Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes

Abstract

Results on stress concentration in thin shells with curvilinear holes subject to plastic deformation and finite deflections are reviewed. The holes (circular, elliptical) are reinforced with thin-walled elements (rings, rods) of different stiffness. A numerical method of solving doubly nonlinear problems of statics for shells of complex geometry is outlined. The stress distribution near curvilinear holes in spherical, cylindrical, and conical shells under statical loading is studied. The numerical results are analyzed Introduction. Thin shells and plates are used as structural members in various fields of modern engineering. In most cases, they have complex geometry (geometry variations, holes, notches, and inclusions of various shapes) for the purposes of design or technology. High loads on structural members made of homogeneous isotropic materials (metals, alloys) may cause structural changes (plastic or creep strains) near stress concentrators and microcracks, which may lead to failure of structural members and then the whole structure. There are also large or finite displacements or strains in the zone of high stresses. One of the basic and most important tasks in the mechanics of shell structures is to analyze the distribution of stresses and strains in structural members of complex form. Therefore, in designing and manufacturing load-bearing structures or their elements with high strength and minimum weight, the need arises to take into account the real operation conditions of structural members and the real properties of structural materials (plastic strains) and their deformation (large or finite displacements).Results of theoretical and experimental analysis of the stress distribution in shells (plates) obtained by solving linear elastic boundary-value problems (linear Hooke's law, small displacements, strains) are discussed in a great many publications most fully generalized in the monographs [2,21,22,25,26,58,65]. They mainly discuss solutions of linear (elastic) problems for thin (spherical, cylindrical, conical, etc.) shells weakened by curvilinear holes of various shapes and made of advances metallic materials.The available theoretical and applied results on the stress distribution in thin and nonthin anisotropic (composite) shells made of materials obtained by solving linear elastic problems (generalized Hooke's law; Kirchhoff-Love or Timoshenko hypotheses) are presented in [27,50].To solve this class of problems in linear elastic formulation, analytic, variational, and numerical methods are used. Methods and results of solving specific classes of physically and geometrically nonlinear problems for some metal shells are discussed in the monographs [20,32,38,42,67,70]. The issue of stress concentration in shells (plates) involving the development of methods of solving nonlinear problems is analyzed in the reviews [24,89,93,94].Note that most studies on nonlinear stress concentration in shells, including problem formulation, development of methods for solving certain classes of nonlinear problems, a...

Stress concentration in a transversely isotropic spherical shell with two circular rigid inclusions

“…Despite their long history [1][2][3], studies of stress concentration in plates and shells with holes and inclusions (rings, washers, covers) are still relevant [5][6][7][8]. A similar situation exists with other problems in the mechanics for multiply connected and structurally inhomogeneous bodies [4,9].…”

“…To separate variables in the unknown functions in the qth coordinate system, we will follow Guz who proposed in [3,6] to use Graf's theorem for cylindrical functions in (5), (7), (9) and to expand each term of the power part of solution (6), (8) into the Laurent series. Doing so give the expressions …”

Elastoplastic deformation of conical shells with two circular holes

“…Study of the interaction between the boundaries of holes in thin-walled conical shells is an important task of solid mechanics. Most results on stress concentration around curvilinear and rectangular holes in conical shells made of metals and composites were obtained by solving linear elastic problems with analytic, variational, and numerical methods and are most fully reported in [2,3,9,13].Also of great interest are the solutions of two-dimensional problems for thin shells with holes of various shapes under high surface and boundary loads, which makes it necessary to take into account both the real properties of materials (plastic strains) and deformation behavior (finite deflections).Physically nonlinear cases of stress-concentration in conical shells with holes were examined in few publications. This is because the system of governing equations becomes very complicated when nonlinear factors are allowed for and there are holes in the lateral walls.…”

Elastoplastic state of flexible conical shells with a circular hole under axial tension

“…The numerical results are presented as plots and tables Introduction. The major results on the stress-strain state around curved and rectangular holes in conical metallic and composite shells were obtained with analytic, variational, and numerical methods and are most fully covered in the monographs [1, 4, 5] and reviews [9][10][11].Only few publications address nonlinear problems of stress concentration in conical shells with holes. This is because of the complexity of the system of governing equations for conical shells with holes when nonlinear factors are taken into account.…”

“…The numerical results are presented as plots and tables Introduction. The major results on the stress-strain state around curved and rectangular holes in conical metallic and composite shells were obtained with analytic, variational, and numerical methods and are most fully covered in the monographs [1, 4, 5] and reviews [9][10][11].…”