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Vol 238, No 1 (2019)
- Year: 2019
- Articles: 6
- URL: https://journals.rcsi.science/1072-3374/issue/view/14995
Article
Boundary-Value Problems with Birkhoff Regular but not Strongly Regular Conditions for a Second-Order Differential Operator
Abstract
We study the self-adjoint problems whose operators split in the invariant subspaces induced by the involution operator Iy(x) = y(1− x). We construct nonself-adjoint perturbations of these problems that are Birkhoff regular but not strongly regular and, for some values of the coefficients of the boundary conditions transform into nonspectral problems in Dunford’s sense. We study the spectral properties of operators corresponding to these perturbations and, in particular, determine the eigenvalues and root functions and analyze the completeness and basis property of the system of root functions. We find the families of boundary conditions that generate essentially nonself-adjoint problems and contain the nonlocal Samarskii–Ionkin conditions.
1-21
22-31
On Semitopological Bicyclic Extensions of Linearly Ordered Groups
Abstract
For a linearly ordered group G , we define a subset A ⊆ G to be a shift-set if, for any x, y, z ϵ A with y < x, we get x · y-1 ··z ϵ A. We describe the natural partial order and solutions of equations on the semigroup B(A) of shifts of positive cones of A . We study topologizations of the semigroup B(A). In particular, we show that, for an arbitrary countable linearly ordered group G and a nonempty shift-set A of G , every Baire shift-continuous T1-topology τ on B(A) is discrete. We also prove that, for any linearly nondensely ordered group G and a nonempty shift-set A of G , every shift-continuous Hausdorff topology τ on the semigroup B (A) is discrete.
32-45
Method of Direct Cutting-Out in the Problems of Piecewise Homogeneous Bodies with Interface Cracks Under Longitudinal Shear
Abstract
The earlier developed method of direct cutting-out is extended to the class of problems of elastic equilibrium of piecewise homogeneous bodies with internal and interface crack-like defects under antiplane deformation. This method is based on modeling of the initial problem for a body with thin inclusions (in particular, cracks) by a simpler problem of elastic equilibrium of piecewise homogeneous space with elevated number of thin defects, which, in fact, form new boundaries of the analyzed body. The reliability of the proposed approach is checked on examples of problems of longitudinal shear for a piecewise homogeneous wedge, a piecewise homogeneous half space, and a two-layer strip with interface crack subjected to the action of homogeneous loads and concentrated forces.
46-62
Asymptotic Distributions of Stresses and Displacements Near the Edge of a Contact Zone
Abstract
We find the asymptotic distributions of stresses and displacements near the edge point of a contact zone for problems of friction contact and contact with complete adhesion of an elastic body with a die, as well as for the problems of friction contact of two elastic bodies and contact of the faces of interface cracks. The distributions of elastic fields are obtained by using one of the Kolosov–Muskhelishvili complex potentials in the form of a Cauchy-type integral whose density is given by a complex function of contact forces.
63-82
Antiplane Shear of an Elastic Body with Elliptic Inclusions Under the Conditions of Imperfect Contact on the Interfaces
Abstract
We study the problem of аntiplane shear of an elastic body containing a finite array of arbitrarily located and oriented elliptic inclusions under the conditions of imperfect mechanical contact on the interfaces. The analytic solution of the problem is obtained by the method of multipole expansions with the use of the technique of complex potentials. By expanding the disturbances of the field of displacements caused by inclusions in a series in the system of elliptic harmonics and using the formulas for their reexpansion and exact validity of all contact conditions, we reduce the boundary-value problem of the theory of elasticity to an infinite system of linear algebraic equations. It is also proved that the reduction method is applicable to the indicated system, the rate of convergence of the solution is investigated, and the accumulated results are compared with the data available from the literature. The presented numerical results of parametric investigations reveal the presence of a strong dependence of stress concentration on the conditions of contact on the interfaces, as well as on the sizes, shapes, and relative positions of the inclusions.
83-95