Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

Mathematical studies of nematic liquid crystals address in general two rather different perspectives: that of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter focuses on stationary ones. The two are usually studied with different mathematical tools and address different questions. The aim of this brief review is to give the practitioners in each area an introduction to some of the results and problems in the other area. Also, aiming to bridge the gap between the two communities, we will present a couple of research topics that generate natural connections between the two areas. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.

Fundamentals of electro-mechanically coupled cohesive zone formulations for electrical conductors

Motivated by the influence of (micro-)cracks on the effective electrical properties of material systems and components, this contribution deals with fundamental developments on electro-mechanically coupled cohesive zone formulations for electrical conductors. For the quasi-stationary problems considered, Maxwell’s equations of electromagnetism reduce to the continuity equation for the electric current and to Faraday’s law of induction, for which non-standard jump conditions at the interface are derived. In addition, electrical interface contributions to the balance equation of energy are discussed and the restrictions posed by the dissipation inequality are studied. Together with well-established cohesive zone formulations for purely mechanical problems, the present developments provide the basis to study the influence of mechanically-induced interface damage processes on effective electrical properties of conductors. This is further illustrated by a study of representative boundary value problems based on a multi-field finite element implementation.

DYNAMIC PROBLEM FOR A THIN-WALLED BAR WITH A MONOSYMMETRIC PROFILE

The paper presents an analytical solution to the dynamic problem for a thin-walled elastic rod, thecross-section of which has one axis of symmetry. The solution is constructed for an arbitrary dynamic load and two types of boundary conditions: hinged support in constrained torsion and free warping of the end sections of the rod; rigid fastening with constrained torsion and absence of warping. The peculiarity of the mathematical model lies in the fact that the differential equations of motion contain a complete system ofinertial terms. Spectral expansions obtained as a result of using the method of integral transformations are represented as an effective method for solving linear non-stationary problems in mechanics. The structuralalgorithm of the method of finite multicomponent integral transformations proposed by Yu.E. Senitsky is used.

Structure of parasite cenosis of endoparasitoses of ruminants

The infestation was studied and the structure of the parasite cenosis of ruminants of the Udmurt Republic was characterized. Over the period of several years of research (2016-2020), a diverse parasitophauna was identified. According to the results of the research, it was established that cattle is mainly invaded by Eimeria spp., nematodes of the Strongylata suborder, dicroceliosis dominates from trematode infestations. The maximum infection rate was noted by eimeriosis invasion, during the entire time of the studies it varied from 87.84 % to 77.73 %, which indicates a tendency to increase invasion and stationary problems of farms. The features of competitive relationships are clearly traced in hepatic flukes: dicrocoeliums and fascioles. Dicrocoeliotic invasion prevails on average by 32 %, which indicates a partial manifestation of antagonism.When analyzing the generic structure of the parasite complex of cattle, on average for five years, it was found that the largest share is occupied by Eimeria spp - 44.35, Strongylata - 30.52, Dicrocoelium - 17.52, Fasciola - 6.43, Eurythrema – 0.67 and Trichostrongylus – 0.51 are less represented. A retrospective epizootic analysis of parasitoses showed that sheeps, like cattle, have a diverse parasitophauna, but in the first case, the detection of monoinvasions is 3.5 times higher. In the structure of sheep parasite cenosis, similar indicators of infection with Eimeria spp, Strongyloides, Moniezia were found, generic indices ranged from 24.18 to 27.61. Strongylatosis of the gastrointestinal tract are replaced by Strongyloides. A significant change was also observed in the group of trematode infestations. Unlike cattle, fasciolosis prevails in sheep with a maximum value of 9.5 times in 2019, while dicroceliosis was not registered at all in 2017. In the sheep parasite complex, moniesiosis occupies the main share – 27.61, which may be related to the conditions of keeping, since sheeps are grazing for a long time and a meeting with intermediate hosts - oribatid mites is inevitable.

Quasistatic elastic-plastic loading and unloading of rods with coulomb dry friction

Loading of elastic-plastic rods with account on Prandtl’s scheme is considered in the paper, and nonlinear interactions are taken into account under the law of Coulomb friction. Exact analytical solutions of quasi-stationary problems of interaction of elastic-plastic rod and undeformable media surrounding it are given.

New operational ratios and their application to non-stationary tasks for rods based on S.P. Timoshenko theory

Relevance. In order to study transient wave processes of deformation in rods on the basis of S.P. Timoshenko theory, it is necessary to have accurate analytical solutions to non-stationary problems in general form. Each accurate solution within this analytical model is an accurate description of the real process, serves as a criterion in assessing the accuracy of approximate solutions. When using operational calculus to analyze traveling waves, it is the inverse Laplace - Carson transformation that poses the greatest difficulty. It follows from the published works that the available solutions to some private problems either have a structure that does not allow to judge the main features of the investigated process, or their efficiency in calculations is achieved only in some rather limited areas of coordinate and time. This problem, which requires resolution, determined the purpose of this article. The aim of the work. The article is devoted to the development of new operational ratios and their application to the construction of accurate analytical solutions to the non-stationary problems of S.P. Timoshenko's theory for rods in a general form, in a physically visible and convenient form for practical calculations. Methods. The work uses methods of function theory of complex variable, operational calculus based on the integral Laplace - Carson transformation, methods of structure dynamics. Results. In general form three types of non-stationary tasks for semi-infinite rod based on Timoshenko theory are formulated. New operational ratios have been obtained. Based on these ratios, a method of inverse transformation without using a general conversion formula has been developed. Solutions of problems are recorded in the form of integrals from Bessel functions and, unlike solutions available in the literature, clearly show the wave nature of the studied processes, have a visual and compact appearance. An example of calculation is reviewed.