wl C pl + wv C pv )∇ T −

wv ρv

⋅ ∇ pv = −∇ ⋅ q − Ll → v µ l → v − Ls → l dɺ

337

[3]

where C pl and C pv are the mass heat capacities of the free (bulk) water and vapor; the heat capacity of concrete is expressed by c ( S l , d ) = mds C ds + φρl S l C pl +φρv (1 − S l )C pv + ( d 0 − d )TCbw , with mds the mass of cement and aggregate per unit volume, Cds its heat capacity and Cbw is the heat capacity of the bound water. The terms of heat transported by fluid convection ( wl C pl + wv C pv )∇T and heat dissipation due to the liquid phase compressibility wv ρ v ⋅ ∇pv are both negligible, according to (Bazant and Kaplan, 1996). Ll → v = T ( sv − sl ) is the heat of vaporization and, by analogy, Ls →l = T ( ss − sbw ) is the heat of dehydration (Ulm, Coussy et al., 1999). The heat flux is classically expressed via the Fourier law, q = −λ ( S l , d )∇T with λ ( S l , d ) = λ60 + ∆λ ( S w S wo ) − kd d the experimental thermal conductivity law (Ranc et al., 2003), where ∆λ = λ20 − λ60 , λ20 and λ60 are saturated and unsaturated concrete thermal conductivity (Bary et al., 2008).

4. Numerical Results We solve this system of non-linear equations numerically by a fixed point method in the finite element code Cast3M developed by the French Atomic Energy Agency (CEA). A partitioned procedure is used: each equation is solved via an iterative procedure for a given variable of the system while keeping the others constant. The main concrete parameters are measured experimentally, see (Galle and Sercombe, 2001; Ranc, Sercombe et al., 2003), except the liquid permeability Kw, the evolution of gas permeability as a function of dehydration Kg(d) = Kgo (20°C) exp[αdβ] and the ratio of liquid and gas permeability Kw/Kg, which are calibrated on experimental results. The thermo-hydro-mechanical characteristics of MAQBETH concrete are: initial porosity φo = 10%, gas intrinsic permeability Kg (d) = 10-17 exp[0.2 d 1.0] m2, liquid intrinsic permeability Kw (d) = 5⋅10-4. Kg (d), saturated concrete thermal conductivity λ 20 = 2.2 W/m⋅°C, unsaturated concrete thermal conductivity λ60 = 1.9 W/m⋅°C, thermal capacity of anhydrous cement Cc = 750 J/kg⋅°C, thermal capacity of aggregates Ca = 800 J/kg⋅°C, thermal capacity of water C w = 4184 J/kg⋅°C, thermal capacity of bound water Cbw = 3760 J/kg⋅°C, enthalpy of dehydration L w→gw = 2500 kJ, steel thermal conductivity λ s = 46 W/m⋅°C, thermal capacity of steel Cs = 450 J/kg⋅°C, concrete elasticity modulus (linear evolution) Ec(T) = 38.9GPa (20°C) to 20.4GPa (250°C), Poisson coefficient of concrete ν s = 0.22, steel elasticity modulus Es(T) = 190GPa (20°-250°C), Poisson coefficients of steel ν s = 0.30, concrete thermal dilatation coefficient α = 11.0×10 -6 °C-1.

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THM Behaviour of an Annular Reinforced Concrete Structure Heated Up to 200°C

The boundary conditions (temperature and vapour pressure) are indicated on Figure 1b. At the internal face, we impose a temporal evolution of temperature (heater source, see Figure 1c) and a constant vapour pressure condition (pv = pvst ⋅hr(Sl)), while a constant exterior temperature and an equivalent convective condition (heq = 10W/m/C) is prescribed at the external surface. Due to symmetry, only one quarter of the specimen is modelled in a 2D axisymmetric mode with 4 nodes linear elements. Our preliminary results show a good ratio quality/CPU time (over 1500 time iterations - 200 hours of time simulation - ≈5 hours).

Figure 2. Profiles of relative humidity HR (%) (left) and liquid saturation Sl (%) (right) as function of radial distance r (m) for different times: relative humidity experimental data (symbols) and numerical results (lines). Figure 2 presents the radial profiles of relative humidity (left) and liquid saturation degree (right) at mid height of the structure (symmetry plane). The experimental data of relative humidity are plotted at both graphs. In the second one, this representation is only indicative of moisture migration into concrete matrix. The slight drop of saturation appearing between about 0.30 and 0.55 m from the inner surface results from the presence of a radial macrocrack in this zone. In Figure 2 left, the experimental drying front appears to migrate quicker than numerical one. The gas pressure and relative humidity are strongly dependent on mechanical damage and dehydration (leading to an increase of porosity) of concrete. The parameter α of gas/liquid intrinsic permeability function has an antagonist impact on gas pressure and relative humidity. Indeed, concrete dries more rapidly for great α parameter (>0.30), but liquid pressure drops consequently. The compromise adopted here is to perform the calibration of parameters on gas pressure rather than relative humidity. We consider that the experimental difficulties for measuring relative humidity at varying temperature may lead to measurement uncertainties. Moreover the large sensor dimensions could be responsible for local perturbations in both gas and liquid

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phases, and weakening of mechanical properties so that the sensor zone is a privileged site for crack initiation.

Figure 3. Profiles of gas pressure Pg (bar) (left) and temperature T (°C) (right) as a function of radial direction r (m) at different times: experimental data (symbols) and numerical results (lines). 0,0

ε (mm/m)

JB0c

-0,1

orthoradial JB1c

-0,2

vertical

-0,3 -0,4 -0,5

JB1c ²

-0,6 JB0c

-0,7 -0,8 -0,9

t (hours) -1,0 0

20

40

60

80

100

120

140

160

180

200

Figure 4. Temporal evolution of vertical and orthoradial strains at a point situated in the middle plan of the heating face. We compare on Figure 3 the radial profiles of gas pressure (left) and temperature (right) at structure mid-height for different times. The numerical results of gas pressure represent quite well the experimental gas pressure in terms of position and magnitude of the peaks. Note that this satisfying confrontation is a consequence of the fact that the adjustment of the permeability parameters is performed precisely on gas pressure curves. Nevertheless some discrepancies appear around r = 0.30 m from

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THM Behaviour of an Annular Reinforced Concrete Structure Heated Up to 200°C

the internal surface. The numerical results of temperature radial profiles also correspond reasonably to the experimental data, but we observe some discrepancies near the steel reinforcement (at r = 0.43m). Finally, we present the evolutions of vertical and orthoradial strains as function of time at a point situated on the heating face in the middle plan of the structure on Figure 4. The numerical results are in reasonable good agreement compared to experimental ones. The simulated vertical strains show the appearance of cracking just after 20h, similarly to experimental data (JB1c). Globally, the present mock-up is subject to a strong mechanical damage, principally in the constant temperature stage (after 60 h). Some differences between experimental and numerical strain curves in this phase could be imputed to the absence in our model of compressive-induced damage.

5. Conclusions This work presents the simulation of an annular reinforced concrete structure submitted to heating up to 200°C (MAQBETH mock-up). The numerical results are obtained by means of a coupled THM model developed on the basis of the mechanics of partially saturated porous media. They are compared with experimental data in terms of evolution of temperatures, gas pressures, relative humidity and strains in the median plan of the structure as a function of time. This comparison shows a reasonable agreement. However some problems were found to calibrate the permeability as function of drying due to antagonist effects on relative humidity and gas pressure. The implementation of compressive-induced damage in the THM model will be the next stage of this study. A parametric analysis regarding the impact of thermo-hydrous and mechanical parameters on the global THM response will also be investigated in the future.

Acknowledgments Special thanks are due to EDF, CEA/DDIN/CLTBE and CEA/DRI for their financial support.

References Bary, B., Durand, S., Ranc, G., and Carpentier, O. "A coupled thermo-hydro-mechanical model for concrete subjected to moderate temperatures." Accepted for publication, International Journal of Heat and Mass Transfer, 2008. Bazant, Z. P. and Kaplan, M. Concrete at high temperatures. Material properties and mathematical models. Harlow, Longman Group Limited, 1996.

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Coussy, O. Poromechanics. New York, John Wiley & Sons, 2004. Galle, C. and Sercombe, J. "Permeability and pore structure evolution of silicocalcareous and hematite high-strength concretes submitted to high temperatures." Materials and Structures 34(10): 619, 2001. Gawin, D., Majorana, C. E. and Schrefler, B. A. "Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature." Mechanics of CohesiveFrictional Materials 4(1): 37-74, 1999. Lewis, R. and Schrefler, B. A. The finite element method in the static and dynamic deformation and consolidation of porous media. New York, John Wiley & Sons, 1998. Ranc, G., Sercombe, J. and Rodrigues, S. "Comportement à haute température du béton de structrure – Impact de la fissuration sur les transferts hydriques." Revue Française de Génie Civil 7(4): 397-424, 2003. Ulm, F. J., Coussy, O. and Bazant, Z. P. "The chunnel fire. I: chemoplastic softening in rapidly heated concrete." Journal of Engineering Mechanics 125(3): 272-282, 1999.