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Institut de Recherche en Génie Civil et Mécanique (GeM), UMR-CNRS 6183, Ecole Centrale de Nantes,. Nantes, France. M. Matallah. RiSAM, Université de ...
Bicanic Mang Meschke de Borst

editors

The book is of special interest to researchers in computational concrete mechanics, as well as industry experts in complex nonlinear simulations of concrete structures.

Computational Modelling of

Conference topics and invited papers cover both computational mechanics and computational modelling aspects of the analysis and design of concrete and concrete structures: * Constitutive and Multiscale Modelling of Concrete * Advances in Computational Modelling * Time-Dependent and Multiphysics Problems * Performance of Concrete Structures

CONCRETE STRUCTURES

The EURO-C conference series (Split 1984, Zell am See 1990, Innsbruck 1994, Badgastein 1998, St Johann im Pongau 2003, Mayrhofen 2006, Schladming 2010, St. Anton am Alberg 2014) brings together researchers and practising engineers concerned with theoretical, algorithmic and validation aspects associated with computational simulations of concrete and concrete structures. The conference reviews and discusses research advancements and the applicability and robustness of methods and models for reliable analysis of complex concrete, reinforced concrete and pre-stressed concrete structures in engineering practice.

Computational Modelling of

CONCRETE STRUCTURES

Editors: Nenad Bicanic, Herbert Mang Günther Meschke, René de Borst

Volume 1

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EURO-C 2014

Computational Modelling of Concrete Structures – Bic´anic´ et al. (Eds) © 2014 Taylor & Francis Group, London, ISBN 978-1-138-00145-9

Multi-scales computation of creep deformation of concrete at very early-age M. Farah, F. Grondin & A. Loukili Institut de Recherche en Génie Civil et Mécanique (GeM), UMR-CNRS 6183, Ecole Centrale de Nantes, Nantes, France

M. Matallah RiSAM, Université de Tlemcen, Tlemcen, Algeria

ABSTRACT: The strength of the cement-based materials increases with time due to the increase of the formed hydrates. During to the hydration process, micro-stresses occur due to the shrinkage of the cement paste. The characterization of the restrained shrinkage deformation at very early ages could be done by applying an equivalent load (creep type). Based on a method for the calculation of the effective visco-elastic response for a non-aging material, an extension to aging materials has been applied. The visco-elastic strains were calculated in a representative elementary volume which allows determining the effective creep compliance tensor. First of all, the “intrinsic” visco-elastic parameters of the C-S-H were calculated by an inverse approach. By considering that the aging creep is due to the evolution of the volume fractions of the concrete components and that they keep intrinsic visco-elastic coefficients, the competition between the concrete solidification and the constant loading was established. 1

INTRODUCTION

The study of the creep of concretes has an important role in characterizing the shrinkage of cement-based materials (A. Bentur & al., 2001). Several studies link the creep phenomenon to the main hydrates of calcium silicate hydrates (C-S-H) (Bazant, 1988) which deform under a constant load. At early ages the creep loading of concrete is more influenced by the cement hydration process than mature ones, the creep deformation decreases significantly as the age of loading increases (Briffaut, 2012). Modelling of macroscopic creep deformation of concrete needs a good knowledge of its microstructure. A number of multi-scales models were developed to model the effective behaviour of concrete from the nanometer-scale (Sanahuja & al., 2009), (Ulm & al., 2004). These models use the homogenization techniques to pass from one scale to another upper. Indeed, for a visco-elastic heterogeneous material, the correspondence principle between elasticity and linear visco-elasticity through the Carson-Laplace transform is used. This allows using classical homogenization schemes (MoriTanaka, self-consistent scheme, etc.), directly in the transformed space (Christensen, 1969) and (Hashin, 1962) to calculate the effective viscoelastic properties. But these techniques are limited to non-aging materials and the extension to aging

materials is rarely dealt (Briffaut & al. 2011) due to difficulties of inverting the Laplace-Carson transform. Within this framework, the aim of this study is to introduce an original multi-scales method for creep of concrete at early-ages by taking into account the strength evolution of the microstructure. This method operates directly in the time domain (Tran & al., 2011) and avoids difficulties of inverting the Laplace-Carson transform. Three scales have been considered to model the effective behaviour of concrete. The choice of the scales was made thanks to the definition of the representative elementary volume (REV): 1) concrete with a mortar matrix and aggregates, 2) mortar with cement paste matrix and sand grains, 3) cement paste with a visco-elastic hydrates matrix and residual clinkers and “elastic” hydrates. At the cement paste scale a strong assumption was made that only C-S-H undergo the creep phenomena while the other constituents have an elastic behaviour. This assumption was made by considering that creep is due to the viscous behaviour of C-S-H restrained by the elastic phases in the cement paste, as the residual clinkers (Bazant & Prasannan, 1988, Acker 2001) which was considered available at early ages in this study. The other hydrates have been considered elastics with properties determined by nanoindentation (Ulm, 2004). So, the matrix of the cement paste scale was formed by the C-S-H

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and the capillary porosity. First of all, the viscoelastic parameters of C-S-H—which could be considered intrinsic thereafter—were estimated by an inverse approach starting from an experimental comparison made on mature concrete (for which we have considered that the overall porosity can not evolve) of two months by applying a constant tensile load (Saliba & al., 2013). An analytical formula (Ricaud & al., 2009) was used to calculate the visco-elastic parameters of CSH independent from the porosity. Then, the overall porosity was deduced vs. age and the visco-elastic parameters of the matrix of the REV at the cement paste scale were calculated for different ages from the setting time of the cement paste (Grondin & al., 2010). As previously reviewed, the strength of concrete increases with age due to the growth of the volume fraction of solidified matter. So, to take into account the competitiveness between the applied load and the solidification of the material during the creep test, we have applied three types of loading: constant, progressive and bearing load. This paper is organized as follow: in the first section, the algorithm and the equations used to formulate and solve the local problem are reviewed briefly; then, the calculation of the visco-elastic parameters of the matrices at the three scales is presented in the second section. Finally, the last section presents the evolution of the creep deformation of concrete at two different ages of loading. 2 2.1

THE PROBLEM FORMULATION The local visco-elastic problem

The local problem is defined over a REV. We consider a volume V formed by two distinct phases: a matrix Vm and n inclusions Vi (i = 1,n). The matrix has a visco-elastic behaviour, where its compliance (J ) is defined by a Kelvin-Voigt model with three chains (Figure 1).

A constant load F is applied on one of the surface boundary Γ1 of V according to the unit normal vector n, and the surface boundary Γ2 is fixed. These conditions imply local displacements fields u(y), local strain fields ε (y,t) and local stresses fields σ (y,t) in each point y of V. The visco-elastic isotropic problem is written as follow: divσ ( y, t ) = 0

(1)

σ ( y, t )

(2)

1 ε ( y, t ) = (∇u( y, t)) 2 F σn u=0

Generalized bounded Kelvin-Voigt model.

∇ ( y, t ))

(3) (4) (5)

where C(y,t) represents the elastic stiffness tensor depending on time. The relation between the local strains and stresses fields is given by the following relation:

ε v ( y, ) J ( ) σ v ( y, t )

(6)

where ε v(y,t) represents the local visco-elastic strains field and σ v(y,t) the local visco-elastic stresses field. According to the Kelvin-Voigt model, J(t) is given as follow: J (t ) =

1 i =3 1 +∑ ( E i =1 ki

−t

e i)

(7)

where E represents he elastic modulus, ki the stiffness and τi the characteristic time. The resolution of the local problem gives a relation between the average stresses 〈σ 〉V and the average strains 〈ε 〉V linked by the effective creep tensor Jhom: εV 2.2

Figure 1.

(t, y ) : (ε ( y, t ) ε ( y, t ))

J hom

σV

(8)

Digital concrete model

In order to take into account the random size distribution of heterogeneities, the digital model has been used to give a realistic representation to the cement-based materials (Mounajed 2002; Grondin & al. 2007). At a given scale, the REV is constituted of two phases (a matrix and inclusions) spatially distributed in a random way. Each phase is characterized by a set of physical and geometrical properties. A specific algorithm has been developed to make a spatial and random distribution of these phases on the basis of a finite element grid. First, all grid elements have matrix properties. Then, aggregates are placed in the grid from the biggest to the smallest, according to the inclusion

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size distribution, such that no overlapping is obtained with other particles already placed. For this, the distance between the center of gravity of the new inclusion (with a radius of Ri2) and the inclusion already validated (with a radius of Ri1) should be superior to the distance dmin defined as the mean radius of the inclusions: d min =

Ri

Ri 2 2

(9)

The properties of the grid elements located in an aggregate area are those of aggregate. The number of inclusions is given by: V Ni = tot Vi

the creep of concrete. So, a creep compliance Jh(t) is affected to the matrix at the cement paste scale. The homogenization calculation gives the effective creep compliance of the cement paste Jphom. At the upper scale, the visco-elastic coefficients of the mortar matrix Jp(t) is equivalent to Jphom. The homogenization calculation gives the effective creep compliance of the mortar Jmhom. At the upper scale, the visco-elastic coefficients of the concrete matrix Jm(t) is equivalent to Jmhom. The homogenization calculation gives the effective creep compliance of the concrete Jchom. 3

(10)

where Vtot is the total volume of elements and Vi the volume of an inclusion. The presented model is implemented in the finite element code Cast3M (Verpaux & al.1988). The resolution of the algorithm was as follow (Figure 2). At the lower scale, assuming that the visco-elasticity of the cement paste is strictly dependent on that of the C-S-H and that the cement paste is formed by a matrix constituted by of C-S-H and pores and inclusions composed by the other hydrated phases and residual clinkers (CH, Ettringite, gypsum, C3AH6, FH3, C3S, C2S, C3 A, C4 AF), the modelling at this scale was simplified by considering only one type of particle with a unique size and elastic characteristics derived from the average of the elastic characteristics of the different hydrates (excluding C-S-H) and clinkers. This assumption was made because we were interested by the restraining effect of the elastic phases on the C-S-H and it was based on the different theories made on the role of C-S-H on

Figure 2. Three levels of the homogenization of concrete.

DETERMINATION OF THE VISCOELASTIC PARAMETERS OF C-S-H

Tensile creep tests on mature concrete (Saliba & al., 2013) (Figure 4) have been used to determine the visco-elastic parameters of the C-S-H. The material

Figure 3.

The digital model for the three scales.

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Figure 4.

Tensile creep test setup.

properties and the experimental procedure were given in detail in (Saliba & al., 2013). The figure 5 shows the damage localisation after 15 days of loading of specimens loaded at 70% and 85% of the maximal strength. We observe that the specimen loaded at 70% did not show damage in the simulation. Thanks to the acoustic emission measurements (figure 5), this result has been confirmed. However, an exponential increase of the number of hits during the tertiary creep has been observed for the specimen loaded at 85%. An inverse approach was applied to calculate the visco-elastic parameters of C-S-H. At each scale the visco-elastic coefficients of the matrix were calibrated to adjust the compliance of the material volume. At the first step, the visco-elastic coefficients of mortar Jm(t) were adjusted from experimental results on tensile creep compliance of concrete (figure 6) to obtain the best fitting of the strain of concrete given by Jchom. At the second step, the visco-elastic coefficients of the cement paste Jp(t) were adjusted to obtain Jmhom at the mortar scale (figure 7), equivalent to Jm(t) defined above. Then, at the lower scale, the visco-elastic coefficients of the matrix Jh(t) were calibrated to obtain Jphom (figure 8), equivalent to Jp(t). The parameters used at the three scales are recapitulated in (table 1). An analytic formula (Ricaud and Masson, 2009) was used to extract the “intrinsic” coefficients of C-S-H kiint by excluding the part of the pores in the case of mature concrete where their volume fraction (fp) can be considered constant and equal to 43.3%: kint

kbci

3.A A fp )

(11)

4

where A f p ) =

Figure 5. Damage localization in digital concrete specimens loaded at 70 and 85% of the maximal strength after 15 days; and measurement of the acoustic emission activity in concrete specimens under the same conditions.

fp 1− fp

Figure 6. Comparison between the measured concrete compliance and that calculated with the calibrated viscoelastic coefficients of mortar (Jm(t)).

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Table 2.

4

kiint

Values GPa

k1int k2int k3int

85.5 28.5 37.05

CALCULATION OF THE AGING CREEP OF CONCRETE

4.1 Figure 7. Comparison between the creep compliance of mortar obtained at the previous step and that calculated with the calibrated coefficients of the cement paste (Jp(t)).

The intrinsic values of C-S-H.

Calculation of the volume fractions of components with hydration

The hydration process which determined the volume of each component in the microstructure is based on the Arrhenius’ equation:

τi

d ξi  = A(ξi ) dt

(12)

where A and τ represent the normalized affinity and the characteristic time, respectively, ξi the hydration coefficient of the clinker i. The hydration process leads to the dissolution of clinkers and to the formation of hydrates with a volume defined by: n ⎛ nP M ρ ⎞ VkP (t ) = ∑ ⎜Vi 0 kR k C ⎟ ξl (t ) k = 1, m nl M l ρk ⎠ l ⎝

Figure 8. Comparison between the creep compliance of the cement paste obtained at the previous step and that calculated with the calibrated coefficients of the matrix (Jh(t)).

Table 1. The visco-elastic parameters of matrices (CP indicates the Cement Paste).

Mortar Aggregates Cement paste Sand grains Matrix of CP Inclusions in CP

E GPa

ft MPa

k1bc GPa

k2bc GPa

k3bc GPa

22.4 60 18 80 12 45

2 6 1.8 4 1.6 3

776.7 – 300 – 150 –

472 – 100 – 50 –

98.1 – 90 – 65 –

where Vi0 represents the residual clinkers in the cement paste, Vkp(t) the new formed hydrates, M the molar mass, ρ the mass density and n the mole. The index k represents the products (clinkers), l the reactants and c the cement. In this study, the hydration model is used to calculate the volume fraction of C-S-H and the volume fraction of the assembly hydrates-clinkers vs. time. Also, it gives the evolution of the volume fraction of the capillary porosity (Grondin & al., 2010). By applying the self-consistent scheme, the effective Young’s modulus of the assembly hydrates-clinkers is calculated with the intrinsic values of each component given in (Bernard & al., 2003). 4.2

The values of the Kelvin-Voigt coefficients for C-S-H are given in (table 2).

(13)

Calculation of the viscoelastic parameters with hydration

With the evolution of hydration the volume fraction of the main hydrates increases and fp decreases (water is consumed and the voids are filled with the formed hydrates). The evolutions of the C-S-H, the fp and the other phases are given in figure 9.

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− a constant load equal to 70% of the strength of the material at the loading age, − a progressive load to maintain the load-tostrength ratio equal to 70% which considers the solidification of the material, − a bearing load to reach the load-to-strength ratio equal to 70% at defined steps of time, describing an experimental procedure.

Figure 9. Evolution of the volume fractions of C-S-H, the porosity and (the clinker and other hydrates) with the age. Table 3. Evolution of the visco-elastic parameters of the matrices with the age.

Matrix of CP Cement paste Mortar

Age H

E GPa

k1bc GPa

k2bc GPa

k3bc GPa

fP %

16 24 >48 16 24 >48 16 24 >48

9.5 10.5 12 13.12 15.85 16.7 17 21 24

66.46 142.74 148.24 170.43 286.18 300 483.91 745.46 776.7

22.15 47.58 49.41 56.81 95.39 100 294 453.02 472

28.8 61.84 64.24 51.13 85.85 90 61.12 94.15 98.1

63 44.2 43.29 43 31 30 28.38 20.46 19.8

The values of fp with the age were used to simulate the aging creep of concrete. For that, at the cement paste scale, the relation (14) was applied to calculate the visco-elastic parameters of Jh(t): i kbc =

4 kini t 3A f p )

The figure 11 shows the three different loads. With the three types of loading, we obtain three different creep deformations, including instantaneous strain combined with the basic creep. The simulation has been performed for two ages of loading (16h and 24h). The results are shown in the figures (12 and 13). First at all, we observe the effect of the age of loading on the creep deformation. At 16h the creep deformation (obtained by deducing the instantaneous deformation) is more important than that observed at 24h of loading especially for the progressive and the bearing load. The difference is

Figure 10. Evolution of the kifp of the matrix of the cement paste with the age.

(14)

The evolution of kibc vs. age is represented on figure 10. The homogenization algorithm was applied to calculate the visco-elastic coefficients of the matrices at each scale (table 3). One can observe that when the age increases, the porosity decreases, and the bulk coefficients increases showing the solidification of the material. 4.3

The early-creep deformation results

To characterize the competition between the solidification and the creep of concrete, three types of load have been applied:

Figure 11.

The three types of loading.

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Figure 12.

The creep results of concrete loaded at 16h.

Figure 13.

The creep results of concrete loaded at 24h.

As a result, the kinetic of the creep deformation is weak during the loading. By applying a bearing load, we try to keep the same level of loading at each point of bearing change, so creep load reaches the solidification at these points. The kinetic of creep deformation increases as the creep load increases at the change of the bearing. Finally, with the progressive load, the load creep exceeds the solidification of the material and the kinetic of creep deformation is very important even after 15 days of loading for the two age of loading. By analysing the curves (figure 13), we observe that the bearing load reaches the progressive one at the points of bearing change when the specimen is loaded at 24h. However, when the age of loading is 16h, we observe a margin between the two curves (Figure 12) which remains acceptable. This difference is influenced by the effect of the age of loading. After some days of loading (2 days figures 12 and 13), the concrete undergoes an important deformation at 16h compared to that at 24h. At 16h the material is viscous; its mechanical properties are still weak to carrying load. But at 24h of loading the material is more rigid; the hydration process reaches an advanced stage (the number of bonds between the hydrates are important). Therefore, the choice of the bearing load in the creep test remains a reliable experimental method. 5

related to the hydration process (called the aging effect); the deformation decreases as the age of loading increases (Briffaut & al., 2012). Based on the Bazant’s solidification theory (for the mature concrete) which we assume to be valid for the young concrete, this result can be explained by the growth of the bearing-load volume fraction of hydrated cement. This fraction continues to growth progressively due to the formation of the bonds between the hydrated particles. The growth of the bonds is due to the phenomenon of polymerization of tricalcium silicates hydrates. Thus, with the increasing of the bonds, the volume fraction of cement hydrates (which are capable of carrying load) increases. The choice of the type of loading is very important at very early ages. As clearly shown in the figures (12 and 13), there is competitiveness between the creep and the solidification of the material. With a constant load, the mechanical parameters of concrete continue to increase especially its strength due to the growth of the bearing load volume fraction. So, in this case the solidification process is more important than the creep load.

CONCLUSION

The multi-scales method proposed in this work has a major interest to model the global behaviour by taking into account the evolution of its microstructure. Indeed, with the evolution of the hydration process, the volume fraction of the main hydrates (C-S-H) increases by evolving the microstructure. The calculation of the porosity vs. age by the hydration model allowed to obtain the aging viscoelastic parameters of the matrices and thus to simulate the creep of concrete at different ages. However, it remains to verify the intrinsic values of C-S-H calculated in this paper. To do this will be the objective of second study by performing a series of the flexural creep tests on a new concerete formula. The application of the bearing load will be chosen to take into account the evolution of flexural strength with the age of concrete. REFERENCES Acker P. (2001). “Micromechanical analysis of creep and shrinkage mechanisms. Creep, Shrinkage, and Durability Mechanics of Concrete and Other QuasiBrittle Materials,” Proceedings of ConCreep-6@ MIT, Elsevier, London, 15–25.

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Bazant Z.P., Prasannan S. (1988). “Solidification theory for aging creep”, Cement and Concrete Research, 18, 923–932. Bernard O, Ulm F, Lemarchand E (2003) “A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials”, Cement and Concrete Research, 33, 1293–1309. Briffaut M., Benboudjema F., Torrenti J.M., Nahas G. (2011). “Numerical analysis of the thermal active restrained shrinkage ring test to study the early age behavior of massive concrete structures”, Engineering Structures, 33, 1390–1401. Briffaut M., Benboudjema F., Torrenti J.M., Nahas G. (2012). “Concrete early age basic creep: Experiments and test of rheological modelling approaches”, Construction and Building Materials, 36, 373–380. Christensen R.M. (1969). “Viscoelastic properties of heterogeneous media”, Journal of Mechanics and Physcis of Solids, 17, 23–41. Grondin F., Dumontet H., Ben Hamida A., Mounajed G., Boussa H. (2007). “Multi-scales modelling for the behaviour of damaged concrete”, Cement and Concrete Research, Volume 37, Issue 10, Pages 1453–1462. Grondin F., Bouasker M., Mounanga P., Khelidj A., Perronnet A. (2010). “Physico-chemical deformations of solidifying cementitious systems: multiscale modeling”, Materials and Structures, 43 (1), 151–165. Hashin Z. (1962). “The elastic moduli of heterogeneous materials”, Journal of Applied Mechanics, 29, 143–150. Mounajed G. (2002). Exploitation du nouveau modèle Béton Numérique dans Symphonie: Concept, homogénéisation du comportement thermomécanique

des BHP et simulation de l’endommagement thermique, Cahiers du CSTB n° 3421, septembre. Ricaud J.M., Masson R. (2009). “Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours”, International Journal of Solids and Structures, 46, 1599–1606. Saliba J., Grondin F., Matallah M., Loukili A., Boussa H. (2013). “Relevance of a mesoscopic modelling for the coupling between creep and damage in concrete”, Mechanics of Time-Dependent Materials, DOI 10.1007/s11043-012-9199-4. Sanahuja J., Dormieux L., Le pape Y.,Toulemonde C. (2009). “Modélisation micro—macro du fluage propre du béton”, 19éme Congrès Français de Mécanique, Marseille, 24-28 août. Smilauer V., Bazant Z.P. (2010). “Identification of viscoelastic C-S-H behavior in mature cement paste by FFT-based homogenization method”, Cement and Concrete Research, 40, 197–207. Tran A.B., Yvonnet J., He Q.-C., Toulemonde C., Sanahuja J. (2011). “A simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials”, Computer Methods in Applied Mechanical Engineering, 200, 2956–2970. Ulm F.-J., Constantinides G., Heukamp F.H., “Is concrete a poromechanics materials?—A multiscale investigation of poroelastic properties”, Materials and Structures, Volume 37, Issue 1, pp 43–58. Verpaux P., Charras T., Millard A..(1988). “Une approche moderne du calcul des structures. In: Fouet J., Ladevèze P., Ohayon, R. (eds.) “Calculs des Structures et Intelligence Artificielle”. Pluralis, Paris.

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