To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

Thermally induced dynamic instability of laminated composite conical shells

Thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method. The laminated composite conical shells are subjected to static and periodic thermal loads. The linear instability approach is adopted in the present study. A set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs. The formulation begins with three-dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible. Using the method of differential quadrature and Bolotin's method, and imposing the orthonormality and solvability conditions on the present asymptotic formulation, we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner. The principal instability regions of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

The Schrödinger equation in theory of plates and shells with orthorhombic anisotropy

This work is the continuation of the discussion of Refs. [1-5]. In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection for orthorhombic misotropic thin shells or orthorhombic anisotropic thin plates on Winkler’s base are classified as several of the same solutions of Schrodmger equation, and we can obtain the general solutions for the two above-mentioned problems by the method in Refs. [1] and [3-5].[B] The. von Karman-Vlasov equations of large deflection problem for shallow shells with orthorhombic anisotropy (their special cases are the von Harmon equations of large deflection problem for thin plates with orthorhombic anisotropy) are classified as the solutions of AKNS equation or Dirac equation, and we can obtain the exact solutions for the two abovementioned problems by the inverse scattering method in Refs. [4-5].The general solution of small deflection problem or the exact solution of large deflection problem for the corrugated or rib-reinforced plates and shells as special cases is included in this paper.     

Analysis of a strengthened weakly conical shell using a complete system of orthogonal functions on an arbitrary contour of its transverse cross-section

Thin-walled weakly conical and cylindrical shells with arbitrary open, simply or multiply closed contour of transverse cross-sections strengthened by longitudinal elements (such as stringers and longerons) are used in the design of wings, fuselages, and ship hulls. To avoid significant deformations of the contour, such structures are stiffened by transverse elements (such as ribs and frames). Various computational models and methods are used to analyze the stress-strain states of such compound structures. In particular, the ground stress-strain states in bending, transverse shear, and twisting of elongated structures are often analyzed with the use of the theory of thin-walled beams [1] based on the hypothesis of free (unconstrained) warping and bending of the contour of transverse cross-sections. In general, the computations with the contour warping and bending constraints caused by the variable load distribution, transverse stiffening elements, and the difference in the geometric and rigidity parameters of the shell cells are usually performed by the finite element method or the superelement (substructure) method [2, 3]. In several special cases (mainly for separate cells of cylindrical and weakly conical shells located between transverse stiffening elements, with the use of some additional simplifying assumptions), efficient variation methods for computations in displacements [4–8] and in stresses [9] were developed, so that they reduce the problem to a system of ordinary differential equations. In the one-and two-term approximations, these methods permit obtaining analytic solutions, which are convenient in practical computations. But for shells with multiply closed contours of transverse cross-sections and in the case of exact computations by using the Vlasov variational method [4], difficulties are encountered in choosing the functions representing the warping and bending of the contour of transverse cross-sections. In [10], in computations of a cylindrical shell with simply closed undeformed contour of the transverse section, warping was represented in the form of expansions in the eigenfunctions orthogonal on the contour, which were determined by the method of separation of variables from a special integro-differential equation. In [11], a second-order ordinary differential equation of Sturm-Liouville type was obtained; its solutions form a complete system of orthogonal functions with orthogonal derivatives on an arbitrary open simply or multiply closed contour of a membrane cylindrical shell stiffened by longitudinal elements. The analysis of such a shell with expansion of the displacements in these functions leads to ordinary differential equations that are not coupled with each other. In the present paper, by using the method of separation of variables, we obtain differential and the corresponding variational equations for numerically determining complete systems of eigenfunctions on an arbitrary contour of a discretely stiffened membrane weakly conical shell and a weakly conical shell with undeformed contour. The obtained systems of eigenfunctions are used to reduce the problem of deformation of shells of these two types to uncoupled differential equations, which can be solved exactly.     

Torsional,flexural, and torsional-flexural buckling modes of a cylindrical shell under combined loading

Stability problems for cylindrical shells under various loading modes were considered in numerous papers. A detailed analysis of such problems can be found, e.g., in the monograph [1]. We refer to the solutions presented in this monograph as classical.For long cylindrical shells in axial compression, one of the buckling modes is the purely beam flexural mode similar to the classical buckling mode of a straight rod. It is well known that it can be studied by using the nonlinear or linearized equations of the membrane theory of shells. In [2], it was shown that, on the basis of such equations constructed starting from the noncontradictory version of geometrically nonlinear elasticity relations in the quadratic approximation [3], under the separate action of the axial compression, external pressure, and torsion, there are also previously unknown nonclassical buckling modes, most of which are shear ones.In the present paper, we show that the use of the above equations for cylindrical shells under compression and external pressure with simultaneous pure torsion or bending permits revealing the earlier unknown torsional, beam flexural, and beam torsional-flexural buckling modes, which are nonclassical, just as those found in [2]. The second of these buckling modes is realized when axially compressing forces are formed in the shell with simultaneous torsion, and the third of them is realized under compression combined with pure bending.It was found that, earlier than the classical buckling modes, the torsional buckling modes can be realized for relatively short shells with small shear rigidity in the tangent plane, while the second and third buckling modes can be realized for relatively long shells.     

Transient solutions for longitudinally impacted inhomogeneous conical shells

Here we study transient elastic wave propagation in inhomogeneous conical shells. The uniaxial theory is employed and two separate techniques are used for extracting information about the stress field for impact problems. Firstly the formal Karal-Keller method is used enabling us to directly determine asymptotic wavefront expansions for the stress field. A transform technique, based on Eason's [1], is then used to obtain conditions on the physical parameters of the medium which give solutions to the governing equation in terms of Bessel functions and some particular problems are discussed. Previously unknown simple closed-form solutions are obtained. Our results have application to the propagation of waves in filamentary cones of constant wall thickness or in homogeneous cones having an axial temperature gradient.     

旋转壳的数值传递函数方法

应用数值传递函数方法建立一种用于分析旋转壳静力、动力响应的截锥壳单元,在本方法中,单元的位移在环向展开为Fourier级数的形式,应用薄壳理论可以得到解耦的微分方程,通过Laplace变换可以将方程转化为频域内的常微分方程,将其表示为状态空间形式后,可以应用数值传递函数方法求解,对复杂的系统可以应用与有限元类似的方法,划分多个单元组合求解,文中给出了几种旋转壳的动力、静力问题的数值算例,并与其它方法进行了比较,表明本文方法具有精度高,计算方便等特点。     

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.     

圆锥壳自由振动传递函数解

本文在线性弹性理论基础上，给出了一种求解圆锥薄壳自由振动的渐进传递函数解法，壳体的三个位移分量，外力和边界条件首先沿环向展开的Ｆｏｕｒｉｅｒ级数，然后关于时间变量进行Ｌａｐｌａｃｅ变换，这样就将壳体的控制方程化为一系列含复参数ｓ的变系数常微分方程组，通过定义状态变量。得到了壳体动力学问题的状态空间控制微分方程，引入一小参数，并利用摄动技术就可以得到微分方程的渐进传递函数解，将各于锥段的解进行综合，     

Nonlinear vibrations of orthotropic shallow shells of revolution

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated.following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.     

Thermoelastic stability of bimetallic shallow shells of revolution

This article considers the thermoelastic stability of bimetallic shallow shells of revolution. Basic equations are derived from Reissner’s non-linear theory of shells by assuming that deformations and rotations are small and that materials are linear elastic. The equations are further specialized for the case of a closed spherical cup. For this case the perturbated initial state is considered and it is shown that only in the cases when the cup edge is free or simply supported buckling under heating is possible. Further the perturbated flat state is considered and the critical temperature for buckling is calculated for the case of free and simply supported edges. The temperature–deflection diagrams are calculated by the use of the collocation method for shallow spherical, conical and cubic shells.     

Free vibration of functionally graded truncated conical shells under internal pressure

The influence of internal pressure on the free vibration behavior of functionally graded (FG) truncated conical shells are investigated based on the first-order shear deformation theory (FSDT) of shells. The initial mechanical stresses are obtained by solving the static equilibrium equations. Using Hamilton’s principle and by including the influences of initial stresses, the free vibration equations of motion around this equilibrium state together with the related boundary conditions are derived. The material properties are assumed to be graded in the thickness direction. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the governing equations and the related boundary conditions. The convergence behavior of the method is numerically investigated and its accuracy is demonstrated by comparing the results in the limit cases with existing solutions in literature. Finally, the effects of internal pressure together with the material property graded index, the semi-vertex angle and the other geometrical parameters on the frequency parameters of the FG truncated conical shells subjected to different boundary conditions are studied.     

Non-linear dynamic analysis of symmetric and antisymmetric cross-ply laminated orthotropic thin shells

In this paper, the governing equations for non-linear free vibration of truncated, thin, laminated, orthotropic conical shells using the theory of large deformations with the Karman-Donnell-type of kinematic nonlinearity are derived. Applying superposition principle and Galerkin’s method, these equations are reduced to a time dependent non-linear differential equation. The frequency-amplitude relationship for the laminated orthotropic thin truncated conical shell is obtained using the method of weighted residuals. In the particular case, we can obtain the similar relationships for the single-layer and laminated orthotropic cylindrical shells, also. The influence played by geometrical parameters of the conical shell and physical parameters of the laminate (i.e. material properties, staking sequences and number of layers) on the non-linear vibration behavior of the conical shell is examined. It is noticed that the non-linear vibration of shells is highly dependent on laminate characteristics and, from these observations, it is concluded that specific configurations of laminates should be designed for each kind of application. Present results are compared with available data for special cases.     

楔形杆中弹塑性波的一种渐近展开式

据我们所知,楔形杆中弹塑性波尚未有很好的分析方法。对弹性波有文献[1,2]等,其中文献[1]研究了圆锥壳轴向撞击的波动问题,发现楔形杆是其很好的近似,故后者的研究对圆锥壳具有重要意义。文中采用拉氏变换方法求得两种特殊情况下(波阵面和冲击端附近,的渐近解,而一般情形下的解未能得到。也有人用WKB方法讨论了类似问题,但仅限于波长很短的情形,局限性很大。另外,文献[5]用正则摄动法研究了楔形杆的自振问题。 本文针对楔形杆(和圆锥壳)的特点建议了一种渐近展开式,并求解了弹性波和弹塑性波问题,并与其他一些方法及其结果进行了比较。     

Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales

A three-dimensional (3D) free vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with closed-circuit surface conditions is presented using the method of perturbation. By means of the direct elimination, we firstly reduce the twenty-nine basic equations of 3D magneto-electro-elasticity to ten differential equations in terms of ten primary variables of magnetic, electric and elastic fields. The method of multiple scales is introduced to eliminate the secular terms in various order problems of the present formulation so that the present asymptotic expansion to the primary field variables leads to be uniform and feasible. Through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of governing equations for various order problems. The coupled classical shell theory (CST) is derived as a first-order approximation to the 3D magneto-electro-elasticity. Higher-order modifications can be further determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Some benchmark solutions for the free vibration analysis of FG elastic and piezoelectric plates are used to validate the performance of the present asymptotic formulation. The influence of the material-property gradient index on the natural frequencies and corresponding modal field variables of the FG shells is mainly concerned.     

碳纳米管增强复合材料圆锥薄壳非线性自由振动

考虑碳纳米管复合材料作为功能梯度材料的不均匀性,基于连续介质理论以及哈密尔顿变分原理,建立了功能梯度碳纳米管增强复合材料开口圆锥薄壳结构的非线性运动偏微分控制方程,然后利用Galerkin法,将非线性偏微分控制方程转化为常微分控制方程,进而采用谐波平衡法求解了开口圆锥壳的非线性自由振动问题,并探讨了圆锥薄壳几何参数、碳纳米管参数对结构非线性自由振动的影响．数值研究表明结构的无量纲非线性自由振动频率与线性自由振动频率的比值随圆锥薄壳长厚比的增大而变小、并随圆锥角的增大而变大．     

Stress-strain analysis of closed nonthin orthotropic conical shells of varying thickness

The approach developed to solve two-dimensional static problems for nonthin conical shells of varying thickness is used to examine the effect of the geometrical parameters on the stress-strain state of shells. The approach is based on spline-approximation and a stable numerical method of solving one-dimensional problems __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 46–58, June 2008.     

Contact problems for a two-layer elastic wedge

We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex. The asymptotic and other methods were earlier used to solve similar plane contact problems for a homogeneous wedge [1, 2]. In the case of sliding fixation of one of the sides of a plane homogeneous wedge, the closed solution of the contact problem is known for a die entering the corner point [3, p. 131]. Two-dimensional contact problems were studied for a truncated wedge [4] and for a wedge supported by a rod of equal resistance [5]. The out-of-plane shear vibrations were studied for wedge-shaped composites [6, 7]. The spatial contact problems were considered for a homogeneous wedge [8]. The plane contact problem was analyzed for a continuously inhomogeneous wedge one of whose sides was rigidly fixed (the shear modulus continuously depends on the angular coordinate and the Poisson coefficient is constant). For a two-layer composite, which is studied in the present paper, the kernel symbol has different asymptotic properties, which are used in asymptotic methods for solving the problem. A similar distinction of the symbol properties takes place in contact problems for a continuously inhomogeneous layer and a layered packet.     

Asymptotic Analysis of the Eversion of Nonlinearly Elastic Shells II. Incompressible Shells

This paper treats the eversion of axisymmetric, strictly convex, incompressible nonlinearly elastic shells within a general geometrically exact theory in which the shell can suffer flexure, shear, and both midsurface and transverse extension. The governing equations differ considerably from those for compressible shells. We first formulate the governing equations carefully, showing how to handle the 3-dimensional notion of incompressibility, and paying special attention to the constitutive equations. We prove that when a thickness parameter is sufficiently small, there is an everted state, having a lip near the edge, that can be approximated effectively by an asymptotic series whose error we estimate.