PSI - Issue 47

L.A. Almazova et al. / Procedia Structural Integrity 47 (2023) 417–425 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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mm; 8 mm}. Thus, over 60 initial geometries are considered to investigate the effect of different geometrical parameters on the pit growth. The vessel material is linearly elastic structural steel with Young's modulus = 2.1 ∙ 10 11 Pa and Poisson's ratio = 0.3. Boundary conditions are as follows: symmetry conditions are imposed along the slice of the half of the sphere; a pressure of 1 MPa is applied to the inner surface of the sphere. Since the rate of pits growth is supposed to be linearly dependent on the stresses at the bottom points, the depth of each pit at every time step increases depending on the stress values obtained at the bottom points of the corresponding pit at the previous time step. The growth rate of each pitting at step k ≥ 1 is calculated according to the following rule: = ( + ∙ ) , (1) where is the maximum principal stress (in MPa) at the depth of the corresponding pit for step k; a and m are experimentally determined kinetics constants (Pavlov, 1987 and Gutman, 1994). Increment of each pit depth and the corresponding total pit depth at step k>1 are: +1 = ∙ ∆ = ( + ∙ ) ∙ ∆ , ℎ +1 =ℎ + +1 (2) where ∆ t is the time step. At the first time step, k=1, the total depth of each pit equals to the initial depth of the pit and 1 =0 . Thus, the following algorithm is implemented for each vessel of a specified initial geometry: • construction of the geometrical model, setting material properties and boundary conditions; • meshing and solving; • determination of the maximum principal stress at the bottom points of the pits; • calculation of the rates of pit growth at the current time step by Eq.(1); • calculation of the depth of pits at the next time step by Eq.(2); • construction of the geometry at the next time step – with changed pitting depths and applying boundary conditions; • back to the second step. Calculations according to this scheme for each initial geometry continue until the depth of one of the pits reaches 90% of the width of the vessel (in our case 9 mm). Additionally, to ensure the convergence of the numerical solution (its sensitivity to the mesh parameters), multiple calculations with different sizes of elements are carried out for a few CAD-models and for all the considered models, the mesh was built in such a way to be more refined at the pit region, with a smooth transition to a coarser mesh in regions far from pits. The convergence of the solution with regard to the time step ∆ t was checked as well; ∆t =4 [ ] was chosen for the further calculations, where is chosen unit of time. 3. Results and discussion At the first part of the research, all possible combinations of the initial pit depths, h1 and h2, from the set {0; 2 mm; 4 mm; 8 mm} and distances D between them from the set {2 mm, 4 mm, 6 mm, 8 mm} were considered on the surface of the shell with radii R=350 and r=340 mm. After analysis of the first part, smaller distances D and the sphere radii were considered as well. Kinetics parameters, and , used for calculations are as follows: = . [mm/ ] and = . [mm /(MPa ) ]. 3.1. Comparison of the growth of a single pit and two pits The obtained results revealed, that for the considered parameters, the presence of a second pitting slightly affects the growth rate of the first pit (for example, for initial geometries with R=350, r=340 mm and initial pits depths h1=0, h2=2 mm and h1=2 mm, h2=2 mm, at D = 2 mm). Since, on average, the instant pit growth rates, at the same time, for the geometries with one and two pits differ by no more than 2%, and the depth increments by no more than 1%, that does not significantly change the time t* when the depth of one of the pits reaches 90% of vessel wall thickness. Nevertheless. it turned out that the presence of the second defect extended the life of the structure by 6-7%; for