Issue 29

N. Cefis et alii, Frattura ed Integrità Strutturale, 29 (2014) 222-229; DOI: 10.3221/IGF-ESIS.29.19 225 1 σ σ bp  (9) The effective stress on the concrete skeleton is related to the total strain ε by an elastic law with isotropic damage D, d is tensor of the elastic properties of the homogenized skeleton: εd σ : ) 1( D  (10) The pressure of the expansive phase depends on the volumetric deformation v ε and on the volumetric expansion ζ due to ettringite formation, M being the Biot’s modulus     ζM ε bM D p v    1 (11) The expansion term ζ can be related to chem v ε for the unconstrained material by the Skempton coefficient ) /( 2 Mb K Mb  : chem v ε Mb Mb K ζ 2   (12) Only tensile damage is considered in this work and its evolution is governed by the loading-unloading conditions proposed in [14], expressed in term of the inelastic effective stress 1 σ σ pβ  , with β ( ) b β  a material parameter which tunes the level of material degradation: 0 ;0 ;0    Df D f D D   (13) The activation function f D depends on the first invariant of inelastic effective stress tensor I 1 and on the second invariant of deviatoric stress tensor J 2 : 0 ) ( ) ( ) ( 2 3 12 2 11 2      Dha DhIa Ia J f D σ (14) where a 1 , a 2 , a 3 are non-negative parameters to be identified through experimental tests. The function h(D) governs the hardening and softening behavior of the material and its expression is                                                    0 75.0 0 0 0 2 0 0 for 1 1 for 1 1 1 ) ( 4 DD D DD DD D D σ σ Dh a e (15) In the above equation e σ is the elastic limit stress, 0 σ is the peak stress, D 0 is the damage corresponding to the peak stress. The parameter a 4 governs the slope of the softening branch of the stress-strain curve. In the finite-element implementation, the exponent a 4 is used to scale the fracture energy density of the material in such a way that each finite element can dissipate the correct amount of energy, independently of its size. This so called ‘‘fracture energy regularization” prevents the occurrence of spurious mesh dependency in the structural global response. R ESULTS he numerical solution of the diffusion-reaction problem and of the subsequent chemo-damage problem are obtained by an ad-hoc developed finite element code. Fracture energy pseudo-regularization is adopted. The finite element internal length for the constant strain tetrahedral elements is assumed to be the cubic root of the volume. To validate the approach proposed in this paper we simulate the external sulfate attack experiments on mortar prism reported in [15] and also simulated in [16]. The experimental campaign was carried out on 25×25×285 mm 3 mortar prisms immersed in a solution with 3 mol/m 2.35 of Na 2 SO 4 . Simulations of the reaction-diffusion process were performed in 3D considering three symmetry planes and solving the two ions formulation (6). The reaction and diffusion parameters used T