Issue 29

N. Cefis et alii, Frattura ed Integrità Strutturale, 29 (2014) 222-229; DOI: 10.3221/IGF-ESIS.29.19 223 concentration and the amount of formed ettringite from a diffusive-reaction equation, taking into account the aluminate depletion due to the reaction. The ettringite formation implies a volume increase and, once the initial porosity is filled, it induces a volumetric deformation. Several proposal exist in the literature to describe the mechanical consequences on concrete of this swelling. In [7] the volumetric expansion is treated as an eigenstrain and a simple uniaxial stress-strain law is used. In [10] Basista and Weglewski develop a micromechanical model based on the Eshelby solution of the equivalent inclusion method to determine the eigenstrain of the ettringite crystals in cement paste. In [11] a poroelasticity approach is followed and the Mazars’ damage model is used to describe concrete microcracking. Idiart et al. in [9] present finite element simulations of the phenomenon at the meso-scale, considering concrete as a two phase composite, constituted by aggregates and cement matrix and describe degradation by cohesive-crack interface elements. In the present work we follow a weakly coupled approach, similar to that proposed in [12, 13] for concrete affected by alkali-silica reaction. In the context of the Biot's theory of porous media, the concrete subject to sulfate attack is represented as a continuous medium consisting of two phases: the solid skeleton of concrete and the expansive products of the reaction. A phenomenological isotropic damage model, [14], describes the material degradation. The reactive-diffusion model and the mechanical model have been implemented in a finite element code and used to simulate the experimental tests concerning ESA reported in [15, 16]. C HEMO - TRANSPORT MODEL he reactions that take place inside the concrete when in contact with sulfate solutions are briefly reviewed in this section. The usual cement notation will be used: C ≡ CaO; A ≡ Al 2 O 3 ; S ≡ SO 3 ; H ≡ H 2 O. Driven by a concentration gradient, the sulfates present in the environment penetrate into the material and, reacting with the calcium hydroxide ( CH ) and with the calcium silicate hydrates ( C-S-H gel), form gypsum 2 HSC . The gypsum thus produced reacts with the calcium aluminates which are present in the cement paste, forming ettringite 32 3 6 HSAC [6-7]. The set of chemical equations that describe the gypsum formation are 2 2 2 4 2 2 4 2 ) ( 2 HSC OH SO HSC NaOH HSC OH SONa CH          (1) The reactions between the gypsum and the aluminates i P (  1 P 12 4 HSAC mono-sulphoaluminate, AC P 3 2  unreacted tricalcium aluminate, 13 4 3 AHC P  tetra-hydrated aluminate and AF C P 4 4  alumino-ferrite) read respectively:     3 32 3 6 2 4 32 3 6 13 4 32 3 6 2 3 32 3 6 2 12 4 , 2 ) (4 12 3 17 3 2 26 3 16 2 HFA HSAC Hα HSC AF C HSAC H S CH AHC HSAC H HSC AC HSAC H HSC HSAC               (2) As proposed in [9], the reactions (2) can be lumped in a single reaction 32 3 6 HSAC Sq C eq  (3) where C eq is the equivalent grouping of calcium aluminates i i i eq Pγ C     4 1 ,    4 1 i i i i c c  (4) c i is the molar concentration of the single species of calcium aluminate i P and 4 3 2 1 4 3 3 2 γ γ γ γ q     is the stoichiometric weighting coefficient of the sulfate phase. The molar concentration ), ( t s x of sulfate S , varying in space x and time t , can be computed taking into account the diffusion process and the consumption of sulfates due to the ettringite formation, through the following reactive-diffusion equation, D s being the diffusion coefficient for the sulfate concentration and k the rate of take-up of sulfates: T