PSI - Issue 47

Alexander Ilyin et al. / Procedia Structural Integrity 47 (2023) 290–295 A. Ilyin and Y. Pronina / Structural Integrity Procedia 00 (2019) 000–000

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The purpose of the study is to assess the service life of the toroidal shell depending on different geometrical parameters. 3. Method of solution According to the solutions presented by Jordan (1962) and Kydoniefs and Spencer (1967) and our computational results for thick shells , maximum normal stress σ 1 in a toroidal shell under internal pressure is the circumferential stress along the cross section. Stress distribution in the toroidal shell is non-uniform along the cross section; therefore, the rate of dissolution varies from one point of the cross-section to another. For the sake of minimization of computational time, the problem is reduced to the 2-D axisymmetric problem for the cross section of the torus loaded with a constant pressure on the evolving internal boundary. The 4-node 2-D finite element (FE) PLANE182 is used. The linear elastic isotropic material model was considered. In contrast to the work of Pronina (2017) where non-uniform wear was modeled under the assumption about the shape of the corroding surface, affecting the results, in this study we do not make any such assumptions, neglecting, however, the change in the normal to the surface in the process of corrosion before its localization. A new APDL code for solving boundary value problems with a priori unknown boundaries was created and launched in the ANSYS FE software. The algorithm consists of the following steps. (I) At initial time t = 0, the inner contour of the torus cross section is a circle with radius r i , approximated by a chain of rectilinear segments. The number of these segments is equal to the number of finite elements along this surface. (II) Pressure p is applied along the normal of each segment. (III) Initial value of the principal stress, σ 1 , at each element is determined by solving the boundary value problem stated in (I)–(II) by FEM. (IV) For a chosen time increment dt, the new position of each node on the inner surface (and thus its new shape) is determined by the use of equation (1). (V) Normal pressure is applied on the evolved inner contour. (VI) The principal stress, σ 1 , along the updated contour is determined by solving a new boundary value problem stated in (IV)–(V) by FEM. Steps (IV)–(VI) are repeated for the next time steps until one of the critical conditions is met. The service life, t* , of the shell is defined as the time at which either an effective stress (here it is σ 1 ) reaches the maximum allowable stress, σ* , or the minimum thickness becomes equal to the minimum allowable thickness, h* (here it is half of the initial thickness). The choice of the time step must be in agreement with the chosen step in space. Calculations were carried out for various time and spatial step sizes to ensure the obtained solution is mesh-insensitive. Note that relatively strong non uniformity of the stress distribution may lead to localization of the corrosive wear when the involved model of the corrosion kinetics doesn’t hold and, moreover, numerical solution may become mesh-sensitive. Our calculations revealed that in sufficiently thick-walled shells, localization of corrosion occured before any of the mentioned critical conditions was met. The problems of stability of the corroding surface morphology were addressed in the works of Pronina (2017) and Zhao and Pronina (2019). Interaction of multiple corrosion pits or cracks is considered in the works of Okulova et al. (2023) and Abakarov and Pronina (2022). In the next Section, only cases with mesh-insensitive solutions are presented – before the strong localization of the corrosion begins. 4. Results and discussion Series of calculations was conducted for different dimension parameters (the thickness and the distance parameter of the torus) and values of internal pressure. For all the presented figures, the outer radius of the toroidal shell, r o ,