Effect of Frictional Slipping on the Strength of Ribbon-Reinforced Composite

A numerical–analytical approach to the problem of determining the stress–strain state of bimaterial structures with interphase ribbon-like deformable inhomogeneities under combined force and dislocation loading has been proposed. The possibility of delamination along a part of the interface between the inclusion and the matrix, where sliding with dry friction occurs, is envisaged. A structurally modular method of jump functions is constructed to solve the problems arising when nonlinear geometrical or physical properties of a thin inclusion are taken into account. A complete system of equations is constructed to determine the unknowns of the problem. The condition for the appearance of slip zones at the inclusion–matrix interface is formulated. A convergent iterative algorithm for analytical and numerical determination of the friction-slip zones is developed. The influence of loading parameters and the friction coefficient on the development of these zones is investigated.

A UNIQUE PERFECT POWER DECAGONAL NUMBER

Let $\mathcal {P}_s(n)$ denote the nth s-gonal number. We consider the equation $$ \begin{align*}\mathcal{P}_s(n) = y^m \end{align*} $$ for integers $n,s,y$ and m. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$ . We consider the case $s=10$ , that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number greater than 1 expressible as a perfect mth power with $m>1$ is $\mathcal {P}_{10}(3) = 3^3$ .

The problem of the partial delamination of the elastic interface thin inclusion in the conditions of longitudinal shear of the bimaterial

The problem of longitudinal displacement of a bi -material with a thin inclusion of arbitrary physical and mechanical nature at the interface of the matrix materials is considered. The bulk is loaded by normal compression and various force factors in the longitudinal direction. The possibility of partial delamination of a part of the boundary between the inclusion and the matrix, where dry friction slip occurs, is assumed. A complete system of equations for the formulated problem is constructed. It is proposed to construct the solution using the structural modular method of jump functions, a description of which is given. A condition for the appearance of a slip zone on the inclusion-matrix boundary is founded. A convergent iterative algorithm for numerically analytical determination of the size of this zone is developed.

A Modular Method for Mechanical Error Analysis of Planar Linkages Composed of Class II Assur Group Kinematic Chains

This paper presents a modular method for the mechanical error analysis of complex planar linkages. The topology of the linkage under investigation is decomposed into several class II Assur group kinematic chains (AGKCs) combined in a given sequence. Therefore, the mechanical error of the whole linkage can be analyzed by investigating the error propagations of adopted AGKCs in successive order. Because class II AGKCs are first served as modules, the mechanical error equations of these AGKCs in terms of each error in link lengths and joint variables can be pre-formulated and embedded in form of subroutines in any programmable language. Once the AGKCs constituting the linkage topology is identified, the corresponding subroutines are introduced to compute the error propagations in the linkage. Therefore, the presented modular approach can facilitate the analysis by concentrating on the topology decomposition instead of the algebraic derivation. Numerical examples are provided to illustrate the advantage and flexibility of the modular approach.

Analysis of calculations when creating a theoretical drawing of a tug by the interpolation method

The development of a theoretical drawing by manual methods is notable for considerable laboriousness, in this regard, the use of methods that significantly reduce the development time and increase its quality is relevant. In this work, a comparative analysis of methods for obtaining ordinates of a theoretical drawing is carried out. The existing methods of computer-aided design and methods of forming a theoretical drawing are considered: classical, interpolation, affine transformation method, modular method. The process of development of the surface of the tug by the interpolation method is shown, the substantiation of its application is carried out. On the basis of the developed model, a program for calculating the ordinates of the theoretical drawing of tugboats was developed. The results of the software package operation are presented, namely, the ordinates of the theoretical drawing of the tugboat and the hull of the theoretical drawing, as well as the analysis of the calculation accuracy. The adopted approach to the development of the ship's surface can significantly reduce the time and cost of design work on the development of the ship's surface, can be used for its further automation and use as scientific, industrial and educational purposes.

Modular Clause Composition in Plautus

The iambic trimeters of Plautus are analyzed by syntactic boundaries and shown to be composed in a very narrow range of clause-measures using regular termini points in trimeters—line-end and the two caesuras. The five most frequently used syntactic measures account for half of trimeter composition. Plautus composed in modular units of syntax. This paper demonstrates: 1) the most frequent clause-type in Plautus’ trimeters is a trimeter in length, 2) the most frequent clause-type involving enjambment is exactly two trimeters in length, 3) certain clause-types appear with greater frequency in certain plays of Plautus, 4) clause-types can be shown to have distinctive, rhythmic cadences associated with each type. This modular method of clause composition must have been the product of its functional service to the playwright as he generated plays, to the actors who memorized them, and to the audience who heard discourse delivered in regular clause-packets.

Lace Canvas: Method of Modular Design of Costume

In this article, the author examines the modular methodology for designing a suit as a way to transform the shape and structure of a suit with minimal expenditure of time and financial resources, which determines the great potential of the modular methodology in the field of suit design. The main task of the study is to provide analytical substantiation of the existing modular design of a suit from lace fabrics and the development of modular design of a suit from lace, lace fabrics and lace-like structures. The adaptation of a modular methodology in designing a suit using lace, lace fabrics and lace-like structures was carried out for the subsequent development and improvement in the design of costumes from lace, which was chosen as the basis of the empirical design method. This allows the research results to be included in the designer’s scientific base for developing a suit from modules using additive technologies. In the article, the author discloses the principles of designing a suit from lacy modular elements by changing the configuration of modules (flat module, volumetric module), which, in turn, are divided into subcategories. The systems for connecting modules into a canvas (in accordance with the geometry of the ornamental mesh, based on the theory of A.V. Shubnikov; according to the principle of unsystematicity and the principle of linear arrangement of modules) and the types of connecting modules in a suit made of lace and lace-like structures (“butt”, “overlay”, “оn the edge”, “figure”). As a result of the study, an algorithm for the method of modular design of costumes from lace, lace fabrics and lace-like structures is proposed, on the basis of which the author's sketches are presented using a modular method of designing costumes from lace fabrics.

Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields

In this paper we prove that for every integer [Formula: see text], there exists an explicit constant [Formula: see text] such that the following holds. Let [Formula: see text] be a number field of degree [Formula: see text], let [Formula: see text] be any rational prime that is totally inert in [Formula: see text] and [Formula: see text] any elliptic curve defined over [Formula: see text] such that [Formula: see text] has potentially multiplicative reduction at the prime [Formula: see text] above [Formula: see text]. Then for every rational prime [Formula: see text], [Formula: see text] has an irreducible mod [Formula: see text] Galois representation. This result has Diophantine applications within the “modular method”. We present one such application in the form of an Asymptotic version of Fermat’s Last Theorem that has not been covered in the existing literature.