Deformation of an Isotropic Plate with Periodic System of Curvilinear Holes and Plasticity Bands
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DEFORMATION OF AN ISOTROPIC PLATE WITH PERIODIC SYSTEM OF CURVILINEAR HOLES AND PLASTICITY BANDS V. S. Kravets1,2 and М. P. Savruk1
UDC 539.3
By the method of singular integral equations, we solve a plane elastoplastic problem of fracture mechanics for an isotropic plane with an infinite series of curvilinear holes and plasticity bands at their tips. We study the influence of the shapes of smooth holes and the radii of rounding of the contours at their tips on the opening displacements and lengths of the plasticity bands. The solutions of the corresponding problems for semiinfinite bilateral rounded notches whose tips serve as the origins of plasticity bands are obtained by the limit transition in the case where the relative distance between the holes tends to zero. On the basis of the deformation criterion of fracture and solutions of the periodic elastoplastic problem, we approximately determine the strength of rectangular specimens with bilateral U-shaped notches. Keywords: plane elastoplastic problem, periodic system of holes, singular integral equations, plasticity bands, crack-opening displacements.
Introduction In the fracture mechanics of solid bodies with cracklike defects, it is customary to solve elastoplastic problems under the assumption that plastic strains near the tips of the defects in the first stage of their development are localized in thin layers (plasticity bands) [1–6], which are modeled by the surfaces of discontinuity of displacements. The conditions of plasticity are satisfied on these surfaces. Outside these surfaces, the body is regarded as elastic. Some solutions of the corresponding periodic elastoplastic problems for bodies with cracks and plasticity bands on their continuation were obtained in [3, 4]. The lengths of plasticity bands were determined for plates with periodic systems of circular holes and plasticity bands in [7] and, for plates with edge cracks and plasticity bands on their continuation, in [8]. In what follows, by the method of singular integral equations (SIE), we solve an elastoplastic problem (within the framework of the model of plasticity bands) for an isotropic plate stretched at infinity (plane stressed state) and containing a periodic system of curvilinear holes with plasticity bands originating from the opposite tips of these holes (mode I deformation). Assume that the material of the plate is perfectly elastoplastic and that initial plastic strains are localized in thin layers (on the continuation of the cracks and from the tips of notches and narrow holes). These layers are simulated by the jumps of normal displacements [5, 9, 10]. Thus, we reduce the solution of the problem for the plate containing a periodic system of holes with plasticity bands (Fig. 1) to the solution of the elastic problem for a plate with periodic system of holes and edge notches with normal stress σ Y (yield strength of the material of the plate) applied to their faces. The length of plasticity bands l is unknown and should be found in solving the problem. 1 2
Karpenko Physicomechan
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