Density functional taking into account nuclear thermal motion of compressed matter

Bogolyubov's inequality is used to generalize the density functional method to take into account nuclear thermal motion. The results apply to a two-component system in which the nuclei satisfy classical statistics. Applications to a cell model describing dense heated states of matter are considered.All-Union Research Institute of Opticophysical Measurements of the USSR State Standards. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 154–157, July, 1992.     

Separation of a surface crack in a spherical shell taking account of elastic deformation

We give the statement and solution of the problem of limiting equilibrium of a spherical shell weakened by a surface crack. We study the dependence of the separation of the edges of the crack on the geometric and physical parameters of the shell.     

The stationary motion of a weakly deformed drop in the flow of a viscous fluid

The flow of a drop of viscous, incompressible fluid in another viscous, incompressible fluid is studied with values of the viscosities, densities, surface tension, and drop size which guarantee small deviation of the shape of the drop from spherical. The effect of the deviation on the character of the flow, the drag coefficient, etc. is considered. The cell model for the motion of a weakly deformed drop is investigated. An approximate formula is given for finding the drag coefficient in constrained flow. Some cases of nonisothermic motion of a spherical drop are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 157–170, 1977.     

Analysis of a unilateral contact problem taking into account adhesion and friction

In this paper, we investigate a contact problem between a viscoelastic body and a rigid foundation, when both the effects of the (irreversible) adhesion and of the friction are taken into account. We describe the adhesion phenomenon in terms of a damage surface parameter according to Frémond?s theory, and we model unilateral contact by Signorini conditions, and friction by a nonlocal Coulomb law. All the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators, whence the highly nonlinear character of the resulting PDE system. Our main result states the existence of a global-in-time solution (to a suitable variational formulation) of the related Cauchy problem. It is proved by an approximation procedure combined with time discretization.     

The opening of a nonthrough longitudinal crack in a cylindrical shell taking account of elastoplastic deformation

In the context of an analog of the _k —model the problem of the stress-strain state of a shallow cylindrical shell with a nonthrough longitudinal crack is reduced to a system of singular integral equations that is solved by the small-parameter method. For a shell under an internal pressure we study the dependence of the opening of the crack on the geometric and physico-mechanical parameters of the shell.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 66–70.     

Optimal production control problem taking into account environmental pollution

We consider a simplified production control model taking into account environmental pollution. We obtain constructive conditions under which each optimal control is Lebesgue equivalent to an optimal bang-bang control.     

Methods of taking into account the nonlinearity of the deformation properties of the material in calculating thick-walled composite rings

The results of an investigation of the deformation properties of unidirectional composites are presented, and engineering models which take into account the effect of the nonlinearity of the properties in the radial direction on the state of stress of rings under pressure are proposed. The relations obtained are checked theoretically — by means of a refined calculation based on the nonlinear theory of elasticity — and experimentally — by loading rings with internal and external pressure by means of rubber cuffs in specially designed apparatus.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 73–84, January–February, 1976.     

Thermo-mechanical modeling of turbine blades taking into account structural defects

In turbine blades of aero-engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non-physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non-penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node-to-segment approach proposed in [3] is extended to ten-node tetrahedral elements with quadratic shape functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theory of potential scattering,taking into account spatial anisotropy

We get new tests for the existence and completeness of wave operators under perturbation of a pseudodifferential operator with constant symbol P() by a bounded potential v(x). The term anisotropic is understood in the sense that the growth of P() as and the decrease of v(x) as x can depend essentially on the direction of the vectors and x respectively. This permits us to include in the sphere of applications of the abstract scattering theory of a nonelliptic unperturbed operator the D'Alembert operator, an ultrahyperbolic operator, nonstationary Schrödinger operator, etc. In view of the anisotropic character of the assumptions on the potential, the results obtained are new even in the elliptic case. As an example we consider a Schrödinger operator with potential close to the energy of a pair of interacting systems of many particles.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 35–51, 1977.     

Method for taking into account gravity in free-surface flow simulation

A numerical algorithm that correctly takes into account the force of gravity in the presence of density discontinuities is constructed using unstructured collocated grids and splitting algorithms based on SIMPLE-type methods. A correct hydrostatic pressure field is obtained by explicitly extracting the gravity force contribution to the pressure equation and computing it using the solution of the gravity equilibrium problem for a two-phase medium. To ensure that the force of gravity is balanced by the pressure gradient in the case of a medium at rest, an algorithm is proposed according to which the pressure gradient in the equations of motion is replaced by a modification allowing for the force of gravity. Well-known free-surface problems are used to show that, in contrast to previously known algorithms, the proposed ones on unstructured meshes correctly predict hydrostatic pressure fields and do not yield velocity oscillations or free-surface distortions.     

Deformation of a drop in a viscous stream and conditions of existence of its equilibrium form

The difference method is used for obtaining a solution of the problem of unsteady motion of a drop in a stream, taking into account its deformation under conditions of axial symmetry. The fluid inside and outside the drop is assumed viscous and incompressible. The stable forms of drop are represented for various Reynolds and Weber numbers of external stream. By analyzing the conditions for normal stresses at the drop boundary, the critical Weber number was obtained, which establishes the conditions of existence of equilibrium form of the drop.