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We suggest a method for the solution of the elastoplastic problem of biaxial tension-compression of a thin cracked plate under the assumption that plastic strains at each tip of the crack are localized along three slip lines. The plastic zone is simulated by three inclined slip strips at the tips of the crack. It is assumed that the material of the plate is absolutely elastoplastic and the yield criterion is satisfied along the slip lines. The strips lying on the continuation of the crack are simulated by segments of discontinuity of normal stresses, while the other strips are simulated by segments of discontinuity of tangential stresses Thus, the elastoplastic problem under consideration is reduced to the boundary-value problem of the linear theory of elasticity for a body weakened by branched cuts with unknown lengths and orientations of side branches. These parameters are determined in the process of numerical solution of the problem by the method of singular integral equations. Numerical results are presented for the semiinfinite plane with edge crack and under the conditions of uniform tension at infinity. We present numerical values of some parameters of nonlinear fracture mechanics, such as crack tip opening displacement and lengths and angles of inclination of plastic strips, for various combinations of the components of external load.

By using the method of singular integral equations, we develop a general approach to the solution of static problems of the two-dimensional theory of elasticity for thin-walled structural elements with curvilinear cracks reinforced by elastic patches. We analyze two basic types of fastening patches to plates, namely, continuous fastening (via the adhesive layer) and discrete fastening (riveting). It is assumed that both the cracked plate and reinforcing patches are characterized by the generalized two-dimensional stressed state. We constructed integral equations for an infinite plate with curvilinear crack reinforced by linear or two-dimensional elastic patches. We also present a brief survey of the literature devoted to the solution of problems of the indicated type.

We studied stresses near the tip of a sharp V-notch in a material with power-law hardening under the action of antisymmetric loading. The problem is reduced to the solution of a fourth-order differential eigenvalue equation. We present the results of numerical analysis of the order of singularity of displacements and the angular distribution of stresses. We also construct the plastic zone near the tip of the notch for various values of the opening angle of the notch and the hardening index of the material. In the case where the material is linearly elastic, our numerical results are in good agreement with known results.

A general method has been proposed for constructing integral representations of general solutions and boundary integral equations of multidimensional boundary value problems of mathematical physics for regions with cuts. It involves the use of the theory of generalized functions, and in particular of the surface delta function. At first, the boundary value problems of Dirichlet and Neumann were studied for n-dimensional Poisson and Helmholtz equations in a space with cuts along piecewise-smooth surfaces. After that the method is extended to the case of a system of differential equations. In this way the basic spatial and plane problems of elasticity theory were considered for an anisotropic infinite body with cracks under static and dynamic loading. The corresponding axisymmetric problems were also studied.

The fundamental heat conduction problems for axisymmetric bodies with cracks are considered. Integral representations of temperature functions are constructed in terms of temperature jumps and heat flows in axisymmetric surfaces and cuts. By means of these representations, boundary-value problems for a space with cuts at whose edges are specified either a temperature, heat flow, or heat conduction conditions are reduced to singular integral equations. Integral equations are also found for boundary-value problems for a conical body of revolution with cavities and cracks. Numerical results are presented for a space with simply or doubly connected nonplanar cracks.

A general approach is given for planar treatments (planar state of stress and planar strain) in the elastoplastic equilibrium of a body containing cracks when the plastic strain is localized in thin layers. The plasticity region is represented as rectilinear cracks arising from the vertex of the main crack, on which the plasticity conditions are satisfied. The elastoplastic treatment is thus reduced to a treatment in the linear theory of elasticity for a body weakened by a kinked or branched crack with unknown lengths and orientations for the side branches. The latter are derived during the solution obtained by means of singular integral equations. Numerical results have been obtained for the stretching of an infinite plate containing an arbitrarily oriented rectilinear crack. Values are given for the crack opening at the vertex and for the length and orientation angle of the plasticity band.

The state of stress and strain is examined for a plate containing a curvilinear crack reinforced by a finite patch. The elastic patch covers the crack completely and is rigidly connected to the infinite plate only along its edge. It is assumed that the plate and patch are in a general state of planar strain. The boundary-value problem is reduced to a system of three singular integral equations, which is solved by mechanical quadrature. Numerical results are given for a plate containing a crack in the form of an arc of a parabola and reinforced with an elliptical patch for various orientations of the tensile forces at infinity. The stress intensity coefficients at the crack vertices have been calculated along with the contact forces at the junction.

A survey is presented on theoretical solutions to elastoplastic treatments in failure mechanics, in which the plastic zones are simulated as slip bands. Some experimental studies are discussed on plastic-strain localization in thin layers near cracks. Two-dimensional treatments of plastic band growth in bodies with cracks are considered for the conditions of planar stress, planar strain, and longitudinal shear, together with the corresponding axisymmetric and three-dimensional treatments. A note is also made of papers in which the plasticity band model is used for the bending of plates and shells containing cracks.

Elastoplastic equilibrium is considered for a thin semiinfinite plate containing an arbitrarily oriented rectilinear edge crack. It is assumed that the plastic strains are localized along a narrow band arising from the crack tip. The elastoplastic treatment is then reduced to a planar treatment in the theory of elasticity with an edge kinked crack whose edges are subject to defined forces. The solution is derived by means of singular integral equations. Numerical results have been obtained for a tensile load acting at infinity for a constant pressure acting on the crack edges. An estimate is made of how the free edge of the plate affects the magnitude and orientation of the plasticity band.

Abstract: The stressed state of an elastic plane weakened by a semiinfinite branching crack whose branches are either shear cracks or cracks of a mixed type is determined by the method of singular integral equations for the case where the stressed state at the tip of the semiinfinite crack without branches is characterized by stress intensity factorsK 10 andK II0 . The values of these factors are obtained for the cases of one, two, and three branches and different values of geometric parameters.

Using the method of singular integral equations we solve a two-dimensional problem of the theory of elasticity for an infinite plate containing an elastic inclusion of arbitrary configuration and a system of curvilinear incisions. The numerical solution is found by the method of mechanical quadratures for the case of an elliptic inclusion and a single polygonal crack.

Experimental investigations indicate that in many cases under plane deformation, the initial plastic strains in the vicinity of a separation crack are localized along two narrow bands at an angle of approximately 45° to the line of the crack [9]. Methods of solving elastoplastic problems in which these plasticity bands (slip bands) are modeled by fracture lines of tangential displacements have been developed in fracture mechanics; in that case, tangential stresses equal to the yield point in shear ty are assigned to these lines [2]. The problem of the development of slip bands then reduces to solution of the plane problem of the theory of elasticity for a branched notch; in that case, the dimensions and orientation of lateral branching corresponding to plasticity bands are determined during the solution. A series of results [1, 5, 6, 8, 10–16] for an infinite plane with a separation crack, where the slip bands are located symmetrical bout the line of the crack, have been obtained in this segment. In our study, we solve the plane elastoplastic problem (plane deformation) for a half space with an arbitrary oriented edge crack under an arbitrary load. Numerical results are obtained for a constant pressure on the crack or tensile forces at infinity.