On local strength of a spherical vessel with pits distributed along the equator

A BSTRACT . The effect of multiple shallow corrosion pits on the strength of a spherical vessel subjected to internal pressure is studied. The pits are considered both randomly and evenly distributed along the equator on the outer surface of the vessel. The dependencies of the stress concentration factor on the number of the pits are compared for linearly elastic and elastic-plastic material with hardening. The behavior of the vessels made of elastic and elastoplastic materials turns out to be qualitatively different. The approximation of periodic pits arrangement is discussed.


INTRODUCTION
ressure vessels are common devices for various industrial applications [1][2][3][4]. The most preferred vessel shape to hold internal pressure is spherical [5]. Moreover, temperature conditions of the environment have the least effect on the contents of the spherical vessel due to the smallest surface area per unit volume. Pressure vessels are usually exploited in aggressive environments that may result in general or local corrosion of their parts. General mechanochemical corrosion of elastic or elastic-plastic thin-and thick-walled spherical vessels was modeled, e.g., in the works [6][7][8][9][10]. Pitting corrosion may be initiated by a local damage of a protective coating or an oxide film (which also may possess protective properties), as well as by many other factors which lead to chemical or physical heterogeneity at the metal surface [11,12]. In natural conditions the likelihood of pitting corrosion is higher compared to uniform corrosion [13]. Despite having minor metal loss, the structures subjected to localized corrosion have significantly reduced service life. Once appeared, local damages cause stress concentration in their adjacency. This may result in crack nucleation, and, consequently, the premature failure of a structural component. Therefore, assessment of the stress level in structural P elements with corrosion defects is an important issue to prevent the abrupt failure of the structures in an aggressive environment. Shallow corrosion pits may show a spherical cap morphology with a constant pit depth/diameter ratio [14,15]. Corrosion pits may also have more complex geometry [16][17][18]. However, most researchers used either simplification of pit forms or local thinning of structural elements [19][20][21]. Corrosion pits are usually idealized as conical [19], semi-elliptical [22][23][24], circular [18,[25][26][27], cylindrical [26,[28][29][30], and rounded rectangular shapes [31,32]. Authors of [26,27,33] showed that the form of the defects (circular, conical, elliptical or cylindrical) has a slight effect on the ultimate strength of plates under different loading. Stress fields around defects in structural elements are often analyzed using finite element method [34][35][36][37]. There are a number of papers focused on the effect of a single surface pitting on the service life of structural components. Stress distribution in the spherical shell with a single corrosion defect was addressed in [22,38]. Many authors considered structures weakened by multiple surface defects. However, most of these studies focused on pipes [29,39], plates [21,27,30,40,41], and shells [42][43][44][45]. Sedova [46] studied a spherical vessel with multiple defects uniformly located on its inner equator. The numerical simulations have shown that the maximum stresses in the vicinity of defects increase with the growing number of defects within a certain range, but further increase in the number of defects leads to a slight decrease in the stress values. Similar qualitative results were obtained in Liao,et. al. [39] using the Ramberg-Osgood plasticity model: the interaction between defects enhances with decreasing distance up to a certain limit when defects become close to intersection, while further decrease in spacing between defects provides the reduction of the interaction between defects. In [29], it is shown that the effect of multiple surface defects on the strength of structures depends on the distribution pattern of pitting corrosion. Since in reality, the defects appear randomly, it is reasonable to use random patterns when modeling multiple pits on the surface of structural components. The influence of pittings randomly distributed over a small area of the surface of a sphere on the collapse pressure was analyzed in [47,48]. Authors of [28] showed that the outer defect is more harmful to the pressure vessel than the inner one. A linearly elastic spherical vessel with several notches evenly distributed along its equator, subjected to internal pressure was considered in [49]. The present paper is devoted to the stress analysis of a spherical vessel under internal pressure, weakened by multiple shallow pits located along the equator on its outer surface. Results are compared for the bilinear plasticity hardening model and the linearly elastic model of the vessel material. Different numbers of defects with random arrangement and periodic arrangement are considered. The limiting case of toroidal notch is also studied.

DESCRIPTION OF THE PROBLEM
spherical vessel with multiple shallow pits along the equator on its outer surface is under study. Let r and R be the inner and outer radii of the vessel. All the pits have a spherical cap shape of the same size with curvature radius δ and depth h = δ/2 ( Fig. 1). A pseudo-random arrangement of the defects along the equator of the sphere and their different numbers n are considered. Along with separate notches, the "limiting" case of notches configuration is analyzed. This case corresponds to the complete covering of the equator of the sphere with a continuous toroidal notch with the same minimal curvature radius δ and depth h (Fig. 2, the surface of the notch is highlighted with another color). Pressure p is applied to the inner surface of the vessel. The problem is studied both within the framework of linearly elastic and bilinear elastic-plastic models. For example, 304 stainless steel is chosen as the vessel material. The inner and outer radii of the sphere are r = 340 mm and R = 350 mm, respectively. The notches' curvature radius is δ = 6 mm, and the notch depth is h = 3 mm. The number of the defects, n, varies from 2 to 260; additional geometries with n up to 320 are considered for the case of uniformly (periodically) located notches. Note that numbers n > 212 correspond to the complete covering of the equator by the defects (their intersection), in the case of their periodical distribution.

NUMERICAL ANALYSIS
o perform the finite element analysis, an array of 3D CAD-models of different geometries was built in ANSYS SpaceClaim. Each of the models represents the hollow sphere with the notches. Since the probability of the defects is supposed to be the same at every point on the equator, the uniform probability distribution was accepted. In order to create the random patterns of the defects along the equator of the vessel, the Python-function «random()» was used. Since the model of the sphere with the notches located on the equator is symmetric, only a half of the sphere was built as a CAD-model. The boundary condition on the face of symmetry was set to "Frictionless support" which prevents moving or deforming in the normal direction. For mesh generation a ten-node tetrahedral element SOLID187 was utilized. In order to enhance the accuracy of the solution, the finite element mesh was refined in the vicinity of the defects. The geometrical model with the continuous torus-shaped notch on the equator is axisymmetric; therefore, the cross section of the geometry was built in ANSYS DesignModeler. Before performing axisymmetric analysis for the geometry with the toroidal notch, special meshing methods with mesh refinement settings were set in the vicinity of the notch to achieve sufficient mesh quality. Fig. 3 shows mesh element quality on the fragment of the geometrical model capturing the vicinity of the notch. For linearly elastic analysis, the material is assumed to behave according to Hooke's Law with the elastic constants given above. Internal pressure is set to p = 1 MPa to stay in the framework of the elastic problem. For elastic-plastic analysis, the bilinear plasticity model "Bilinear Isotropic Hardening" available in ANSYS Workbench material properties library is used. The vessel is subjected to internal pressure p = 6 MPa which causes the plastic deformations of the material in the vicinity of the pits. 3D CAD-models are meshed using "Sizing" meshing options. The mesh is more refined at the weakened region and a smooth transition to a coarser mesh in regions far from notches is implemented. To ensure the convergence of the solution (its sensitivity to the mesh parameters), multiple calculations with different sizes of elements are carried out for each CADmodel. As a result, sizes of elements on the face of symmetry are set to 1.6 mm, while the sizes on the surface of notches are 0.5 mm. The estimated error value is about 2%. For bilinear plasticity hardening model, about 30 iterations were needed for each geometry model to reach the convergence of the nonlinear solution.
Since linearly elastic and elastic-plastic analyses are performed with the same geometries and mesh parameters, there is no need to run two standalone calculations for each set of material properties. Two linked projects that share geometries and mesh parameters are used instead.

RESULTS AND DISCUSSION
he maximum principal stress in the notched vessel was analyzed for various n from 2 to 260, and five different random distributions of pits for each n being considered. Let the maximum value of this stress for a certain geometry be denoted by σ max .
It was observed that for the configurations where all the defects were either far enough from each other or, conversely, their centers were exceedingly close to each other (resulting in their extensive overlapping so that there were no thin ligaments between them), the maximum stress values were reached at the bottom of a certain pit or pits (Fig. 4). Concerning the configurations where the pits slightly overlap or close to overlapping, the maximum stress values were observed at the cusps formed by the adjacent boundaries (Fig. 5). This observation is in accordance with the results of [36,39,46]. For the toroidal notch, the maximum stress value was observed along the bottom of the notch (Fig. 6). Let the stress concentration factor (one of all the possible factors) be denoted by K:   As can be seen from these figures, interaction of multiple defects may result in the significant increase in the maximum stresses with n increasing. The dependencies of K on n in the considered interval are qualitatively the same for both the models: at the beginning of the graphs, the values of K rise steeply and then a certain plateau (with a weak minimum inside) is formed at n > 200. The observed plateau may be explained by the overlaps of neighboring pits for large n. Obviously, the frequency of such overlaps grows with an increase in the number of the defects. However, the increase in K for the material in the elastic state is much higher than in the elastic-plastic: the value of K in the elastic sphere with multiple defects can be more than two times higher than that for a single defect, while in the elastic-plastic sphere the increase does not exceed 30%.
At first glance, it may seem strange that the maximum stresses in the elastic sphere with multiple pits may be larger than in the sphere with the toroidal notch (what is observed for n ≥ 32 in our case), while for the elastoplastic sphere the situation is opposite. Such behavior of the elastic sphere with multiple pits may be explained by the fact that the maximum stresses are observed at beak-like bridges between closely spaced defects (Fig. 5), which are absent in the sphere with a toroidal notch. The higher stresses at the bottom of the toroidal notch in the elastic-plastic vessel -compared to the stresses in the vicinity of the pits -are explained by the following reasoning. In the vessel with the continuous girdle notch, the vessel wall bends along the notch (see Fig. 6). Moreover, it is obvious that the bending deformations in the elastoplastic vessel are greater than in the elastic one, however, stresses in the last are not limited. Large deformations in the elastoplastic vessel initiate a hardening effect, resulting in an increase in stresses along the equatorial notch. In the vessel with separate random pits, such large bending deformations (which can lead to a hardening effect) do not occur. Therefore, stresses in the vicinity of individual pits in the elastoplastic vessel are smaller, since limited by a lower yield strength. Moreover, it is expected that as the number of pits increases, the stresses in the vicinity of the pits should tend to the stresses in the vicinity of the torus. It is really true and confirmed by Figs. 9 and 10. These figures show the values of K for the elastic and elastic-plastic vessels with multiple uniformly (i.e. periodically) located pits. As can be seen from Fig. 9, for the elastic material, the value of K experiences a sharp increase when thin ligaments or cusps form between adjacent pits (numbers n > 212 correspond to pit overlapping). Starting from a certain number n (in Figs. 9 the peak is reached at n = 228), cusps between the pits in the elastic vessel become more obtuse resulting in the gradual decrease in the maximum stresses approaching to the value corresponding to the toroidal notch. Numerical experiments with other pit sizes lead to qualitatively the same dependencies of maximum stress values on the number of notches, n: for relatively small numbers of notches, the maximum stress grows slightly with growing n; then (when thin ligaments form between adjacent pits) a sharp increase in the maximum stress is observed; as n grows further, the maximum stress slightly decreases. The results in Fig. 9 are consistent with ones obtained in [50] for linearly elastic sphere with multiple uniformly located pits with different values of notches sizes (both for δ < 6 mm and δ > 6 mm). Fig. 10 demonstrates a nearly S-shaped dependency of K on n in the elastic-plastic vessel. A small jump in the value of K at n = 276 is explained by the formation of a relatively smooth notch along the equator of the vessel and its bending, resulting in large deformations and hardening effect. For n > 275, the difference between the values of K for multiple defects and for torus is less than 3.5%. Thus, Figs. 9 and 10 confirm that the maximum stresses in the vessel with multiple pits periodically distributed along the equator, tend to that for the toroidal notch. It is obvious that this tendency should be preserved for randomly located defects. Nevertheless, there is a difference in the behavior of dependencies K(n) for random and periodical distribution of the defects. The values of K for the random defects rise significantly faster than for the periodical pits, due to the random formation of dangerous cusps at various and even very small n. The same effect for large n also leads to the slower decrease in K values for random pits in the elastic sphere. The difference between the peak value of K in Fig. 9 (K = 4.399 at n = 228) and the maximum values observed in Fig. 7 (which are greater than 4.5 for large n) is explained by the fact that for periodically distributed pits, the cusp angle changes stepwise (since the distance between the pits changes stepwise as L/n -d, where L is the length of the sphere equator and d is the pit diameter) and the angle values corresponding to the maximum K observed in Fig. 9 are just skipped. For certain ratios d/L at a certain n, the distance between the neighboring pits can become very close to that which corresponds to the maximum possible value of K, then the maximum values of K for periodical pits will become closer to that for random pits.    The opposite effect is observed for elastoplastic material: maximum values of K for randomly located pits (Fig. 8) are less than for periodical pits (Fig. 10). This is explained by the fact that at large n (starting from n=276), multiple periodic pits form a relatively deep girdle notch, along which the wall of the sphere bends (almost the same as in the sphere with the toroidal notch, see Fig. 6), resulting in a hardening effect and an increase in stresses. Randomly located pits do not form a continuous notch (as there may be relatively large gaps between the pits at random locations) where large bending deformations may occur. Therefore, the hardening effect does not manifest itself as noticeably as in a sphere with periodic pits. Thus, the mutual arrangement of defects on the vessel surface may have a greater effect on its local strength than the total volume of metal loss.

CONCLUSIONS
ased on the study, the following common conclusions can be drawn: Interaction of multiple defects on the vessel surface may result in the significant increase in the maximum stresses with the number n of the defects increasing. However, behavior of the vessels with multiple defects is markedly different for linearly elastic and elastic-plastic models. First, it was observed that for the considered data, the value of maximum stresses in the elastic sphere with multiple defects can be more than two times higher than for a single defect, while in the elastic-plastic sphere this increase does not exceed 30%. Moreover, smoothing the surface due to the damage accumulation may lead to a little decrease in the stress concentration for elastic materials. Note that for other parameters, behavior of the vessels weakened by multiple pits was qualitatively the same. The maximum stresses in the vessel with multiple pits located along the equator tend to that for the toroidal notch, as n increases. However, the maximum stresses in the elastic vessel with multiple pits may be significantly higher than that for the toroidal notch. This means that it is unacceptable to calculate the strength of elastic vessels with individual defects by considering vessels with a thickness smoothly reduced along a certain area. At the same time, for an elastic-plastic material, the maximum stresses in the vessel with a toroidal notch is the upper limit for the stresses in the vessel with multiple defects; therefore, for the strength analysis of the latter, the vessels with a smoothly reduced thickness can be considered. Stress concentration in the vessels with randomly distributed defects can be much higher than in the vessel with the same number of periodical defects, due to the random formation of thin ligaments or sharp cusps between the adjacent defects. However, for elastoplastic vessels with large numbers of defects, the opposite phenomenon may be initiated by the hardening effect. This means that the mutual arrangement of defects on the vessel surface may have a greater effect on its local strength than the total volume of metal loss and that periodic solutions do not always provide good approximations for random defects. B T