In this paper, a sequential multi-scale framework to solve mass (air or water) transfer problems is described. Numerical results are checked against mechanical and permeation experimental datas from a reinforced concrete specimen under tensile load designed by C. Desmettre and J.P. Charron [2

Benboudjema, F.

Colliat, J.B.

De Sa, C.

Gatuingt, F.

Jourdain, X.

Vallade, A.

English

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1618MESO-MACRO NUMERICAL APPROACH TOMACROSCOPIC PERMEABILITY OF FRACTUREDCONCRETE – COMPLAS XIX. Jourdain, A. Vallade, J.B. Colliat, C. De Sa, F. Benboudjema andF. GatuingtLaboratoire de Technologie Mécanique (LMT)École Normale Supérieure de Cachan, Université Paris VI, CNRS61 avenue de Président Wilson, 94235 Cachan Cedex, Francee-mail: jourdain@lmt.ens-cachan.fr, web page: http://www.lmt.ens-cachan.fr/Key words: Permeability, Heterogeneous Material, Strong DiscontinuityAbstract. In this paper, a sequential multi-scale framework to solve mass (air or water)transfer problems is described. Numerical results are checked against mechanical andpermeation experimental datas from a reinforced concrete specimen under tensile loaddesigned by C. Desmettre and J.P. Charron [2]1 IntroductionThe durability of reinforced concrete structures is strongly linked to its porosity andmoreover to its cracks pattern which can be anisotropic in many realistic contexts. Createdby mechanical loading or unfavorable thermo-hydrical environment, those cracks increasethe penetration of water and aggressive agents within the material which can severelyweaken its mechanical behavior. It is therefore obvious that being able to compute the flowgoing through concrete structure is a big issue for numerous applications : concrete madebridges or CO2 storage as well as civil nuclear industry. Thanks to a mechanical model ableto represent the cracks opening in heterogeneous materials [3], we then compute a mass(air or water) transfer problem within concrete and deal with permeability on isotropicas well as anisotropic crack patterns. The experimental permeability datas obtained ona tie-specimen under uniaxial loading described by C. Desmettre and J.P. Charron in”novel water permeability device for reinforced concrete under load”[2] is then checkedagainst our numerical results.The section 2 of this paper focuses on the sequential multi-scale framework, dealingwith the cracks FE modelling and the hydro-mechanical coupling, both at a meso scale.The section 3 briefly sum up the experimental study of C. Desmettre and J.P. Charron[2] and checks the experimental datas against our numerical results.1XI International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XI E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) 1619X. Jourdain et al.2 Sequential multi-scale frameworkAs said previously, the aim is to compute a flow problem on isotropic as well asanisotropic crack patterns on cement base materials. Firstly, it means that the mechanicalmodel must be able to deal with heterogeneous materials. Actually, reinforced concreteis composed of steel bars and concrete, which itself is a mix of mortar and aggregates.Secondly, the flow is strongly influenced by the crack openning u as the Poiseuille flowQP going through a u gap between two planes is proportionnal to the cube of the open-ing (QP ∝ u3). Therefore, the crack opening is a crucial value for the mass transfercomputation.Greatfully, the model used [3] has those two specificities :• it represents multi-phasic materials (cement paste, aggregates and steel have differ-ent mechanical and transfer caracteristics),• it gives the value of the opening u of each broken element whatever the sollicitationis.The next part focuses on those two special features of the mechanical model beforedescribing the permeation computation.2.1 Crack representation : Strong discontinuitiesWhen a concrete specimen (ie: made of cement paste and aggregates) is under a tensilesollicitation, the stress in the cement paste quickly reaches its rupture value (ft ≈ 3MPa)resulting in a crack initiation before its propagation. This is a dissipative phenomenonwhere the so-called ”fracture energy” (Gf [J.m−2]) represents the energy dissipated by aone square metre crack. Practically, it can be easily deduced from a light experimentaldevice like a three points bending test and is therefore one of the most used caracteristicsof concrete as the tensile limit stress (σf ) or the Young modulus (E). Thence, those threeparameters seems judicious to be the basis of a mechanical element able to represents thebehaviour of a brittle material like concrete is.Dealing with brittle and quasi-brittle material in computational mechanic is still abig issue and numerous models exist, each one with its benefits and its drawbacks. Onone hand, discrete mesh models [6] are able to represent the crack opening, but theyusually need intensive re-meshing in order to compute the crack direction. On the otherhand, with a Finite Element basis, several approach exist like the smeared crack models[5]. Their main drawback is their mesh dependency, problem which can be bypassesintroducing a length scale wich is mesh objective.Recently [7], [8], [9], an elegant crack model based on FE theory has been developped.This method is mesh independent and doesn’t need re-meshing but introduce a so-calledstrong discontinuity in the displacement field. The strong discontinuity is activated thanksto a yield function Φ written as :21620X. Jourdain et al.Figure 1: Strong discontinuity crack representation and crack opening processΦ(t, q) = t− (σf − q) (1)where t is the traction vector at the discontinuity and σf the limit stress. The softening isintroduced through the variable q = k(u) by considering the exponential form (cf figure1),k(u) = σf(1− exp(− σfGf(u)))(2)The next part focuses on the permeability part.2.2 Mesoscopic scale flow: The Poiseuille lawOnce the cracks initiates, the second step starts : the aim is to compute a masstransfer problem with the damaged mesh. Here is the key point, the material is composedof a ”double porosity”. At a micro-scale, the cement paste porosity is isotropic. Itrepresents the undamaged part of the permeability (pore diametre ¡ 10µm). The flowcan be computed with the Darcy law (eq 3). This equation links the mass flow densityq [kg.s−1.m−2] and the pressure p.q = ρv = − 1µρ.k.1 grad(p) (3)The permeability (k) unit is m2.As soon as the first crack initiates, a bigger porosity appears and the permeabilityrises.If the crack pattern is isotropic, one can analytically links damage to permeability [4].In case of anisotropy, which represents most of the realistic studies, the problem is there-fore more complex to solve. Taking into accounts the anisotropy of the crack pattern isinherent to our model. It is based on a local implementation of the constitutive equationwhich gives the Poiseuille flow between two planes (cf figure 2 and equation 4).qEdge = ρ vEdge = ρw3Edge12µLEdge∆pL(4)31621X. Jourdain et al.XYZBroken BarLbwSpeed Figure 2: speed between two planes defined by the Poiseuille lawwhere µ is the viscosity of the studied fluid (µwater = 8, 90.10−4Pa.s at 20˚C) and ρ is aconstant as this equation 4 is given for incompressible fluid.Once the elementary permeability matrices written, the ”double porosity” is easy tobe written as the sum of the isotropic permeability and the anisotropic permeability dueto the possible crack : K = Kiso+Kaniwhere Kiso= −kµ1 and Kani= w3Lµ(1− n⊗ n)As the anisotropic permeability matrix shows it, there is no flow in the colinear directionof the bar while in the two perpendicular direction, the permeability is equal to w3Lµ.Once the assembly of the elementary matrices done thanks to a classical FE software,the problem can easily be solved and, as a result, the macroscopic flow can be computed.This method is an elegant way to automatically takes into account the tortuosity and theconnectivity of the cracks.2.3 Permeability of RC element under loadThe multiscale framework we present here is clearly suitable dealing with mass transferswithin concrete structures or their components. To that aim we focus on experimentalresults from [2] who designed a coupled tensile – permeability test on Reinforced Con-crete (RC) specimens (610× 90× 90 mm3 concrete element including a 11 mm diameterreinforcement bar). Here we aim at comparing their experimental measures to numericalresults and focus on the permeability evaluation along the failure process.The fine scale mechanical analysis of this RC tie is based on a spatial truss represen-tation [3] built with non-adapted meshes. In order to fit to the experimental conditions,displacements are prescribed at both ends of the steel reinforcement bar. Such increasingload leads to progressive cracking of the concrete element and Figure 3 shows a typicalcrack pattern obtained from a numerical analysis. Three main macroscale cracks are dis-tributed along the tie which is in accordance with experimental observations. Moreover,41622X. Jourdain et al.Figure 3: RC tie: numerical crack patternswe show on Figure 4 that there is a quite good agreement between the measured values ofthose cracks openings during the loading process and the corresponding numerical values.Figure 4: RC tie: maximum crack opening versus stress in reinforcement (experimental [2] vs computa-tional results)On the permeability assessment point of view (here the test deals with water), Figure 5shows the norm of the mass flow vector at the end of the failure process. Although theconcrete specimen contains a large number of mesoscale cracks, it is clear that the masstransfer takes place within a subset of those cracks, corresponding to several percolatedpaths. Thus, on the mass transfer point of view, the former may be seen as a set of51623X. Jourdain et al.macroscale cracks. Hence, it is worth noting that, apart from considering the openingvalue as the pertinent criterion, this mass transfer analysis leads to an other way todetermine this set of macroscale cracks. This analysis being linear and so quite simple todrive, it is a very convenient way to characterize macroscale cracks.Figure 5: RC tie: numerical values of the massflow vector norm Figure 6: RC tie: numerical crack patternsFinally, Figure 6 shows the permeability coefficient – here in [m.s−1] – evolution alongthe load increase. The permeability increases from approximatively 2.10−10m.s−1 to1.10−5m.s−1 both for numerical and experimental studies. It is also worth noting thatthe increasing rate is also quite well represented. Yet, the permeability rise appears forsmaller loading values in the numerical study than in the experimental one. Consideringthe concrete heterogeneity as well as the experimental discrepancy, those results are quitepromising.3 Concluding RemarksThis paper presents the results given by a mesoscopic model able to compute the flowgoing through a specimen under load. Three important points have been validated :• Cracks spacing (3 main cracks on the 610mm long specimen),• Cracks opening (0,35mm maximum opening when the yielding stress is reach in thereinforced bar),• Permeability values during loading (rise of 5 orders of magnitude from the originalpermeability for this study).The numerical results are therefore quite promising, especially knowing that more at-tention can be paid on the boundary conditions and loading to fit to the experimentalstudy.61624X. Jourdain et al.REFERENCES[1] X. Jourdain, J.B. Colliat, C. De Sa, F. Benboudjema and F. Gatuingt. Meso-macro numerical approach to macroscopic permeability of cracked concrete quan-tification.submitted.[2] C. Desmettre, J. P. Charron. Novel water permeability device for reinforced concreteunder load. Mat. and Struct., 1-11, (29 mars 2011).[3] N. Benkemoun, M. Hautefeuille, J. B. Colliat, A. Ibrahimbegović. Failure of hetero-geneous materials: 3D meso-scale FE models with embedded discontinuities. Int Jalfor Numerical Methods in Engineering, 82, 1671-1688, (2010).[4] L. Dormieux, D. Kondo. Approche micromcanique du couplage permabilit-endommagement. C.R. Mecanique, 332, 135-140, (2004).[5] A. Hillerborg. Application of the fictitious crack model to different types of materials.International Journal of fracture, 51, 95-102, (1991).[6] Toto. Tu connais un model discret qui a besoin de remaillage ou nimporte lequel faitlaffaire.[7] G.N. Wells, L.J. Sluys. Application of embedded discontinuities for softening solids.Engineering Fracture Mechanics, 65, 263-281, (2000).[8] D. Brancherie,A. Ibrahimbegovic. Novel isotropic continuum-discrete damage modelcapable of representing localized failure of massive structures. Engineering Compu-tation, 26 . under press (2009).[9] N. Moes, J. Dolbow, T. Belytshko. A finite element method for crack growth withoutremeshing. International Journal of Numerical Methods in Engineering, 46, 131-150.(1999).7

Meso-macro numerical approach to macroscopic permeability of fractured concrete – COMPLAS XI

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Meso-macro numerical approach to macroscopic permeability of fractured concrete – COMPLAS XI

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