Stressed state of an elastoplastic isotropic shell with a through crack with regard for material hardening

Abstract

We have solved the problem of determining the stressed state of an elastoplastic isotropic shell of arbitrary curvature with a through crack with regard for material hardening. We have obtained a system of singular integral equations and solved it numerically by the method of mechanical quadratures. We have also studied the influence of material hardening on the general characteristics of the stressed state.The theoretical foundations and applied problems of the mechanics of strain hardening were considered in [1-3, 6, 9, 11, 12, 14]. The results of present-day investigations in this field are described in [8,10,13,15]. The problem of the stressed state of gently sloping isotropic cylindrical and spherical shells with a through crack with regard for material hardening was considered in [5]. Here, the δ c -model generalized for materials with hardening [4] was used. In the present work, we study a similar problem for an isotropic shell of arbitrary Gaussian curvature.Consider a gently sloping isotropic shell of arbitrary curvature and constant thickness h , which contains a through crack of length 2 0 along one of the lines of principal curvatures (Fig. 1).We assume that the crack sizes are great as compared with the shell thickness but small as compared with other linear sizes. This enables us to consider the problem of the equilibrium of a thin shell with a crack on the basis of the two-dimensional theory of shells. Within the framework of this theory, we model cracks by mathematical cuts of the shell median surface.We refer the shell to a system of orthogonal coordinates Oxyz , which is chosen so that the Ox and Oy axes are oriented along the lines of principal curvatures of the shell median surface, and the Oz axis is directed along a normal to it. We assume that the shell and crack faces are loaded with forces T and moments M symmetric with respect to the crack line. In the course of shell deformation, the crack faces do not contact between themselves.We assume that the crack sizes, external load, and properties of the material are such that plastic deformations take place in a narrow strip near the crack, over the entire depth. Furthermore, according to the δ c -model, we simulate the zones of plastic deformation by the surfaces of discontinuity of the displacements and rotation angles and assume that the response of the material of plastic zones to unknown normal force T and bending moment M is distributed linearly [4,5] by the law

Influence of hardening of a material on the stressed state of an elastoplastic shell with internal crack

“…145-152] and a surface crack [5]. Theoretical foundations of strain hardening were presented in [4,6,9,[11][12][13], and modern applied problems of the mechanics of strain hardening of materials were described in [1].…”