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We investigate the initial stage of quasistatic development of plastic deformations in the vicinity of the tips of a rigid rectangular inclusion whose pair of faces do not contact with a medium. Deformation is caused by shear forces that act at infinity in parallel with this pair of faces of the inclusion. The cases of plastic deformations localized in bands that develop from the tips of the inclusion and distributed continually are investigated. The characteristics of plastic zones for loads much smaller than the yield strength are obtained.The investigation of the stress-strain state in the vicinity of inclusions of finite sizes is necessary to study the deformation behavior of structural elements. It is important for fracture mechanics as a complement to the theory of thin inclusions [1,7,8] and mechanics of composite materials [9]. It is also necessary to take into account the almost unavoidable presence of imperfections of the contact of an inclusion with a medium as a result of an operation or a manufacturing process. Moreover, in contrast to a body with notches, in a body with inclusions, stress concentration is caused by not only corner points, but also points at which the contact with the medium undergoes qualitative changes. In investigating the stress-strain state of a body with an inclusion, one must foresee changes in its contact with the medium in the process of loading and, first of all, the emergence and development of plastic exfoliation on the interface.The problems of investigating the stress-strain state of a body with an inclusion that admit its plastic exfoliation remain inadequately studied [6].Let us investigate plastic exfoliation under conditions of antiplane deformation of a rigid rectangular inclusion − a ≤ x ≤ a , − b ≤ y ≤ b , − ∞ ≤ z ≤ + ∞ in an infinite ideally elastoplastic medium. Consider the case of incomplete mechanical contact of the medium and inclusion. Let the vertical faces x = ± a , − b ≤ y ≤ b , − ∞ ≤ z ≤ + ∞ of the inclusion be in ideal contact with the medium before loading, and assume that the horizontal faces − a ≤ x ≤ a , y = ±b , − ∞ ≤ z ≤ + ∞ do not contact at all. The medium deforms under the action of a quasistatically monotonically increasing shear load τ xz = 0 , τ yz = τ ∞ , which acts at infinity in parallel to the faces of the prism that are free from stresses. We assume that the applied load is fairly small and use the linear model of a plastic zone (LMPZ) for an analysis of development of plastic strains [5].Let us consider two possible schemes of development of plastic strains: namely, a scheme for strains localized in bands that emanate from the inclusion and a continual scheme with a continuous distribution in the vicinity of the tips of the inclusion.The thin-band scheme of development of plastic strains corresponds to the assumptions on the number of bands and their mutual location [4]. We assume that two plastic bands develop from each vertex. These are a vertical band of length d 1 , which leads to the slip of the medium over the surface of the in...

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