On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527-3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.     

Complete system of two-dimensional invariants for formulation of triangular finite element of composite shells

The paper describes a system of invariants of symmetric two-dimensional tensors defined on a plane or a surface. The system comprises the well-known first and second invariants and a new quantity called the combined invariant of two tensors. The focus is on the expression for the invariants in terms of normal components of the tensors determined in three different directions on the surface. The system of invariants is used to construct a triangular finite element for geometrically nonlinear analysis of shear deformable anisotropic shells subject to the Reissner–Mindlin assumptions. The relations obtained allow one to readily determine the strain energy of the element for the normal components of the stress and strain tensors in the direction of the element edges. Numerical examples are given to demonstrate some nonlinear capabilities of the element.     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.     

Solution of boundary-value problems in the theory of analytic functions of several variables in Vladimirov algebras

A solution is given for the Riemann problem for tubular domains in Vladimirov algebras in closed form by means of an integral representation of Bochner-Vladimirov type which is constructed here. In particular, the Schwartz problem is solved. The statement of the Hilbert problem in Vladimirov algebras is examined and its solution is given by a reduction to the Riemann problem, and in one case by a reduction to the Schwartz problem.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 51–60, July, 1977.     

Stress state of a composite shell with a sizable opening

The stress-strain state of a nonshallow cylindrical shell of a composite material is investigated. The shell is weakened by a circular hole and loaded with internal pressure. For solving the problem, the variational-difference method is used. The calculations are carried out for an orthotropic shell with a sizable hole, with account of the reduced shear stiffness of the material.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 49–56, January–February, 2005.     

Yield condition for circular cylindrical shells made of a fiber-reinforced composite

A yield condition is obtained for circular cylindrical shells made of a definite class of fiber-reinforced composite material whose components possess plastic properties. It is shown that, in the plane of generalized stresses — the axial bending moment and the circumferential force (when the axial force is absent) — the yield curve consists of two linear and four curvilinear sections. By approximating the curvilinear sections, we get a piecewise linear yield condition described by a hexagon in the plane indicated. The nonlinear equations and the corresponding piecewise linear equations of the yield condition for particular cases are given in the form of tables. In solving specific boundary-value problems, we consider a circular cylindrical shell simply supported at its ends and loaded with a uniform internal pressure, for which the load-carrying capacity is determined in relation to the mechanical properties of composite components and some characteristic geometrical parameters. The results of numerical calculations are represented in the form of graphs. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 655–666, September–October, 2006.     

Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method

This paper presents a novel finite element formulation for static, free vibration and buckling analyses of laminated composite plates. The idea relies on a combination of node-based smoothing discrete shear gap method with the higher-order shear deformation plate theory (HSDT) to give a so-called NS-DSG3 element. The higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The numerical examples demonstrated that the present element is free of shear locking and shows high reliability and accuracy compared to other published solutions in the literature.     

Limit theorems in the theory of multipoint boundary-value problems

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system. Kamenets-Podol'sk Pedagogic University, Kamenets-Podol'sk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 519–531, April, 1999.     

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.     

A homogeneous limit methodology and refinements of computationally efficient zigzag theory for homogeneous,laminated composite,and sandwich plates

The Refined Zigzag Theory (RZT) for homogeneous, laminated composite, and sandwich plates is revisited to offer a fresh insight into its fundamental assumptions and practical possibilities. The theory is introduced from a multiscale formalism starting with the inplane displacement field expressed as a superposition of coarse and fine contributions. The coarse displacement field is that of first‐order shear‐deformation theory, whereas the fine displacement field has a piecewise‐linear zigzag distribution through the thickness. The resulting kinematic field provides a more realistic representation of the deformation states of transverse‐shear‐flexible plates than other similar theories. The condition of limiting homogeneity of transverse‐shear properties is proposed and yields four distinct variants of zigzag functions. Analytic solutions for highly heterogeneous sandwich plates undergoing elastostatic deformations are used to identify the best‐performing zigzag functions. Unlike previously used methods, which often result in anomalous conditions and nonphysical solutions, the present theory does not rely on transverse‐shear‐stress equilibrium constraints. For all material systems, there are no requirements for use of transverse‐shear correction factors to yield accurate results. To model homogeneous plates with the full power of zigzag kinematics, infinitesimally small perturbations in the transverse shear properties are derived, thus enabling highly accurate predictions of homogeneous‐plate behavior without the use of shear correction factors. The RZT predictive capabilities to model highly heterogeneous sandwich plates are critically assessed, demonstrating its superior efficiency, accuracy, and a wide range of applicability. This theory, which is derived from the virtual work principle, is well‐suited for developing computationally efficient, C⁰ a continuous function of (x₁,x₂) coordinates whose first‐order derivatives are discontinuous along finite element interfaces and is thus appropriate for the analysis and design of high‐performance load‐bearing aerospace structures. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Homogenization of problems of elasticity theory on composite structures with thin armoring

The paper considers the problems of elasticity theory on a flat slab armored by a periodic thin mesh or in a three-dimensional body armored by a periodic thin box structure. The composite medium depends on two small mutually related geometric parameters; one of them controls the periodicity cell and the other controls the thickness of the armoring structure. It is proved that the homogenization of the indicated problems is classical. In doing so, one applies V. V. Zhikov’s approach (“Zhikov measure approach”) together with the two-scale convergence method. Preliminarily, the paper studies the peculiarities of the two-scale convergence with the variable composite measure and also the Sobolev spaces of elasticity theory with variable composite measure. The obtained compactness principle (an analog of the Rellich theorem) in these spaces made it possible to prove the Hausdor. convergence of the spectrum of the problem studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic circular cylindrical shells

Buckling and vibration analysis of functionally graded magneto-electro-thermo-elastic (FGMETE) circular cylindrical shell are carried out in the present work. The Hamilton principle, higher order shear deformation theory, constitutive equation considering coupling effect between mechanical, electric, magnetic, thermal are considered to derive the equations of motion and distribution of electrical potential, magnetic potential along the thickness direction of FGMETE circular cylindrical shell. The influences of various external loads, such as axis force, temperature difference between the bottom and top surface of shell, surface electric voltage and magnetic voltage, on the buckling response of FGMETE circular cylindrical shell are investigated. The natural frequency obtained by present method is compared with results in open literature and a good agreement is obtained.     

Solution of heat-conduction problems for thin shells and plates by application of a generalized variational approach

By applying a generalized variational approach we construct an approximate system of equations for heat conduction for thin shells and plates and develop a method of solving them. We give the results of numerical studies for a particular problem.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 66–70.     

Free vibration analysis of two-dimensional functionally graded cylindrical shells

In this paper, the free vibration of a two-dimensional functionally graded circular cylindrical shell is analyzed. The equations of motion are based on the Love’s first approximation classical shell theory. The spatial derivatives of the equations of motion and boundary conditions are discretized by the methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ). Two kinds of micromechanics models, viz. Voigt and Mori–Tanaka models are used to describe the material properties. To validate the results, comparisons are made with the solutions for FG cylindrical shells available in the literature. The results of this study show that the natural frequency of the material can be modified in order to meet the expected results through manipulation of the constituent volume fractions. A comprehensive comparison is then drawn between ordinary and 2-D FG cylindrical shells.     

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

Mixed boundary-value problems in the theory of elasticity of thin laminated bodies of variable thickness consisting of anisotropic inhomogeneous materials

Starting from the three-dimensional equations of the theory of thermoelasticity, two-dimensional equations for thin laminated bodies are derived in a general formulation and solved by an asymptotic method. The bodies and layers, consisting of anisotropic and inhomogeneous materials (with respect to two longitudinal coordinates), bounded by arbitrary smooth non-intersecting surfaces, also have variable thicknesses. Recursion formulae are derived for determining the components of the stress tensor and the displacement vector when the kinematic or mixed boundary conditions of the static boundary-value problem of the theory of thermoelasticity are specified on the faces of the body, assuming that the corresponding heat conduction problem is solved. An algorithm for constructing of the analytical solutions of the boundary-value problems formulated is developed using modern computational facilities.     

Numerical solution of boundary value problems in the theory of thin shells and plates with a curvilinear hole

The efficiency with which coordinate systems are chosen and resolvents are constructed is illustrated for the example of linear stress concentration near a curvilinear (elliptical) hole in a circular plate and in a spherical shell. A method is proposed for bunching the grid when solving these problems numerically by a variational-difference method. The rate of convergence of solutions on uniform and nonuniform grids is studied and the results are compared with analytic values. The proposed coordinate transformations are shown to provide a substantial improvement in the rate of convergence of the numerical results on linear (nonlinear) stress concentration near curvilinear holes. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 117–126, 1999.     

Solvability of geometrically nonlinear boundary-value problems for the Timoshenko-type anisotropic shells with rigidly clamped edges

In the nonlinear theory of shells all known existence theorems are based on the Kirchhoff-Love model. We prove a new existence theorem using the displacement model proposed by S. P. Timoshenko.