Arch. Mech., 69, 4–5, pp. 305–316, Warszawa 2017 Antiplane strain (shear) of orthotropic non-homogeneous prismatic shell-like bodies N. CHINCHALADZE, G. JAIANI Iv. Javakhishvili Tbilisi State University Faculty of Exact and Natural Sciences & I. Vekua Institute of Applied Mathematics 2 University st. 0186 Tbilisi, Georgia e-mails: chinchaladze@gmail.com, george.jaiani@gmail.com Antiplane strain (shear) of orthotropic non-homogeneous prismatic shell- like bodies are considered when the shear modulus depending on the body projection (i.e., on a domain lying in the plane of interest) variables may vanish either on a part or on the entire boundary of the projection. We study the dependence of the well- posedness of the boundary conditions (BCs) on the character of the vanishing of the shear modulus. The case of vibration is considered as well. Key words: antiplane strain, degenerate elliptic equations, weighted spaces, Hardy’s inequality. Copyright c  2017 by IPPT PAN 1. Introduction The antiplane shear (strain) is a special state of strain in a body. This state is achieved when the displacements in the body are zero in the plane of interest but nonzero in the direction perpendicular to the plane. If the plane Ox 1 x 2 of the rectangular Cartesian frame Ox 1 x 2 x 3 is the plane of interest, then (1.1) u α (x 1 ,x 2 ,x 3 ) ≡ 0, α =1, 2; u 3 (x 1 ,x 2 ,x 3 )= u 3 (x 1 ,x 2 ), (x 1 ,x 2 ) ∈ ω, where u j , j =1, 2, 3, are the displacements, ω is a projection of the prismatic shell-like body Ω on the plane Ox 1 x 2 , correspondingly ∂ω is a projection of the lateral boundary S of Ω. The relations (1.1) mean that all the sections of the body parallel to the plane of interest Ox 1 x 2 will be bent as its section by the plane Ox 1 x 2 . Ω may have either Lipschitz (see Figs. 1–4) or non-Lipschitz boundary (see Fig. 5), ω has Lipschitz boundary (see Figs. 6, 7). Below Einstein’s summation convention is used. A bar under one of the repeated indices means that this convention is not used.