Equilibrium and stability of non-linearly elastic bodies with cavities containing fluid

Boundary conditions are formulated on the surface of a cavity filled with a compressible fluid or gas for the equilibrium problem of an elastic body experiencing large deformations. A formulation is given of the stability problem for the equilibrium of a non-linearly elastic body with fluid inclusions. The stability problem is solved for a thick-walled closed spherical shell filled with gas and loaded by external pressure. © 1988

Diagrammatic method for theory of magnetic and resistive properties of manganites

Effective field theory of magnetic and resistive properties of manganites with account of strong Hund exchange coupling and electron-phonon interactions has been evolved under the strong Hund coupling condition. In parallel with Lang-Firsov unitary transformation of the zeroth Hamiltonian, we have realized the diagonalization of Hund's Hamiltonian neglecting the upper triplet. The diagram techniques taking into account the quantum spin fluctuations of lower quintet and hole state with spin S=3/2 was developed. The magnetic structure of the ground state and an influence of electron-phonon interaction have been analyzed using the first nonvanishing approximation of perturbation theory. The calculated temperature dependence of resistivity agrees well with experimental data including these obtained in applied magnetic field.Comment: 44 pages, 14 figure

ТОЧНЫЕ РЕШЕНИЯ ЗАДАЧ ТЕОРИИ МНОГОКРАТНОГО НАЛОЖЕНИЯ БОЛЬШИХ ДЕФОРМАЦИЙ ДЛЯ ТЕЛ, ОБРАЗОВАННЫХ ПОСЛЕДОВАТЕЛЬНЫМ СОЕДИНЕНИЕМ ДЕФОРМИРОВАННЫХ ЧАСТЕЙ

Large strains of composite solids made of incompressible isotropic nonlinear-elastic materials are analyzed for the case in  which the parts of these solids are preliminarily strained. The  approaches to exact analytical solutions of these problems are given  and developed in cooperation with V.An. Levin. He is a professor at  the Lomonosov Moscow University. The solution of these problems is  useful for stress analysis in members containing preliminarily stressed parts. The results can be used for the verification of industrial software for numerical modeling of additive  technologies. The problems are formulated using the theory of  repeated superposition of large strains. Within the framework of this  theory these problems can be formulated as follows. Parts of a  member, which are initially separated from one another, are subjected to initial strain and passes to the intermediate state. Then  these parts are joined with one another. The joint is performed by  some surfaces that are common for each pair of connected parts.  Then  the body, which is composed of some parts, is strained as a  whole due to additional loading. The body passes to the final state.  It is assumed that the ideal contact conditions are satisfied over the  joint surfaces. In other words, the displacement vector in the joined  parts is continuous over these surfaces. The exact solutions for  isotropic incompressible materials are obtained using known  universal solutions and can be considered as generalizations of these solutions for superimposed large strains. The following problems are considered in detail:— the problem of stress and strain state in two hollow circular elastic cylinders (tubes) one of which is preliminarily strained and inserted into another cylinder (the Lam´e-Gadolin problem);— the problem of torsion of a composite cylinder;— the problem of large bending strains of a composite beam consisting of some preliminarily strained parts (layers). The  mathematical statements of these problems are given, the methods  of solution are presented, and some results of solution are shown.  The impact of preliminary strains on the state of stresses and strains is investigated, and nonlinear effects are analyzed.В статье приведены и развиты разработанные совместно с профессором МГУ им. М.В.  Ломоносова В.Ан. Левиным подходы к точному аналитическому решению задач о больших  деформациях составных изделий (тел) из несжимаемых изотропных нелинейно-упругих  материалов, части которых предварительно деформированы. Решение этих задач  представляет интерес при анализе напряжений в элементах конструкций, изготавливаемых  из предварительно нагруженных частей. Результаты могут быть использованы для  тестирования промышленного программного обеспечения, предназначенного для  численного моделирования аддитивных технологий. Постановка задач осуществляется на  основе теории наложения больших деформаций и в рамках этой теории может быть  сформулирована следующим образом. Части изделия, первоначально не связанные между  собой, подвергаются начальному деформированию и переходят в промежуточное состояние. Затем эти части соединяются между собой. Соединение происходит по некоторым поверхностям, общим для каждой пары соединяемых частей. Далее тело, составленное из нескольких частей, деформируется как единое целое под действием приложенной к нему  дополнительной нагрузки и переходит в конечное состояние. Предполагается, что на  поверхностях, по которым соединены части тела, выполняются условия идеального  контакта, т.е. векторы перемещений в соединяемых частях изделия на этих поверхностях  совпадают. Точные решения для изотропных несжимаемых материалов найдены с  использованием известных универсальных решений и могут быть рассмотрены как  обобщение этих решений на случай наложения больших деформаций. В статье детально  рассмотрены следующие задачи:— задача о напряженно-деформированном состоянии в  двух полых круговых упругих цилиндрах (трубах), один из которых был предварительно  деформирован и вставлен в другой цилиндр (задача Ламе-Гадолина);— задача о кручении  составного цилиндра;— задача о больших деформациях изгиба составного бруса, состоящего из нескольких предварительно деформированных частей (слоев). Приведены  математические постановки этих задач, методы и некоторые результаты их решения.  Исследовано влияние предварительных деформаций на напряженно-деформированное  состояние, анализируются нелинейные эффекты

Quantum phase transitions and thermodynamic properties in highly anisotropic magnets

The systems exhibiting quantum phase transitions (QPT) are investigated within the Ising model in the transverse field and Heisenberg model with easy-plane single-site anisotropy. Near QPT a correspondence between parameters of these models and of quantum phi^4 model is established. A scaling analysis is performed for the ground-state properties. The influence of the external longitudinal magnetic field on the ground-state properties is investigated, and the corresponding magnetic susceptibility is calculated. Finite-temperature properties are considered with the use of the scaling analysis for the effective classical model proposed by Sachdev. Analytical results for the ordering temperature and temperature dependences of the magnetization and energy gap are obtained in the case of a small ground-state moment. The forms of dependences of observable quantities on the bare splitting (or magnetic field) and renormalized splitting turn out to be different. A comparison with numerical calculations and experimental data on systems demonstrating magnetic and structural transitions (e.g., into singlet state) is performed.Comment: 46 pages, RevTeX, 6 figure

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function

Fast Control Systems: Nonlinear Approach

International audienceThis chapter treats the problem of fast control design for nonlinear systems. First, we discusses the question: which nonlinear system can be called fast? Next, we develop some tools for analysis and design of such control systems. The method generalized homogeneity is mainly utilized for these purposes. Finally, we survey possible research directions of the fast control systems