PSI - Issue 33

Francesco Freddi et al. / Procedia Structural Integrity 33 (2021) 371–384 Author name / Structural Integrity Procedia 00 (2019) 000–000

377

7

Ψ� � � � � G � � � � ℓ ��� ℓ � d ℓ � � ℓ ‖∇ ‖ � �

(20)

Therefore, the total energy functional is given by Π ℓ � � � � � � �� � � � �� � �� d � � � � � � � d � � � � � � � Ψ� � � � d � The evolution of the fracture within the solid is obtained by solving the minimization problem of the total energy functional, given by min � ��∈� Π ℓ � � � � � �� � ∈ � � �Ω ℝ � � � � � �Ω �0 1�� � � on � � (22) By differentiating the energy functional with respect to the strain tensor and the phase-field the Euler-Lagrange equations are obtained � di� � � 0 � � � � � ℓ ��� ℓ � � � G � d ℓ � � ℓ � � �1 � �ℂ � � ∙ � � 0 (23) where T is the Cauchy stress tensor equal to � � �� �� �� � �� �� � � �� � �� � �� � � � �� � � �� � (24) An irreversibility condition to the damage field is considered to avoid the healing of the material. The Euler Lagrange equation for the phase-field changes to the following Karush-Kuhn-Tucker conditions ⎨⎩ ⎧ � � � � � ℓ ��� ℓ � � � G � d ℓ � � ℓ � � �1 � �ℂ � � ∙ � � 0 � � � 0 � � � � � ℓ ��� ℓ � � � G � d ℓ � � ℓ � � �1 � �ℂ � � ∙ � � � � 0 (25) 3. Numerical tests At each time step, the carbonation process is solved via the set of equations (4, 7, 8, 10) with a fully implicit finite element scheme for the different concentrations, keeping the damage parameter fixed. Once the pH at any point of the rebar reaches values lower than 9, the corrosion phenomenon is described using the equations (14, 15, 16). Lastly, using the rebar swelling value obtained from equation (17), the damage state is obtained solving the Euler-Lagrange equation (23)a coupled with the constrained evolution equations (25). All the numerical simulations have been performed using the open-source finite element computing platform for solving partial differential equations FEniCS (Logg et al., 2012). 3.1. Model validation The proposed model has been validated against results from the literature. Following (Papadakis et al., 1991), the carbonation process of a rectangular specimen of L = 300 mm and H = 100 mm of ordinary portland concrete with a w/c ratio equal to 50% at 65% RH has been considered. Carbon dioxide with a 50% concentration at a temperature (21)