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539.375 In the framework of the model of plasticity bands, we analyze an elastoplastic fracture-mechanics problem of propagating plasticity bands near the tip of a semiinfinite crack of mixed type. It is assumed that, under the conditions of plane stressed state, plastic strains near the tip of a semiinfinite crack are localized along three plasticity bands (L I , L 2, L 3 ) and, under the conditions of plane deformation, they are localized along two bands (L 1, L 2). One of these bands (L 3 ) is simulated by a line of discontinuity of normal and tangential displacements. The remaining two bands (L I, L 2) are simulated by the lines of discontinuity of tangential displacements. Their lengths and orientations are determined in the process of numerical solution of the problem by the method of singular integral equations. We also present the values of crack tip opening displacements.In the vicinity of crack tips, materials always undergo plastic deformation. To analyze plasticity zones, it is customary to use the model of plasticity bands regarded as surfaces of discontinuities of displacements satisfying the conditions of plasticity. Beyond these surfaces, the body is supposed to be elastic [ 1,2]. However, within the framework of this model, it is mathematically difficult to describe the onset of plastic deformation (small-scale yield) for a crack of finite length [3][4][5][6][7]. In the present work, by using the method of singular integral equations, we develop a general approach to the solution of the problem of propagation of slip bands near a semiinfinite crack of mixed type (mode I + II) under the conditions of plane stressed state or plane deformation. We determine crack tip opening displacements and the lengths of plasticity bands under general loading. For some cases, we also present approximate formulas in the explicit form. The solution of this problem can be regarded as the asymptotics of the corresponding solution of the problem of propagation of plasticity zones near the tip of a crack of finite length under the conditions of small-scale yield.Plastic deformation near the crack tip in the case of small-scale yield was studied by using various approaches [8][9][10][11]. Most often, these investigations deal with tensile cracks for which the plasticity zone is symmetric about the crack line [8,9]. The problems of propagation of the plasticity zone near the tip of a semiinfinite crack of mixed type in a homogeneous body [10] and along the interface of rigid and elastic materials [11] were studied only under the assumption that the crack tip serves as the origin of a single slip band (simulated by a jump of tangential displacements). In [5,6], the parameters of nonlinear fracture mechanics are evaluated for a crack of mixed type. Plane Stressed StateConsider a semiinfinite crack L 0 with plasticity bands in an infinite plate. We assume that three bands (L 1 , L 2, L 3 ) of different lengths and orientations originate from the tip of this crack (Fig. 1). The crack L 0 is directed along the Ox-axis (x...

539.375 Within the framework of the model of plasticity bands, we consider a two-dimensional elastoplastic problem of fracture mechanics for a body with one circular hole and two collinear cracks whose ends lie on the contour of the hole under the conditions of plane stressed state and plane deformation. It is assumed that plastic strains near each tip of the edge cracks are concentrated along three (for the plane stressed state) or two (under the conditions of plane deformation) plasticity bands. Their lengths and orientations are determined as a result of the solution of the problem. Numerical results are obtained for the cases of a biaxial tensile load acting at infinity and constant pressure applied to the contour of the hole and edge cracks (under the conditions of plane deformation).Plastic zones are formed near stress concentrators (cracks and holes) under sufficiently high loads. In nonlinear fracture mechanics, these zones are often analyzed by using model of plasticity bands according to which plastic zones are localized in thin layers [1,2]. The first experimental investigations of these bands were performed by testing plates with cracks or circular holes for uniaxial tension [3][4][5]. The shapes of plastic zones were also studied near the tips of edge cracks originating from a circular hole [6]. It was shown that, in uniaxially stretched thin plates weakened by a circular hole and two collinear radial cracks, the process of formation of the first narrow bands on the continuations of the cracks is followed by the appearance of oblique bands and a system of bands parallel to the plane of the crack [6]. A similar picture is observed near the tips of a crack (in the absence of a circular hole) [3][4][5].Since plastic zones have the shape of narrow bands, we can neglect their thickness and simulate these zones by the surfaces of discontinuity of displacements satisfying the conditions of plasticity. Outside these surfaces, the body is regarded as elastic [4]. The model of plasticity bands made it possible to solve numerous problems for cracked bodies [ 1,2]. Thus, for a circular hole with two edge collinear cracks, plastic strains localized along the plasticity bands lying on the lines of the cracks were investigated in [7]. This situation corresponds to the first stage of deformation (plane stressed state). In the present work, we analyze the development of plasticity bands near the tips of two collinear radial cracks originating from a circular hole under the conditions of plane stressed state (with regard for oblique bands) and plane deformation. Plane Stressed StateConsider an inf'mite plane with a Cartesian coordinate system xOy and a circular hole L 0 of radius R centered at the origin of coordinates. Two radial cracks L l and L 2 of the same length 2l originate from the contour of the hole. Assume that plastic strains near tips of edge cracks in a perfectly elastoplastic body are localized along three narrow rectilinear slip bands L 3, L 5, and L 6 and L4, L7, and L 8 , respectively. The plasti...

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539.375 Within the framework of the model of plasticity bands, we consider an elastoplastic problem of fracture mechanics of the development of plasticity bands near the tips of a central crack in a disk. We assume that, in the plane stressed state, plastic strains in the vicinity of the crack tip are localized along three plasticity bands (L I , L 2 , and L~ ) one of which is located on the continuation of the crack and the other two make nonzero angles with the direction of the crack and that, under the conditions of plane deformation, plastic strains are localized along two plasticity bands (L I and ~ ). The band (~) is modeled by a line of discontinuity of normal and tangential displacements and the bands (L l and L 2) are modeled by lines of discontinuity of tangential displacements. The lengths and orientations of these lines are determined in the process of numerical solution of the problem by the method of singular integral equations. The values of the crack tip opening displacement are also determined.Disk specimens with notches or cracks are used in the investigation of the processes of initiation and development of cracks under static and cyclic loading [1][2][3][4]. The level of local strains near a concentrator characterized by the jump of normal displacements (opening displacement) at its tip 6 can be regarded as a parameter that controls these processes in elastoplastic materials. To find this parameter, it is necessary to solve the problem of determination of the strain-stress state of the disk taking into account elastoplastic strains near the tip of the concentrator. For this purpose, we use the model of plasticity bands (see, e.g., [5-1 1]) according to which the plasticity zone is localized along narrow bands lying on the continuations of the crack. As follows from the experimental data presented in [5][6][7], this model reflects the actual process of deformation in the first stage of the development of plastic strains in thin plates. However, in many cases (in particular, in disks with central cracks [6]), under loads of certain types, we observe the appearance of new bands inclined to the line of continuation of the crack. Neglecting these inclined bands, we get underestimated values of the displacement ~ [12].In the present work, within the framework of the model of plasticity bands, we solve the problem of elastoplastic equilibrium of a disk with central crack and study the influence of inclined plasticity bands on the crack tip opening displacement. Plane Stressed StateA thin disk of radius R containing a central crack L 1 of length 2/1 is related to a Cartesian coordinates xOy. The disk is subjected to tension by two concentrated forces F applied at points Zo = + ih (Fig. 1). The material of the disk is perfectly elastoplastic.In the disk subjected to tension, near the crack tips, we first observe the appearance of plastic bands directed along the continuation of the crack. Further, as the load increases, we observe the appearance of two additional bands at each tip making nonzero an...