Vol 28, No 3-4 (2022)

In this article, we study the inverse problem of determination of time-dependent coefficient in the parabolic equation. We prove existence and uniqueness theorem for the solution of the inverse problem with nonlocal boundary conditions and integral observation. The proof is based on a priori estimates obtained in this article and the results on solvability of corresponding direct problem for the equarion under consideration.

The subject of this paper are the systems of vectors and subspaces in finite dimensional spaces admitting the recovery of an unknown vector-signal by modules of measurements. We analyze the relationship between the properties of recovery by modules of measurements and recovery by norms of projections and the properties of alternative completeness in Euclidean and unitary spaces. The theorem on ranks of one linear operator is considered, the result of which in some cases can be regarded as another criterion for the possibility to restore a vector-signal. As a result of this work, the equivalence of the alternative completeness property and the statement of the rank theorem for Euclidean space is proved. It is shown that the rank theorem in the real case can be extended to the systems of subspaces.

The questions about the minimum number of vectors admissible for reconstruction by modules of measurements are considered. The results available at the moment are presented, which are summarized in the form of a table for spaces of dimension less than 10. Also the known results to the question of the minimum number of subspaces admitting reconstruction by the norms of projections are briefly given.

Modern mechanical engineering sets the tasks of calculating thin-walled structures that simultaneously combine sometimes mutually exclusive properties: lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of orthotropic materials and plastics seems quite justified.

The article demonstrates the complex representation method of the equations of orthotropic shells general theory, which allowed in a complex form to significantly reduce the number of unknowns and the order of the system of diferential equations. A feature of the proposed technique for orthotropic shells is the appearance of complex conjugate unknown functions. Despite this, the proposed technique allows for a more compact representation of the equations, and in some cases it is even possible to calculate a complex conjugate function. In the case of axisymmetric deformation, this function vanishes, and in other cases the influence of the complex conjugate function can be neglected.

Verification of the correctness of the proposed technique was demonstrated on a shallow orthotropic spherical shell of rotation under the action of a distributed load. In the limiting case, results were obtained for an isotropic shell as well. 

The article defines stress fields near the tips of mathematical cracks in an isotropic linearly elastic plate with two horizontal collinear cracks lying on a straight line of different lengths under the uniaxial tensile condition, using two approaches - experimental, based on the method of digital photomechanics, and numerical, based on finite element calculations. To represent the stress field at the tip of the section, the Williams polynomial series is used - the canonical representation of the field at the top of the mathematical section of a two-dimensional problem of elasticity theory for isotropic media. The main idea of the current study is to take into consideration the regular (non-singular) terms of the series and analyze their impact on the holistic description of the stress field in the immediate vicinity of the top of the section. The first fifteen coefficients of the Max Williams series were preserved and determined in accordance with experimental patterns of isochromatic bands and finite element modeling. To extract the coefficients of the Williams series used a redefined method designed to solve systems of algebraic equations, the number of which is significantly greater than the unknown - amplitude multipliers. The influence of the non-singular terms of the Williams series on the processing of the experimental pattern of interference fringes is demonstrated. It is validated that the preservation of the terms of a high order of smallness makes it possible to expand the area adjacent to the tip of the crack, from which experimental points can be selected. The finite element study was carried out in the SIMULIA Abaqus engineering analysis system, in which experimental samples tested in a full-scale experiment were reproduced. It is revealed that the results obtained by the two methods are in good agreement with each other.