Напружено-деформований стан ортотропної пружно-пластичної замкненої циліндричної оболонки з внутрішньою поздовжньою тріщиною довільної конфігурації

Abstract

Задачу про напружений стан і граничну рівновагу циліндричної ортотропної оболонки з внутрішньою тріщиною довільної конфігурації зведено до системи нелінійних сингулярних інтегральних рівнянь. Запропоновано алгоритм числового розв’язання отриманої системи. Вивчено вплив ортотропії, навантаження та геометричних параметрів на розкриття тріщини і розміри пластичних зон поблизу неї.

The strength of real solids is determined not only by their physical and chemical nature, but also depends significantly on the defectiveness of their structure. The solving of the problem on fracture of materials can be successful only in the case of accounting the defects contained in the body, as well as structural cuts and inclusions. During the deformation of the solid body in the vicinity of such concentrators high stress intensity occurs, which leads to the yielding flow of the material, to cracks initiation and propagation, i.e. to a local or complete fracture of the body [1, 2]. To evaluate the effect of different types of defects on the stress state and the limit equilibrium of the body, it is advisable to perform an investigation of stress concentrators that are subject to analytical study. Such stress concentrators, for example, in thin-walled structures are through and internal cracks. The anisotropic closed cylindrical elastic-plastic shell, weakened by internal cracks of arbitrary configuration is studied. The three-dimensional elastic-plastic problem on the stressed state and limiting equilibrium of closed orthotropic cylindrical shell with inner longitudinal crack of arbitrary configuration using analogue of the δc- model is reduced to a two-dimensional problem on elastic equilibrium of the same shell with a through crack of unknown length, and the last problem (in the refined Timoshenko-type shell theory) is reduced to the tenth order system of five key differential equations with unknown boundaries of integration and discontinuous functions in the right-hand side containing the unknown values. Using the fundamental solution and operation of convolution the integral representations of key functions of forces and moments for the closed orthotropic cylindrical shell are constructed, which, in turn, are reduced to the system of two singular integral equations. The algorithm for the numerical solution of the obtained system together with the conditions of plasticity of thin shells, with conditions of the uniqueness of displacements and conditions of boundedness of forces and moments in the vicinity of the crack is proposed. This algorithm is implemented for the case of a parabolic crack. The numerical analysis of dependence of crack opening displacements and sizes of the plastic zone is carried out for shells made from such materials: a composite material on epoxy basis reinforced by unidirectional graphite fibers, composite material on epoxy basis reinforced by unidirectional fibers of S-glass; bor-epoxy composite. The comparison of obtained results with the results for the corresponding rectangular internal crack, bounded by lines parallel to the coordinate lines is given.