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CONCRACK 3 – RILEM-JCI International Workshop on Crack Control of Mass Concrete and Related Issues Concerning Early-Age of Concrete Structures, 15-16 March 2012, Paris, France

133

THERMO-HYGRO-CHEMO-MECHANICAL MODELLING OF THE BEHAVIOR OF A MASSIVE BEAM WITH RESTRAINED SHRINKAGE

Giuseppe Sciumè (1), Bernhard A. Schrefler (1) and Francesco Pesavento (1)

(1) Department of Civil, Building and Environmental Engineering, University of Padua, Italy

Abstract

The behaviour of concrete during hydration is very complex especially in the early hours of curing. The relevant parameters depend in a non-linear manner on the hydration degree. We present a thermo-hygro-chemo-mechanical model for concrete at early age and its numerical solution. The mathematical model is coupled (thermo-hygro-chemical behavior inducing the mechanical one) and has been implemented in Cast3M, the finite elements code developed by the CEA (the French Atomic Agency). The primary variables of the model are: the capillary pressure Pc, the gas pressure Pg, temperature T and the displacements vector u. The model parameters depend on two internal variables: hydration degree and mechanical damage, which are associated to the evolution of hydration and mechanical damaging.

Résumé Le comportement du béton pendant l’hydratation est très complexe surtout dans les

premières heures après la mise en œuvre. Le problème est gouverné par beaucoup de paramètres, dépendant du degré d’hydratation du ciment. Dans cet article, une modélisation thermo-hydro-chimio-mécanique du béton au jeune âge et les résultats des simulations numériques sont présentés. Le modèle mathématique est couplé (comportement thermo-hydro-chimique impactant le comportement mécanique), et a été implémenté dans Cast3M, le code aux éléments finis développé par le CEA (Commissariat à l’Energie Atomique). Les variables primaires du modèle sont : la pression capillaire Pc, la pression du gaz Pg, la température T et le déplacement u. Les paramètres du modèle dépendent de deux variables internes : le degré d’hydratation du béton et l’endommagement mécanique.

1. INTRODUCTION Concrete at early ages is subject to various physical and chemical phenomena which

determine thermal, hygrometric, chemical and viscous strains. Therefore, during hydration the behavior of concrete can’t be predicted using simple models based on phenomenological studies since those can typically introduce errors when changing from laboratory specimens to real applications. In this paper a thermo-hygro-chemo-mechanical (THCM) model for concrete at early age, inspired by the multi-physics model developed by D. Gawin, F.


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Pesavento and B.A. Schrefler [1, 2], is presented and the most important aspects are highlighted.

2. SOLIDIFICATION MODEL The degree of hydration Γ is defined as the ratio between the amount of water consumed

for the hydration of concrete at time t and the quantity of water consumed at time t= ∞ :

( )( )

hydrt

t hydr

mГ

m∞

= (1)

The hydration process rate depends on the chemical affinity of the reaction, relative humidity and temperature. To calculate the hydration process rate, the following equation is used:

( ) ( ) exp aГ h

EdГ Adt RT

β ⎛ ⎞= −⎜ ⎟⎝ ⎠

(2)

where A(Γ) = chemical affinity; Ea = hydration activation energy; R = universal gas constant and β(h) is a function of relative humidity (h).

3. HYGRO-THERMAL MODEL EQUATION The balance equations are written by considering concrete as a multi-phase porous material. The model equations are obtained by means of the hybrid mixture theory (HMT) originally proposed by Hassanizadeh and Gray [3–5], and then applied for geomaterials by Lewis and Schrefler [6] and Schrefler [7], and for building materials by Gawin et al. [1-2]. For the sake of brevity, only the final form of the macroscopic balance equations is given here. The hygro-thermal part of the model is governed by the dry air mass balance equation (Equation (3)), the water mass balance equation (Equation (4)) and the enthalpy balance equation (Equation (5)).

( )2 0gw rggwa w

ga g g ga gg g g

P KkM Mm D PM P

ρ ρμ

⎡ ⎤⎛ ⎞ ⎡ ⎤+∇⋅ ∇ −∇⋅ ∇ =⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

& (3)

( )

( ) ( )( )

2gw rggwa w

gw w g g gw gg g g

relw c g hydr

w

P KkM Mm m D PM P

Kk P P m

ρ ρμ

ρμ

⎡ ⎤⎛ ⎞ ⎡ ⎤+ −∇⋅ ∇ −∇⋅ ∇ +⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦⎡ ⎤

+∇⋅ ∇ −∇ = −⎢ ⎥⎣ ⎦

& &

&

(4)

( ) ( )c eff hydr vap vapeff

C T T L H mρ λ⎡ ⎤−∇ ⋅ ∇ = Γ −⎣ ⎦& & & (5)


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For more details on the parameters of the numerical model and on the meaning of used symbols see the reference papers of Gawin et al. [1, 2]. The desorption isotherm is intimately linked with the microstructure of the cement paste and for this reason its dependence on the degree of hydration is not negligible. Thus the classical analytical expression proposed by Van Genuchten [8] is used and adapted to take into account the cement matrix degree of hydration.

1

1

11

b bc b

pcw

p

Г ГPSa Г

−

− −⎧ ⎫

⎡ ⎤⎪ ⎪⎪ ⎪⎢ ⎥+⎨ ⎬⎢ ⎥⎪ ⎪⎣

⎛ ⎞+= ⎜

⎦⎪ ⎪⎩ ⎭

⎟⎜ ⎟+⎝ ⎠ (6)

where a and b are the classical parameters of the Van Genuchten law, c and Γp are the newly introduced parameters.

4. MECHANICS OF CONCRETE AT EARLY AGE The mechanical behaviour of concrete is modeled by an elastic damage model coupled

with creep, which includes the evolution of the mechanical properties with respect to the hydration degree and damage. The relationship between apparent stressesσ , effective stresses σ% (in the sense of damage mechanics), damage D, elastic stiffness tensor E, elastic strains εel, creep strains εcr, shrinkage strains εsh, thermal strains εth and total strains ε reads:

( )1 D= −σ σ% (7)

( ) ( ) ( )Γ Γ= = − − −el sh th crσ E ε E ε ε ε ε&% & & & & & (8)

The Young’s modulus and the tensile strength increase due to hydration as follows (De Schutter & Taerwe [9]):

( )

0

01E E

γ

∞Γ+

Γ −Γ=

−Γ ( )

0

01t tf fβ

∞Γ+

Γ −Γ=

−Γ (9)

in which Γ0 = mechanical percolation threshold i.e. the degree of hydration below which the concrete has negligible mechanical properties (Young’s modulus, strength, etc.); it is kept equal to 0.1, which corresponds to the usual value reported in the bibliography, γ and β are constants and < >+ is the positive part operator.

The damage criterion is given by Mazars [10], and is linked to the elastic equivalent tensile strain

ˆ :el elε ε ε

+ += .

The thermal strain εth is related to the temperature variation: TTα=thε 1&& , where αT is the thermal dilatation coefficient.


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4.1 Modelling of concrete shrinkage The equation used to calculate the instantaneous elastic shrinkage reads:

( )( ) ( ),3

wcPcT S P

Kα Γ Γ

Γ

= −sh1ε (10)

where α is the Biot’s coefficient, KT is the bulk modulus for the skeleton, Sw is saturation degree; 1 is the unit tensor.

4.2 Modelling of concrete creep by means of the effective stress principle A Kelvin-Voigt chain and three dashpots combined in serial way are used (Fig. 1) to

compute the creep deformation. The first and second cells are used to compute the basic creep strain. The last two cells are used to calculate the drying creep and the thermal transient creep.

Creep and shrinkage are strictly coupled. Even if total stresses are used (see Equation (8)), in the equilibrium equations it is of primary importance to also account for the creep strain induced by the capillary pressure. For this reason in the rheological model the effective stress tensor σeff is used (see Fig. 1).

Figure 1: Creep cells for the prediction of the creep strain caused by the effective stresses

This creep model was developed in Cast3M by F. Benboudjema and J.M. Torrenti. For details see the reference paper [11].

5. CALIBRATION OF THE NUMERICAL MODEL The concrete used in the analyzed structure is a C50/60 concrete cast with a CEM I 52.5N

cement. The composition is given in the following table:

Table 1: Mix design of the concrete used in the calculation

CONSTITUENTS Quantities (kg/m3) CEM I 52,5N CE CP2 NF Couvrot 400 Sand 0/4 GSM LGP 785 Gravel 4/20 GSM LGP 980 Superplasticizer Axim 4019 5.4

Total water 185


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5.1 Hydration adiabatic test The hydration adiabatic test has the objective to follow the concrete’s temperature during

the cast and maturation phase. The fresh concrete has been placed in a 300 mm sealed cubic container, thermally isolated (adiabatic conditions). There is good agreement between the experimentally measured temperature and the numerical one (see Fig. 2). The only available experimental result is the temperature, but the numerical simulation gives also the expected evolution of the saturation degree and of the gas pressure during the adiabatic test. The cement’s hydration is accompanied by a decrease in volume which is equal to about 8,7% of the formed hydrates volume (Le Chatelier 1900). Before the development of a rigid mineral skeleton, this contraction produces only a small decrease in the external volume of cement paste. After the transition from semi-fluid state to semi-solid state (Г > Г0), the decrease in volume due to hydration is incompatible with the admissible deformation of the solid skeleton. Therefore in the capillary pores initially almost saturated by water, the volume of gas increases and consequently the gas pressure shuts down (see Fig. 2). The decrease of the saturation degree causes the autogenous shrinkage. The stoichiometric approach used in the mathematical formulation of the hydration model allows us to compute the autogenous shrinkage mechanically without the introduction of additional constitutive equations.

Figure 2: Results during the three days after the casting

5.2 Loss of mass and shrinkage test Autogenous and total shrinkage are determined by a refractometer with 70x70x280 mm

specimens. Three specimens have been made up for each kind of shrinkage.

Figure 3: Comparison of the simulation results with the experimental data for the test of loss of mass and shrinkage


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After the cast, the specimens have been protected by a plastic film and kept at 20° for the first 24 hours. Then the specimens have been removed of their form and transported to the laboratory. The autogenous shrinkage has been measured without water interchange between the specimen and the environment. For the total shrinkage, the specimens are subjected to the environmental conditions of the laboratory (20°C and 50±5% RH). For the specimen subjected to the desiccation, the loss of water mass has also been measured.

Figure 4: Comparison of the simulation results with the experimental data for the test of loss of mass (left). Curve loss of mass - drying shrinkage (right).

5.3 Mechanical properties during hydration The mechanical properties of concrete have been measured at several ages in order to

investigate the variation of the mechanical characteristics during the hydration. Fig. 5 shows the good agreement between experimental measures and numerical results.

Figure 5: Evolution of Young’s modulus (left) and tensile strength (right) during hydration

5.4 Viscous properties To investigate the delayed behaviour of concrete some cylindrical 110 x 220 mm

specimens were cast in the laboratory (initial temperature 17.3 °C) and saved at 20 °C. After 48 hours, they have been loaded at 60 % of the compressive strength measured on the specimens (Rc = 20.3 MPa => load = 12.2MPa). The test was performed without water interchange between the specimen and the environment (basic creep). The delayed creep deformation has been calculated by deduction of the instantaneous elastic strain from the total strain measured. Comparison of the simulation results with the experimental data for the creep test is displayed on Fig. 6.


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Figure 6: Comparison of the simulation results with the experimental data for the creep test

6. NUMERICAL SIMULATION OF THE MASSIVE BEAM BEHAVIOUR The analyzed test is a large beam specimen realized for ConCrack [12]: the international Benchmark for Control of Cracking in Reinforced Concrete Structures (Fig. 7).

Figure 7: Experimental specimen (left) and finite elements mesh (right)

The test is divided in two phases. During the first 48 hours after casting, the structure is thermally isolated and protected from drying. Then after two days the isolation and the formwork are removed and the structure is conserved during two months in the environment. During the two phases of the test the longitudinal strains of the structure are globally restrained by two metallic struts. After two months the structure is loaded by a static bending test. For more details on the analyzed test see the reference web page of the benchmark [12]. The mesh of concrete is made of 3D elements (Fig. 7). To model the steel, truss elements rigidly linked with the concrete 3D mesh are used. Two truss elements are also used to model the two struts that contrast shrinkage.

6.1 Boundary and initial conditions The boundary conditions are assumed to be of convective type for both heat and mass

exchange. Thus, the convective heat flux qt (Wm−2) and convective water mass flux qh (kg s-1 per m2) are defined as:

( )t t s extT Tϕ= −q n ( )*

c ch h s extP Pϕ= −q n (11)


140

where φt and φh are the thermal and hygral convective coefficients; Ts = temperature on the surface (K); Text = ambient temperature (K); Pc

s= capillary pressure on the surface; Pcext* =

fictitious capillary pressure related to the ambient relative humidity and temperature (calculated using the Kelvin equation), and n is the unit vector normal to the surface (oriented towards the exterior). The initial conditions are summarized in Table 2.

Table 2: Thermo-Hygro-Chemical initial conditions

INITIAL CONDITIONS symbol value unit Temperature T 290.15 K Relative humidity h 0.998 - Hydration degree ξ 0.002 -

6.2 Material parameter To define the material many parameters are required. In Tables 3 to 5 some of the material

data and parameters used for the THCM analysis are classified by type.

Table 3: Thermal parameters

PARAMETER symbol value unit Conduction coeff. (dry) λ 1,5 W m-1 K-1 Latent heat of hydration Lhydr 117,0 ·106 J m-3 Activation energy / R Ea/R 5369 K-1

Table 4: Hygral parameters

PARAMETER symbol value unit Parameter a in Equation (7) a 2,3·107 Pa Parameter b in Equation (7) b 2,1 — Parameter c in Equation (7) c 1,1 — Parameter Γs in Equation (7) Γp 0,1 % Intrinsic permeability (when Γ= 1) K∞ 6·10-22 m2

Table 5: Mechanical parameters

PARAMETER symbol value unit Young modulus (when Γ= 1) E∞ 39,4 GPa Tensile strength (when Γ= 1) ft∞ 4,65 MPa Poisson ratio ν 0,19 — Thermal expansion coefficient (when Γ=1) α 1,2·10-5 1/K

Creep cell 1: spring (when Γ= 1) 1bck −∞

24 GPa Creep cell 1: retardation time 1

bcτ 20 days


141

6.3 Thermo-hygro-chemical results from the casting until sixty days During the test, the temperature has been measured in several points of the specimen. For

the first phase (structure isolated) to take into account the thermal bridge of the lateral isolation [13], two different equivalent convective coefficients are used (0.73 W K-1m−2 and 3.9 W K-1m−2). For the hygral part sealed conditions are assumed. For the second phase a uniform thermal convective coefficient (10 W K-1m−2) is used and solar radiation is taken into account. To compute the convective water mass flux the hygral convective coefficient is set equal to 5.10-14 kg s-1 m−2 Pa-1.

Figure 8: Temperature in the central point of the beam (left). Hydration degree in the central point of the beam (right)

Figure 9: Specimen orientation and surrounding (left). Temperature after 2.25 days (right)

6.4 Mechanical results and four point bending test During the first and second phases the longitudinal displacements of the specimen have

been globally restrained by two metallic structures. Therefore during hydration the thermal extension of the concrete has been restrained and so in this phase the section of concrete was compressed. After the hydration, on the other hand, is the shrinkage of the beam that has been contrasted.

After two months a static four point bending test until rupture has been carried out. Compression will be assured by eight jacks and live controlled by a pressure sensor with an


142

independent data registration system. The load is applied with increments of 50 kN and each loading step is kept during 20 min. For more details on the analyzed test see the reference web page of the benchmark [12].

Figure 10: Position of the displacement measurement points (and coordinates)

Figure 12: Relative displacement between the point C and D (left). Axial force in the two metallic truss that restrain shrinkage (right)

Figure 11: Deformed configuration (x 500) and damage after 60 days. Face exposed to the sun

(left) and face not exposed to the sun (right)


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Figure 12: Comparison between the numerical and experimental crack pattern (left). Curve force vs displacements during the bending test (right)

7. CONCLUSION The analyzed case confirms that the factors influencing mainly the behavior of concrete at

early age are: installation and environmental conditions, the material properties, geometry and boundary conditions. Creep has a very important role because it relaxes the tensile stress and moderates cracks phenomena.

The presented application case is useful to show which can be the potentialities of this THCM numerical model for concrete at early ages. However for the model to succeed in simulating real cases accurately, the knowledge of the concrete is essential. Mechanical and thermal parameters do not pose important problems because they are easily found experimentally but hygral parameters are more difficult to be estimated; in addition, all the properties vary with respect to the hydration degree of cement. To solve this problem some micro-scale and meso-scale THCM models are studied (with respect to the hydration chemical processes) to obtain an estimate of the input parameters for the numerical model simply from concrete's mix data (w/c ratio, aggregates, additives). In this way the numerical tool will also be easier to use, even for users without specialized knowledge of the used multiphysics approach.

REFERENCES

[1] Gawin, D., Pesavento, F. and Schrefler, B.A., 'Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part I: Hydration and hygro-thermal phenomena', International Journal for Numerical Method in Engineering, 67 (3) (2006) 299-331.

[2] Gawin, D., Pesavento, F. and Schrefler, B.A., 'Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part II: Shrinkage and creep of concrete', International Journal for Numerical Method in Engineering, 67 (3) (2006) 332-363.

[3] Hassanizadeh, S.M., Gray, W.G., 'General conservation equations for multi-phase systems: 1. Averaging procedure', Advances in Water Resources, 2 (1979) 131-144.

[4] Hassanizadeh, S.M., Gray, W.G., 'General conservation equations for multi-phase systems: 2. Mass, momenta, energy and entropy equations', Advances in Water Resources, 2 (1979) 191-203.

[5] Hassanizadeh, S.M., Gray, W.G., 'General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow', Advances in Water Resources, 3 (1980) 25-40.

[6] Lewis, R.W., Schrefler, B.A., 'The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (2nd edn)'. Wiley: Chichester, 1998.


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[7] Schrefler, B.A., 'Mechanics and thermodynamics of saturated-unsaturated porous materials and quantitative solutions', Applied Mechanics Reviews (ASME), 55 (4) (2002) 351-388.

[8] Van Genuchten, M.Th., 'A closed-form equation for predicting the hydraulic conductivity of unsaturated soils', Soil Science Society of America Journal, 44 (5) (1980) 892-898.

[9] De Schutter, G., Taerwe, L., 'Degree of hydration based description of mechanical properties of early-age concrete', Materials and Structures, 29 (6) (1996) 335-344.

[10] Mazars, J., 'A description of micro and macroscale damage of concrete structures', Engineering Fracture Mechanics, 25 (5-6) (1986) 729-737.

[11] Benboudjema, F., Torrenti, J.-M., 'Early-age behaviour of concrete nuclear containments', Nuclear Engineering and Design, 238 (10) (2008) 2495-2506.

[12] ConCrack: International Benchmark for Control of Cracking in R.C. Structures, in 'www.concrack.org', 2011.

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THERMO-HYGRO-CHEMO-MECHANICAL …demo.webdefy.com/rilem-new/wp-content/uploads/2016/... · the capillary pressure Pc, the gas pressure Pg, ... Le comportement du béton pendant l’hydratation