Dynamic properties of some polymer materials in loading and unloading

A series of quasi-static tests have been carried out on kapron fiber, vinyl plastic, and fluoroplastic specimens with small cross sections. Stress-strain diagrams have been obtained at strain rates from 1 · 10^–3 to 1 · 10² sec^–1. In the experiments the strain rate changed sign, which corresponded to loading and unloading. In each regime the strain rate remained constant. The experimental data are analyzed using the model of a standard linear viscoelastic solid.Mekhanika Polimerov, Vol. 3, No. 1, pp. 118–122, 1967     

Impact strength of high-density polyethylene with various molecular weight distributions

The Charpy impact strength of high-density polyethylene specimens prepared under low pressure with average molecular weight from 60·10³ to 1.5·10⁶ and differing substantially in molecular weight distributions has been studied at room temperature and at –190°C. It is shown that, both at room temperature and at temperature considerably below the glass-formation temperature, the impact strength of polyethylene in the range of molecular weights mentioned is determined mainly by the content of fractions with molecular weights about 10⁵ and more and does not depend on the width of the molecular weight distribution."Plastpolimer" Scientific-Research Association, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 919–921, September–October, 1973.     

The dynamic properties of polyethylene and an acrylonitrile — Butadiene — Styrene material in a broad range of temperature and strain-rate variation

Conclusions The mechanical properties of an acrylonitrile-butadiene-styrene material and polyethylenes of densities =0.92 and 0.96 g/cm³ are studied within the temperature range of from 100 to –196°C and at strain rates of from 10^–3 to 1.3 · 10³ sec^–1. It is observed that under low-temperature and high-strain-rate conditions, the laws governing the mechanical properties of the acrylonitrile-butadiene-styrene material and polyethylenes are similar. Regions of strain rates and temperatures are found for which the behavior of the test materials is in good agreement with the Ree-Eyring equation, and, consequently, conforms to a temperature-time analogy. No embrittlement of the polyethylene is observed during tests at strain rates to 1.3 · 10³ sec^–1 and temperatures down to –196°C. In approximating these values of and T, the mechanical properties of polyethylenes will cease to be dependent on initial density.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, 1027–1033, November–December, 1978.     

Effect of the molecular weight of monodisperse specimens of high-density polyethylene on the dynamic modulus of elasticity

The effect of the molecular weight on the instantaneous dynamic modulus of elasticity of monodisperse specimens of linear polyethylene has been studied. It is shown that above a critical value of the molecular weight the modulus of elasticity is constant and equal to 4·10⁹ N/m², while below that value it increases to 7.28·10⁹ N/m².M. Azizbekov Azerbaidzhan Institute of Petroleum and Chemistry, Baku. Translated from Mekhanika Polimerov, No. 3, pp. 531–533, May–June, 1976.     

Effect of the molecular weight of polystyrene on the viscosity of concentrated solutions

The viscosity of solutions of polystyrene of various molecular weights (from 1.04 · 10² to 3.8 · 10⁵) in a poor solvent (decalin) and a good solvent (ethylbenzene) has been measured at temperatures from 15 to 70°C over a broad range of shear stresses from 10² to 10⁶ dyne/cm². The nature of the solvent has a considerable influence on the critical molecular weight and the absolute value of the viscosities of the solutions over the entire range of molecular weights and on the form of the flow curves of decalin solutions of polystyrene as a function of temperature. The heat of activation of viscous flow increases with increase in molecular weight and shear stress on the interval 20–80°C. The results obtained are explained in terms of the effect of the molecular weight of the polymer, the nature of the solvent, stress and temperature on structure formation in the solution and on the orientation of the macromolecules and structures in the flow process.Ural Gor'kii State University, Sverdlovsk. Translated from Mekhanika Polimerov, No. 5, pp. 920–926, September–October, 1970.     

Strength of glass laminates at low temperatures and high strain rates

The tensile strengths of glass laminates of various compositions under standard, high-speed and impact loading have been experimentally investigated in relation to the effect of low-temperature and stress raisers. As the loading rate increases (from 0.7 to 253 · 10³ kgf/cm². · sec) and the temperature falls to –196° C, the effective stress concentration coefficient decreases, which indicates that the glass laminates will perform adequately under these conditions.All-Union Scientific-Research Institute of Aviation Materials, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 836–841, September–October, 1970.     

Effect of cycle asymmetry on the cyclic strength of polymers

Cyclic strength diagrams have been obtained for polymethyl methacrylate and kaprolon at different degrees of loading asymmetry (from a pure compression to a pure tension cycle) on the interval from –100 to +80° C at 2·10³ and 3·10⁵ cycles.M. I. Kalinin Leningrad Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 948–951, September–October, 1968.     

On the existence of certain cyclic difference families and difference matrices

A theorem is proved to the effect that if there exists a BIB-schema with parameters (p^m–1,k, k–1), where k¦(p^m–1), p is prime, and m is a natural number, then there exists a BIB-schema (p^mn–1),k, k–1). A consequence is the existnece of a cyclic BIB-schema (p^mn–1, p^m–1, p^m–2) (p^m–1 is prime) that specifies each ordered pair of difference elements at any distance = 1, 2, ..., p^m–2 (cyclically) precisely once. Recursive theorems on the existence of difference matrices and (v, k, k)-difference families in the group Z_v of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 114–119, July, 1992.     

Experimental study of the effect of laser beams on transparent dielectrics

The destruction of transparent polymer blocks by laser action is investigated. It is established that microscopic breakdown regions, whose size remains constant in time, are formed in the zone of action of the laser pulse. The intensity of the radiation from these microregions varies with time and corresponds to a temperature of (2–4) · 10³°K. The microregions are located near, but not at the centers of the cracks. The previously irradiated region of the polymer is shown to be in a state of stress. The possibility of thermal explosion with subsequent crack formation is evaluated. The continued development of the cracks as a result of repeated irradiation is studied. The temperature of the gas in the large cracks has been measured and found to exceed room temperature by 15°. The molecular weight has been measured before and after irradiation and found to decrease. The effect of laser beams of various wavelengths is considered. The rate of growth of the disk-shaped cracks has been measured.Institute of Problems of Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 827–835, September–October, 1969.     

Failure of water-filled cylindrical glass-reinforced epoxy shells under internal impulsive loading

Conclusions 1. The explosive failure of water-filled, geometrically similar glass-reinforced epoxy shells, whose dimensions increase by a factor of 1.5–4.4, is not associated with an energetic scale effect. This is attributable to the imperfect similarity of the shells, since although their dimensions changed, the diameter of the glass reinforcing fibers remained the same.2. The relative mass of explosive required to fracture the shells is approximately 0.4%. With respect to this index, glass-reinforced epoxy is comparable to structural steels 20 and 17Mn1Si at R₀160 mm.3. The circumferential deformation of the shells at failure is about 4% and does not depend on the thickness of the shell wall, the dimensions of geometrically similar shells, or the initial strain rate on the interval 0.21 · 10³–1.2 · 10³ sec^–1. Thin-walled shells begin to fail from the outside surface, thick-walled shells from the inside surface.4. The modulus of elasticity of the GRE is 2.4 · 10⁵ kgf/cm² and does not depend on the strain rate on the interval 10^–3–1.5 · 10³ sec^–1. The material deforms elastically up to failure.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 283–289, March–April, 1978.     

Dynamic modulus of polymethylmethacrylate saturated with carbon dioxide

The authors have devised a method for measuring the complex modulus of elasticity of non-plasticized polymethylmethacrylate of grade ST-1-110 in an atmosphere of carbon dioxide. The pressure of the saturated gas in the working chamber was varied between 4·10⁵ and 60·10⁵ N/m², and the experiments were performed at 18–90°C.S. M. Kirov Ural Polytechnic Institute, Sverdlovsk. Translated from Mekhanika Polimerov, No. 2, pp. 370–372, March–April, 1974.     

The maximum number of Hamiltonian paths in tournaments

Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n ^3/2 · n!/2 ^n–1, wherec is a positive constant independent ofn.Research supported in part by a U.S.A.-Israel BSF grant and by a Bergmann Memorial Grant.     

Mass-spectrometric investigation of solid lubricating coatings subjected to friction in a deep vacuum

The life of solid lubricating coatings of the VNII NP type, based on molybdenum disulfides and various binders, has been experimentally investigated under deep vacuum conditions (10^–8–5 · 10^–9 torr) together with the composition of the gas released in the friction process. It is shown that both under atmospheric conditions and in a deep vacuum the life of the coatings depends on the chemical nature of the film-former. The depth of the vacuum also has an important influence on the life of the coatings, both the mechanism and the end result of this effect depending to a large extent on the physicochemical properties of the bind. On the interval 10^–1–10^–2 torr there is a sudden change in the life of the coating.Physicotechnical Institute of Low Temperatures, Academy of Sciences of the Ukrainian SSR, Khar'kov. Translated from Mekhanika Polimerov, No. 6, pp. 1070–1075, November–December, 1970.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

Study of the role of intermolecular interactions (hydrogen bonds) in the initial stage of orientation of polyvinyl alcohol and vinyl alcohol copolymers

The effect of hydrogen bond rupture on the process of destruction of the starting crystallite structure of polyvinyl alcohol and copolymers of vinyl alcohol and N-vinylpyrrolidone and -vinylcaprolactam has been investigated. The differential spectrum method is used to establish the frequency (3200 cm^–1) of the band corresponding to the vibrations of the bound hydroxyl groups; the temperature dependence of the optical density of the 3200 cm^–1 band has been recorded. The strength of the starting structure is calculated from the stress-strain diagram and its temperature dependence is plotted. It is concluded that the strength of the starting structure is chiefly determined by the presence of hydrogen bonds between the polymer macromolecules.Leningrad Kirov Institute of Textile and Light Industry. Translated from Mekhanika Polimerov, No. 2, pp. 327–329, March–April, 1971.     

Spherical multipliers

It is proven in the paper that if functionf(x)L^p(Rⁿ), where 1/p> 1/2 + 1/(n + 1), then the restriction of the Fourier transform f() to the unit sphere S^n–1 lies in L²(S^n–1). As was shown by Fefferman [1], it follows from this that, when > (n –1)/(2(n + 1)), the Riesz-Bochner multiplier acts in LP(Rⁿ) if (n –1–2)/(2n) <1/p < (n + 1 + 2)/(2n).Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 105–112, January, 1978.The author wishes to thank B. S. Mityagin for his attention to this work.     

Maximal length of circuit in a unitary n-dimensional cube

In a unit n-dimensional cube a circuit is constructed of length const · 2ⁿ. Thus, the order is found of the maximal length of a circuit.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 309–329, September, 1969.The author is very grateful to V. V. Glagolev for the constant interest he has shown in this work.     

On the positivity of certain Cotes numbers

It is shown that the Cotes numbers are nonnegative for some interpolatory quadratic rules with nodes at the zeros of the ultraspherical polynomialsC _n , when integrating with respect to the weight functions w_µ(x)=(1–x²)^µ–1/2.     

Partitioning arrangements of lines II: Applications

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m ^2/3 n ^2/3 · log^2/3 n · log^/3 (m/n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where is a constant <3.33; (ii) anO(m ^2/3 n ^2/3 · log^5/3 n · log^/3 (m/n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n ^4/3 log^(+2)/3 n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n ^4/3 log^{( + 2)/3} n) algorithm to count red-blue intersections between two sets of segments, and (v) anO(n ^3/2 log^/3 n) algorithm to compute spanning trees with low stabbing number for a set ofn points. We also present an algorithm that, given set ofn points in the plane, preprocesses it, in timeO(nm log^+1/2 n), into a data structure of sizeO(m) forn lognmn ², so that the number of points ofS lying inside a query triangle can be computed inO((n/m) log^3/2 n) time.Work on this paper has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation. A preliminary version of this paper appears in theProceedings of the 5th ACM Symposium on Computational Geometry, 1989, pp. 11–22.