PSI - Issue 33

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

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The corrosion process has been modelled via the electrochemical kinetics equations. The first step consists in the determination of the electric potential for each of the semi-reaction involved in the corrosion process via the Nernst equation � � � � � ����� � �� � ��� � � �� � �� � (14) From the electric potential distribution on the surface of the steel rebar is possible to determine the corrosion current density rearranging the Butler-Volmer equation at the equilibrium condition, obtaining I ���� � ��� � �� � �� �� �� � � ����� �� ��� � ����� �� � � � � � � � � (15) Once the corrosion current density is known, it is possible to calculate the associated radius expansion due to the formation of rust deposits on the surface of the rebar via the Faraday law dr � � � ���� ������� �� ��� �� � � � � � �� dt (16) Lastly, the rebar swelling is obtained dividing the change in the radius by its initial value � � � � � (17) 2.3. Cracking As the corrosion of the steel rebar progresses, the swelling process generates overpressures on the surrounding concrete. Once the concrete traction resistance is reached, cracks nucleate and propagate. Multiple cracks might form, leading to spalling phenomenon in which the concrete cover is expelled from the element. The damaging process in the concrete cover is modelled via phase-field approach to fracture, where cracks nucleation and propagation are described via the minimization of an energy functional which depends on the displacement field u and the damage field α. Here, only a brief description useful for the comprehension of the paper is presented. A complete description of the method could be found in (Bourdin et al., 2008). The total potential energy functional is obtained as the sum of two contributions depending on the strain energy density and fracture energy density. The strain energy density has been divided into two terms to consider the different materials. The concrete strain energy has been split into its positive and negative part to account for the concrete asymmetric damaging behavior and is given by �� � � � ��� � � � � � � � ∙ � � � d � � � � 〈tr 〉 �� � d � � � � � ∙ � � �� � � � � � � d � � � � 〈tr 〉 �� � d � � � � � ∙ � � (18) The steel strain energy accounts for the swelling obtained from the corrosion process. To do so, only the elastic part of the strain tensor is considered and the strain energy density for the steel rebar is given by � � � � � � � � � � � ∙ ℂ � ∙ � � � � (19) The fracture energy term, which is proportional to the fracture toughness G c , approximates the Griffith’s energy term as follows