Limit equilibrium of a closed transversally isotropic cylindrical shell with a nonthrough longitudinal crack

Abstract

In the context of an analog of the Leonov-Panasyuk-Dagdeil model we consider the problem of limit equilibrium of a nonshaltow transversally isotropic cylindrical shell weakened by a nonthrough surface longitudinal crack. Based on the equations that take account of the initial stresses, we reduce the problem to a system of two singular integral equations with unknown limits of integration. We carry out a numerical analysis of the dependence of the opening of the edges of the crack on the load and the geometric and physico-mechanical parameters of the shell.The enlargement of the doma.n of practical use of cr teria for the mechanics of fracture involves a need to study the stress-strain state near cracks in bodies with different configurations and loading conditions. At present the stress distribution has been well studied near through cracks in shell structural elements [2,7] made of isotropic materials. Significantly less study has been devoted to anisotropic materials, in particular to transversally isotropic shells with through cracks [6]. In the case of a shell with a nonthrough crack the problem becomes three-dimensional, and taking account of the plastic deformations that develop in the vicinity of the crack, that is, the study of a shell with a nonthrough crack in the elastoplastic formulation yields a very difficult problem. For that reason in the formulation of such problems it is worthwhile to pay attention to simplified models that are consistent with experimental data. Thus, in application to plates in a plane stressed state, whose fracture is preceded by the development of significant zones of plastic deformation, the 5k-model of Leonov-Panasyuk-Dagdeil [1, 9] is quite effective. It is natural to assume that one can use an analog of the 5k-model also for thin shells with cracks, when the membrane stresses are significantly larger than the bending stresses, in solving the corresponding problems. Under such assumptions the limit equilibrium state of shallow [10,11] and nonshallow [3,5] isotropic cylindrical shells with through and nonthrough longitudinal cracks has been studied. We give below a construction of a solution of the problem of limit equilibrium of a closed transversally isotropic cylindrical shell (with low shear stiffness) with a nonthrough surface crack.Suppose a closed transversally isotropic cylindrical shell whose middle surface is referred to the lines of curvature a, ~3 is weakened by a longitudinal nonthrough crack lal < a0, fl = 0 (h -2d) < 3' < h, and is subject to forces and moments that are symmetric with respect to the crack. Suppose that ao = lo/R, 2/0 and 2d are the length and depth of the crack, 2h and R are the thickness of the shell and the radius of its middle surface, and V is the coordinate normal to the middle surface. The behavior of the material, the load level, and the size of the crack are assumed to be such that plastic deformations develop as a narrow strip over the entire thickness of the shell in the vicinity of the crack. Here over the entire depth of the crack...