BENHUR AND Vi(CA) 2 T, TWO TOOLBOXES TO MODEL CONCRETE AS AN HETEROGENEOUS MATERIAL COMBINING ANALYTICAL AND NUMERICAL APPROACHES

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1 BENHUR AND Vi(CA) 2 T, TWO TOOLBOXES TO MODEL CONCRETE AS AN HETEROGENEOUS MATERIAL COMBINING ANALYTICAL AND NUMERICAL APPROACHES Y. Le Pape (1), C. Toulemonde (1), R. Masson (1) and J. El Gharib (2) (1) EDF R&D MMC Dpt, Moret-sur-Loing, France (2) EDF R&D AMA Dpt., Clamart, France Abstract It is recognized that modelling concrete as an heterogeneous material is an arduous task due to several factors: (i) the wide range of shapes and scales from the smallest particles within the cement paste to the largest aggregates; (ii) scale separation, that is a main theoretical basis of upscaling techniques, is hardly retrieved since the basic principle for concrete formulation is the continuity of the sieving curve in order to obtain the best compact medium, (iii) even at the microscopic level, the cement paste and the so-called interface transition zone (ITZ) properties are still not characterized, (iv) each concrete is a kind of a prototype. However, considerable progress was achieved in the field of testing and numerical computation that allows to challenge accurately multiscale modelling. For practical purpose, modelling concrete as a true heterogeneous material should lead to: (i) a better understanding of the material behaviour; (ii) derivation of macroscopic properties; (iii) the possibility to optimize numerically new formulations. In order to address these questions, EDF Research and Development Division has been developing two toolboxes: Vi(CA) 2 T (Virtual Cement and Concrete Ageing Analysis Toolboxes) and BENHUR (Numerical Homogenized Concrete). This paper presents the current level of their developments. 1. INTRODUCTION As the owner of numerous large facilities, EDF is to warrant their safety and their serviceability. Industrial plants face ageing induced by the environment, the plant usage itself or unexpected pathologies. Thus, the properties of the materials are evolving with time. In addition, in the prospect of building new facilities, the responsibility of a future owner is to prescribe the exposure class and the required properties of a set of concretes depending on their location inside the plant. In both cases, the assessment of the structural integrity requires to handle monitoring, in some cases additional non destructive method, structural computation in combination with the knowledge or the estimation of the history of loadings whether they fall into mechanical, thermal or hydraulic loading. In any case, mastering the 147

2 properties of the material is a crucial matter. Indeed, it must be kept in mind that when decades-old structures are handled, concrete prescription at the time of construction can not be expected to provide explicit knowledge of the relevant parameters required for ageing analysis. It has come clear in the last few years that alternative strategies such as upscaling techniques could bridge the gap between material parameters required for structural assessment and their actual knowledge through building data or non destructive tests. A less conventional approach is to be considered: it can be referred as virtual testing. Before dealing with this specific subject in some more details, it must be understand that virtual testing will not substitute the here above mentioned strategies. It is however an additional trail that may overcome some drawbacks and difficulties. 2. TOOLBOX REQUIREMENTS Virtual testing consists in building numerical toolboxes to model the life of a concrete sample starting from its hydration to degraded states induced by external or internal damaging processes. Model inputs are the constitutive materials (i.e. cement composition, water amount, aggregate...). If successful, the numerical model should be easily handled by the whole concrete community or practical engineers since it does not focus on any specific concrete formulation. Two strategies are pursued in parallel: - Analytical and semi-analytical approaches called OD strategy: EDF R&D has been involved in a constant modelling effort for two years to develop a toolbox called Vi(CA) 2 T Virtual Cement and Concrete Ageing Analysis Toolbox under Scilab platform ( This toolbox already models cement hydration, evolution of poroelastic properties during hydration, interfacial transition zone modelling and hydration heat release. Basic creep and shrinkage are currently under development. This toolbox devoted to practical applications needs to be completed to account for strength evolution, and possibly to more complex problem such as coupled damage and creep problems, leaching modelling, internal sulphate attacks (DEF) and alkali-silica reaction (ASR) problems. - Numerical approach called 3D strategy in the sequel: EDF R&D has also developed a numerical toolbox namely BENHUR Homogenised Numerical Concrete which allows to describe within a finite element framework, concrete as a true heterogeneous materials containing aggregates and paste matrix. Current applications to simple elastic and viscous (pseudo-elastic) cases demonstrated the accuracy of the approach that still needs to be extended to more practical problems. It is to be mentioned that this numerical strategy is limited to R&D applications though its results can be usefully inserted for practical purpose inside Vi(CA) 2 T. Each approach is described in some more details hereafter. 3. 0D STRATEGY Bridging morphological features to practical effective properties has long been the focus of prior research. In the field of cementitious materials, pioneer works [1][2][3] had good 148

3 success in relating concrete formulation and compressive strength. In parallel, concrete deformability as a result of aggregates and hardened cement paste properties is a question that has been addressed since the 60 s. But it is only rather recently that an important key point was finally raised thanks to nano-indentation techniques that allowed to characterize the mechanical properties of hydrated and anhydrous phases, and thus, open the door to upscaling techniques starting from the very microscopic scale [4][5]. Other valuable scientific advances were also addressed in the field of micromechanics of unsaturated porous media [6]. However, there exists no actual practical toolbox that gather all these theoretical results. Figure 1: Vi(CA) 2 T overall architecture Vi(CA) 2 T has been developed toward this goal. Its salient features are: (i) object-oriented concept, so that at any scale, cementitious materials can be seen as heterogeneous materials made of other composite materials or new intrinsic compounds; (ii) mutiphysics toolbox to model hydration, diffusion-related or mechanical problems, (iii) capacity to handle not only deterministic quantities but statistical distribution of properties. Vi(CA) 2 T has been developed using GPL Scilab for its simplicity, its set of implemented numerical toolboxes and graphics and HMI libraries. Vi(CA) 2 T overall architecture is presented on figure 1. Hydration is modelled through a set of chemical equations (see e.g. [7]) chosen by the user. So far, only CEM-I hydration is available. Intrinsic hydrates and anhydrous phases properties are stored in the database and handled automatically to give as an output cement paste compound volume fractions as a result of time or hydration degree. Hydration kinetics follow Tennis and Jennings assumptions. Nucleation-growth process and diffusion controlled kinetics are under current developments [8]. 149

4 Figure 2: sample of hydration equations used by Vi(CA) 2 T. Aluminates hydration equations are handled in successive steps: gypsum reaction, ettringite dissolution, and AFm natural reaction. After this initial stage, mechanical properties can be computed at any step of the hydration process. Various homogenization schemes have been implemented (Mori-Tanaka, selfconsistent, generalized self-consistent, Hashin & Shtrikman bounds, Voigt-Reuss bounds). An output sample is shown on figure 3. Figure 3: Chooz BR1 modulus as a function of the hydration degree (assumed constant paste volume during hydration). Legend : plus: cementitutious matrix using self-consistent scheme, triangle: cementitutious matrix using Mori-Tanaka scheme, diamond: mortar using Mori- Tanaka scheme, circle: concrete using Mori-Tanaka scheme, square: experimental data Granger[10], p. 66 & 83. Note that MT scheme can not account for cement paste setting time. 150

5 ITZ modelling has also been made available thanks to original developments funded by EDF and using asymptotic analysis [9]. It allows to compute the equivalent poro-mechanical properties of a composite made of aggregates and ITZ. If ITZ thickness is assumed constant, it results in a distribution of equivalent properties being a function of the particles size see Fig. 4. As a result of ITZ weak properties, concrete young modulus exhibits a drop. Experimental results and theoretical estimations are consistent though it appeals for extensive experimental investigation about the effective ITZ properties. Figure 4: Aggregate-ITZ equivalent mechanical properties as a function of the particles sizes. Legend: solid line: equivalent Young modulus; dashed line: elastic bulk energy density function; diamond: particle density function. 151

6 Figure 5: influence of ITZ thickness on concrete and mortar Young modulus. 4. 3D strategy Modelling concrete, or more generally heterogeneous materials, by numerical 3D approaches is a rather old concern see e.g. [11].Is has always been limited by the current computer capacities. Though tremendous progress was made in the last decades, modelling concrete is still an arduous task as a result of the following constraints: (i) the large volume fraction of aggregates (70% of the whole, and 40% when upscaling from mortar to concrete), (ii) the practical sieving curve is derived with the idea to obtain the best aggregate compact (Apollonian mix) that induces very small matrix thickness, and thus, aggregates are distributed along a wide range of sizes (approx. 100 µm to a few centimetres). In the sequel, the debate is restricted to upscaling from mortar to concrete. Making first the crude assumption that all particles are spherical, and fulfilling the above mentioned constraints lead to a set of nearly 2500 spheres, with diameters ranging from 5 to 25 mm to be positioned in a representative concrete cube of 15 cm. Putting them randomly within the box is quite an easy task (Monte Carlo or Poisson type method). But, the main problem to overcome remains: how to mesh the aggregates and the connected matrix properly. The common approach focuses on a regular meshing of the particles themselves and then, on a free meshing of the matrix [12][13]. Two drawbacks arise soon: first, the size of the mesh becomes quickly huge (millions of tetrahedron with strong hypotheses made on the sieving curve and distribution process) and thus, can hardly be handled by computer for practical purpose; and, second, it often leads to strongly distorted elements. For practical purpose, an alternative is suggested [13]. It can be summarized as a fuzzy representation of the heterogeneities. The REV is meshed initially and independently from the aggregates. It can be free or regular. Once this task is completed, particles are embedded within this prior mesh. Elements are then categorized whether they fully belong to an aggregate or to the matrix, or 152

7 partially to both. In the later case, homogenized properties are given to the mixed-materials elements. This approach reduces drastically the mesh size that is totally controlled by the user. Of course, the larger is the initial mesh, the most accurate will be the prediction. Two operating questions arise as soon as the basement of the approach is set: (i) does it converge to the real solution when reducing the mixed-materials element volume fraction toward zero; (ii) and if so, what kind of homogenized rule provides the optimal convergence rate. These issues were addressed in a former paper [13] that it is asked to refer for details. Moreover, the main conclusion of the paper is that this method can be used to determine very tight bounds of mechanical effective properties for composite materials with complex microstructures such as concrete. Regarding the practical use of the method a first strategy has been developed to make the choice of the operational scheme. Theoretical developments are made to derive automatically the characteristics of the optimal scheme. These strategies allow the derivation of effective properties for cruder mesh sizes, they give results even for large contrast problems such as diffusion as well as steady creep. Figure 6 showed an elastic REV compressed unidirectionally under vertical load. Using only 200,000 tetrahedra for the free mesh led to an estimation of Young's modulus of 33 GPa in comparison to 33.7 GPa obtained by experimental tests. Matrix and mixes-elements properties were computed by Vi(CA) 2 T on the basis of concrete formulation. Figure 6: Code_Aster ( FE elastic computation showing both stresses (upper and lateral faces) and strains (front face) patterns. Stress paths follow aggregates column-like skeleton. Strain pattern indicates strong horizontal compressed bands. 5. CONCLUSIONS Virtual testing is an alternative trail to derive effective properties of cementitious materials. It is helpful for practical purpose in the following cases: (i) life management cycle: bridge the gap between the qualitative knowledge and the need for quantitative estimation, (ii) building new facilities: accept/reject contractor proposal, (iii) time saving with experimental procedures when performed at a lower scale (creep, diffusion...), (iv) expertise tool: discriminate singular behaviour, (v) understanding the salient microscopic parameters or mix 153

8 quantities that impact the macroscopic properties, (iv) understanding complex chained/coupled multiphysics problems. In order to gather scientific advances in the field of micromechanics, EDF has been developing a toolbox, namely Vi(CA) 2 T, and pursues the aim to make it a practical tool for material engineers. Due to the foreseen limitation of analytical approaches when complex coupled cases will be addressed, EDF has also been developing a 3D numerical tool, namely BENHUR, that allows to make realistic computation with respect to the granular spectrum representation. REFERENCES [1] Féret, R., Sur la compacité des mélange hydrauliques, in Annales des Ponts et Chaussées, 4 (1892) [2] Baron, J. et Lesage, R., La composition du béton hydraulique, du laboratoire au chantier, Rapport 64 LCPC, [3] De Larrard, F. Structures granulaires et formulation des bétons, OA 34 LCPC, [4] Boumiz, A. Sorrentino, D., Vernet, C. and Cohen-Tenoudji, F., Modelling the development of the elastic moduli as a function of the degree of hydration in cement pastes and mortars, In 2nd workshop of hydration and setting: why does cement set? An interdisciplinary approach, RILEM, Ed. A. Nonat, Dijon, France, [5] Bernard, O. Ulm, F.-J. and Lemarchand, E., A mutiscale micromechanics-hydration model for early elastic properties of cement based materials, Cem. Con. Res., 33 (9) (2003) [6] Dormieux, L., Kondo, J. and Ulm, F.-J., Microporomechanics, 1st Edn (John Wiley & Sons, 2006). [7] Jennings, H.M. and Tennis, P.D., Model for developing the microstructure in Portland cement paste, J. Am. Ceram. Soc., 77 (12) (1994) (corrections in J. Am. Ceram. Soc., 78 (9) (1995) 2575). [8] Bezjak, A. and Jelenic, I., On the determination of rate constants for hydration processes in cement pastes, Cem. Con. Res., 4 (10) (1980) [9] Bouby, C., He, Q.C., Gu, S.T. and Pensée, V. A Coordinate-Free Asymptotic Approach to Modelling Curved Interfacial Transition Zones in Cement-Based Materials, in 9th International Conference on Computational Plasticity, Barcelona, Spain, [10] Granger, L., Comportement différé des enceintes de confinement de centrales nucléaires, analyse et modélisation, OA 21, LCPC, [11] Wittmann, F.H., Roelfstra, P.E. and Sadouki, H., Simulation and analysis of composite structures, Mat. Sc. Eng., 66 (2) (1985) [12] Guidoum, A. and Navi, P., Numerical simulation of thermomechanical behaviour of concrete through a 3D granular cohesive model, In Micromechanics of concrete and cementitious materials, Presses Polytechniques Universitaires Romandes, Lausanne, (1993) [13] Wriggers, P. and Moftah, S.O., Mesoscale models for concrete: homogenisation and damage behaviour, Finite Elements in Analysis and Design, 42 (2006) [14] Toulemonde, C., Masson,, R. and El Gharib, J., Modeling the effective behavior of composite: a mixed Finite Elements and homogenisation approach, Compte-Rendu Mécanique (Ac. Sc.), accepted for publication,