MODELLING ELASTICITY OF A HYDRATING CEMENT PASTE

Transcription

1 MODELLING ELASTICITY OF A HYDRATING CEMENT PASTE J. Sanahuja (1,2), L. Dormieux (2) and G. Chanvillard (1) (1) Lafarge Centre de Recherche, France (2) École Nationale de Pont et Chauée, France Abtract Concrete i a complex multi-cale compoite involving multi-phyic procee. A it i the only evolving component of concrete, the cement pate ha a major influence on the development of the mechanical propertie of concrete at early age. Thi contribution focue on the increae of the elatic propertie of a cement pate during it hydration. Upcalling technique are able to predict the effective mechanical characteritic of material from their microtructure. Numerou micro-to-macro model have been developed on metallic alloy, rock, oil, and even biomaterial. Thee homogenization approache are now more and more ued in the fie of buiing material uch a concrete (ee, for example, [1] which addree the hrinkage of concrete). Due to it computational efficiency, the Ehelby-baed homogenization theory for diordered media ha been choen to bui a multi-cale model of cement pate. Thi clearly require a microtructure model decribing the mot important pattern of the geometrical arrangement of the different phae. However, the exact patial ditribution of phae i not neceary. The model alo need a input the elementary elatic characteritic of the phae and their volume fraction. The latter can be computed by a hydration model preented in a joint paper [2]. 1. INTRODUCTION During the hydration reaction of cement, hydrate (product of the reaction) make bond between the diolving anhydrou grain (reactant). The pore pace, initially filled by water, i progreively reduced a hydrate precipitate. The mechanical characteritic of the pate thu evolve during hydration. Thi communication propoe to focu on the tiffne evolution, etimated by Ehelby-baed homogenization of random media. Thi require a model repreenting the morphology of the pate, at leat in a implified way. 2. OBSERVATIONS AND MODELLING OF MORPHOLOGY On polihed ection of hydrated cement pate (figure 1), three phae can be ditinguihed, from the leat tiff to the tiffet: low-denity hydrate (LD), high-denity hydrate (HD), 783

2 anhydrou. From a morphological point of view, it eem reaonable to conider that the low-denity hydrate (alo called outer product) form a matrix in which are embedded compoite incluion made up of an anhydrou core urrounded by a layer of high-denity hydrate (alo called inner product). The eparation between thee two type of hydrate ha been propoed by many author [3, 4, 5] a far a C-S-H are concerned. Figure 1: Polihed ection of a cement pate at w/c = 0.5: microcope obervation and typical nano-indentation grid [6] At a lower cale, thee two type of hydrate are porou media whoe morphology now need to be precied. The growth of C-S-H over the urface of a C3S crytal eem to happen by aggregation of mall flattened particle [7]. The oberved particle have a typical thickne of 5 nm, the larget ide (meauring about 60 * 30 nm) being parallel to the crytal urface. Thee particle are referred to a elementary brick in the equel. In the implified approach preented here, we conider that the whole hydrate of a cement pate are made up of thee elementary brick, packed in a way that need to be decribed. One can reaonably conider that on top of the firt layer, the elementary brick depoit themelve in a much le ordered way. To implify the technical derivation of the model, we conider an iotropic ditribution of the orientation of thee elementary brick in high-denity hydrate. The ize of the pore, which are found inbetween the brick, then range from 5 to 60 nm, which roughly correpond to the gel pore (1 to 50 nm according to [8]). Moreover, the poroity of high-denity hydrate i uppoed to be uniform and contant throughout hydration. Only the volume fraction of thee hydrate in the pate increae a the reaction proceed. Low-denity hydrate contain the pore pace which ha not been included in high-denity hydrate, that i both gel and capillary pore, the latter ranging from 50 nm to 20 m [8]. A way to reach uch large ize i to conider the packing of everal elementary brick, to make platelet, of larger width. Another way, propoed in [9] and which will not be dealt with here, i to introduce a cale eparation between capillary pore and both gel pore and the platelet. A hydration proceed, more and more platelet precipitate in the pace initially filled by water. Thu, the poroity of low-denity hydrate evolve. However, the platelet are uppoed to keep the ame hape along hydration. From a mechanical point of view, the platelet of low-denity hydrate and the brick of high-denity hydrate are aumed to have the ame propertie. Thee remark lead to the morphological model chematically depicted on figure 2. Thi implified repreentation i now tranlated into a homogenization cheme yieing etimate of the elaticity of the pate. 784

3 Figure 2: Schematic repreentation of the propoed morphological model of cement pate 3. DESCRIPTION OF THE HOMOGENIZATION SCHEME The high-denity hydrate material i conidered a a porou polycrytal whoe olid particle are the elementary brick meauring 5 * 30 * 60 nm. It thu eem natural to reort to a elf-conitent cheme to etimate the effective tiffne, denoted by C. The olid phere involved in the uual elf-conitent cheme [10] ha to be replaced by an oblate pheroid (whoe apect ratio i etimated a r = 5 / 30 * ). In fact, we conider an infinite et of pheroid, which differ only by orientation, with an iotropic ditribution. The effective tiffne i itelf iotropic: the effective bulk and hear moduli are denoted by k and g. The etimation of the moduli require the reolution of the auxiliary problem of elaticity depicted on figure 3. The average train of the pore pace i etimated a the train which etablihe in a pherical cavity embedded in an infinite domain whoe elaticity i characterized by the homogenized tiffne C, and with uniform train ( E 0 ) boundary condition at infinity (ee left part of figure 3): (1) ph where S i the Ehelby tenor [11] of a phere in the material of iotropic tiffne C. The average train in a particular oblate pheroid, whoe orientation i characterized by the Euler angle (θ,φ ), i etimated a the uniform train which etablihe in the ame oblate pheroid, embedded in an infinite medium of tiffne C, and with uniform train ( E 0 ) boundary condition at infinity (ee right part of figure 3). Then, the average train in the olid phae i etimated a an angular average, conidering the iotropic ditribution of the pheroid orientation: (2) obl where P ( θ, φ) i the Hill tenor of an oblate pheroid in a media of tiffne C whoe axi i oriented by (θ,φ ), and C i the tiffne of the elementary brick. The elf-conitent tiffne of the high-denity hydrate then read: (3) which yie two calar nonlinear equation whoe k and g are olution. Thu, the elatic moduli of high-denity hydrate are etimated a function of the elatic moduli of the 785

4 elementary brick, the poroity ϕ and the apect ratio r of the brick. Thee etimation can be compared to finite element computation [12] on 3D image of microtructure built putting 21 * 3 * 3 voxel parallelepiped into a cube (ee reult on figure 4). A very good eff agreement i oberved, not only on the effective Young modulu E (calculated from the eff olid elatic characteritic E = 45.7 GPa and ν = 0.33), but alo on the Poion ratio ν (calculated from three different olid Poion ratio ν = 0, 0.2 and 0.33: ee the left part of the curve, at ϕ = 0). The elf-conitent cheme involving prolate pheroid a olid particule thu provide intereting etimation of the effective elaticity of porou polycrytal made up of elongated particle. We admit that it i alo the cae when the olid particle have a flattened hape, uch a the elementary brick of high-denity hydrate or the platelet of low-denity hydrate. Figure 3: Auxiliary problem of elaticity to olve in order to implement the elf-conitent cheme propoed to model the high-denity hydrate Figure 4: Effective Young modulu and Poion ratio of a porou polycrytal made up of elongated particle, etimated by a elf-conitent cheme (for three value of the apect ratio r of the olid pheroid), and numerically computed by FEM [12] Low-denity hydrate are alo regarded a porou polycrytal with flattened olid particle (platelet). Thu, the homogenization cheme which ha been decribed in the previou paragraph i reued. The pate i regarded a a compoite with a matrix (low-denity hydrate) in which are embedded incluion (compoite phere: an anhydrou core urrounded by a layer of higenity hydrate). It thu eem natural to reort to a generalized Mori Tanaka cheme. It i a generalized cheme ince the homogeneou incluion of the uual Mori Tanaka cheme [13] ha to be replaced by a compoite phere. The development of thi homogenization cheme typically require the reolution of the auxiliary problem of elaticity depicted on figure 5 (ee [9] for detail). 786

5 Figure 5: Auxiliary problem of elaticity to olve in order to implement the generalized Mori Tanaka cheme propoed to model the pate 4. INPUT DATA OF THE MODEL The homogenization cheme (built from a morphological model) of cement pate developped in the previou ection need a input the volume fraction and the elementary elatic propertie of the different phae. The volume fraction can either come from an advanced hydration model uch a [2], of from a impler one. Here, we briefly preent a quite baic hydration model baed on the Power model [14]. For completene, five phae have to be conidered: anhydrou (volume fraction in pate denoted by f a ), olid phae of low ( f ) p p and high ( f ) denity hydrate, and pore pace of low ( f ) and high ( f ) denity hydrate. Thee five volume fraction have to be etimated a a function of both the water to cement ratio w/c and the degree of hydration α. The Power hydration model [14] provide the volume fraction of anhydrou (ubcript a), hydrate (h), water (w) and pore (p, gathering the capillary pore whoe volume fraction i 1 f a f h, and the maller gel pore): (4) The hydration reaction i expected to be complete at the ultimate degree of hydration ult α = α, that i when f a = 0 or f w = 0 : (5) Note that thi implified tatement obviouly neglect any kinetical effect. Tenni and Jenning [3] propoed a model to plit the hydrate olid part between low and high-denity: (6) We now write the total poroity, the total volume fraction of hydrate olid ( f h, alo obtained from the Power model) and the poroity of high-denity hydrate (contant by hypothei, ee ection 2): (7) The four equation (6) and (7), with, furthermore, the expreion of a f extracted from (4), allow to derive the five volume fraction of interet. Then, the quantitie which are neceary a input of the homogenization cheme developed in ection 3 are derived a: 787

6 (8) Apart from the volume fraction of the phae, we need the elatic moduli of thee one. The elatic propertie of anhydrou and high-denity hydrate have been meaured by nanoindentation [15, 16] (ee left part of table 1). The effective propertie of low-denity hydrate evolve with time, through the poroity ϕ. The latter depend on the w/c ratio and on the degree of hydration α according to (8). Thu, nano-indentation can only provide a naphot of the tiffne of low-denity hydrate. Thi i why a homogenization cheme dedicated to thee hydrate ha been developed in ection 3. The effective elaticity of the low-denity hydrate depend on the apect ratio of the platelet, the poroity and the tiffne C of the olid making up the platelet. Thi tiffne cannot be aeed by nano-indentation a the elementary brick are too mall. However, a practical way to etimate it i to perform an invere analyi of the elatic characteritic of high-denity hydrate, contant through time and uccefully meaured by nano-indentation. Thi procedure ue equation (3), the apect ratio r of the elementary brick etimated in ection 3, and an etimation of the poroity ϕ = 0.30 (one can find 0.35 and 0.30 according to Tenni and Jenning [3], or 0.28 according to Power). The Young modulu and the Poion ratio of the olid brick are then derived a E = 71.6 GPa and ν = IMPLEMENTATION AND EXPERIMENTAL COMPARISONS In order to implement the model, only one parameter mie: the apect ratio of the olid platelet contituting the low-denity hydrate. The critical poroity above which the effective tiffne of low-denity hydrate vanihe depend on thi apect ratio (ee left part of figure 4). Thu, we can reaonably think that the optimum apect ratio can be calibrated on exp experimental data on etting of the pate. The critical degree of hydration α 0, under which the mechanical trength of the pate i negligible ha been experimentally etimated a a function of the w/c ratio [17]. Thee experimental data are plotted together with model prediction on figure 6. The optimum value of the platelet apect ratio i r = (ee left part of figure 6). The degree of hydration at etting hardly depend of the poroity ϕ of high-denity hydrate (ee right part of figure 6). The arbitrary choice ϕ = 0.30 only ha a light influence. The optimum apect ratio r = of the platelet making up the lowdenity hydrate doe not correpond to the one of the high-denity hydrate ( r = ). Thi jutifie a poteriori the concept of platelet introduced to model the low-denity phae, and defined a a packing of elementary brick. The whole et of data required to implement the model i gathered on table 1. Firtly, the model prediction are compared to the experimental reult of [18]. In thi tudy, Haecker et al. meaured the degree of hydration and the Young modulu of pate hydrated during 14, 28 and 56 day, with two different cement and w/c ratio between 0.25 and 0.6. The agreement i very good on the whole range of w/c ratio (ee the firt two plot of figure 7). Note that the imulation are baed on parameter from the literature, without any adjutment with repect to data from [18]. 788

7 Table 1: Input data of imulation E (Gpa) ν method anhydrou (a) nano-indentation [15] r 0.12 high-denity hydrate () nano-indentation [16] r olid phae of hydrate () invere analyi of ϕ 0.30 Figure 6: Self-conitent etimate of the degree of hydration at etting, and experimental etimation [17] Figure 7: Young modulu of cement pate during hydration and fully hydrated, etimated by the propoed homogenization model (line), and experimentally meaured (dot) [18, 19] We now move on to mature pate, conidering that hydration ha reached the ultimate tate (5). In [19], the Young modulu i meaured on three material (pate elaborated from two Portland cement and nearly pure C3S), everal w/c (from 0.3 to 0.6) and everal time (6, 7, 8, 14 and 24 monthe for cement and 8 and 14 monthe for C3S). The (reaonable) dicrepancie between model and experiment (ee 3rd graph on figure 7) cou be linked to an etimation too imprecie of the real degree of hydration. It i important to note that thi model i only a firt approach: the ditinction between high-denity and low-denity C-S-H i not enough to decribe the whole complexity of cement pate, made up of many hydration product which differ in term of denity, chemical and mechanical characteritic. Thi 789

8 wou be epecially required to ae the mechanical conequence of chemical degradation mechanim, uing a input the reult of durability model uch a the one preented in [20]. REFERENCES [1] Thierry, J.-Ph. and Chanvillard, G., 'Autogenou and drying hrinkage modeling: from pate to concrete', in Concrete Modelling 2008 (Delft, 2008). [2] Guillon, E., 'Phyical and chemical modeling of the hydration kinetic of a OPC pate uing a emi-analytic approach', in Concrete Modelling 2008 (Delft, 2008). [3] Tenni, P.D. and Jenning, H.M., 'A model for two type of calcium ilicate hydrate in the microtructure of portland cement pate', Cement and concrete reearch 30 (2000) [4] Richardon, I.G., 'The nature of the hydration product in hardened cement pate', Cement and Concrete Compoite 22 (2000) [5] Diamond, S., 'The microtructure of cement pate and concrete a viual primer', Cement and Concrete Compoite 26 (2004) [6] Contantinide, G., 'Invariant mechanical propertie of calcium ilicate hydrate (C-S-H) in cement-baed material: intrumented nanoindentation and microporomechanical modelling', PhD thei (Maachuett Intitute of Technology, 2005). [7] Garrault-Gauffinet, S., 'Étude expérimentale et par imulation numérique de la cinétique de croiance et de la tructure de hydroilicate de calcium, produit d hydratation de ilicate tricalcique et dicalcique', PhD thei (Univerité de Bourgogne, 1998). [8] Oberholter, R.E., 'Pore tructure, permeability and diffuivity of hardened cement pate and concrete in relation to durability: tatu and propect', in 8th International congre on chemitry of cement (Rio de Janeiro, 1986) [9] Sanahuja, J., Dormieux, L. and Chanvillard, G., 'Modelling elaticity of a hydrating cement pate', Cement and Concrete Reearch 37 (2007) [10] Hill, R., 'A elf conitent mechanic of compoite material', Journal of the Mechanic and Phyic of Solid, 13 (1965) [11] Ehelby, J.D., 'The determination of the elatic fie of an ellipoidal incluion, and related problem', Proc. R. Soc. Lond. A 241 (1957) [12] Meille, S. and Garboczi, E.J., 'Linear elatic propertie of 2D and 3D model of porou material made from elongated object', Modelling Simul. Mater. Sci. Eng. 9 (2001) [13] Mori, T. and Tanaka, K., 'Average tre in matrix and average elatic energy of material with mifitting incluion', Acta Metallurgica 21 (5) (1973) [14] Power, T.C. and Brownyard, T.L., 'Studie of the phyical propertie of hardened Portland cement pate (nine part)', J. Amer. Concr. Int. 43 (oct to april 1947). [15] Velez, K., Maximilien, S., Damidot, D., Fantozzi, G. and Sorrentino, F., 'Determination by nanoindentation of elatic modulu and hardne of pure contituent of portland cement clinker', Cement and Concrete Reearch 31 (4) (2001) [16] Velez, K. and Sorrentino, F., 'Characterization of cementitiou material by nanoindentation', in Kurdowki Sympoium 'Science of cement and concrete' (Krakow, 2001) [17] Torrenti, J.M. and Benboudjema, F., 'Mechanical threho of cementitiou material at early age', Material and tructure 38 (2005) [18] Haecker, C.-J., Garboczi, E.J., Bullard, J.W, Bohn, R.B., Sun, Z., Shah, S.P. and Voigt, T., 'Modeling the linear elatic propertie of portland cement pate', Cement and Concrete Reearch 35 (2005) [19] Helmuth, R.A. and Turk, D.H., 'Elatic moduli of hardened portland cement and tricalcium ilicate pate: effect of poroity', in Symp. Struct. Portland Cem. Pate Concr. (1966) [20] Barbarulo, R., 'Modeling chemical degradation of cement pate in contact with aggreive olution', in Concrete Modelling 2008 (Delft, 2008). 790