Stability of shear-compliant cylindrical shells connected with an elastic foundation in a geometrically nonlinear formulation

The effect of transverse shear strains on the stability "in the large" of a cylindrical transversal-isotropic shell with an elastic filler under the effect of axial compression is investigated. The length of the cylindrical shell is assumed to be greater than the diameter. The approximate solution is obtained by the Bubnov-Galerkin method. Such a problem for an isotropic shell was considered earlier in [2, 4] on the basis of the equations of the classical Kirchhoff-Love theory.L'vov Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR. Ternopol Branch, L'vov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 113–117, January–February, 1972.     

Nonlinear equations of a timoshenko-type theory of laminated anisotropic shells

In recent years analysis of the stress—strain state of shell structures made out of composite materials has been based on refined shell theories which take into account strains in the direction normal to the reference surface. There are several approaches to the formulation of the refined theories. One can point to shell theories developed on the basis of variational principles (e.g., [1, 2]) as well as theories created with the help of iterational processes (e.g., [3–6]). A resolving system of nonlinear equations for laminated anisotropic shells has been derived in the proposed research based on the Reissner variational principle [7, 8]. A similar linear theory which takes into account the strain e₃₃ also has been developed in [1]. If the shear stiffnesses of the layers differ greatly from each other in the transverse direction, then one can treat the shell structure as a single-layer shell of nonuniform structure. In this case it is advisable to solve a problem of the type of a uniform shell with minimal stiffnesses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 501–507, May–June, 1979.     

Torsional stability of a glass-reinforced plastic cylindrical shell with an elastic core

The problem of the stability of a glass-reinforced plastic cylindrical shell with an elastic core subjected to twisting moments applied to the edges of the shell is considered. As in various other studies [4–6], the glass-reinforced plastic is treated as an elastically orthotropic material. The core is treated as an isotropic elastic cylinder, whose outer surface is bonded to the shell. Expressions for the critical stresses are obtained for an infinitely long shell and a shell of finite length.Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1082–1086, November–December, 1970.     

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.     

Creep stability of glass-reinforced plastic cylindrical shells under long-term external pressure

The behavior of a glass-reinforced plastic cylindrical shell under long-term hydrostatic pressure is investigated using the geometrically nonlinear equations of Timoshenko-type shell theory, which permit transverse shear strains to be taken into account. A system of nonlinear differential equations for describing the variation of the state of the shell with time under load is obtained and solved on a BÉSM-3M computer using a program written in Algol-60 and a "Signal" translator. Values of the critical time are obtained for various load levels.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 81–85, January–February, 1970.     

On the applicability of two-dimensional applied theories to problems of the stability of axially loaded cylindrical shells made of materials with low shear stiffness

The applicability of two-dimensional applied theories of the Kirchhoff-Love, Timoshenko, and Ambartsumyan types to problems of the stability of shells with low shear stiffness [1] is discussed. The critical loads given by these theories are compared with those recently obtained [6–8] by solving the problem of the stability of a cylindrical shell on the basis of general solutions [3, 4] of the three-dimensional linearized equations of the theory of elasticity [5].Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 141–143, January–February, 1970.     

Stability of transversally isotropic shells coupled with an elastic foundation

The stability of shells coupled with an elastic Winkler foundation is investigated. It is assumed that the shell is made of a material (glass-reinforced plastic) with low resistance to shear, as a result of which generalized theories that take transverse shear strains into account [1–4] must be used in the stability calculations. The solution obtained is compared with the corresponding solution obtained on the basis of the classical Kirchhoff-Love theory [8].Lvov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 669–673, July–August, 1969.     

Stability of a laminated spherical shell

The stability "in the small" of a spherical shell consisting of alternating stiff and soft layers is investigated. The spectra of the bifurcation values of the load are found for a closed shell subject to hydrostatic pressure and their dependence on the buckling mode is studied. The effect of the relative stiffness of the layers on the buckling of the shell is investigated.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 3, pp. 459–464, May–June, 1976.     

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.     

Dynamic stability of glass-reinforced plastic shells

The problem of the dynamic stability of circular-cylindrical glass-reinforced plastic shells subjected to external transverse pressure is examined in the nonlinear formulation. After the Lagrange equations have been constructed, the problem reduces to the integration of a system of ordinary differential equations with aperiodic coefficients. The integration has been carried out numerically on a computer for various loading rates and shell parameters. Analogous problems for isotropic metal shells were examined in [1–4]. A review of the subject may be found in [5].Mekhanika Polimerov, Vol. 4, No. 1, pp. 109–115, 1968     

Stability of glass-reinforced plastic shells with allowance for shear stresses

The stability of a cylindrical glass-reinforced plastic shell subjected to external pressure is considered in the geometrically nonlinear formulation with allowance for initial irregularities. The refined shell theory [6, 7], which enables transverse shear strains to be taken into account, is employed. A general algorithm of the solution has been written in ALGOL-60. A numerical solution of the problem has been obtained on a BÉSM-3M computer. Critical loads have been determined over a wide range of variation of the geometrical and physical parameters of the shell. It is established that the difference between the results of the classical and refined theories depends on the thickness, length, and physical parameters of the shell. The classical theory is asymptotically exact as the thickness of the shell tends to zero or the interlaminar shear modulus tends to infinity.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 857–862, September–October, 1969.     

Optimization of reinforced cylindrical shells with nonuniform thickness

An optimum multilayer shell is designed whose stack of elementary layers has a nonuniform thickness. This optimization problem is solved numerically for the special cases of three-layer cylindrical shells with dynamic and static stability. The optimum variants of layer distribution in this model are compared with the optimum solutions in [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 298–303, March–April, 1976.     

Effect of a local load on an orthotropic glass-reinforced plastic shell

The deformation of a layered orthotropic cylindrical shell under a local normal load is investigated on the basis of equations that do not depend on the hypothesis of straight normals. The solutions of the analogous classical problems were analyzed in [3]; a solution based on equations that take transverse shear strains approximately into account was proposed in [4]. The high degree of variability of the state of stress created by local loads indicates that it is quite important to take transverse shear strains rigorously into account in problems of this class. An attempt is made to estimate the error introduced by the hypothesis of straight normals and to calculate the load leading to debonding of the shell.S. Ordzhonikidze Moscow Aviation Institute. Translated from Mekhanika Polimerov, No. 1, pp. 95–101, January–February, 1970.     

Concerning the static-geometric analogy and the complex equations of Timoshenko-type shell theory

The static-geometric analogy of the classical Kirchhoff-Love shell theory, established by Gol'denveizer, is extended to the theory (Timoshenko type) in which transverse shear strains are taken into account [1, 2, 7, 8]. The construction of the complex equations of this theory is discussed.Franko L'vov State University; L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 5, pp. 942–945, September–October, 1969.     

A generalized constitutive equation for creep of polymers at multiaxial loading

Summary This paper introduced a unified formulation for generalized deformation models including load dependent effects (2nd order effects). It is given in more detail for stationary creep of isotropic, orthotropic, and anisotropic material behavior. A further generalization of the introduced 6-parameter constitutive equation is possible by coupling creep and damage. These generalizations include the classical theory of creep damage [13]. The proof of the proposed theory is given in [20–22] for special cases with a reduced number of material parameters. The results of calculations show a good agreement with results from multiaxial tests.Dedicated to our colleague and friend Vitauts Tamus on the occasion of his sixtieth birthdayPublished in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6, pp. 723–733, November–December, 1995.     

A note on a result of Zolotarev and Bernstein

In this note we obtain error bounds to x²ⁿ⁺¹–x²ⁿ on [–1, 1] by polynomials of degree at most (2n–1). The result proved here improves and extends some of the known results of Zolotarev and Bernstein. The proof presented here is different (and simple) from the one adopted by Zolotarev and Bernstein.     

Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.     

Method of functional integration and an eikonal approximation for potential scattering amplitudes

Conclusions We have obtained an exact closed expression for the potential scattering amplitude of particles with spin o and 1/2 as a functional integral with respect to trajectories. This has made possible a relatively simple expansion of the amplitude in powers of the small parameter 1/E. The first term of the expansion is an eikonal approximation for the amplitude for scattering through any angle and in the case of dynamically small angles (pR)^–1/2 is identical with the Glauber representation.We have found the asymptotic form of the scattering amplitude for two particles that exchange virtual mesons. A similar result was obtained in [7] by functional integration with respect to the external fields of the exact Green's functions and a subsequent eikonal expansion on the mass shell. The equivalence of this more accurate method to the approximation described in the present paper (see also [8, 9]) is connected with the interesting problem of the commutativity of the operations of eikonal approximation and second quatization.If the latter do commute, the Glauber representation for the amplitude (18) in quantum field theory is a consequence of the eikonal approximation in quantum mechanics.Joint Institute of Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 4, No. 1, pp. 22–32, July, 1970.     

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.     

Stress state of a composite shell with a sizable opening

The stress-strain state of a nonshallow cylindrical shell of a composite material is investigated. The shell is weakened by a circular hole and loaded with internal pressure. For solving the problem, the variational-difference method is used. The calculations are carried out for an orthotropic shell with a sizable hole, with account of the reduced shear stiffness of the material.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 49–56, January–February, 2005.