Basic equations of the theory of reinforced media

The author derives the basic equations of the theory of composite elastic media obtained by reinforcing some elastic medium with a large number of linear or planar elastic elements with high strength and deformation resistance. The argument is based on macrostructural considerations. The stress-strain state of each of the reinforcing elements is considered with allowance for interaction with the matrix material. In addition, the "smoothing" principle introduced in [1–3] is applied. This corresponds to approximating the reinforced medium with some equivalent quasi-homogeneous anisotropic medium.The case of a fibrous medium in which the reinforcing elements are rods or filaments [4] is discussed in detail. Allowance for moment effects leads to equations analogous to the equations of the Voight-Cosserat moment theory and its later generalizations. Similar equations are obtained for the case of laminated media, where the reinforcing elements are membranes or plates. On the basis of the viscoelastic analogy [7], the equations of the theory of reinforced media are extended to include the case in which the matrix and/or reinforcing materials are linear viscoelastic.Mekhanika Polimerov, Vol 1, No. 2, pp. 27–37, 1965     

Determination of regular discrete relaxation spectra from the viscoelastic functions

A procedure for determining regular discrete relaxation spectra from the viscoelastic functions is presented as a generalization of a previously proposed procedure for determining the conditional rheological characteristics in a piecewise-exponential approximation of the experimental creep, relaxation, and – curves. The calculated data are in good agreement with the experimental results.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 801–807, September–October, 1971.     

Stress relaxation in physically nonlinear polymer materials under simultaneous tension and torsion

Conclusions 1. Application of our method of determining the parameters of physically nonlinear polymer materials from tests in uniaxial tension or in torsion ensures a satisfactory agreement between theoretical and experimental stress-relaxation curves corresponding to other modes of loading.2. Theoretical stress-relaxation curves for shear (tension) calculated on the basis of parameters which have been determined from approximations of stress-relaxation curves for tension (torsion) fall closely within the confidence interval for an individual measurement (with =95%).3. The agreement between theoretical and experimental stress-relaxation curves for a plane state of stress is 2–3 times worse, in terms of the mean-square relative difference, than the agreement between the respective curves for uniaxial tension.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 797–803, September–October, 1977.     

Failure of laminar fiberglass-plastics in compression

Conclusions 1. The most general form of failure of laminar fiberglass-plastics in compression, as revealed on the basis of experimental studies, is chipping along an inclined area.2. A numerical comparison of the theoretical and experimental values of the chip angles for the materials studied supports the suggestion of interpreting the chip bands as slippage bands in an ideal elastoplastic anisotropic body.3. On the basis of this suggestion, valuable equations (11)–(13) have been obtained for calculating the strength of fiberglass-plastics in compression as a function of the strength properties of the matrix and of the reinforcement, the percent reinforcement, and the cutting angle .M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 2, pp. 240–243, March–April, 1978.     

Creep on an initial small interval

The creep processes of hereditary media on a small initial interval are described, using as influence functions the resolvent functions for the generators of the Dirac function, which are employed as relaxation kernels. Equations are constructed for determining the instantaneous modulus and the parameters of these functions from experimental stress-strain diagrams obtained at three different constant loading rates.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 5, pp. 945–950, September–October, 1970.     

Investigation of longitudinal wave propagation in a polyethylene rod

Longitudinal wave propagation in a polyethylene rod has been investigated. A computer has been used to obtain a numerical solution of the problem of wave propagation in a rod, whose material is described by the Boltzmann-Volterra equation. The results of the computations are compared with the experimental data. The method of characteristics is employed to construct the dynamic – diagrams for the case of wave propagation in a prestressed rod on the basis of the model of a viscoelastic standard solid.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 59–67, January–February, 1970.     

Generalized concentration dependence of the viscosity of concentrated polymer solutions

The construction of a generalized concentration dependence of the viscosity of concentrated solutions is analyzed. It is shown that there should be a single dependence for different polymers in different solvents of the form: (/_o)^1–2k=1+(1–2k)c[], where k is the Huggins-Martin constant, and [] is the intrinsic viscosity. Deviations from this relation may be observed in the presence of structure formation in the solution or when the experimental temperature is close to the glass-transition temperature of the system."Plastpolimer" Okhtinsk Research-Production Association, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 172–175, January–February, 1976.     

Sur les notions de quasi-homogénéité de feuilletages holomorphes endimension deux

In this note, we recall the different notions of quasi-homogeneity for singular germs of holomorphic foliations in the plane presented in [6]. The classical notion of quasi-homogenity allude to those functions which belong to its own jacobian ideal. Given a foliation in the plane, asking that the equation of the separatrix set is a classical quasi-homogeneous function we obtain a natural generalization in the context of foliations. On the other hand, topological quasi-homogeneity is characterized by the fact that every topologically trivial deformation whose sepatrix family is analytically trivial is an analytically trivial deformation. We give an explicit example of a topological quasi-homogeneous foliation which is not quasi-homogeneous in the sense given above.

     

Sur des notions de n—categorie et n—groupoide non strictes via des ensembles multi-simpliciaux (On the notions of a nonstrict n–category and n–groupoid via multisimplicial sets)

In this paper we first give a simplicial approach to the definition of a nonstrict n–category that we call a n–nerve following the idea that a category could be interpreted as a simplicial set (its nerve). Then we prove that for n=2 our construction is equivalent to the usual nonstrict 2–category (bicategory). Next,we give a simplicial definition of a nonstrict n–groupoïd, and we associate to any topological space a n–groupoïd n (X) which generalises the famous Poincaré groupoïd 1 (X) and embodies the n–truncated homotopy type of . Conversely, we construct for each n–groupoïd a geometric realisation and we show that the functors geometric realisation and Poincaré n–groupoïd induce an equivalence between the category of n–groupoids and the category of n–truncated topological spaces, when we localise both categories by weak equivalence.     

Correlation between static and fatigue strength of reinforced plastics

The conventional estimation of the fatigue strength of glass-fiber-reinforced plastics from the static strength in tension is shown to be inefficient by analysis of experimental data. It was found that a more reliable correlation is observed between their static and fatigue strengths as a result of referring _–1 to *, where * is the smaller of the stresses in tension or compression, determined from the inflection points on the static deformation curves. Views are expressed on the use of the correlation found for reducing the amount of fatigue tests.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 468–474, May–June, 1973.     

Moisture and low temperature impact on the properties of a polyether glass-reinforced plastic

The cyclic action of water and the combined action of water and low temperatures on the physical-mechanical properties of a polyether glass-reinforced plastic are studied. For experimental investigations, methods of sorption, thermogravimetry, and mechanical testing in static bending were used. It was found that the most aggressive external factor for the glass-reinforced plastic was the cyclic action of water, but not the change in its phase state at cooling. The process of sorption-desorption of the glass-reinforced plastic is irreversible.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Institute of Non-Metallic Materials, Siberian Branch of the Russian Academy of Sciences, Yakutsk, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 519–524, July–August, 1998.     

Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford

We develop a cohomology theory for Deligne–Mumford stacks, adapted to Hirzebruch–Riemann–Roch formulas. For this, we define the cohomology with coefficients in the representations and a Chern character, and we prove a Grothendieck–Riemann–Roch formula for the associated Riemann–Roch transformation.     

Determination of coupled creep functions from experimental creep curves

A method is given for constructing the coupled creep functions g(t) from the experimental longitudinal creep curves and the known Poisson's ratio. It is shown that for this purpose it is sufficient to substitute for one of the parameters of the singular kernel a quantity containing the Poisson's ratio and the parameter determined from the solution of the elastic problem.I. M. Gubkin Moscow Institute of the Petrochemical and Gas Industry. Translated from Mekhanika Polimerov, No. 2, pp. 216–220, March–April, 1974.     

Buildup of failures in polymers on micro- and macrolevels

The buildup of damage in polymers has been studied by mechanical and structural—analytical methods. A method of programmed loading has been developed which makes it possible to make an estimate of damage buildup in macroscopic experiments. A comparison has been performed of kinetic curves of failure buildup on molecular and macroscopic levels. It is shown that the kinetic equations where the rate of failure is basically initiated by true stress qualitatively follow the experimental curves for failure buildup on micro- and macrolevels."Plastpolimer" Scientific-Manufacturing Association, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 263–268, March–April, 1976.     

Temperature-frequency reduction of the dynamic functions of black-filled rubbers in the nonlinear region

The possibility of using temperature-frequency reduction of the dynamic functions of black-filled rubbers in the nonlinear region is demonstrated and an additional applicability criterion — similar slopes of the isotherms of the amplitude dependences of the dynamic functions — is formulated. In the case of heavily filled rubbers it may be necessary to determine the constants of the Williams-Landel-Ferry equation and the reference temperature T_S from the experimental data — they may not coincide with the universal values. Master curves of the dynamic functions are obtained for a series of shock-absorbing rubbers at a shear strain amplitude =0.01.A. V. Topchiev Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow. Scientific-Research Institute of the Rubber Industry, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 751–754, July–August, 1973.     

Kinematic approach to the description of autowave processes in active media

The kinematic approach formulated in the paper is very general. The main results obtained above also remain valid for active media of a more complicated nature than the simple ones described by Eqs. (1). To construct the kinematic theory, it is necessary to know only a few phenomenological parameters such as the propagation velocity of a plane front and the critical curvature. In principle, these parameters can be calculated using the particular equations of the active medium, for example, (1).The value of the phenomenological parameters for an active medium can also be obtained experimentally. For example, for a medium with the Belousov-Zhabotinskii chemical reaction [1] the propagation velocity of a plane front is V₀=2–3 mm/min, D is the diffusion coefficient for the solution and equal to D=1.8·10^–5 cm²/sec, and the rotation frequency is =5 min^–1. Then, as follows from (16), the critical curvature at the free end must be 70 cm^–1, in good agreement with the experimental data.     

Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames

This paper is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin [17]. The paper [17] derives, by means of a three-scale matched asymptotics, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis Number – i.e. the ratio between thermal and molecular diffusion – to be strictly less than unity. If is the inverse of the – reduced – activation energy, the idea underlying the construction of [17] is that (i) the time scale of the radius motion is ^-2, and that (ii) at each time step, the solution is -close to a steady solution.In this paper, we give a rigorous proof of the validity of this model under the restriction that the Lewis number is close to 1 – independently of the order of magnitude of the activation energy. The method used comprises three steps: (i) a linear stability analysis near a steady – or quasi-steady – solution, which justifies the fact that the relevant time scale is ^-2; (ii) the rigorous construction of an approximate solution; (iii) a nonlinear stability argument. Mathematics Subject Classification (2000) Primary 80A25, Secondary 35K57, 47G20     

经济学中拟齐次生产模型的几何刻画

齐次和拟齐次生产函数作为经典的生产函数模型,在经济分析中具有广泛的应用.从几何学的角度研究齐次和拟齐次生产函数的分类具有重要理论意义和应用价值.该研究利用几何不变量方法,完全分类了拟齐次生产函数所对应的平坦拟齐次生产曲面,并且也给出了极小拟齐次生产曲面的完整分类.     

A model of the deformation of a thick-walled composite ring under the action of pressure

A model is examined for the deformation of a ring of unidirectional composite under the action of pressure which permits one, to a first approximation, to take account of the nonlinearity of the - relationship on compression of the composite perpendicularly to the fibers. It is shown that this feature of the material leads to a considerable overloading of the ring coils from the side of the applied pressure. The results obtained qualitatively explain some of the known experimental evidence on the strength of thick-walled rings.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1087–1092, November–December, 1972.     

Contribution to the theory of forced high-elastic deformation of polymer materials

The large reversible and irreversible deformations of network polymers are examined. Expressions are derived for the stress tensor and for the conventional tensile stress. The uniaxial homogeneous extension curve has a maximum and a minimum if it is assumed that the coefficient of friction depends exponentially on the applied stress. The "necking" condition is derived for uniaxial tension. The results are compared with the experimental data.Mekhanika Polimerov, Vol. 4, No. 2, pp. 255–262, 1968