• #### 539.378:678.0 Mathematical model for finding finite deformations of rubber-like materials

##### Duishenaliev T. B. (NRU "MPEI"), Merkuryev I. V. (NRU "MPEI"), Duishembiev A. S. (Razzakov KSTU)

doi: 10.18698/2309-3684-2020-2-325

The article considers finite (geometrically nonlinear) elastic deformations of rubber-like materials and structures. Such deformations are described by a mathematical model developed on the basis of a non-classical approach to solving boundary-value problems of statics. Formulas are given for determining the final deformations of elastic rubber-like bodies based on the elements of spatial and material displacement gradients. Comparison of definitions for these two approaches is given. The validity of the above conclusions is confirmed by the example of one-dimensional, two-dimensional and three-dimensional transformations in the MathCad system. An example of determining the elements of the spatial displacement gradient is considered. It is known that a static boundary value problem has two formulations. The first one is put forward during its formulation and is used to derive the fundamental relations of the mechanics of a deformable body (Betty's theorem, general solution in the form of Somiliani formulas, etc.). The second is used in solving such problems. It is believed that the problems of both statements have the same solution. A non-classical solution of the boundary value problem of statics is proposed. It strictly corresponds to the generally accepted statement. The Chezaro method of representing the
displacement field using the deformation components is given. Further, this method is developed, it becomes possible to express the field of displacements also through the stress components. The problem of equilibrium of a rectangular plate of rubber-like material is solved. The expressions obtained determine the components of deformations, stresses, and displacements at any point of the plate. In all these expressions, only the coordinates of the finite region of the elastic body are present. There is no usual coordinate misunderstanding: in displacements and stresses are the same coordinates. This problem is also represented by the Navie equations. The uniqueness of its solution is proved.

Дуйшеналиев Т.Б., Меркурьев И.В., Дуйшембиев А.С. Математическая модель для оценки конечных деформаций резиноподобных материалов. Математическое моделирование и численные методы, 2020, № 2, с. 3–25

• #### 539.3 Modeling nonlinear dielectric properties of composites based on the asymptotic homogenization method

##### Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Zubarev K. M. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2020-2-2645

The paper is devoted to the development of a method for calculating the nonlinear dielectric properties of composites with a periodic structure. Methods for predicting of the nonlinear dielectric properties of composites play an important role in the design of dielectric materials with specified properties, in particular for heterogeneous ferroelectrics, which are widely used to create various devices and electrical devices, for example, to create memory storage devices for computers. A quasi-static problem of the distribution of an electric charge in an inhomogeneous polarizable medium with a periodic structure and nonlinear dielectric properties is considered. To solve this nonlinear problem, the asymptotic homogenization method proposed by N.S. Bakhvalov, E. Sanchez-Palencia, B.E. Pobedria. As a result, local nonlinear problems of electrostatics on the periodicity cell are formulated, an algorithm for calculating effective nonlinear constitutive relations for dielectric properties, and an averaged problem for a composite with effective properties are proposed. For the case of a composite with a layered structure, the solution of local problems is obtained, and effective defining relations for the nonlinear dielectric properties of the composite are constructed. It is shown that a laminated composite is a transversely isotropic nonlinear dielectric material if it is isotropic materials. A numerical example of calculating the nonlinear properties of a 2-layer composite based on barium titanate and ferroelectric ceramic varicond VK4 is considered. A model is proposed that describes the nonlinear dependence of the dielectric constant of these materials on the vector of the electric field strength. It is shown that the nonlinear dependence of the dielectric constant tensor of the composite on the strength vector differs significantly for the direction of the field in the plane of the layers and in the transverse direction. It is shown that the developed technique can serve as a basis for designing nonlinear dielectric composite materials with anisotropic properties.

Димитриенко Ю.И., Губарева Е.А., Зубарев К.М. Моделирование нелинейных диэлектрических свойств композитов на основе метода асимптотической гомогенизации. Математическое моделирование и численные методы. 2020. № 2. с. 26–45

• #### 532.2 Modeling of droplets Coalescence

##### Fedyushkin A. I. (Ishlinsky Institute for Problems in Mechanics), Rozhkov A. N. (Ishlinsky Institute for Problems in Mechanics)

doi: 10.18698/2309-3684-2020-2-4658

The paper considers the dynamics of the coalescence of two drops of Newtonian fluid. The changing forms drops in time for the different properties of liquids it is shown using numerical simulation for two-phase system of «liquid – air». The results of the numeri-cal simulation are compared with experimental data.

Федюшкин А.И., Рожков А.Н. Моделирование коалесценции капель. Математическое моделирование и численные методы, 2020, № 2, с. 46–58.

• #### 517.927.4:614.841.1 Математическое моделирование тушения лесного пожара путем доставки воды в его очаг с помощью капсул с термически активной оболочкой

##### Kataeva L. Y. (Nizhny Novgorod State Technical University/Samara State Transport University), Ilicheva M. N. (Nizhny Novgorod State Technical University), Loshchilov A. A. (Nizhny Novgorod State Technical University)

doi: 10.18698/2309-3684-2020-2-5980

В работе проведен численный анализ процессов тушения крупных лесных пожаров с применением капсул воды в термически активной оболочке. Предложена интегральная характеристика для капсул, позволяющая учесть процесс разрушения оболочки при перемещении ее в горячей среде. Предложен простой алгоритм, позволяющий учесть последовательное движение капсул друг за другом с учетом процессов распыления жидкости и процессов тепло- и массообмена. Распыление жидкости происходит в виде высвобождения дисперсных частиц жидкости и подчинено нормальному закону. В работе исследуется динамика процессов тушения лесного пожара при разных сценариях сброса капсул и интегрального параметра термо- устойчивости оболочки. Показано, что полученные результаты хорошо согласуются по количеству тушащего состава, необходимого для тушения с результатами Гундар и Абдурагимова. Выполнен анализ таких ключевых параметров как термоустойчивость и количество последовательно сбрасываемых капсул. Анализ результатов численного моделирования показал, что значение интегрального параметра термоустойчивости является ключевым при тушении лесных пожаров, так как именно он определяет зону распыления дисперсных частиц тушащего состава. Если значение термоустойчивости слишком высокое, то капсулы пролетают зону уязвимости пожара и распыление тушащего состава происходит близко к поверхности земли. В случае слишком маленького значения параметра термо- устойчивости - капсулы начинают распылять воду, не достигая зоны уязвимости пожара, и уносятся конвективными потоками, сформированными пожаром. Сброс капсул последовательно - позволяет более равномерно распределить тушащий состав по вертикали, покрывая зону уязвимости пожара. На основе полученных результатов можно с уверенностью сказать, что более эффективное тушение лесных пожаров можно осуществлять, используя «умную» термически активную оболочку, позволяющую доставить тушащий состав в зону уязвимости пожара.

Катаева Л.Ю., Ильичева М.Н., Лощилов А.А. Математическое моделирование тушения лесного пожара путем доставки воды в его очаг с помощью капсул с термически активной оболочкой. Математическое моделирование и численные методы. 2020. № 2. с. 59–80.

• #### 519.6 Synthesis of optimum control of vertical landing of returned space units

##### Mozzhorina T. Y. (Bauman Moscow State Technical University), Osipov V. V. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2020-2-8194

In this paper, we consider one of the possible feedback algorithms for the vertical landing of the returned first stage of the spacecraft for its reuse in the future. It is proposed to use for thrust correction not correction engines, but the main engine of the power plant of the spacecraft, which can be throttled to 60% of the maximum thrust value. A numerical experiment using the Monte Carlo method is performed to evaluate the performance of this algorithm. A soft landing is a landing with a speed of zero or no more than a few meters per second. The last section of the vertical landing is subject to investigation. The optimal program control in this problem statement in terms of minimum fuel consumption is free fall, then turning the engine on at full power until the moment of landing. It is assumed that such parameters as the speed and mass of the spacecraft's return module at an altitude of 2000 m, the specific impulse, as well as the air density and the coefficient of aerodynamic drag can be randomly deviated from the calculated values. It is assumed that these random variables are distributed according to the normal law, are independent, and their deviations from the calculated values do not exceed 1% for the engine pulse and 5% for all other variables. The landing speed is a random value for which the distribution parameters are calculated. The results of the calculation are analyzed.

Мозжорина Т.Ю., Осипов В.В. Моделирование и синтез оптимального управления вертикальной посадкой возвращаемых космических модулей. Математическое моделирование и численные методы. 2020. № 2. с. 81–94.

• #### 519.6 Computer construction of an equidistant network of complex non-smooth curves on the terrain

##### Valishin A. A. (Bauman Moscow State Technical University), Tumanov I. A. (Bauman Moscow State Technical University), Akhund-zade M. R. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2020-2-95106

The paper considers the problem of constructing a network of equidistant surveys of coastal waters during the design of marine terminals, as well as for landing amphibious assaults on the coast. A computer algorithm is proposed for constructing equidistant complex irregular curves on the ground. The algorithm is implemented as a computer program. The program is tested on an example of a curve of a real coastal strip.

Валишин А.А., Туманов И.А., Ахунд-заде М.Р. Компьютерное построение сети эквидистант сложных негладких кривых на местности. Математическое моделирование и численные методы. 2020. № 2. с. 95–106.

• #### 519.8 Probabilistic model of the battle of two similar combat units against two different types

##### Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University), Anisova T. L. (Bauman Moscow State Technical University)

doi: 10.18698/2309-3684-2020-2-107116

On the basis of the theory of continuous Markov processes, a model of the battle of two of the same type of combat units of the side against two of different types has been developed. The areas of application of various tactics of fighting by the side are shown. It is established that the use of the correct tactics of combat by a party can significantly increase the probability of preserving its two combat units. The developed battle model can be used to evaluate the combat effectiveness of multi-purpose weapons systems.

Чуев В.Ю., Дубограй И.В., Анисова Т.Л. Вероятностная модель боя двух однотипных боевых единиц против двух разнотипных. Математическое моделирование и численные методы. 2020. № 2. с. 107–116.