Multilayered Anisotropic Shells Research Articles

Theory of the multilayer thin anisotropic shells, based on the asymptotic analysis of the general equations for the elasticity theory

A new theory of thin multilayer anisotropic elastic shells is proposed, based on the application of the asymptotic analysis over small geometric parameter to the general 3-dimensional equations of the elasticity theory for curvilinear coordinates. In deriving the equations of the shell theory no assumptions are made concerning to the distribution of displacements, deformations, or displacements trough thickness. Recurrent consequences of local problems of the elasticity theory for shells are formulated and an analytical solutions are obtained. The averaged equations of the asymptotic theory of shells that are of the same type as the classical equations of the Kirchhoff-Love theory of shells are derived. It is shown that the asymptotic theory makes it possible to obtain in explicit form the distributions of all 6 components of the stress tensor over the thickness of the shell. An example of the calculation of a cylindrical shell under axisymmetric bending is considered. The case of monoclinic shell layers, which have no more than 13 independent elastic constants, is considered. An algorithm is proposed for obtaining explicit analytical equations for calculating the distribution of components of the total stress tensor over the cylindrical shell. The effect of the geometric d imensions of the shell on the character of the distribution of displacements and stresses in the shell is analyzed.

Coupled Thermoelasticity Problem for Multilayer Composite Shells of Revolution. II. Applied Problems

We deduce a system of nonclassical nonlinear differential equations of coupled thermoelasticity for multilayer composite anisotropic shells of revolution in a coordinate system connected with the lines of curvature of the reference surface. The constructed nonclassical model of deformation of a multilayer shell and the nonlinear model of distribution of heat fluxes over the thickness of the shell enable us to take into account the transverse shear strains and guarantee the validity of the conditions of thermal and mechanical conjugation of the layers and the conditions of thermomechanical loading on the faces of the shell. We deduce the linearized differential equations of the axisymmetric coupled problem of thermoelasticity for a conic reinforced multilayer shell and solve the quasistatic problem of thermoelasticity for a two-layer cylindrical shell cross-reinforced in the direction of helical lines. By using the structural approach to the formulation of the criteria of strength for composite materials, we determine the loads of the onset of fracture of the binder and reinforcing elements of a two-layer metal-composite cylindrical shell.

Application of nonclassical models of shell theory to study mechanical parameters of multilayer nanotubes

Stress-strain state of multilayer anisotropic cylindrical shells under a local pressure is studied. Such a problem may model the bending of an asbestos nanotube under the action of a research probe. In earlier works, these authors showed that the application of classical shell theories yields results far from experimental data. More accurate results are obtained by taking into account additional factors, such as the change of the transverse displacement magnitude (according to the Timoshenko-Reissner theory) or the layered structure of asbestos and cylindrical anisotropy (according to the Rodinova-Titaev-Chernykh theory). In the present paper, yet another shell theory, the Palii-Spiro theory, is applied to solve the problem; this theory was developed for shall of average thickness and is based on the following assumptions: (a) the rectilinear fibers of the shell perpendicular to its middle surface before deformation remain rectilinear after deformation; (b) the cosine of the angle between the shell of such fibers and the middle surface of the deformed shell equals the averaged angle of the transverse displacement. Deformation field are studied with the use of nonclassical (the Rodinova-Titaev-Chernykh and Palii-Spiro) shell theories; a comparison with results obtained for three-dimensional models with the use of the Ansys 11 package is performed.

Non-linear strain–displacement equations exactly representing large rigid-body motions. Part I Timoshenko–Mindlin shell theory

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko–Mindlin (TM) shell elements is considered. This consideration requires the development of the strain–displacement relationships of the finite deformation TM shell theory with regard to their consistency with the large rigid-body motions. For this purpose a refined TM theory of multilayered anisotropic shells undergoing finite deformations is elaborated. The transverse shear and transverse normal deformation response and bending–extension coupling are included. The fundamental unknowns consist of six displacements and 11 strains of the bottom and top surfaces of the shell, and 11 stress resultants. On the basis of this theory the simple and efficient mixed models are developed by using the incremental total Lagrangian formulation in conjunction with the Newton–Raphson method. The elemental arrays are derived applying the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed four-node shell elements and to compare their performance with other non-linear finite elements reported in the literature.

Efficient mixed Timoshenko–Mindlin shell elements

AbstractThe precise representation of rigid body motions in the displacement patterns of curved Timoshenko–Mindlin (TM) shell elements is considered. This consideration requires the development of the strain–displacement relationships of the TM shell theory with regard to their consistency with the rigid body motions. For this purpose a refined TM theory of multilayered anisotropic shells is elaborated. The effects of transverse shear deformation and bending‐extension coupling are included. The fundamental unknowns consist of five displacements and eight strains of the face surfaces of the shell, and eight stress resultants. On the basis of this theory the simple and efficient mixed models are developed. The elemental arrays are derived using the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed 4‐node shell elements and to compare their performance with other finite elements reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.

Implications of damaged interfaces and of other non-classical effects on the load carrying capacity of multilayered composite shallow shells

A dynamic geometrically non-linear third-order theory of multilayered anisotropic and sandwich shallow shells featuring damaged interfaces is presented. The theory (i) accounts for an arbitrary distribution of the tangential displacements through the shell thickness, (ii) fulfils a priori the continuity conditions on the transverse shear stresses at the interfaces between the layers and their vanishing on the top and bottom surfaces, (iii) allows for jumps in the in-plane displacements when interlayer slips are present, (iv) incorporates the initial geometric imperfections. The pertinent equations of motion and variationally consistent boundary conditions are derived by means of the virtual work principle. Kirchhoff-Love's and first-order shallow shell models are obtained, among others, as special cases. A numerical assessment of the relative merits of the developed shallow shell model, of the effects of interface damage, shallowness parameter and degree of movability of the edge on the non-linear response under transverse loading and a uniform temperature change of sandwich and five-layered, symmetric cross-ply cylindrical panels is presented. The developed theory can contribute to a better understanding of the load-carrying capacity and failure of shell structures exhibiting damaged interfaces.

Refined Global Approximation Theory of Multilayered Plates and Shells

A refined global approximation theory of thin multilayered anisotropic shells is developed. The effects of the laminated material response, transverse shear, and transverse normal strains are included. The material of each layer of the shell is assumed to be linearly elastic, anisotropic, homogeneous, or fiber reinforced. As unknown functions the tangential displacements of the face surfaces and transverse displacements of the face or middle surfaces of the shell are chosen. It is an important feature of the proposed theory. This fact simplifies, for example, an analysis of the contact problems and allows to elaborate universal numerical algorithms. An exact solution for the problem of the bending of homogeneous isotropic rectangular plates subjected to a sinu- soidal load has been found. Numerical solutions for the problem of the bending of multilayered composite plates and cylindrical shells have been also obtained. The influence of anisotropy, and transverse shear and transverse normal deformation response of the stress state of the shell is examined.

Elastic Buckling of Multilayered Anisotropic Conical Shells

On the basis of 3D elasticity, asymptotic solutions for buckling analysis of multilayered anisotropic conical shells under axial compression are presented. By means of proper nondimensionalization, asymptotic expansion, and successive integration, the classical shell theory is derived as a first-order approximation to the 3D theory. Because the governing equations for various orders consist of partial differential equations with variable coefficients, the use of analytical techniques is restricted. The method of differential quadrature is adopted in the present study. The modifications of the buckling loads and associated buckling modes can be determined in a consistent and hierarchic manner by considering the solvability and normalization conditions for various orders. The critical loads of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.

GENERAL EQUATIONS OF ANISOTROPIC PLATES AND SHELLS INCLUDING TRANSVERSE SHEAR DEFORMATIONS, ROTARY INERTIA AND INITIAL CURVATURE EFFECTS

The present work deals with a generalization of geometrically linear shear deformation theory for multilayered anisotropic shells of general shape. No assumptions are made other than to neglect the transverse normal strain. The results, which include the effects of shear deformations and rotary inertia as well as initial curvature (included in the stress resultants and assumed transverse shear stresses) are deduced by application of the virtual work principle, with displacements and transverse shear as independent variables. These equations are applied to different shell geometries, such as revolution, cylindrical, spherical and conical shells as well as rectangular and circular plates.

Non-linear analysis of multilayered shells under initial stress

A refined non-linear first-order discrete-layer theory of initially stressed composite shells is developed. The material of each layer of the shell is assumed to be linearly elastic, anisotropic, homogeneous or fiber-reinforced. The transverse shear and transverse normal effects are included. It is also assumed, that the well-known Novozhilov's three-dimensional partially non-linear strain–displacement relationships are valid. As unknown functions the tangential and transverse displacements of external surfaces of the shell and layer interfaces are selected. A computational model for solving the non-linear problems of the axisymmetric deformation of initially stressed multilayered anisotropic shells of revolution is presented. The joint influence of anisotropy, initially stressed state response, geometrical non-linearity and laminated material response on the stress state of the shell is examined. Results show, that neglecting the effects of anisotropy and geometrical non-linearity leads to an incorrect description of the stress field in cross-ply toroidal shells made of cord-rubber composites.

Analysis of mid-surface symmetric layered anisotropic shells by the two-surface truss model

The present article considers the generalization of the recently developed truss-model computational scheme, formerly restricted to the calculation of single-layered isotropic shells, so as to encompass mid-surface symmetric multi-layered anisotropic shells. Following a review of the relevant constitutive relations of layered anisotropic shells, the application of the truss model to such structures is then investigated; in particular, it is shown how the simple scheme for the imposition of boundary conditions involving stress and strain measures can be extended in order to cater for anisotropic material properties of the type presently considered. Finally, the truss model is applied, for purposes of illustration, to a layered orthotropic shell in the form of a paraboloid of revolution with square plan subjected to two cases of loading.

Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression

Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression

Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear

The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic (nonorthotropic). A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unknown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfaces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered. It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thickness without an expensive software based on the 3D anisotropic theory of elasticity.

HEAT CONDUCTION EQUATIONS FOR MULTILAYER ANISOTROPIC SHELLS

We derive nonstatic two-dimensional differential equations of heat conduction in multilayered anisotropic shells with corresponding boundary conditions assuming a cubic and linear temperature distribution in the thickness of each shell. The nonideal thermal contact (thermoresistance) between layers is considered. As an example we study the problem of temperature field distribution in a two-layered cylindrical shell of finite length under local heating.

Theory of Multilayered Anisotropic Shells Based on an Asymptotic Variational Formulation

ABSTRACTA general theory for multilayered anisotropic elastic shells is developed in an asymptotic variational framework of 3-D elasticity. The generic shell continuum considered is heterogeneous through the thickness. It is shown that the classical laminated shell theory based on Love's assumption arises naturally as the first-order approximation to the 3-D theory. Higher-order corrections can be determined by solving the 2-D shell equations hierarchically. The associated edge conditions at each level of approximation are derived. Various types of shells such as shells of revolution, conical shells, spherical shells, circular cylindrical shells can be treated within the context.

THERMOELASTICITY OF MULTILAYER HIGHLY ANISOTROPIC SHELLS

The complete system of basic equations of linear refined dynamic theory of thermal stresses in multilayer shells is developed with allowance for transverse anisotropy of the material for each layer. It is shown that a number of general theorems of linear elasticity theory have their analogues in refined theory of thermoelastic multilayer anisotropic shells. These theorems allow one to construct methods for solving the corresponding boundary value problems for these shells.

Discrete-layer models for multilayered shells accounting for interlayer continuity

Based on adiscrete-layer approach, in a recent series of papers, the first author has developed the equations governing the elastodynamic behaviour of moderately thick multilayered anisotropic plates by making use of a displacement field which allows a non-linear variation of the in-plane displacements through the laminate thickness and fulfilsa priori the static and geometric continuity conditions at the interfaces between two adjacent layers. Based on this approach, in the present paper we derive the equations of motion and variationally consistent boundary conditions of moderately multilayered anisotropic shells. To show the accuracy and reliability of the proposed approach, closed-form solutions are given and compared with results from three-dimensional elasticity and other approximate bi-dimensional models with and without continuous interlayer stresses. Based on this numerical investigation, the proposed approach appears to work very well.

Nonstationary heat transfer in multilayer anisotropic shells of revolution

On the basis of the variational principle of “least scattering” in nonequilibrium thermodynamics we obtain an approximate equation and boundary conditions describing the process of nonstationary heat conduction in multilayer anisotropic shells of arbitrary shape. The structure of the equation obtained is independent of the number of layers and the types of boundary conditions. We give an example of the computation of a one-dimensional nonstationary temperature field for an infinite hollow homogeneous cylinder.

Thermally forced loading of multilayered anisotropic shells

Thermally forced loading of multilayered anisotropic shells

Nonaxisymmetric deformation of tangentially loaded multilayered anisotropic shells of revolution

Nonaxisymmetric deformation of tangentially loaded multilayered anisotropic shells of revolution