Issue 29

N. Cefis et alii, Frattura ed Integrità Strutturale, 29 (2014) 222-229; DOI: 10.3221/IGF-ESIS.29.19 224 sk s D t s s )) (grad div(     (5) During the process of delayed ettringite formation also the calcium aluminates concentration ), ( t c c eq eq x  decreases and the phenomenon affects the diffusion process of the sulfates. A more accurate description can be obtained by considering a second order chemical reaction in a two-ions formulation and computing the evolution of both the concentration of sulfate and aluminates from the following system:              s c q k t c s ck s D t s eq eq eq s )) (grad div( (6) Note that no diffusion term is present in the second equation since aluminates can not move in the cement paste. To integrate the system of differential Eq. (6), one should specify proper boundary conditions fixing the sulfate concentration s or its normal gradient and initial conditions for both sulfate and aluminate concentrations. The simplified single-ions formulation (5) is obtained for eq ck k  , with eq c constant. The two-ions formulation (6) predicts a faster penetration of the sulfate with respect to the one-ion formulation (5), since the depletion of aluminates reduces the sulfate consumption due to the reactive term s ck eq in (6a) and thus results in a higher diffusion. In the present work the two-ions formulation has been implemented. Assuming that ettringite is the only reaction product governing the expansion of the concrete, the volumetric strain of chemical nature chem v ε is obtained from the amount of reacted calcium aluminates eq eq reac c c c   0 (i.e the difference between the initial aluminates molar concentration and the current one) and the volume change associated with the reaction. For any of the individual reactions described above, Eq. (2), the volume change for unit volume associated to the formation of one mole of ettringite can be calculated as in [7]: 1 Δ    gypsum i i ettringite i i ma m m V V (7) where ettringite i mm , and gypsum m are the molar volumes (m 3 /mol) of the aluminate phase, of the ettringite and of the gypsum, respectively, and a i is the stoichiometric coefficient involved in the reaction. To obtain the overall volume change per unit volume one has to multiply the change related to one mole by the number of reacted moles ) ( 0 i i i reac i i c cm cm   and then take the sum of the contributions coming for the different aluminates. Using the lumped formulation (3) one finally obtains 0 0 Φ ) ( f c c ε eq eq chem v    α with    4 1 Δ i i i i i γm V V α (8) In the above equation  denotes the positive part of  and 0 Φ f is a fraction of the initial porosity 0 Φ , introduced to take into account the fact that the reaction products partially fill the initial porosity, without producing any macroscopic expansion. M ECHANICAL BEHAVIOR OF CONCRETE he mechanical response of the material to the chemical expansion expressed by Eq. (8) is computed in this work by a poroelastic-damage model. Within the framework of the Biot’s theory [17], the concrete is described as a two- phase material: the homogenized skeleton phase, including cement paste and aggregates, and the expansive phase of the products of the reaction. The total stress σ is the sum of the effective stress acting on the solid skeleton σ  and of the stress on the reaction products phase, p being the pressure at the microscale, b the Biot’s coefficient related to the concrete porosity and 1 the unit second-order tensor: T