Limiting equilibrium of a crack-weakened spherical shell on an elastic base

Abstract

In the deformation of a crack-weakened solid, as a rule, plastic deformation appears in the vicinity of the crack tips. Experimental investigation shows that, in ideally elastoplastic bodies, plastic deformation is localized in layers of small thickness in comparison with their extent. The Leonov-Panasyuk-Dugdale 6 k model proves satisfactory for taking account of the plastic deformation in front of the crack in a thin plate [I, 3].In the present work, on the basis of an analog of the 8k model, the influence of plastic deformation on the stress and limiting-equilibrium state of a hollow spherical shell made from ideally elastoplastic material, with a meridional through crack of length 2s is considered.It is assumed here that the shell is on an elastic base for which the Winkler hypothesis holds.The shell median surface is referred to a Cartesian coordinate system XOY, with its origin at the crack center and the OX axis directed along the line of the crack. Suppose that the stress-strain state of the shell is symmetric relative to the line of the crack. According to the 6k model, the plastic zones in front of the crack arising in the course of shell loading are replaced by lines of discontinuity of the elastiG displacements and angles of rotation, at which a plasticity condition must be satisfied by the normal azimuthal force N and the bending moment M. The Tresk condition is taken as the plasticity condition here, in the form [5] N 3M. 2/~ ,, + ~ = I, (i)where oT is the yield point; 2h is the shell thickness.Thus, within the framework of the given models the plastic deformation from the front of a crack of length 2s 0 is taken into account by introducing and considering a new crack of length 2s at the edges of which there acts a load of the form ~%(x,o)= I -m~ IXl<to;[N--N ~ , lo~lXl~l; M2(X, O) = l -M~ IXl<go;1M--M~ &~IXl<~t.(2)Here (s -~0) is the length of the plastic zone; N2 ~ and M2 ~ are the normal force and bending moment in a shell without a crack; N and M are the unknown force and moment acting in the plastic region and tending to close the crack.The system of resolving differential equations for the hollow spherical shell, taking account of the presence of a crack and the reaction of the elastic medium, is written in the form Institute of Applied Problems in Mechanics and Mathematics, Academy of Sciences of the Ukrainian SSR, Lvov.

Efficient Model of the Interaction of Elastomeric Filler with an Open Shell and a Chrome-Plated Shaft in a Dry Friction Damper

“…Similar problems for loading by simultaneous strain and bending are solved in the works [71][72][73][74][75]. The elastic and boundary equilibrium of cut thin-walled structures resting on an elastic foundation was studied in [76][77][78][79][80].…”