The problem of the limiting equilibrium of a closed cylindrical shell with evenly spaced longitudinal cuts, stated in the contest of the analogy with the ~k-model of Leonov-Panasyuk-Dagdeil, is On the basis of the 5k-model of Leonov-Panasyuk-Dagdeil we use the initial equations of the general moment theory of shells to study the stress-deformed state and limiting equilibrium of a closed cylindrical shell of an ideal elastoplastic material with a system of k evenly spaced longitudinal cracks of equal length 2/0. We assume that the stress-deformed state caused by externM action in the shell without cracks is axisymmetric, and that the edges of the cracks are free of load. In this case because of the cyclic symmetry it suffices to consider a cylindrical panel [/31 _< r/k with a crack lal _< a0, /3 = 0 (a0 = lo/R), where c~ and/3 are the distances along the generator and directrix measured in units of the radius R of the middle surface. In what follows we shall limit ourselves to the case when the stressed state is symmetric with respect to the center of the crack a = 0.In accordance with the 5k-model we replace the plastic zones that form ahead of the cracks in the process of loading the shell by the lines of discontinuity of the elastic displacements and angles of rotation at which the unknown normal strain N and the bending moment M must satisfy a plasticity condition, say that of Tresk:where 2h is the thickness of the shell and aT is the yield point. Thus in the context of the model adopted the following conditions must hold on the line of a new crack that is enlarged in comparison with the original by the amount of the plastic zone:where al = ll/R, ll = lo + p, p is the length of the plastic zone ahead of the crack, and N ~ = N~ and M~ = M~ are the strain and moment in the shell without cracks. Using the method proposed in [2], we can reduce the problem of determining the stressed state of this panel with a crack of length 2ll to a system of singular integral equations:
539.3On the basis of an analog of the Leonov-Panasyuk-Dugdale ~-model and with the use of the initial equations of the general moment theory of shells the article deals with the limit equilibrium of a closed infinite cylindrical shell which has a transverse blind crack. On the basis of the generalized dislocation method the problem is reduced to a system of two singular integral equations whose solution is constructed with a view to the conditions of finiteness of the stresses in the vicinity of the crack. For a shell uniformly tensioned at infinity the article investigates the dependence of the dimension of the plastic zone and of the crack opening on the external load.We deal with an infinite closed cylindrical shell whose middle plane is referred to the lines of curvature a,/3. The shell is weakened by a transverse blind crack [/3 [ 30, a = 0, (h --2d) _< 3' < h, where/30 = /0/R; 2•0, 2d is the length and depth of the crack, respectively; 7 is the coordinate normal to the middle plane; R is the radius of curvature of the median surface; 2h is the thickness of the shell. The crack lips are free of any external load. We assume that the external load applied to the shell is such that the state of stress and strain is symmetrical about the line of the crack and its center.The given problem is three-dimensional, and when the plastic deformations developing in the vicinity of the crack are taken into account, it is difficult to solve. In stating such problems it is therefore expedient to use simplified methods that are in good agreement with the experimental data. One such method is the analog of the Leonov-Panasyuk-Dugdale cSk-model for thin shells. In accordance with this model the plastic zones in front of the crack/30 < [ /3 ] -< /31 (/31 = ll/R; ll = l0 + p, p is the length of the plastic zone in front of the crack) are replaced by the lines of the jumps of the elastic displacement and of the angle of rotation which are acted on by the unknown finite normal force N and the bending moment M. We also assume that at the continuation of the crack into its depth, i.e., with [/3 [ _< fi0, -h _< 3' < h --2d, there act membrane stresses only, which are constant and equal to the yield strength Cry of the material (Fig. 1). With these assumptions the authors of [1] investigated the limit equilibrium state of a gently sloping, the authors of [2] of an abruptly sloping cylindrical shell with blind longitudinal cracks, the authors of [3] also investigated a gently sloping shell with a blind transverse crack. Below we obtain the solution of the equations of the general moment theory for a closed cylindrical shell with a transverse blind crack.Thus, within the framework of the adopted model, plastic deformations are taken into account by replacing the blind crack by a through crack with unknown length 2l 1 on whose lips the following conditions are fulfilled:(1) f --M~ + 2oral (h --d), 181 ~< 80; M,(O, 8)=/M_M °, ~0<181..<8~, where N1 °, M1 ° are, respectively, the normal force and the bending moment in a shell without crac...
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