Shear and Flexural Buckling Modes of a Spherical Sandwich Shell in a Centrosymmetric Temperature Field Inhomogeneous across the Thickness

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.     

Research in the flexural buckling modes of a cylindrical sandwich shell under the action of a temperature field inhomogeneous across its thickness

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.     

Stability of a transversely isotropic cylindrical shell in axial compression

The problem of the stability of a transversely isotropic cylindrical shell subjected to axial compression is considered with the aid of three-dimensional linearized equations for small subcritical strains. Transcendental equations are obtained for the critical loads corresponding to axisymmetric and nonaxisymmetric deformation. It is shown that in the case of the axisymmetric buckling mode the Kirchhoff-Love hypothesis is asymptotically exact irrespective of the properties of the material. The dependence of the critical load on the properties of the shell material is investigated numerically. Graphs of the variation of the critical load with relative shear modulus are presented.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1064–1068, November–December, 1969.     

Optimization of cylindrical composite shells with regard to their critical mode of imperfections

This study deals with the optimum design of composite shells under external pressure with material strength and loss of stability according to the critical mode of imperfections taken as the failure criterion. The problem of optimum design is solved and the critical mode is obtained by nonlinear optimum programming for which the geometric and initial imperfection parameters are treated as variables. Numerical results are obtained for a cylindrical composite shell supported freely at its ends. The effect of shear forces between layers on the load-carrying capacity of the shell is also investigated.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 613–620, September–October, 1998.     

Diamond折纸管状结构轴向冲击性能分析

为了降低薄壁管状结构受轴向冲击时的初始峰值载荷,将diamond刚性折纸模型引入到薄壁管状结构的设计中.利用有限元分析方法,以方形截面为例,分析了diamond折纸管状结构的轴向冲击性能.结果表明:相比于传统方形薄壁管,diamond折纸管状结构具有较低的初始峰值载荷和更加平稳的变形过程.得出了diamond折纸管状结构按折纸预折形式变形的临界条件.分析了diamond折纸管状结构在轴向冲击载荷作用下,底角对初始峰值载荷和平均冲击载荷的影响.     

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.     

Dynamic stability of an elastic orthotropic cylindrical shell with allowance for transverse shears

The resolvents for the dynamic stability of an elastic orthotropic cylindrical shell are obtained in accordance with the Ambartsumyan and Timoshenko-type refined theories. The regions of instability given by the classical and refined theories are compared. The dependence of the refinements on the shell parameters, the shear moduli of the material, and the buckling modes are investigated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 312–320, March–April, 1973.     

Study of the effect of asymmetric arrangement of the layers on the stability of three-layer cylindrical shells in axial compression

The stability of a three-layer cylindrical shell under axial compression is examined in a three-dimensional linearized formulation. The load-bearing layers are assumed to be orthotropic, while the binder is modeled by an isotropic body. The effect of asymmetric arrangement of the layers and the mechanical characteristics of the material on the critical strains is determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 103–106, 1988.     

Effect of transverse shear on the stability of an orthotropic cylindrical shell with an elastic core in axial compression

The effect of transverse shear on the stability of a composite shell in axial compression has been investigated. The core is treated as an isotropic elastic cylinder bonded to the inner surface of the shell. The effect of the tangential shearing forces between the shell and the core is taken into account.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 267–273, March–April, 1973.     

Optimal winding of filament-wound reinforced plastic shells designed to resist buckling under external pressure or axial compression

The optimal method of winding the reinforcement of thin cylindrical shells subjected to external pressure or axial loading is investigated. Various types of winding — normal, simple oblique, crossed, and isotropic — are considered. An approximate formula is obtained for the critical external pressure. This formula is used to analyze the possibility of raising the critical load by adopting oblique winding. In the case of axial compression different buckling modes are examined and the optimal winding determined.Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 864–875, September–October, 1968.     

Creep strength of anisotropic plates taking account of transverse shear strains

Cylindrical creep buckling of an orthotropic glass-reinforced plastic plate is investigated with consideration of shear creep in the plane xz. In this case the rheology of buckling is not described by the Kirchhoff-Love model, therefore the investigation is carried out using the modified theory of plates [3]. The long-time and instantaneous critical forces are obtained.Mekhanika Polimerov, Vol. 1, No. 5, pp. 114–117, 1965     

Stability of composite shells with an elastic core

The stability in axial compression of orthotropic cylindrical shells with an elastic core is investigated with allowance for the following factors: transverse shear strains in the shell material, precritical state of stress of the core, and the moments due to the surface forces exerted on the shell by the core. The numerical results obtained are compared with the results of approximate methods in which the above-mentioned factors are disregarded.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 544–547, May–June, 1976.     

考虑横向剪切的对称圆柱正交异性层合扁球壳的热屈曲

研究了对称层合圆柱正交异性扁球壳在温度场和均布压力联合作用下的热屈曲问题．考虑横向剪切的影响,应用修正迭代法获得了温度场影响下的临界荷载的解析表达式,讨论了剪切刚度和温度场对临界荷载的影响．     

Compressive stability of multilayered delaminations in composites

The behavior of defects of the near-surface crack type was investigated in laminated composites. Multilayered delaminations in compressed elements were examined on the assumption that cylindrical bending of the delamination is accompanied by shear within the framework of a model similar to the Timoshenko model. The effect of the mechanical properties and thickness and number of layers in the delaminated section on the critical buckling strain and strain corresponding to the beginning of growth of the delamination was investigated. It was shown that when the delaminated section contains several bearing layers and the rigidity of the composite with respect to interlaminar shear is small, consideration of the shear in the delaminated section can lead to a significant change in the critical compressive strains and/or critical size of the delamination.Translated from Mekhanika Kompozitnykh Materialov. Vol. 33, No. 3, pp. 312–320, May–June, 1997.     

Stability of cylindrical glass-reinforced plastic shells

The stability of cylindrical shells in axial compression has been experimentally investigated. The glass-reinforced plastic shells were fabricated in various ways: by winding glass strand, from glass cloth, and by a combination of the first two methods. The buckling mode and critical load were determined as functions of the ratio of shell radius to shell thickness. A statistical analysis of the test data leads to a formula with an empirical coefficient for use in practical stability calculations.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 5, pp. 886–891, September–October, 1969.     

Numerical analysis of diamond buckles

The “diamond” modes of collapse are studied with finite element methods. Both linear and nonlinear analyses are performed on the buckling of a cylindrical shell under axial compression. Among the postbuckling shapes of the cylindrical shell, a number of diamond modes (cost nθ; N = 0, 14, 18, 10, 24 and 28) are found to be possible. The analysis is compared to those conducted by Maewal and Nachbar, Crisfield, and Yoshida et al. Agreement is established in conceiving the deformed shape with circumferential number of 14 as the stable postbuckling mode of the cylindrical shell.

The transition from the axisymmetric mode to a diamond mode of collapse is shown to be an instantaneous process triggered in the proximity of the critical state by a small perturbation of the load increment.     

Creep buckling of glass-reinforced plastic cyclindrical shells

The buckling modes of cylindrical shells made of a polymeric composite with creep properties are considered. The initial imperfections of the shell are characterized by a Fourier series. The time dependence of a large number of harmonics of the Fourier series is investigated by means of nonlinear equations of the Timoshenko type. It is established that some are damped, while others grow at an ever-increasing rate. It is shown that in the course of time the effective shape of the deflections is transformed and at the critical moment the shell buckles in a mode that differs from the shape of the initial deflections.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 697–703, July–August, 1971.     

Deformation of cylindrical composite shells under combined dynamic loading

Conclusions A procedure has been shown for calculating the stress-strain state of cylindrical multilayer shells made from composite materials under the combined action of dynamic axial compression and dynamic external pressure, as well as with different variants of combined loading with static and dynamic forces. An investigation has been made of the effect on the mode of the buckled shell surface of the ratio of the application rate of dynamic loads; ranges of loading rates have been established in which stresses predominate caused either by axial compression or external pressure. It has been shown that, as a result of preliminary static loading, a marked change occurs in the initial imperfections of the shell mode which affects subsequent dynamic buckling. To calculate the time when the first defect occurs and its location in the shell body, a procedure has been devised for layer-by-layer strength analysis employing a tensor-polynomial criterion. It was demonstrated that the level of preliminary static loading noticeably affects the time until the first failure of the layer, not only a reduction of this time being possible with an increase in the static loads, but also an increase in it.We should also point out the work in [10] where it is shown that it is possible to weaken the susceptibility of the shell to initial imperfections when internal pressure is applied.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 461–473, May–June, 1981.     

Surface buckling of a layered medium

The surface buckling of a layer-reinforced medium with the layers arranged at right angles to the surface is investigated on the basis of Bolotin's theory of layered media. The critical load is determined and the buckling mode studied. By means of a simplified approach simple expressions are obtained for the critical loads corresponding to surface buckling.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 4, pp. 648–654, July–August, 1973.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.