Nonlinear elastic state of thin-walled toroidal shells made of orthotropic composites

A study is made of problems on the statics of circular toroidal shells made of nonlinear elastic orthotropic composites. The study is conducted on the basis of the method of successive approximation, the variational-difference method, and the method of Lagrangian multipliers. The parameters of the circular torus are varied within broad ranges of values in the calculations. Numerical results are presented in the form of tables and graphs and are analyzed. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 49–55, December, 1999.     

Dependence of the deformation of flexible cylindrical shells on their geometry and the orthotropy and nonlinear properties of the composite materials

A study is made of geometrically and physically nonlinear inverse problems concerning the axisymmetric deformation of cylindrical shells into conical shells. Results obtained from the numerical solution of the problems are used to determine the laws of distribution of the surface loads, stresses, strains, and displacements in relation to the initial parameters and nonlinearities of the shells. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 86–91, June, 1999.     

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．     

Vibrations of shallow noncircular membranes with initial stresses

The basic relations for a geometrically nonlinear (quadratic approximation) theory of thin elastic membranes are obtained. The relations are used to develop a variational method for studying free vibrations of initially flat membranes bounded by a stationary piecewise-smooth contour. The membrane is deformed by uniform pressure. Numerical results are given for different types of vibrations of rectangular and elliptical shells. N. V. Gogol' Pedagogical Institute, Nezhin: S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 78–86, August 1999.     

Physically and Geometrically Nonlinear Static Problems for Thin-Walled Multiply Connected Shells

Physically and geometrically nonlinear two-dimensional problems are formulated for multiply connected thin shells (weakened by several curvilinear holes). A technique and algorithm are proposed for their solution with allowance for elastoplastic strains and finite deflections of shells under static loading. Numerical results for a shell with two circular holes are presented and the stress concentration is analyzed     

Stress Concentration Near Openings in Composite Shells

The results of studying the stress–strain distribution in composite shells with curvilinear openings are reported. Nonclassical generalizing formulations and methods for solution of linear and nonlinear problems are stated. Numerical results obtained for thin and nonthin shells are analyzed with regard for features of the deformation of composites     

Stability of conical shells with small curvatures of the generatrix

Numerical solutions are used as the basis for a comparative analysis of the qualitative and quantitative effect of small initial curvatures of the generatrices of conical and cylindrical shells on the critical axial pressures of the shells S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 36–40, February, 1999.     

Nonsteady operating conditions of a multimode cylindrical radiator, taking account of processes in the cable channel

A multimode cylindrical piezotransformer submerged in liquid is considered, in the case of excitation by nonsteady pulses traveling through the extended cable channel of the electroded sections (the other section is short-circuited). The accompanying transient process is modeled on the basis of the theory of thin electroelastic shells, the acoustic approximation, and the telegraph equations. Numerical results are given for an electric signal whose profile is specified by the Heaviside function. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 35–43, August, 1999.     

Stability and efficient design of cylindrical shells of metal composites subject to combination loading

A study is made of the stability of boron-aluminum shells under a combination of axial compression and uniform external pressure. An approximate theoretical model is constructed to describe the deformation of a layer of a fiber composite consisting of elastoplastic components. The model is used to derive the equations of state of multilayered shells reinforced by different schemes. The nonlinear equation describing the subcritical state is solved by the method of discrete orthogonalization with the use of stepped loading. The homogeneous problem is also solved by discrete orthogonalization. It is shown that shells can be efficiently designed for combination loading by plotting the envelope of the boundary curves for specific reinforcement schemes. The envelope is convex for elastic shells and is of variable curvature for elastoplastic shells. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 67–73, June, 1999.     

The stability and load-carrying capacity of incomplete shells

The survey is devoted to problems on the stability and load-carrying capacity of imperfect shells in a nonhomogeneous stress-strain state. Methods recently developed at the Institute of Mechanics (Kiev, Ukraine) are briefly described. The approaches proposed are based on the generalized Euler criterion. Special attention is focused on numerical methods. The stability and load-carrying capacity of shells with symmetric and asymmetric, local and regular imperfections are considered. The data presented are compared with the well-known theoretical results and experimental data. At the end of the review, an analytical method (of reduced stiffness) is presented for predicting the lower bounds of sensitivity to imperfection in elastic buckling of longitudinally compressed stiffened shells with a nearly cylindrical shape. S.P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 36–59, July, 2000.     

椭球壳体液压成形的塑性变形规律的研究

采用大变形弹塑性有限元法对椭球壳体的液压成形过程进行了模拟，结果表明壳体成形时，各点塑性变形不是同时发生的，并对壳体塑性变形的扩展过程和点的加载轨迹进行了研究，证明了壳体在胀形时焊缝区受到弯曲作用的影响，同时还成形后的椭球壳体的壁厚分布及壳体尺寸进行了预测。     

On the stress-strain state of toroidal shells of elliptical cross section formed from nonlinear elastic orthotropic materials

Problems of the statics of toroidal shells with elliptical cross sections, which are formed from nonlinear elastic orthotropic materials, were examined. Investigations were carried out in accordance with a procedure based on the method of successive approximations, the variational-difference method, and the method of Lagrangian multipliers. Calculations were performed by varying the ellipticities of the cross section of the torus and the radii of the circular axis of the torus over a broad range. Numerical and analytical results are presented in tables and graphs. Attention was focused on the accuracy of the computational results obtained, and agreement between the internal and external forces in the cross section was adopted as an integral criterion. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 103–109. January, 2000.     

Allowance for geometrical nonlinearity in dynamic problems of the theory of discretely stiffened cylindrical shells under nonstationary loading

A variant of the two-dimensional equations of the motion of a discretely stiffened cylindrical shell is considered within the framework of the elastic nonlinear Timoshenko-type theory of shells and rods. The initial system of equations of motion is derived based on the Hamilton-Ostrogradskii variation principle. A numerical algorithm for solution of such problems with allowance for discrete nonuniformities is constructed. Some aspects of equation approximation are studied. The effect of geometrically nonlinear factors on the stress-strain state of a structure is analyzed. The scientific results of the present work were obtained during implementation of Project No. 182 of the Ukrainian Scientific and Technological Center. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 120–124, April, 2000.     

Magnetoelastic nonlinear deformation of a conical shell of variable stiffness

A complete system of equations of nonlinear magnetoelasticity of shells is presented. A resolving system of equations of a conical shell of variable thickness is constructed for the axisymmetric case. A numerical example of a flexible shell illustrates the possibility of using the proposed technique to solve this class of problems. T. G. Shevchenko National University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 34–39, November, 1999.     

Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment

Based on the nonlinear large deflection theory of cylindrical shells, this paper deals with the nonlinear buckling problem of functionally graded cylindrical shells under torsion load by using the energy method and the nonlinear strain–displacement relations of large deformation. The material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical state of the shell are also investigated. Numerical results show various effects of the inhomogeneous parameter, the dimensional parameters and external thermal environment on nonlinear buckling of functionally graded cylindrical shells under torsion. The present theoretical results are verified by those in literature.     

Stability of multilayered shells of revolution with allowance for geometric nonlinearity of the subcritical state

The finite-difference method and the Trefftz-Reissner variational principle are used to obtain a system of equations in mixed from to describe the stability and geometric nonlinearity of composite shells of revolution. Methods are developed and an algorithm is proposed to calculate the components of the geometrically nonlinear subcritical stress-strain state and to use those components to determine the “upper” critical values for shells with zero Gaussian curvature loaded by uniform external pressure, an axisymmetric load, or a combination of these loads. The stability of cylindrical, conical, and compound shells under uniform pressure is examined for different support conditions. Linear and nonlinear methods of determining the subcritical stress-strain state are compared and their effect on the critical loads is estimated. Ukrainian Transportation Institute and the Ukrainian Academy of Water Management, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 60–66, June, 1999.     

Nonlinear Response of Cylindrical Shells to Random Excitation

An analytical study of nonlinear flexural vibrations of cylindrical shells to random excitation is presented. Donnell's thin-shell theory is used to develop the governing equations of motion. Thermal effects for a uniform temperature rise through the shell thickness are included in the formulation. A Monte Carlo simulation technique of stationary random processes, multi-mode Galerkin-like approach and numerical integration procedures are used to develop nonlinear response solutions of simply-supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and histograms of probability density. Comparison of Monte Carlo results is made to those obtained by statistical linearization and the Fokker–Planck equation.     

Vibrations of shallow shells that are circular in plan and have local supports

The paper deals with vibrations of doubly curved shallow shells that are circular in plan and are reinforced by local rod-type supporting elements or cylindrical shells. The coefficients of the frequency equation are found by using numerical-analytical methods to solve boundary-value problems for fixed values of frequency. The natural frequencies and modes of vibration of a system composed of a shell and elastic supports are determined in the course of solving the problem. It is shown that it is possible to also account for reactive moments and shearing forces, in addition to the normal reactions of an elastic support. The potential of the approach which is developed is illustrated by the solution of specific problems. Special Design Office of the S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 69–75, March, 1999.     

On the retarded potentials of inhomogeneous ellipsoids and sources of arbitrary shapes in the three-dimensional infinite space

We analyze the infinite space solutions of the three-dimensional inhomogeneous wave equation (the ‘retarded potentials’ or ‘causal propagators’) for ellipsoidal sources and for sources of arbitrary shapes. The ‘short-time characteristics’ of the retarded potential for a spatially inhomogeneous source density of δ-shaped time profile is considered. It is found that, the short-time characteristics is governed by the spatial inhomogeneity of the source density in the immediate vicinity of a spacepoint.Surface integral representations are derived for spatial inhomogeneous source regions of ellipsoidal symmetry. For spherical sources these integral representations yield closed form solutions for the retarded potentials. We find that the wave field inside a spherical source consists of an incoming and outgoing spherical wave package, whereas the external wave field consists of an outgoing spherical wave package only. Characteristic runtime and superposition effects are discussed. Moreover, a numerical technique based on Gauss quadrature is applied to generate the wave field for a cubic source. The integral representations derived for the retarded potentials of inhomogeneous ellipsoidal sources are consistent with results previously derived by the authors for the Helmholtz potentials of homogeneous ellipsoids and ellipsoidal shells [Michelitsch, T.M., Gao, H., Levin, V.M., 2003. On the dynamic potentials of ellipsoidal shells. Q. J. Mech. Appl. Math. 56 (4), 629]. The derived solutions are crucial for many problems of wave propagation and diffraction theory as they may occur in materials science. As an example we give a formulation for the solution of the retarded Eshelby inclusion problem due to spatially and temporally varying eigenfields in the elastic isotropic infinite medium.     

Interaction of boundaries of opening and rigid inclusion under unsteady convective heat exchange on external surfaces of cylindrical shell

The stressed state of a thin cylindrical shallow shell containing a circular opening and a circular undeformable inclusion is investigated using a thermoelasticity equation in complex forces. The boundaries of the opening and inclusion are thermally insulated. The unsteady convective heat exchange between the environment and the shell surfaces is described by Newton's law. Numerical studies of thermal stresses are carried out for various times and reduced distances between the boundaries of the opening and inclusion. Simferopol University. Donetsk University, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 4, pp. 61–66, April, 1999.