On the relation between the speed of sound and the internal energy of polymers

The relation between the speed of sound and the changes of internal energy, entropy, volume, and temperature associated with the deformation of polymers is derived.All-Union Scientific Research and Design-Technological Institute of Asbestos Technical Products, Yaroslavl. Translated from Mekhanika Polimerov, No. 1, pp. 157–159, January–February, 1970.     

非热平衡Schwarzschild de Sitter黑洞的熵

从Schwarzschild de Sitter时空背景下的Klein Gordon方程出发，利用brick wall方法计算了黑洞的自由能和熵。这种黑洞由于存在两个视界面，而且它们的温度不相同，因此用一般方法计算熵存在着一定的困难。采用改进的brick wall模型，认为自由能和熵主要来自于视界附近薄层的贡献，很好地解决了这一困难，并得到了满意的结果。结果表明，这种黑洞的熵为它的两个视界面积之和的1/4，与人们预期的结果相符。由此可见，渐近de Sitter时空中的黑洞熵除了黑洞视界面的贡献之外，还应包括宇宙视界面的贡献。这从一定程度上揭示了黑洞熵与视界面积之间的内在联系，也更进一步地揭示了brick wall模型的本质。     

非热平衡Schwarzschild-de Sitter黑洞的熵

从Schwarzschild—de Sitter时空背景下的K1ein—Gordon方程出发，利用brick—wall方法计算了黑洞的自由能和熵．这种黑洞由于存在两个视界面，而且它们的温度不相同，因此用一般方法计算熵存在着一定的困难．采用改进的brick—wall模型，认为自由能和熵主要来自于视界附近薄层的贡献，很好地解决了这一困难，并得到了满意的结果．结果表明，这种黑洞的熵为它的两个视界面积之和的1／4，与人们预期的结果相符．由此可见，渐近de Sitter时空中的黑洞摘除了黑洞视界面的贡献之外，还应包括宇宙视界面的贡献．这从一定程度上指示了黑洞熵与视界面积之间的内在联系，也更进一步地指示了brick—wall模型的本质．     

Thermodynamics of high-elastic deformation of polyurethanes

The thermoelastic behavior of polyurethanes based on diethylene glycol adipate and tolylene diisocyanate with different network densities has been investigated in connection with their tensile deformation. The change in entropy and internal energy is calculated at strains up to 100%. It is shown that during extension both the entropy and the internal energy decrease. The reduction in internal energy depends on the relative importance of the physical and chemical bonds in the network and is attributable to the redistribution of intermolecular bonds.Mekhanika Polimerov, Vol. 4, No. 3, pp. 450–453, 1968     

On the microcanonical solution of a system of fully coupled particles

We study the Hamiltonian mean field (HMF) model, a system of N fully coupled particles, in the microcanonical ensemble. We use the previously obtained free energy in the canonical ensemble to derive entropy as a function of energy, using Legendre transform techniques. The temperature–energy relation is found to coincide with the one obtained in the canonical ensemble and includes a metastable branch which represents spatially homogeneous states below the critical energy. “Water bag” states, with removed tails momentum distribution, lying on this branch, are shown to relax to equilibrium on a time which diverges linearly with N in an energy region just below the phase transition.     

A possible approach to the construction of a relation between the stress and strain tensors and temperature in rheonomic media

So-called averaging functionals have been determined and used to extend and generalize the local strains theory in relation to the solution of a problem of the mechanics of rheonomic media-the construction of the strain tensor as a function of the stress tensor and temperature in time.Mekhanika Polimerov, Vol. 3, No. 1, pp. 104–110, 1967     

Convergence of Finite Element Approximations and Multilevel Linearization for Ginzburg–Landau Model of d-Wave Superconductors

In this paper, we consider the finite element approximations of a recently proposed Ginzburg–Landau-type model for d-wave superconductors. In contrast to the conventional Ginzburg–Landau model the scalar complex valued order-parameter is replaced by a multicomponent complex order-parameter and the free energy is modified according to the d-wave paring symmetry. Convergence and optimal error estimates and some superconvergent estimates for the derivatives are derived. Furthermore, we propose a multilevel linearization procedure to solve the nonlinear systems. It is proved that the optimal error estimates and superconvergence for the derivatives are preserved by the multilevel linearization algorithm.     

Deformation of teflon in linear and plane states of stress

Teflon specimens have been deformed in accordance with various laws of variation of stress and strain with time under plane stress conditions. Fairly simple expressions describing the relation between the stresses and large (up to 60% relative to the initial dimensions) strains are constructed for a broad class of loading regimes.Scientific-Research Institute of Mathematics and Mechanics, A. A. Zhdanov Leningrad State University. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 742–746, July–August.     

Statistical Theory of Equilibrium Fluctuations

An analysis of the BBGKY hierarchy shows that the fluctuation theory for hydrodynamic variables (i.e., the fluid density, velocity, and temperature) must be constructed based on the hydrodynamic equations. We show that fluctuations in the hydrodynamic variables uniquely determine the fluctuations in all other thermodynamic parameters of a fluid (such as the pressure, entropy, and intrinsic energy). We find the spectral structure of fluctuation waves of the hydrodynamic variables, calculate their amplitudes, and derive the dispersion equation establishing the relation between the phase velocity of the fluctuation wave propagation and its frequency. We explain the effects of the flicker noise generation and of the stability loss on the first-order phase equilibrium lines. We also analyze the fluctuation decay process and show that this process gives rise to the effect of the Mandelstam–Brillouin scattering. We compare the obtained results with the existing fluctuation theories.     

Deformation and sliding friction of polyurethane in the presence of a lubricant

The relation between coefficient of friction and elastic and high-elastic strains is investigated with reference to polyurethane subjected to friction in various liquid media. It is shown that a definite relationship between these parameters does exist. The effect of various liquids on the elastic and high-elastic deformation of polyurethane is studied.Kiev Institute of Civil Aviation Engineers. Translated from Mekhanika Polimerov, Vol. 5, No. 2. pp. 357–359, March–April, 1969.     

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.     

Thermocreep of polyethylene in the cyclic temperature regime

The shear creep and recovery of low-density polyethylene has been investigated in the constant and cyclic temperature regimes. It is shown that in the cyclic regime the relaxation processes are accelerated and the nonisothermal creep strains become progressively greater than the isothermal strains at the maximum temperature of the cycle. In recovery the relaxation spectra are shifted along the time scale relative to the creep spectra, the shifts for simple shear and uniaxial tension being nonidentical.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 227–235, March–April, 1969.     

Relation between the vectors E and S for a plastic material in succesive biaxial tension

The local strains theory is used to determine the values of the components of the vector E{E₁, E₂} in five-dimensional space for a material subjected to successive biaxial tension at degrees of nonlinearity n=3 and n=5. The relation between the vectors E and S and the tangents to the loading and strain trajectories is considered together with the relation between the moduli E of the strains for simple and complex loading.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 975–979, November–December, 1969.     

Photoelastic determination of polymer creep stresses and strains

A physical basis provided for the relation between the measured optical quantities and the unknown mechanical quantities in the photoelastic determination of stresses and strains. Questions relating to the "freezing in" of strains are considered in connection with creep tests on models made of optically sensitive polymeric materials. The determination of the stress concentration at the edge of an opening in a strip tested in creep is considered as an example of the application of the theory.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 739–746, July–August, 1970.     

Nonlinear dynamical problem of a cyclinder with a slowly changing internal boundary

The problem of a hollow viscoelastic cylinder enclosed in an elastic shell is examined. The inside diameter varies. A nonlinear cubic theory is used to describe the relation between the stresses and strains in time. A vibration analysis is carried out for variable internal pressure.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 2, pp. 274–280, March–April, 1973.     

Kinetic nature of polymer microhardness

The dependence of the microhardness of PMMA and PS on indenter penetration time and temperature has been studied. It is shown that the microindentation process is a kinetic process. A relation is established between the kinetic characteristics of the microindentation process and the corresponding characteristics of the fracture process.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 22–28, January–February, 1973.     

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.     

Statistical interpretation of the temperature and stress shift factors

Expressions for the temperature and stress shift factors are derived on the basis of a statistical-probability treatment of the segmental motion of the macromolecules using the theory of energy level transitions. It is shown that in the general case the temperature shift factor should depend on stress, and the stress shift factor on temperature.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 528–531, May–June, 1976.     

Effect of hydrostatic pressure on certain thermodynamic properties of polymeric materials

By means of a thermodynamic approach it is demonstrated that the mechanical properties of homogenous polymeric materials depend only on specific volume. Expressions are derived for the entropy and volume coefficient of thermal expansion as functions of hydrostatic pressure and temperature. It is shown that for both crystalline and amorphous polymeric materials the bulk modulus depends on reduced temperature.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 825–829, September–October, 1970.     

Haldane–Wu Statistic and Rogers Dilogarithm

The Haldane–Wu exclusion statistic is considered from the generalized extensive statistics point of view and certain related mathematical aspects are investigated. A series representation for the corresponding generating function is obtained. Equivalence of two formulas for the central charge derived for the Haldane–Wu statistic via the thermodynamic Bethe ansatz is established. As a corollary, a series representation with a free parameter for the Rogers dilogarithm is found. It is shown that the generating function, entropy, and central charge for the Gentile statistic majorize those for the Haldane–Wu statistic (under an appropriate choice of parameters). This fact is applied in derivation of a dilogarithm inequality. Bibliography: 14 titles.