Solution of the problem of the plane deformation of a tube made of physically nonlinear quadratic viscoelastic material

An exact solution is obtained for the problem of the plane deformation of a tube of quadratically nonlinear viscoelastic material characterized by four kernels. For given boundary pressures the solution contains functions determined from simple creep and relaxation tests for stresses and strains lying in the region of linearity of the relation. In accordance with the method of approximations for nonlinear materials [9], an approximation is given for a series of convolution integrals of increasing multiplicity in transforms and inverse transforms together with an error estimate.Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, No. 1, pp. 117–123, January–February, 1973.     

Creep of polymer materials under periodically varying loads

A method is proposed for constructing the creep curves of a material whose nonlinear memory properties are described by Rozovskii's nonlinear integral equation [2] (with allowance for the stress dependence of the relaxation time) under given periodic loading from known creep curves recorded at constant stress. In deriving the theoretical relation certain simplifying assumptions are made (the creep strain accumulated in 1–2 cycles is small, no vibration [4–6]). An experimental check shows that the proposed method can be used to predict the behavior of a material under periodic loading with an accuracy sufficient for practical purposes.Mekhanika Polimerov, Vol. 2, No. 3, pp. 330–336, 1966     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

On the Hencl's notion of absolute continuity

We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in , J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].     

Quadrature formulas with positive weights

Recent developments in the theory of stability or contractivity of numerical methods for solving ordinary differential equations (see for instance [4], [5], [8]) have renewed the interest for the study of quadrature formulas with positive weights. Nørsett-Wanner [8] and Burrage [2], [3] have given characterisation of such quadrature formulas of order 2m–2 or 2m–3. In this paper we extend these investigations to the case of formulas of order 2m–4 and then to the case where the order is 2m–7. Finally we use these results to characterise the algebraically stable methods out of a 12-parameter family of implicit Runge-Kutta methods of order 2m–4.     

Edges of Canonical Decompositions for 2-Bridge Knots and Links

Let L=C(a1,..., ak) be a hyperbolic 2-bridge knot or link. We give a family of arcs in S3–L which are ambient isotopic to edges of the canonical decomposition of S3–L if each aj is sufficiently large. This result supports the conjecture that the decomposition of S3–L given by Sakuma–Weeks in [12] is the canonical one.     

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.     

Variation of Poisson's ratio in the course of a complete one-dimensional creep cycle

By solving the system of differential equations of a standard solid, expressions are obtained for determining Poisson's ratio with allowance for the characteristics of volume and shear strain. For a material having values of Poisson's ratio in creep (0)<0.5 and ()=0.5, Poisson's ratio in reverse creep (t)^rev>0.5. The volume strains that occur at the beginning of reverse creep have the same magnitude as the creep volume strains, but the opposite sign. The law of disappearance of the volume strains is the same for creep and reverse creep.Mekhanika Polimerov, Vol. 4, No. 2, pp. 227–231, 1968     

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.     

Carrying capacity and deformation properties of glass-reinforced plastic pipes of various construction

Glass-reinforced plastic pipe under internal hydrostatic pressure starts to sweat at loads lower than the mechanical strength of the material. It is established that the life of such pipes is well described by an exponential law. The leakage mechanism is mainly controlled by the deformation properties of the reinforcement and resin. The period of reliable service can be estimated from the time of development of the strain corresponding to the end of the phase of steady-state creep. It is shown that for pipe made with cross-wound glass rovings, the winding angle and the number of roving ends have a considerable influence on the creep and life.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1082–1088, 1967     

Major subgroups of nilpotent-by-finite groups

The major subgroups of a group G and their intersection (G) were introduced in [1] as a variation on maximal subgroups and the Frattini subgroup of G. In particular, we have proved in [1–4] a number of results of the form: if G/(G) has property X then G has property X. The property X may be that of being hypercentral or hypercyclic or of having some finiteness condition. These results, of course, require some restriction on the group G.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 44, Mo. 6, pp. 853–856, June, 1992.     

On the rate of convergence to a stable limit law. II

Summary In [5] Feller states the following assertion: As soon as we leave the domain of applicability of the central limit theorem we find ourselves on practically unknown terrain; the problems receive an entirely new aspect and no systematic tools have as yet been developed for treating the theory, After the appearance of Volume II [6], many papers were published on stable distributions. The reader will find several references in [8].In the same paper [5] Feller states the following assertions: The theory becomes the simpler the fewer moments are finite. This assertion was stated in a paper in which he proved a law of the iterated logarithm for the heavy tail of stable distributions. (See also Chapter 8 in [8].) We shall show that the second assertion also holds in the study of the rate of convergence to a stable limit distribution. In several examples we shall compare our results with the results of Christoph [1–3] and Dubinskaite [4].Department of Mathematics, University of Leiden, Holland. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 26, No. 3, pp. 482–487, July–September, 1986.     

Theoretical and experimental investigation of the state of stress of a cantilevered plate with low shear stiffness

A general solution is obtained for the problem of a cantilevered, transversely isotropic plate in the form of an infinite strip with a clamped edge bent by a concentrated load applied at the free edge. The investigation relies on the equations of the generalized theory of plates [21–23] with allowance for interlaminar shears. In contrast with the Papkovich solution [1], the maximum theoretical bending moment at the clamped edge depends on the relative width of the strip and on the properties of the material. The classical result is asymptotically exact as h 0 or G_Z . The experimental results confirm the principal conclusions based on the theoretical solution.Kamenetsk-Podol'sk General Engineering Department, Khmel'nitskii Technological Institute. L'vov Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR. Translated from Mekhanika Polimerov, No. 3, pp. 476–481, May–June, 1971.     

Creep of thermally cycled polyethylene

A method for determining creep strains of polymer tube specimens (in three mutually perpendicular directions) tested at constant and cyclically changing temperatures is described. The equipment for the programmed variation of specimen temperature and for the application of complex loads is described. The application of the method is illustrated by the results of creep tests on tube specimens of low-density polyethylene (LDP). The creep strain _c of thermally cycled specimens is noticeably larger than that of specimens tested isothermally at the median temperature of a given thermal cycle, and in time it becomes larger than the strain of specimens tested isothermally at the maximum cycle temperature.Mekhanika Polimerov, Vol. 3, No. 4, pp. 676–685, 1967     

Shape-Preserving Approximation of Spatial Data

We present a new method for the construction of shape-preserving curves approximating a given set of 3D data, based on the space of quintic like polynomial splines with variable degrees recently introduced in [7]. These splines – which are C ³ and therefore curvature and torsion continuous – possess a very simple geometric structure, which permits to easily handle the shape-constraints.     

Some algebraic properties of cyclic homology groups

In [10], see also [8], a cyclic homology theory HC _* ^– was introduced. The purpose of this paper is to study algebraically the properties of this version of cyclic homology. First we study its relation to Connes cyclic cohomology theory HC ^* and to the usual cyclic homology theory HC _* studied by Loday and Quillen in [15]. We explain the precise sense in which HC _* ^– is dual to HC ^*. Next we study products and describe a general method for constructing product operations in cyclic homology and cohomology theories. Finally we examine the relation between the theory HC _* ^– and algebraic K-theory.     

Kinetic theory of creep and lifetime of oriented crystalline polymers with allowance for supermolecular structure

A previously proposed model [7] is used as the basis for a study of the kinetics of failure under load of an amorphous-crystalline oriented polymer with allowance for the unequal length of the chains in the amorphous zone. A constant-load creep curve is constructed. An expression is given for the lifetime t of the specimen. The relation _mt = const, where _m is the steady-state creep rate, is proved. The kinetics of free radical concentration in time in a specimen under load are investigated. The results are compared with experimental data.Mekhanika Polimerov, Vol. 3, No. 1, pp. 8–18, 1967     

Investigation of the one-dimensional problem of nonlinear creep

A method of analyzing experimental creep curves of a nonlinear viscoelastic material to obtain relaxation curves is examined. It is assumed that the family of creep curves in question cannot be represented as a product of the stress function and a function of time. The investigation is carried out using the memory theory. The sum of the exponentials for curves not having a singularity at the start of the process is taken as the creep kernel [1]. A method of approximation by the sum of the exponentials is given. For processes with an initial singularity it is proposed to use the corresponding kernel, for which the resolvent is given.Mekhanika Polimerov, Vol. 2, No. 5, pp. 678–687, 1966     

Universal (Co) Homology Theories

It is shown that the homology and cohomology theories on separable C*–algebras given by nonstable E–theory are the universal such theories. By specializing to Abelian C*–algebras, we obtain a family of extraordinary Steenrod homology and cohomology theories on pointed compact metric spaces which are the universal such theories in the same way. For each of the extraordinary Steenrod (co)homology theories considered, we describe an –spectrum which represents the theory.     

nvolutorial Płonka sums

The construction of the sum of a direct (semilattice ordered) system of algebras introduced by J. Plonka – later known as the Plonka sum – is one of the most important methods of composition in universal algebra, having a number of applications in different algebraic theories, such as semigroup theory, semiring theory, etc. In this paper we present a more general way for constructing algebras with involution, that is, algebraic systems equipped with a unary involutorial operation which is at the same time an antiautomorphism of the underlying algebra. It is the sum – involutorial Plonka sum, as we call it – of an involution semilattice ordered system of algebras. We investigate its basic properties, as well as the problem of its subdirect decomposition.