Statistical analysis of some characteristics of cylindrical glass-fiber-reinforced plastic materials

The results are given of a statistical treatment of experimental data on several characteristics of glass-fiber-reinforced plastic cylindrical shells, prepared from glass fabric TS8/3-250 and epoxide binder IF-ÉD-6 (TU-26-59) at the "Électroizolit" factory by winding an impregnated fabric cloth on a mandrel. The experim ental data on the mechanical properties of the glass-fiber-reinforced plastic material were obtained in tests on representative samples cut from the margin of the cylindrical shells. The results of the statistical analysis can be used to evaluate the mechanical reliability of structures made of glass-fiber-reinforced plastic material of the above type.Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 906–910, September–October, 1973.     

Three formulas for eigenfunctions of integrable Schrödinger operators

We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherlandsintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the Etingof–Kirillov tracesof intertwining operators, the second an integral representationof hypergeometric type, and the third is a formula of Bethe ansatz type.The last two formulas are degenerations of elliptic formulasobtained previously in connection with theKnizhnik–Zamolodchikov–Bernardequation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctionsare parametrized by a Hermite–Bethe variety, a generalizationof the spectral variety of the Lamé operator.We also give the q-deformed version of ourfirst formula. In the scalar slN case, this gives common eigenfunctionsof the commuting Macdonald–Rujsenaars difference operators.     

Design of anisotropic structures: Calculation,optimization, and tests

Conclusions A computational procedure based on a limiting state has been worked out which proposes the performance of tests of samples, small-scale models, and full-scale structures and a statistical analysis of the experimental data. Advantages of the method are shown in comparison with the computational method based on fracturing loads which pertain to estimating the accuracy of the computational scheme of structures and determining their safety coefficients. An optimality criterion is formulated within the framework of the method which is related to ensuring a maximum of the limiting (fracturing or critical) load with a constant mass of the structure.Applications of the method are given to estimation of the strength and stability of structures made out of polymeric and metal CM with the use of the Gol'denblat-Kopnov criterion and solutions based on linear shell theory. The conditions for realization of the optimality criterion of structures are determined. Recommendations are made for the rational reinforcement of structures made out of filamentary CM, and a procedure is proposed for the rational design of reinforced structures.As an example of the tests for stability of cylindrical shells made out of aluminum-magnesium alloy under external pressure, the legitimacy of the cause of the disagreement between the calculated and experimental critical loads, which consists of a discrepancy in the nature of wave formation at the instant of stability loss and initial imperfections, and the effectiveness of the computational method based on a limiting state are confirmed. Refined computational formulas for the critical loads of cylindrical shells made out of fiberglass, carbon fiber, and boraluminum under typical kinds of loading are proposed within the framework of the method and on the basis of an analysis of the results of tests, and a comparison is made of the effectiveness of the reinforcement of shells made out of aluminum-magnesium alloy and carbon fiber. The values of the safety coefficients of structures which guarantee their reliability are determined.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 262–271, March–April, 1980.     

Calculation of the natural-vibration spectra of layered shells of moderate thickness

Conclusions Relations from a linear, kinematically nonuniform model of a layered shell were used to construct a system of motion equations for an M-layered shallow shell which considered all components of the stress-strain state and inertia of the shell. It was shown using sample calculations of the natural frequency spectrum of physically uniform and hybrid threelayer hells that this model makes it possible in a linear approximation to calculate the complete natural-frequency spectrum of layered shells. It can be used in engineering calculations of the dynamic characteristics of shells in which relatively thin and stiff bearing layers alternate in the packet with layers of a soft filler (structurally nonuniform hybrid shells).The use of simplified (classical) models, refined kinematically uniform models, and nonuniform models not accounting for compressive strains in the shell layers, etc. (see [1, 5]) is limited to the classes of physically uniform and quasiuniform shells and to cases of calculation of the dynamic characteristics determined by three fundamental frequencies of the shell when regarded as a three-dimensional body.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 298–304, March–April, 1985.     

On Rapidly Convergent Series Expressions for Zeta- and L-Values,and Log Sine Integrals

Regarding the rapidly convergent series expansion for special values of - and L-functions for integer points, there are two approaches.One approach starts from Euler's 1772 formula for (3) and culminates in Srivastava's very recent results via many intermediate results, and the other is due to Wilton's investigation, which was shown by us (Aeq. Math. 59, 2000, 1–19) to be a consequence of Ramanujan's work (Collected Papers of Srinivasa Ramanujan, CUP 1927, reprint Chelsea, 1962, pp. 163–168).More recently, Katsurada (Acta Arith. 90, 1999, 79–89.) has generalized all existing formulas into a rather wide framework of Dirichlet L-functions.Our purpose is to show that even the most general Katsurada's formulas are easy consequences of our fundamental summation formulas for the series with Hurwitz zeta-function coefficients.We give a three-line proof of Katsurada's main theorem, and also we make some remarks on the recent paper of Bradley (The Ramanujan J. 3, 1999, 159–173).     

Experimental-theoretical studies of the stability of orthotropic shells with a filler in axial compression

Conclusions 1. An analysis has been made of the solution to the problem of the stability of multilayer cylindrical shells having a filler and simple calculation formulas have been obtained for determining the critical forces.2. The stability of fiberglass-plastic shells with rubber-like fillers has been studied experimentally.3. Comparative experimental-theoretical studies of critical forces have been made, and the stability coefficients have been ascertained for the shell class under consideration.Translated from Mekhanika Polimerov, No. 3, pp. 485–489, May–June, 1978.     

Evaluating the effect of whiskerization of the fibers of composites on the fundamental frequency of a laminated cylindrical shell

Conclusion The calculations showed that whiskerization of the reinforcement of the structural material of multilaminate shells makes it possible in some cases to increase the fundamental vibration frequency of the structure up to 15–20%. In combination with the well-known [1] effect of improved strength characteristics for a whiskerized composite in the transverse and shear directions, this finding allows us to conclude that whiskerized structural materials are more efficient than ordinary laminated composites in shell-type load-bearing structures. Here, the greatest benefit can be expected in the case of whiskers which have higher elastic moduli than the main reinforcement. Since considerably higher reinforcement intensities can be achieved in whiskerized laminated composites than in composites with a reinforcement characterized by an arbitrary three-dimensional structure, it can be concluded on the basis of the results obtained here that, at least for shells of moderate thickness (10 < R/h 50), whiskerized composites are the optimum structural material for load-bearing shells.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1022–1027, November–December, 1987.     

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.     

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.     

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.     

Iterative method of designing uniform and laminated shells of revolution made of highly elastic materials

Conclusion An iterative method was proposed for numerical solution of problems concerning the axisymmetric deformation of uniform and laminated shells of revolution made of highly elastic materials. Calculations performed for rubber, rubber-fiber, and rubber-layered shells demonstrate the reliability and effectiveness of the method.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 109–116, January–February, 1990.     

Effect of the geometry of the structure on the strength distribution of multilaminate tridirectional reinforced plastics

Conclusion The proposed analytical method makes it possible to predict the strength distribution of an LRP of the type [0/±]_s from its structure (the geometry of the packet, the number of layers, and the strength properties of the layers) in a plane stress state. Allowance is made for the random character of the strength properties of the layers, which makes it possible to evaluate the reliability of the LRP for both determinate and random loading. A criterion was formulated for the optimum design of the structure of an LRP with respect to ensuring maximum reliability for specific loading conditions. We also evaluated the effect of the parameters of the structure and the characteristics of the plane stress state on reliability. According to the results of a numerical analysis performed with the above-developed structural model of the failure of an LRP — with allowance for the random character of the strength properties of the layers — the imbalance of the laminated packet which occurs during failure can be ignored. The method used to predict the strength distribution of the LRP, involving determination of the strength distribution law of an RSE and subsequent examination of the loading of parallel-connected RSEs, is promising for other reinforcement schemes as well.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 805–812, September–October, 1988.     

Analysis of the effect of transverse shear strains on the stability of multilayer cylindrical shells under external pressure

The effect of transverse shear strains on the critical pressure is investigated using the results of the solution obtained for the problem of the stability "in the small" of elastic multilayer cylindrical shells of regular structure with alternating light and stiff layers. Attention is drawn to the need to estimate the state of stress of the shells in the critical-load zone with the object of studying the desirability of taking the shear effect into account in the stability calculations. The results obtained can be used in calculating the stability of shells made from resin-based composites (glass-reinforced plastics, graphite-reinforced plastics, etc.). The numerical calculations were carried out using a computer.Translated from Mekhanika Polimerov, No. 6, pp. 1066–1070, November–December, 1973.     

Accounting for kinematic inhomogeneity in calculations of the subcritical deflection of laminar cylindrical shells

Conclusion The range of application of kinematically homogeneous models 2 and 3 for estimating the stress-strain state of a laminar shell is limited to the class of structures, whose stiffness characteristics of the individual layers differ by one-two orders of magnitude. In this case, the shell's subcritical deflection can be computed from simplest model 1 for relatively long shells (at least for L/R 2 in the cases under consideration). In other cases, the stressstrain state of a laminar shell should be evaluated on the basis of the fracture-line hypothesis (model 4). Consideration of transverse-reduction deformations of the shell's layers does not introduce significant corrections into the results of the computation.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 299–304, March–April, 1987.     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices

The aim of this paper is to find asymptotic formulas for eigenvalues of self-adjoint discrete operators in given by some infinite symmetric Jacobi matrices. The approach used to calculate an asymptotic behaviour of eigenvalues is based on method of diagonalization, Janas and Naboko’s lemma [J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36(2) (2004) 643–658] and the Rozenbljum theorem [G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, (Russian) Trudy Maskov. Mat. Obshch. 36 (1978) 59–84]. The asymptotic formulas are given with use of eigenvalues and determinants of finite tridiagonal matrices.     

Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

Representation of integers by positive ternary quadratic forms (a new modification of the discrete ergodic method)

One gives a detailed presentation of a new modification of the discrete ergodic method outlined in A. V. Malyshev's note (this Zapiski,50, 179–186 (1975)). One gives new proofs for the asymptotic formulas obtained in Chap VI of A. V. Malyshev's monography (Tr. Mat. Inst. Akad. Nauk SSSR,65 (1962). One obtains estimates for the remainder terms of this formulas under the assumption of some hypotheses about the zeros of Dirichlet's L -functions.Translated from Zapiski Nauchnykh Seminarog Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 33–87, 1979.     

Method of estimating the strength of glass-reinforced plastic shells

A combined method of estimating the strength of glass-reinforced plastic shells in the plane stress is described. The method consists in a combination of destructive and nondestructive methods of determining the necessary starting characteristics for use with a criterion of the strength of orthotropic materials in the form of a fourth-degree polynomial.Ritm Scientific-Production Association. Central Scientific-Research Institute of Shipbuilding Technology, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 571–573, May–June, 1972.     

The distribution of eigenvalues of the Dirac operator

The Dirac system is studied on (-, ). Asymptotic formulas are obtained of the distribution for both positive, and negative eigenvalues. The asymptotic formulas, established in Theorem 1, are essentially different from formulas obtained by Sargsyan [1], and permit asymptotic formulas to be written for the distribution of positive (negative) eigenvalues, even in those cases when the negative (positive) spectrum is continuous, if appropriate conditions hold on the potential.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 843–852, December, 1973.In conclusion the author thanks R. S. Ismagilov and B. M. Levitan for valuable instructions and attention to this work.