Invariant Integrals for the Equilibrium Problem for a Plate with a Crack

We consider the equilibrium problem for a plate with a crack. The equilibrium of a plate is described by the biharmonic equation. Stress free boundary conditions are given on the crack faces. We introduce a perturbation of the domain in order to obtain an invariant Cherepanov–Rice-type integral which gives the energy release rate upon the quasistatic growth of a crack. We obtain a formula for the derivative of the energy functional with respect to the perturbation parameter which is useful in forecasting the development of a crack (for example, in study of local stability of a crack). The derivative of the energy functional is representable as an invariant integral along a sufficiently smooth closed contour. We construct some invariant integrals for the particular perturbations of a domain: translation of the whole cut and local translation along the cut.     

Small finite deformations of an amorphous polymer material

Small finite deformations of a polymer material are considered. It is assumed that the volume of the specimen is conserved during deformation and that the strains are affine; furthermore, the equilibrium distance between network nodes is not equal to zero in the specimen, and the unit cell of the network is cubic. Various types of deformation are considered. The expressions obtained are compared with the results of the theory of high elasticity and the experimental data.Branch Institute of Chemical Physics, AS USSR, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 997–1001, November–December, 1968.     

Twistless Tori Near Low-Order Resonances

In this paper, we investigate the behavior of the twist near low-order resonances of a periodic orbit or an equilibrium of a Hamiltonian system with two degrees of freedom. Namely, we analyze the case where the Hamiltonian has multiple eigenvalues (the Hamiltonian Hopf bifurcation) or a zero eigenvalue near the equilibrium and the case where the system has a periodic orbit whose multipliers are equal to 1 (the saddle-center bifurcation) or −1 (the period-doubling bifurcation). We show that the twist does not vanish at least in a small neighborhood of the period-doubling bifurcation. For the saddle-center bifurcation and the resonances of the equilibrium under consideration, we prove the existence of a “twistless” torus for sufficiently small values of the bifurcation parameter. An explicit dependence of the energy corresponding to the twistless torus on the bifurcation parameter is derived. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 135–144.     

The casimir effect at nonzero temperatures in a universe with topologyS 1×S 1×S 1

We investigate the influence of finite-size effects on the free energy of mass and massless scalar fields at nonzero temperatures in a universe with the topology of a three-dimensional torus. The renormalized free energy and the resulting energy density and pressure are found. Their physical consequences are discussed.Republished from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 3, pp. 482–496, September, 1996.     

Thermodynamic constraints on the relation between the stresses and the time derivatives of the strains and temperature for reversible processes

It is assumed that the free energy and entropy are functions of time and time derivatives of the temperature and strain of any order. The laws of thermodynamics of reversible processes are employed to obtain constraints on the relation between the stresses and the time derivatives of the strains and temperature and, moreover, the relation between the free energy and the part of the entropy that does not depend on the first derivative of the strains with respect to time.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 6, pp. 1001–1008, November–December, 1970.     

Return to Equilibrium for Pauli-Fierz Systems

We study ergodic properties of Pauli-Fierz systems – W*-dynamical systems often used to describe the interaction of a small quantum system with a bosonic free field at temperature $ T \geq 0 $. We prove that, for a small coupling constant uniform as the positive temperature $ T \downarrow 0 $, a large class of Pauli-Fierz systems has the property of return to equilibrium. Most of our arguments are general and deal with mathematical theory of Pauli-Fierz systems for an arbitrary density of bosonic field. Communicated by Gian Michele Graf submitted 22/09/02, accepted: 06/05/03     

Stability of incompressible current-vortex sheets

We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28 (2005) 917–945] where an energy a priori estimate for the linearized free boundary value problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in [S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622–629 (in Russian); W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655]. In this paper we derive an a priori estimate in the whole stability domain. The crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss' type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition, i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of derivatives in the energy estimate. The result of this paper is a necessary step to prove the local-in-time existence of stable nonplanar incompressible current-vortex sheets by a suitable Nash–Moser type iteration scheme.     

Supersymmetric p-branes at finite temperature

The regularized free energy of open and closed supersymmetric p-branes in (D – 1)-dimensional flat space is calculated by the method of quasiclassical quantization. The critical temperature, indicating a phase transition, is obtained. It is shown that the quantum properties of bosonic p-branes (open and closed) with p = D – 1 are identical to the properties of a massless scalar particle.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 36–40, 1990.     

Effect of fillers on the structure and properties of cured resins

The elastic (modulus of elasticity and equilibrium high-elastic modulus) and thermal (volume coefficients of thermal expansion below and above the glass transition temperature) properties of compositions based on ÉD-5 epoxy resin cured with polyethylenepolyamine have been investigated. Quartz powder and aluminoborosilicate glass powder were employed as fillers at concentrations from 0 to 0.413. The thermal expansion coefficients of the compositions were studied in a dilatometer, in which the specimen is free of mechanical loads. The Young's modulus at 25°C and the equilibrium high-elastic modulus at 125°C of the compositions were determined in the compression regime in an instrument based on the IZV-2 optical length gage. The thermal expansion coefficients of the polymer matrix were calculated with allowance for the elastic properties of the resin and the filler. It is shown that, as the filler concentration increases, the thermal and elastic properties of the resin in the filled system change. This can be interpreted as a change in the properties of the resin as it approaches the surface of the filler particles. Increased interaction between the filler surface and the epoxy resin tends to stiffen the polymer network.Scientific Research Institute of Precision Technology, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1018–1022, November–December, 1969.     

Bateman variational principle for a class of problems of dynamics and stability of surface waves

A generalization is made of the Bateman-Luke principle for the problem concerning acoustic interaction with the free surface of a bounded volume of fluid. Extremal criteria are presented for the stability of capillary-sound equilibrium forms.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1181–1186, September, 1991.     

Sudden buckling of the outer layer of a noncentrally compressed triple-layer rod

The stability of the outer nonmetallic layer of a triple-layer rod with foam core was studied both experimentally and theoretically by the energy balance method. The buckling of the outer layer occurs by a sudden transition to a nonadjacent equilibrium form and this process should be seen in regard to a set of deformational factors.Moscow State University for Ecology and Environmental Studies, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 525–530, July–August, 1996.     

Effect of powdered mineral fillers on the crosslinking of phenol-formaldehyde oligomers and the mechanical properties of the molding materials

By investigating the effect of various powdered oxides on network formation in phenolformaldehyde oligomers, it has been established that nonionogenic oxides increase the number of network defects in proportion to the surface energy of the powder. Small amounts of ionogenic oxides with a divalent cation cause the formation of new ionic or ionic-coordination linkages in the polymer network. The effect of mineral powders on polymer network formation is correlated with the mechanical properties of molding materials based on phenol-formaldehyde polymers with different network densities.Moscow Aviation Technological Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1033–1042, November–December, 1968.     

Thermodynamic Potential of a System of Electrons and Phonons in a Disordered Alloy

We obtain a cluster expansion for the two-time retarded Green's functions and the thermodynamic potential of a disordered crystal taking the electron–phonon and electron–electron interactions into account. The electron states of the system are described in the framework of a multiband tight-binding model. The calculations are based on the diagram techniques for the temperature Green's functions. The coherent potential approximation is chosen as a zeroth-order one-site approximation in this cluster expansion method. We show that the contributions from the processes of scattering of elementary excitations on clusters decrease as the number of sites in the cluster increases in accordance with certain small parameters. Analytic estimates of the influence of the electron–phonon interaction on the energy spectrum of electrons of an alloy being ordered are obtained in a one-band model. The applicability of these results to describing the influence of strong electron correlations on the electron structure and properties of alloys of transition metals with narrow energy bands is illustrated with the example of the Fe–Ti alloy.     

Logarithmic corrections in the one-dimensional Hubbard model with attraction

The free energy in the vicinity of a half-filled band for the one-dimensional Hubbard model with attraction is calculated as a function of density (chemical potential). The expansion of the elementary excitation energies for a lattice of finite length N is accomplished with an accuracy of the order (N² ln N)^–1. An explicit expression for the free energy and the elementary excitation spectrum as functions of external field or of volume N is necessary for studying the asymptotic behavior of correlation functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 24–36, 1991.     

Refined stability theory for sandwich shells with transversally stiff core. 2. Linearized equations of neutral equilibrium

Neutral equilibrium equations of the refined theory of stability for sandwich shells with a transversally stiff core are constructed and used for studying local mixed forms of stability loss (FSL), as well as admitting different variants of simplification, depending on the type of precritical state and realized FSL. The generalized Reissner variational principle used for deriving the stability equations allows us to refine transverse shear stresses in the core as compared to [1]. A method for a highly accurate definition of these stresses is proposed. Namely, after the integration of three-dimensional equilibrium equations over the transverse coordinate, the number of free constants and the number of static conditions to be satisfied are equalized according to the actual stress distribution across the thickness.Science and Technology Center for Study of Dynamics and Strength. Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 786–795, November–December, 1997.     

On Instability of Equilibrium Figures of Rotating Viscous Incompressible Liquid

We prove that non-axially symmetric equilibrium figures of uniformly rotating viscous incompressible liquid are unstable when the second variation of the energy functional can take negative values. Bibliography: 8 titles.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 69–85.     

Variational formulations of nonlinear boundary-value problems with a free boundary in the theory of interaction of surface waves with acoustic fields

Variational problems equivalent to nonlinear evolutionary boundary-value problems with a free boundary are formulated. These problems arise in the theory of interaction of limited volumes of liquid, gas, and their interface with acoustic fields. It is proved that the principle of separation of motions can be applied to these variational problems. The problem of a capillary-acoustic equilibrium form is given in a variational formulation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1642–1652, December, 1993.     

Research in the flexural buckling modes of a cylindrical sandwich shell under the action of a temperature field inhomogeneous across its thickness

A solution to the problem on the stability according to the flexural buckling mode is given for a cylindrical sandwich shell with a transversely soft core of arbitrary thickness. The shell is under the action of a temperature field inhomogeneous across the thickness, and its end faces are fastened in such a way (in the axial direction, the face sections of the external layer are fixed, but of the internal one are free) that an inhomogeneous subcritical stress-strain state arises in the shell across the thickness of its layers. It is shown that, under such conditions, the buckling mode of the shell is mixed flexural. To reveal and investigate this mode, equations of subcritical equilibrium and stability of a corresponding degree of accuracy are needed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 715–730, November–December, 2004.     

Model of crack opening in a molecular crystal

Conclusion The mechanical behavior of the discrete structure of a solid can be modeled by a linear system of three particles interacting in accordance with the approximate Born-Oppenheimer law with Lennard-Jones coefficients. Such a model manifests the following properties: elasticity (with small strains); plasticity (residual strains in the presence of two different equilibrium positions); viscosity (change in the frequency of the vibrations with an increase in the forces); hysteresis (with the jump of the middle particle from the left to the right equilibrium position and back); fusion (change in the frequency of the vibrations with an increase in strains).The phenomenon of the interaction of three particles, referred to a linear chain of particles and an organic crystal model, offers a representation of the mechanism of crack opening in a discrete solid.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 446–452, May–June, 1986.     

Casimir effect at nonzero temperatures in a closed Friedmann Universe

For massive and massless scalar fields with an arbitrary coupling to gravity, we investigate finite-size (Casimir) effects on the free energy at nonzero temperatures in a slowly evolving closed Friedmann universe. The renormalized expressions for the free energy and the resulting energy density and pressure are found and their physical properties are discussed. The equation of state turns out to have the form P=/3 for radiation with arbitrary coupling.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 2, pp. 307–320, November, 1996.