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1.
V. N. Paimushin V. A. Ivanov S. A. Lukankin A. A. Bushkov 《Mechanics of Composite Materials》2004,40(4):309-330
Problems on buckling modes (BMs) are considered for a spherical sandwich shell with thin isotropic external layers and a transversely soft core of arbitrary thickness in a centrosymmetric temperature field inhomogeneous across the shell thickness. For their statement, the two-dimensional equations of the theory of moderate bending of thin Kirchhoff–Love shells are used for the external layers, with regard for their interaction with the core; for the core, maximum simplified geometrically nonlinear equations of thermoelasticity theory, in which a minimum number of nonlinear summands is retained to correctly describe its pure shear BM, are utilized. An exact analytical solution to the problem on initial centrosymmetric deformation of the shell is found, assuming that the temperature increments in the external layers are constant across their thickness. It is shown that the three-dimensional equations for the core, linearized in the neighborhood of the solution, can be integrated along the radial coordinate and reduced to two two-dimensional differential equations, which supplement the six equations that describe the neutral equilibrium of the external layers. It is established that the system of eight differential equations of stability, upon introduction of new unknowns in the form of scalar and vortical potentials, splits into two uncoupled sets of equations. The first of them has two kinds of solutions, by which the pure shear BM is described at an identical value of the parameter of critical temperature. The second system describes a mixed flexural BM, whose realization, at definite combinations of determining parameters of the shell and over wide ranges of their variation, is possible for critical parameters of temperature by orders of magnitude exceeding the similar parameter of shear BM. 相似文献
2.
Free across-the-thickness vibrations of a closed spherical shell consisting of three rigidly connected layers with arbitrary
physical constants and thicknesses are studied. A closed-form solution in displacements to a one-dimensional (along the radius)
vibration problem for a homogeneous spherical shell is derived and then used in posing a boundary-value problem on free vibrations
of a heterogeneous sphere. Based on the degeneration of the sixth-order determinant of a system of homogeneous equations satisfying
the corresponding boundary conditions, a transcendental equation for eigenfrequencies is found. Transformation variants for
the equation of eigenfrequencies in the cases of degeneration of physical and geometric parameters of the compound shell are
considered. The main attention in investigating the lowest frequency is given to its dependence on the structure of shell
wall, whose parameters greatly affect the calculated values of the high-frequency vibration spectrum of the shell.
Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 839–852, November–December, 2008. 相似文献
3.
R. Douglas Gregory Thomas I. Milac Frederic Y. M. Wan 《Studies in Applied Mathematics》1998,100(1):67-94
A refined shell theory is developed for the elastostatics of a moderately thick spherical cap in axisymmetric deformation. This is a two-term asymptotic theory, valid as the dimensionless shell thickness tends to zero.The theory is more accurate than “thin shell” theory, but is still much more tractable than the full three-dimensional theory. A fundamental difficulty encountered in the formulation of shell (and plate) theories is the determination of correct two-dimensional boundary conditions, applicable to the shell solution, from edge data prescribed for the three-dimensional problem. A major contribution of this article is the derivation of such boundary conditions for our refined theory of the spherical cap. These conditions are more difficult to obtain than those already known for the semi-infinite cylindrical shell, since they depend on the cap angle as well as the dimensionless thickness. For the stress boundary value problem, we find that a Saint-Venant-type principle does not apply in the refined theory, although it does hold in thin shell theory. We also obtain correct boundary conditions for pure displacement and mixed boundary data. In these cases, conventional formulations do not generally provide even the first approximation solution correctly. As an illustration of the refined theory, we obtain two-term asymptotic solutions to two problems, (i) a complete spherical shell subjected to a normally directed equatorial line loading and (ii) an unloaded spherical cap rotating about its axis of symmetry. 相似文献
4.
A. Ya. Grigorenko T. L. Efimova L. V. Sokolova 《Journal of Mathematical Sciences》2012,181(4):506-519
For determining the dynamic characteristics of free vibrations of circular unclosed cylindrical shells of variable thickness
in two coordinate directions, we have used the spline-collocation method together with the method of discrete orthogonalization.
The problem has been solved within the framework of the refined Timoshenko–Mindlin theory. We have also investigated the influence
of different laws of change in the shell thickness on the character of its natural vibrations. Our calculations have been
carried out for different geometrical and elastic parameters of the shell under study and different boundary conditions. 相似文献
5.
V. N. Paimushin V. A. Ivanov S. A. Lukankin A. A. Bushkov 《Mechanics of Composite Materials》2004,40(6):461-472
A solution to the problem on the stability according to the flexural buckling mode is given for a cylindrical sandwich shell with a transversely soft core of arbitrary thickness. The shell is under the action of a temperature field inhomogeneous across the thickness, and its end faces are fastened in such a way (in the axial direction, the face sections of the external layer are fixed, but of the internal one are free) that an inhomogeneous subcritical stress-strain state arises in the shell across the thickness of its layers. It is shown that, under such conditions, the buckling mode of the shell is mixed flexural. To reveal and investigate this mode, equations of subcritical equilibrium and stability of a corresponding degree of accuracy are needed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 715–730, November–December, 2004. 相似文献
6.
Victoria Dzyuba 《PAMM》2005,5(1):605-606
In the present paper a method is proposed to investigate behaviour of the axis symmetric system consisting of an infinite thin elastic cylindrical shell, filled by potential flow of ideal compressible liquid and containing pulsating spherical inclusion. This coupled problem is considered in the framework of classical mechanics models: linear potential flow theory and thin elastic shells theory based on the Kirchhoff - Love hypotheses. The approach suggested consists in possibility of rewriting general solution of the corresponding mathematical physics equations from one to the other coordinate system. It enables to satisfy boundary conditions on both the spherical and the cylindrical surfaces and to get an exact analytical solution (as a Fourier series) of the coupled interaction problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
S. KESAVAN 《数学年刊B辑(英文版)》2000,21(1):1-16
1.IntroductionInthispaper,westudythelimitingbehaviourofeigenvaluesandeigenfunctionsdescribingthevibrationsofathinlinearlyelasticshell,clampedalongitslateralsurface,underageometricassumptiononthemiddlesurfaceoftheshellthatthespaceofinextensionaldisplacements(of.(4.2))isnonzero.Inthestationarycasegunderadditionalassumptionsontheorderofmagnitudeofthebodyforces,thisleadstothetwo-dimensionalmodelofthe"fie-curalshell"asshownbyCiarlet,LodsandMiara[5].Examplesofclampedshellswhichobeytheabovegeometric… 相似文献
8.
Jones G. W.; Chapman S. J.; Allwright D. J. 《The Quarterly Journal of Mechanics and Applied Mathematics》2008,61(4):475-495
The problem of a thin spherical linearly elastic shell perfectlybonded to an infinite linearly elastic medium is considered.A constant axisymmetric stress field is applied at infinityin the matrix, and the displacement and stress fields in theshell and matrix are evaluated by means of harmonic potentialfunctions. In order to examine the stability of this solution,the buckling problem of a shell which experiences this deformationis considered. Using Koiter's nonlinear shallow shell theory,restricting buckling patterns to those which are axisymmetricand using the Rayleigh–Ritz method by expanding the bucklingpatterns in an infinite series of Legendre functions, an eigenvalueproblem for the coefficients in the infinite series is determined.This system is truncated and solved numerically in order toanalyse the behaviour of the shell as it undergoes bucklingand to identify the critical buckling stress in two cases, namely,where the shell is hollow and the stress at infinity is eitheruniaxial or radial. 相似文献
9.
I. Yu. Khoma 《Journal of Mathematical Sciences》1998,92(5):4133-4135
On the basis of a generalized theory constructed using the Fourier-series expansion of the unknowns in Legendre polynomials
of the thickness coordinate we give a representation of the general solution of the equilibrium equations of a transversally
isotropic spherical shell for an arbitrary approximation. On this basis we study the problem of the stressed state of a shallow
spherical shell with a circular cavity on whose boundary surface there are tangential stresses varying nonlinearly over the
thickness.
Bibliography: 3 titles.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 19–24. 相似文献
10.
An analytic solution of the problem concerning the frequencies and shapes of free axially symmetric oscillations of a truncated spherical sector filled with an ideal compressible fluid is constructed. A spherical wall of smaller radius and the radial wall of the sector are absolutely rigid. A thin elastic shell whose edge is clamped in the radial wall is located on a spherical boundary of larger radius. The outer surface of the shell borders vacuum. The phenomenon of an anomalous decrease in the fundamental frequency as the spherical walls approach each other is discovered. An approximate formula for determination of the lowest fundamental frequencies, which are approximately proportional to the square root of the difference of the radii of spherical walls for small values of this difference, is constructed and tested numerically. Bibliography: 13 titles. 相似文献
11.
Christoph Adam 《PAMM》2006,6(1):283-284
This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
12.
We propose a method of computing the natural vibrations of thin elastic multilayer shells of revolution. The generalized mathematical
model makes it possible to take account of the orthotropy of the material of the component layers, the angle of orientation
of the elastic constants relative to the coordinate lines, and attached solid bodies. The solution is constructed for various
boundary conditions on the basis of the linear theory of shells using the Ritz method. We give the results of numerical studies.
Four figures. Bibliography: 5 titles.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 114–120. 相似文献
13.
M. M. Babatin 《Theoretical and Mathematical Physics》2011,169(3):1767-1774
We construct an idealized spherically symmetric relativistic model of an exploding object in the framework of the theory of
surface layers in general relativity and match a Vaidya solution for a radially radiating star to another Vaidya solution
through a thin spherical shell. We reduce the equations of motion and the radiation density of the Vaidya solution given by
the matching conditions to a first-order system and analyze the general characteristics of the motion. We use a post-Newtonian
approximation to find the equation of motion of a spherically symmetric radiating shell moving in a central gravitational
potential. 相似文献
14.
Yu. A. Lavrov 《Journal of Mathematical Sciences》2004,122(5):3519-3522
A solution of the problem of determining the frequencies and mode shapes of free nonsymmetric oscillations in an annular volume filled with an ideal compressible fluid is constructed. The inner tube and the end plane walls are ideally rigid. A thin elastic shell with edges clamped to the end walls is located on the outer tube boundary. A phenomenon of a decrease in the fundamental frequency as the thickness of a fluid layer adjacent to the elastic wall decreases is confirmed. Bibliography: 8 titles. 相似文献
15.
I. V. Panferov 《Journal of Applied Mathematics and Mechanics》2001,65(6):1025-1029
The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably. 相似文献
16.
R. Yu. Amenzadeh G. Yu. Mekhtiyeva L. F. Fatullayeva 《Mechanics of Composite Materials》2011,46(6):649-658
The buckling of a long multilayered nonlinearly elastic shell made of different materials and subject to the action of external
pressure is investigated. The load is not hydrostatic and greatly varies in value and direction. Neglecting the effect of
end fastening of the shell, the problem is reduced to an analysis of the loss of load-carrying ability of a ring of unit width
separated from the shell. The solution is based on a variational method of mixed type formulated for heterogeneous nonlinearly
elastic bodies, taking into account the geometrical nonlinearity, in a combination with the Rayleigh–Ritz method. The initial
analysis is reduced to solving the Cauchy problem for a nonlinear ordinary differential equation resolved for the derivative.
Numerically, using the Runge–Kutta method, the effect of the number of layers and of the parameter of nonuniformity of the
external pressure on the critical buckling force is revealed. The urgency and importance of the problem are connected with
the research of reserves in the saving of materials with a simultaneous possibility of increasing the load-carrying ability
of a structure. 相似文献
17.
18.
A. A. Kuleshov V. V. Mymrin A. V. Razgulin 《Computational Mathematics and Mathematical Physics》2009,49(1):146-171
The problem of transverse vibrations of a thin elastic plate is considered. It is proved that the differential operators of the boundary value problem are regularly elliptic, and weak solutions are estimated. For a previously developed difference method, the solution to the difference problem is proved to converge strongly to a weak solution of the original differential problem and the rate of convergence is estimated. 相似文献
19.
The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2εwith controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickness of the shell goes to zero, to the solution of an exact controllability problem in two dimensions. 相似文献
20.
V. A. Polyakov 《Mechanics of Composite Materials》1996,32(5):406-421
A refined model for bending of a three-layered panel with a soft filler is proposed. The modified model permits us to consider
the asymmetry of elastic properties and thickness of the outer layer relative to the middle plane of the panel in a composite
sandwich structure. In constructing the deformation mechanism, a heterogeneous kinematic model was adopted, which, in contrast
to the assumptions for the deformation of the whole stack of layers, features four degrees of displacement freedom permitting
consideration of the separate nature of the deformation of the outer layers in bending and of the intermediate layer in transverse
compression combined with shear. This approach is postulated according to an energetic evaluation of the deformation of the
layers [2]. The specific features of the stress from point forces in cylindrical bending are considered using the operational
Laplace method, which avoids the additional difficulties in analyzing the solution convergence arising when it is represented
by a series of eigenfunctions of the boundary value problem. The fundamental functions of a twelfth-order set of equations
are used to construct the boundary problem reduced to a Cauchy problem. Various boundary effects of the point stress are described
using a generalized Dirac function. Variants are examined for the limiting transformation of the model parameters leading
to a qualitative change in its kinematics and the corresponding simplified bending models.
Institute of Polymer Mechanics, Latvian Academy of Sciences, LV-1006 Riga, Latvia. Translated from Mekhanika Kompozitnykh
Materialov, No. 5, pp. 588–611, September–October, 1996. 相似文献