A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 231–240, March–April, 2009.     

Numerical solution of the nonlinear problem of motion of a long rectangular plate in a variable magnetic field

The coupled problem of motion of a long rectangular plate in a variable magnetic field is analyzed in the framework of the geometrically nonlinear theory of thin shells. The proposed numerical procedure is applied to estimate the effect of external electromagnetic and mechanical fields on the stress- strain state of the plate.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 37–42, 1990.     

Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations

This paper presents a nonlinear free vibration analysis of corrugated circular plates with shallow sinusoidal corrugations under uniformly static ambient temperature. Based on the nonlinear bending theory of thin shallow shells, the governing equations for corrugated plates are established from Hamilton’s principle. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude for nonlinear vibration of heated corrugated plates are obtained successfully. The comparison with available published results shows that the numerical analysis here is of good reliability. A detailed parametric study is conducted involving the dependency of nonlinear frequency on the depth and density of corrugations along with the temperature change. Effects of these variables on the trend of nonlinearity are plotted and discussed.     

Lax pairs and a new spectral method for linear and integrable nonlinear PDEs

A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

The stress-deformed state of a packet of cylindrical shells taking account of friction

We propose a method of determining the contact pressures between the shells in a packet under the influence of nonlinear internal and constant external pressure. Using the equations of the general moment theory of shells we determine the stress-deformed state of a packet of finite cylindrical shells taking account of frictional forces on the contacting surfaces. One table. Bibliography: 10 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 94–98.     

Axisymmetric deformation of bimetallic shells of revolution in the supercritical region

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 52–56, 1990.     

Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten

The problem of stress determination in the area of cut-outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part C_t1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part C_t2. Starting from the Betti-Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions. The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo-differential operators, the Galerkin method converges. Assuming that curves C_t1 and C_t1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.     

On a method of solving physically nonlinear problems in the bending of thick rectangular plates

We study the three-dimensional boundary-value problem of the physically nonlinear theory of elasticity for bending of a homogeneous isotropic thick plate by transverse forces. We assume that the intensity of the load is such that the connection between the stresses and small strains within the elastic limit can be described by nonlinear relations in the form of H. Kauderer. We develop an approximate analytic method that makes it possible to reduce the original nonlinear boundary-value problem to a recursive sequence of corresponding linear boundary-value problems. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, No. 40, 1997, pp. 30–35.     

Finite deformations of thin shells with piezoelectric sensors and actuators

We treat coupled electromechanical problem of finite deformations of piezoelectric shells with the help of the direct approach. A shell is considered as a material surface with mechanical degrees of freedom of particles and with an additional field variable, namely electric potential on the electrodes. This results both in the nonlinear system of equations of piezoelectric shells and in the appropriate numerical scheme. Application of the direct approach is preceded with the three-dimensional asymptotic analysis of a linear electromechanical problem for a non-homogeneous piezoelectric plate, which provides the constitutive relations for the nonlinear theory. As a sample problem, we present finite element analysis of deformation and local buckling of cylindrical panel, equipped with piezoelectric sensors. The latter influence the mechanical behavior and produce signals, which can be interpreted in terms of structural entities. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

波纹壳的格林函数方法

应用轴对称旋转扁壳的基本方程,研究了在任意载荷作用下具有型面锥度的浅波纹壳的非线性弯曲问题·　采用格林函数方法,将扁壳的非线性微分方程组化为非线性积分方程组·　再使用展开法求出格林函数,即将格林函数展成特征函数的级数形式,积分方程就成为具有退化核的形式,从而容易得到非线性代数方程组·　应用牛顿法求解非线性代数方程组时,为了保证迭代的收敛性,选取位移作为控制参数,逐步增加位移,求得相应的载荷·　在算例中,研究了具有球面度的浅波纹壳的弹性特征·　结果表明,由于型面锥度的引入,特征曲线发生显著变化,随着荷载的增加,将出现类似扁球壳的总体失稳现象·　本文的解答符合实验结果·     

The heat-conduction problem for a shell with a system of diathermanous cuts

Using the two-dimensional Fourier transform and the elementary theory of distributions, we solve the heat-conduction problem for shells with a system of diathermanous cuts. We take account of heat exchange according to Newton's law on the lateral surfaces of the shells. For a spherical shell with two cuts of identical length we carry out numerical studies of the influence of the thermophysical properties of one cut on the jump in temperature of the adjacent cut. Three figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 86–89.     

An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.     

The effect of a reinforcing ring on the stress-strain state of flexible cylindrical shells

We investigate the effect of a reinforcing ring on the stress-strain state of a cylindrical shell in the geometrically nonlinear problem with a nonaxisymmetric load on the edge. The nonlinear boundary-value problem is reduced to a sequence of linear problems by the quasilinearization method. The linear problems are solved by the discrete orthogonalization method. The results obtained using linear and nonlinear theory are compared.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 92–96, 1985.     

Global existence for a nonlinear theory of bubbly liquids

Global existence of smooth solutions is proved for an effective theory of bubbly liquids for either the initial value problem or initial boundary value problem in one dimension. This shows that the theory does not describe shock waves or bubble collapse. Since the analysis is not for the steady boundary value problem, there is no discussion of resonance. The proof uses a semilinear form of the equations to get local existence. A priori bounds resulting from energy conservation and a nonlinear Gronwall-like inequality are then derived to prove global existence.     

Solution of a problem of nonlinear potential theory by the fixed domain method

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 51–56, 1986.     

Variant of the geometrically nonlinear theory of anisotropic shells with allowance for transverse shear

A very simple variant of the geometrically nonlinear theory of anisotropic shells with allowance for the high compliance of the material in transverse shear is proposed. From this theory there follow, as a special case, the equations for an isotropic shell; these differ from the relations of [2] with respect to terms of the order of the ratio of the thickness of the shell to the radii of curvature small as compared with unity. The equations obtained are used to solve the problem of the stability of orthotropic shells of revolution relative to the starting axisymmetric state of stress.Translated from Mekhanika Polimerov, No. 5, pp. 863–871, September–October, 1969.     

Differential-algebraic equations with a small nonlinear term

In the present paper, we generalize some known results of the theory of differentialalgebraic equations to a more complicated case under the assumption that the nonlinear term is small. We give an asymptotic estimate of the behavior of the solution with respect to the parameter μ.     

Calculation of the Thermoelastoplastic Nonaxisymmetric Stress-Strain State of Layered Orthotropic Shells of Revolution

The methods for determining the nonaxisymmetric thermoelastoplastic stress-strain state of layered orthotropic shells of revolution are developed. It is assumed that the layered package deforms without mutual slippage or separation of layers. The problem is solved using the geometrically nonlinear theory of shells based on the Kirchhoff-Love hypotheses. In the isotropic layers, plastic deformations may appear, whereas the orthotropic layers deform in the elastic region. It is assumed that the mechanical properties of the materials are temperature-dependent. The thermoplasticity equations are presented in a form corresponding to the method of additional deformations. The order of the system of partial differential equations obtained is reduced with the help of trigonometric series in the circumferential coordinate. The resulting systems of ordinary differential equations are solved by the Godunov technique of discrete orthogonalization. The nonaxisymmetric thermoelastoplastic stress-strain states of layered shells of revolution are considered as examples.