Stress concentration in the moment theory of elasticity

A multimoment theory of elasticity of inhomogeneous media is used to develop algorithms for determining a sequence of essentially independent fundamental parameters associated with the structure, composition, and characteristics of the medium components. The general form of stress concentration factors is established by analyzing specific models of media. It is found that the mean tangential stresses in a fibrous medium with a transverse stress state are asymmetric. As an example, the methods are used to solve specific problems for plates and shells __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 1, pp. 66–76, January 2007.     

Inequalities of Korn’s Type and Existence Results in the Theory of Cosserat Elastic Shells

This paper is concerned with the linear theory of anisotropic and inhomogeneous Cosserat elastic shells. We establish the inequalities of Korn’s type which hold on Cosserat surfaces. Using these inequalities, we prove the existence of the solution to the variational equations in the elastostatics of Cosserat shells. For the dynamic problems, we employ the semigroup of linear operators theory to obtain the existence, uniqueness and continuous dependence of solution.      

Nonlinear deformation and buckling of elastic inhomogeneous shells under thermomechanical loads

The paper outlines the fundamentals of the method of solving static problems of geometrically nonlinear deformation, buckling, and postbuckling behavior of thin thermoelastic inhomogeneous shells with complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure under complex thermomechanical loading. The method is based on the geometrically nonlinear equations of three-dimensional thermoelasticity and the moment finite-element scheme. The method is justified numerically. Results of practical importance are obtained in analyzing poorely studied classes of inhomogeneous shells. These results provide an insight into the nonlinear deformation and buckling of shells under various combinations of thermomechanical loads     

Stressed state of a composite cylindrical shell with an elliptical hole

The results of investigations of the stress-strained state (SSS) of isotropic cylindrical shells with an elliptical hole are represented in monograph [4]. The modified method of expansion in terms of minor parameter [3] is suggested for calculation of orthotropic shells. The method does not consider, however, lateral shear strains introducing a significant contribution in SSS of composite shells. The procedure for solving problems of SSS calculation near curvilinear holes in shells of arbitrary shape with variable geometrical and physical characteristics is suggested in [1] on the basis of variational-difference method (VDM). Here the relations of the linear theory of anisotropic inhomogeneous shells and the hypothesis of a straight line are taken as the initial ones for all the packet of laminated composite shell as a whole. In the present work we present the numerical results obtained according to the procedure given in [1] for an orthotropic cylindrical shell with an elliptical hole loaded by the axial force and complaint for the lateral shear.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 11, pp. 57–62, November, 1993.     

Contact interaction between prestressed laminated shells of revolution and a flat foundation

A nonclassical model of shells that accounts for transverse shears and reduction is used to develop a method for solving the contact problem for inhomogeneous anisotropic shells of revolution subject to a field of mechanical and thermal loads. The prestresses are described by parametric terms in the linearized geometrically nonlinear equations of the second-order theory of flexible shells. The influence of the prestressed state of shells interacting with a flat surface on the contact area and the distribution of contact pressure is analyzed. Some computational features of the technique are discussed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 67–77, October 2006.     

Stress Analysis of Laminated Shells of Revolution with an Imperfect Interlayer Contact

An approach is proposed to solve problems on the contact interaction of layers in structurally inhomogeneous shells of revolution with allowance for both the normal and tangential stresses in the contact zones. A system of algebraic equations is constructed by using a method for solution of static problems for arbitrary shells of revolution. From this system, the contact zones and contact stresses are determined by the iteration method. The restricted behavior of the adhesion layer in cleavage and shear is taken into account. As an example, the stress state of a two-layer cylindrical shell under a concentrated load is determined     

Some Approaches to the Solution of Problems on Thin Shells with Variable Geometrical and Mechanical Parameters

A number of approaches to the solution of stress problems for anisotropic inhomogeneous shells in the classical formulation are discussed. A review is made of approaches to the solution of one- and two-dimensional static problems for thin shells with variable parameters and to the solution of stress–strain problems for anisotropic shells of revolution under axisymmetric and non-axisymmetric loading, shallow convexo-convex shells, noncircular cylindrical shells, plates of various shapes, and shells of complex geometry     

Effect of Irradiation on the Stability and Natural Vibrations of Containment Shells

An approximate approach is proposed to solve the problem on the natural vibrations and stability of an NPP reactor containment. Radiation alters the mechanical properties of such shells and causes volumetric expansion of their material. The problems are solved within the framework of the Timoshenko kinematic hypotheses by reducing shells inhomogeneous throughout the thickness to homogeneous shells with reduced tension–compression, shear, and flexural rigidities     

轴对称正交异性圆环壳的齐次完全渐近解

承受轴对称载荷的正交异性圆环壳的静力分析,归结为求解一非齐次二阶复变量方程.当所含参数μ较大时,常采用渐近解法.因方程含一阶转点,所以求全域一致有效且达到薄壳理论精度的完全渐近解较为困难.过去,齐次解只求到一级近似.本文采用广义Airy函数方法,求出了高级近似.这样,轴对称正交异性圆环壳的齐次解第一次有了达到薄壳理论精度的完全的渐近展开.     

Vibrations of statically loaded shells of revolution subject to transverse shear and reduction

An approach to studying the vibrations of statically loaded nonthin inhomogeneous anisotropic shells of revolution with complex geometry is developed. The approach is based on a nonclassical theory of shells that takes into account transverse shear and reduction. The expediency of allowing for these factors in the analysis of natural frequencies is confirmed by specific examples     

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.     

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.     

Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure

The nonlinear large deflection theory of cylindrical shells is extended to discuss nonlinear buckling and postbuckling behaviors of functionally graded (FG) cylindrical shells which are synchronously subjected to axial compression and lateral loads. In this analysis, the non-linear strain-displacement relations of large deformation and the Ritz energy method are used. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, by taking the temperature-dependent material properties into account, various effects of external thermal environment are also investigated. The non-linear critical condition is found by defining the possible lowest point of external force. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on non-linear buckling behaviors of combine-loaded FG cylindrical shells. In addition, the postbuckling equilibrium paths are also plotted for axially loaded pre-pressured FG cylindrical shells and there is an interesting mode jump exhibited.     

Stress–Strain Analysis of Smooth and Ribbed Shells under Local Loads

Static problems for smooth and discretely reinforced cylindrical shells under local loads and complex boundary conditions are solved. The stress–strain states of the casing and ribs are determined by the technical theory of shells and the Kirchhoff–Clebsch theory of rods, respectively. The reinforcing elements are arranged eccentrically. They are of equal or different stiffness, which is also variable along the length. The problems are solved using the finite-difference method. Theoretical results obtained from a refined mesh are compared with experimental data     

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.     

A theoretical solution of cylindrical shells for axisymmetric plain strain elastodynamic problems

A method is developed for the transient responses of axisymmetric plain strain problems of cylindrical shells subjected to dynamic loads. Firstly, a special function was introduced to transform the inhomogeneous boundary conditions into the homogeneous ones. Secondly, using the method of separation of variables, the quantity that the displacement subtracts the special function was expanded as the multiplication series of Bassel functions and time functions. Then by virtue of the orthogonal properties of Bessel functions, the equation with respect to the time variable was derived, of which the solution is easily obtained. The displacement solution was finally obtained by adding the two parts mentioned above. The present method can avoid the integral transform and is fit for arbitrary loads. Numerical results are presented for internally shocked isotropic and cylindrically isotropic cylindrical shells and externally shocked cylinders, as well as for an externally shocked, cylindrically isotropic cylindrical shell that is fixed at the internal surface. Contributed by Ding Hao-jiang Foundation item: the National Natural Science Foundation of China (10172075) Biography: Ding Hao-jiang (1934-)     

基于混合变量条形传递函数方法的圆柱壳屈曲分析

提出了一种分析交各向异性圆柱壳和阶梯圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Fluegge薄壳理论，通过定义广义位移变量和对应的广义力变量，建立了圆柱壳混合变量能量泛函；然后通过引入条形单元，定义混合状态变量和采用传递函数方法对超级壳单元求解，得到具有多种边界条件圆柱壳屈曲问题的半解析解；最后通过位移连续和力平衡条件，可以得到阶梯圆柱壳屈曲问题的解。理论解推导过程表明此方法在引入边界条件和进行阶梯圆柱壳求解时非常方便。算例分析的结果验证了本方法的正确性。     

A THEORETICAL SOLUTION OF CYLINDRICAL SHELLS FOR AXISYMMETRIC PLAIN STRAIN ELASTODYNAMIC PROBLEMS

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Nonaxisymmetric Thermoelastoplastic Analysis of Branched Shells of Revolution by the Semianalytic Finite-Element Method

A technique is proposed to solve elastoplastic deformation problems for branched shells of revolution under the action of asymmetric forces and a temperature field. The kinematic equations are derived within the framework of the linear Kirchhoff–Love theory of shells and the thermoelastic relations within the framework of the theory of small elastoplastic strains. The problem is given a variational formulation based on the virtual-displacement principle and the Fourier-series expansion of the unknown functions and loads with respect to the circumferential coordinate. The additional-load method is used to solve a nonlinear problem and the finite-elements method is used to carry out a numerical analysis. As an example, an asymmetric stress–strain analysis is performed for a cylindrical shell reinforced by a ring plate.