Limit equilibrium of a transversely isotropic spherical shell with two surface cracks

Abstract

An analog of the d c -model is used to reduce the limit-equilibrium problem for a transversely isotropic spherical shell with surface cracks to a system of integral equations. An algorithm for numerical solution of this system is proposed Keywords: transversely isotropic spherical shell, surface crack, plastic strains, integral equations, crack opening displacement Introduction. When loaded, a solid with cracks undergoes plastic deformation at their tips. Since the classical solution of elastoplastic problems for cracked bodies involves severe difficulties (because two systems of equations, one in the elastic domain and the other in the plastic domain have to be solved simultaneously, with the boundary between these domain being unknown), simplified models that agree with experimental data take on special significance. For example, the d c -model [7,15] and its analogs [4,17] can effectively be applied to thin plates and shells whose fracture is preceded by the development of large plastic zones. This was demonstrated experimentally in [11] for plates and in [2, 10] for large-scale specimens such as welded joints of structural steels with low and medium strength. The satisfactory agreement of theoretical and experimental data was also observed when the shape of plastic zones in specimens differed from that in the d c -model. An analog of this model was used to study the stress state and limit equilibrium of cylindrical [16] and isotropic spherical shells with part-through cracks [4,5]. An extensive review of relevant studies can be found in [3]. The limit state of an orthotropic plate with periodic cracks is analyzed in [12,13]. The stress-strain state of spherical shells with an eccentric circular hole and prestressed shells was studied in [14,18]. The stress intensity factor for an orthotropic cylindrical shell of medium thickness with an internal surface crack was determined in [9].The present paper uses an analog of the d c -model and the refined Timoshenko-type theory of shells [3] to analyze the limit equilibrium of a transversely isotropic elastoplastic spherical shell weakened by two surface cracks.1. Let us consider a shell with two identical surface cracks arranged along a straight (in plan) line. The length and depth of the cracks are denoted by 2 0 l and 2d, the distance between their centers by 2l d . We choose a coordinate system XOY so that the OX-axis is aligned with the crack line and the origin is at the middle of the segment 2l d . The stress-strain state of the shell is symmetric about the lines X = 0and Y = 0. The external loads, crack length, and the behavior of the material are supposed to be such that the plastic zones originating at the crack front would develop as narrow strips on the crack continuation all the way throughout the thickness of the shell. Denote the length of the plastic zones near the internal (neighboring) tips by l p and near the external tips by l p . In accordance with [4], assuming that tensile stresses act near the crack tips, we model the plastic zones by the disco...