Some applied problems of stability of stochastic models

We consider several applied problems of probability theory. These problems are related to biophysics, mathematical economics, and automated control theory, and are considered in the framework of the theory of stability of stochastic models. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Some fundamental problems of polymer mechanics

A survey of the present state of the science of polymer mechanics is presented with particular reference to the theory of linear and nonlinear viscoelasticity of homogeneous and heterogeneous polymers. Various approaches adopted in studies of this subject are discussed; these include methods based on the theory of creep of metals, on a statistical model composed of viscoelastic elements, and on the representation of functionals which, in a quite general sense, are sufficiently close to being linear.The second of these methods, which is approximate, yields results which always point to the nonlinearity of the properties of solids, even those that consist of linear elements. The third method is a natural generalization of the Boltzmann-Volterra theory.One of the fundamental tasks relating to the linear viscoelasticity of polymers is the application of the temperature-time analogy in studies of nonuniform and nonstationary problems of solids subjected to loading and heating-cooling cycles. The necessity of developing methods of modeling phenomena in solids of complex shape in nonuniform and nonstationary temperature and stress states (particularly in the nonlinear range) is discussed and a new method of solving problems of this type is described.Mekhanika Polimerov, Vol. 1, No. 3, pp 33–42, 1965     

Some properties of generalized epicycloids applied to fracture mechanics

Résumé La réflexion partielle d'un faisceau lumineux sur le revers d'une plaque mince en plexiglas comportant une fissure intérieure symétrique formant un angle quelconque avec l'axe d'application de la charge extérieure, crée des déviations des rayons réfléchis dans la zone contractée qui entoure l'extrémité de la fissure. Ces déviations forment une caustique. On montre, dans cet article, que cette caustique prend la forme d'une courbe épicycloïdale générale, valable pour tous les matériaux qui ont un comportement élastique. Cette courbe est créée par l'extremité du vecteur résultant qui représente les rayons de lumière réfléchis et déviés sur la plaque fissurée. On étudie ensuite les proprietés caractéristiques de ce type d'épicycloïdes en relation avec le facteur complexe d'intensité des contraintesK ^*. Les résultats des essais faits à l'aide des plaques munies des fissures intérieures symétriques d'orientation différente par rapport à l'axe d'application de la charge extérieure sont en parfait accord avec les résultats de la théorie.     

Some problems of the mechanics of polymeric composites

Work on the mechanics of polymer composite materials and structures carried out in the Moscow Power Engineering Institute during the last 5 years is reviewed. The review covers work on the theory of reinforced media and its application to composite sturctures, the theoretical and experimental investigation of the effect of technological factors on the reliability of composite structure, the theory of hardening viscoelastic media, and the mechanical testing of specimens and models of polymer composites.     

Some aspects of the group-theoretic approach in problems of nonlinear mechanics

Two aspects of the application of the asymptotic decomposition method to the decomposition and investigation of nonlinear systems are considered. The first one is the extension of the class of functions in terms of which to form the asymptotic decomposition by applying special functions of mathematical physics. The other aspect is the extension of the asymptotic decomposition method to a new class of differential systems, namely, Pfaffian systems in involution.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 22–33, January, 1992.     

Some mathematical problems related to the nonequilibrium statistical mechanics of infinitely many particles

This review article deals with mathematical problems related to the time evolution of systems of infinitely many particles. The probabilistic aspect of the obtained results is emphasized.Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 14, pp. 5–39, 1977.     

Nonlocal parabolic problems in statistical mechanics

A nonlocal parabolic equation describing the evolution of a cloud of particles is studied.     

Variational problems in space flight mechanics

The effect of weight limitations on the parameters of the optimum motion of a variable-mass body in a gravitational field is considered. An analysis is made of the optimum motion for three possible propulsion systems: limited exit velocity, limited power, and limited thrust. Solutions are presented for some typical cases. The problems of selecting the optimum weight parameters, finding the optimum propulsion system controls, and determining the optimum trajectory are solved simultaneously.     

Averaging initial data in applied problems

A method for averaging experimental data is proposed and formulas for averaged values of these data are given. A two-sided estimate accurate to the order of approximation of the averaged data is obtained.     

Numerical resolvent methods for constrained problems in mechanics

Resolvent methods are presented for generating systematically iterative numerical algorithms for constrained problems in mechanics. The abstract framework corresponds to a general mixed finite element subdifferential model, with dual and primal evolution versions, which is shown to apply to problems of fluid dynamics, transport phenomena and solid mechanics, among others. In this manner, Uzawa’s type methods and penalization-duality schemes, as well as macro-hybrid formulations, are generalized to non necessarily potential nonlinear mechanical problems.     

Three problems in the mechanics of continua in polyhedra

The following problems are considered: the problem of the electromagnetic oscillations of a hollow resonator with an ideally conductive boundary; the steady-state Stokes equations under conditions of slippery walls; the problem of elasticity theory under the boundary condition of a rigid contact. Formally, in polyhedral domains these three problems coincide. It is shown that in a meaningful sense such a coincidence is realized only in convex polyhedra. Related effects are discussed. Bibliography: 10 titles. Dedicated to Olga A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 27–37. Translated by V. I. Ochkur.