On the boundary conditions formulation in the problems of synthesis of woven 3d materials

В статье обсуждаются формулировки определяющих дифференциальных ограничений на поверхности наращивания на случай моделирования процессов формирования 3D материала, характеризующегося дополнительными характерными директорами (направлениями выкладки волокон в тканых материалах, арматуры в бетонных конструкциях). Выведена общая форма тензорного соотношения на поверхности наращивания, при учете дополнительных выделенных направлений. Определить набор совместных рациональных инвариантов тензора напряжений и характерных директоров. Дана инвариантно-полная формулировка определяющих соотношений на поверхности наращивания. Полученные результаты могут быть использованы для постановки и решения краевых задач, моделирующих процессы синтеза тканых 3D материалов. The article discusses the formulation of the defining differential constraints on the buildup surface in the case of modeling the processes of forming a 3D material characterized by additional characteristic directors (directions of laying fibers in woven materials, reinforcement in concrete structures). The general form of the tensor relation on the growing surface is derived, taking into account the additional selected directions. Determine the set of joint rational invariants of the stress tensor and characteristic directors. An invariant-complete formulation of the constitutive relations on the surface of the build-up is given. The results obtained can be used to formulate and solve boundary value problems that simulate the processes of synthesis of woven 3D materials.

PURSUIT-EVASION DIFFERENTIAL GAMES WITH THE GRÖNWALL TYPE CONSTRAINTS ON CONTROLS

A simple pursuit-evasion differential game of one pursuer and one evader is studied. The players' controls are subject to differential constraints in the form of the integral Grönwall inequality. The pursuit is considered completed if the state of the pursuer coincides with the state of the evader. The main goal of this work is to construct optimal strategies for the players and find the optimal pursuit time. A parallel approach strategy for Grönwall-type constraints is constructed and it is proved that it is the optimal strategy of the pursuer. In addition, the optimal strategy of the evader is constructed and the optimal pursuit time is obtained. The concept of a parallel pursuit strategy (\(\Pi\)-strategy for short) was introduced and used to solve the quality problem for "life-line" games by L.A.Petrosjan. This work develops and expands the works of Isaacs, Petrosjan, Pshenichnyi, and other researchers, including the authors.