Development of plastic zones in a body with rectangular slot under the conditions of antiplane deformation
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DEVELOPMENT OF PLASTIC ZONES IN A BODY WITH RECTANGULAR SLOT UNDER THE CONDITIONS OF ANTIPLANE DEFORMATION V. A. Kryven’, M. I. Yavors’ka, and V. B. Valyashek
UDC 539.375
We study the quasistatic development of plastic zones in a perfectly elastoplastic body with rectangular slot caused by the action constant shear forces applied at infinity. The exact analytic solution of the problem is obtained, and it is shown that the plastic zones can be studied in the initial stage of development on the basis of the elastic solution of the problem. The shapes of the plastic zones are analyzed depending on the loads for the cases of cracklike, wide, and square slots.
The analysis of plastic effects in the vicinity of stress concentrators is the principal problem of nonlinear fracture mechanics. In the theory of cracks, the cracks are almost always modeled by mathematical cuts. The crack tips are regarded as cusps. These simplifications enable us to use the efficient and well-developed methods of singular integral equations [1]. The assumptions concerning the appearance of discontinuities in which the distance between the edges is equal to zero disagree with the physics of interatomic interaction in solids, especially for the stress-free crack edges. The stress-strain state and the development of plastic strains near semiinfinite notches of constant width under the conditions of shear are studied in [2]. It is shown that if plastic strains occupy a region whose sizes significantly exceed the width of the notch, then the shape of the plasticity zone approaches the shape of the zone located near the crack tip. In the present work, we study the development of plastic zones in the vicinity of a rectangular slot and compare their shapes near the slot and near the crack for various width-to-length ratios of the slot depending on the load. Consider an infinite perfectly elastoplastic body with rectangular slot – a ≤ x ≤ a, – h ≤ y ≤ h, – ∞ ≤ z ≤ + ∞ subjected to the action of a quasistatic monotonically increasing load τ x y = 0, τ y z = τ∞ applied at infinity (Fig. 1). In the perfectly elastoplastic body, the role of slip lines in the plastic zone is played only by segments of straight lines originating from the angular points of the notch [3, 4]. Hence, under any load, the plastic zones formed near the tips of the rectangular slot do not leave the boundaries of the angles formed by the continuations of the sides of the rectangle. The strips whose width is equal to the width of the slot located on the continuations of the slot remain elastic (Fig. 1). Formalization of the Problem Due to the symmetry of the problem (the displacements w ( x, y ) are even about the x-axis and odd about the y-axis), it sufficient to determine the stress-strain state in the first quadrant. Let L be the boundary of the zone of plastic strains formed near a point of the curve ( a, h ) and let D be a part of the first quadrant bounded by the sides of the rectangle x = a, 0 ≤ y ≤ h and 0 ≤ x ≤ a, y = h and the line L. In the region D, the body is elastic.
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