Elastoplastic orthotropic shell with an internal crack of arbitrary configuration

Abstract

We have developed methods of constructing systems of singular integral equations for solving the problems of the stressed state of orthotropic shells of arbitrary curvature with an internal crack of arbitrary configuration. The influence of elastic and geometric parameters of shells on the size of the plastic zone and crack opening displacement has been considered.Shell structures often contain crack-like defects, and cracks present in them can be through and internal. Methods and models which will be used in the strength analysis of these structures depend on the type of the crack. In calculations, it is important that the chosen method or model be well defined in using and describe adequately the mechanical problem [8].An analog of the δ c -model, using which, one can simultaneously take into account both the type of the crack and plastic deformations that arose in the vicinity of it was developed by Folias and Erdogan [7] and used by many scientists in the investigation of isotropic and orthotropic shells of certain and arbitrary curvature with internal cracks [2-4]. However, in all indicated investigations, the contour of the depth of an internal crack was modeled by a rectangle, which substantially simplified the solution of the problem, but the shape of the real crack was not taken into account.In the present work, the contour of the depth of a crack is described by two smooth curves. We perform a comparative analysis of results obtained under the assumption that the contour of the depth of the crack is rectangular or parabolic. Statement of the ProblemConsider a thin elastic shell of arbitrary curvature and constant thickness h made from an orthotropic material such that, at each point of the shell, the line of principal curvatures of the middle surface coincides with principal directions of elasticity of the material. A system of orthogonal coordinates Oxyz was chosen so that the axes Ox and Oy were oriented along the lines of principal curvatures of the surface of the shell and the axis Oz was directed along a normal to it.The shell is weakened by an internal crack of length 2 along the axis Ox . The depth of the crack is set by two smooth curves D 1 (x) and D 2 (x) (Fig. 1). We assume that, under the action of a symmetric external load, above and under the front of the crack, and on its continuation, zones of plastic deformations formed and propagated over a thin layer through the thickness of the shell. For the investigation of this elastoplastic shell with an internal crack, it is reasonable to use the aforementioned analog of the δ c -model. Solution of the ProblemAccording to the analog of the δ c -model, in plastic zones, on the continuation of the crack where stresses attained the yield strength σ τ , we introduce unknown forces T and a bending moment M , which satisfy the