Calculation of composite structures subjected to follower loads by using a geometrically exact shell element

Based on a 7-parameter shell model, a numerical algorithm has been developed for solving the geometrically nonlinear problem of a multilayer composite shell subjected to a follower pressure and undergoing large displacements and rotations. As unknowns, six displacements of the outer surfaces and addition ally the transverse displacement of midsurface of the shell are chosen. This allows one to use the Green–Lagrange strain tensor, introduced earlier by the authors, which exactly represents arbitrarily large rigid-body displacements of the shell in curvilinear coordinates of a reference surface. A geometrically exact solid shell element is formulated, which permits one to solve the nonlinear deformation problem for thin-walled composite structures subjected to a follower pressure by using a very small number of load steps.     

Analysis of contact stresses in structural joints of composite shell with steel banding

Based on the generalized Timoshenko-type shell theory, a numerical-analytical procedure for determining contact stresses from the interaction between a cylindrical composite shell and rigid bandings is proposed. Specific cases of loading and contact interaction (ideal contact through an adhesive interlayer) are considered. The contact problems are reduced to the solution of a Fredholm integral equation of the second-kind. A calculation analysis is performed. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 109–120, January–February, 2000.     

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 2. Geometrically Nonlinear Statement

Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells subjected to large displacements and arbitrarily large rotations. As the sought-for functions, six displacements and eleven strains of the shell faces are chosen, which allows us to use nonlinear deformation relationships exactly representing arbitrarily large displacements of the shell as a rigid body. The stiffness matrix of a shell element has a proper rank and is calculated based on exact analytical integration. The bilinear element developed does not allow false rigid displacements and is not subjected to the membrane, shear, or Poisson locking phenomenon. The results of solving the well-known test problem on a nonsymmetrically fixed circular arch subjected to a concentrated load and the problem on a locally loaded toroidal multilayer rubber-cord shell are presented.     

The contact problem for a geometrically non-linear Timoshenko-type shell

An algorithm is developed for the numerical solution of the contact problem of an elastic Timoshenko-type shell subjected to arbitrarily large displacements and rotations, using mixed finite-element approximations. It is essential that six displacements of the faces of the shell are chosen as the required functions. This enables one, first, to simplify the formulation of contact problems in the mechanics of thin-walled structures, since functions by means of which the conditions for the non-penetration of the bodies are formulated are chosen as the required functions and, second, to obtain relations for the components of the Green-Lagrange strain tensor in curvilinear, orthogonal coordinates which accurately represent arbitrarily large displacements of a shell as a rigid body.     

Nonlinear dynamic response for functionally graded shallow spherical shell under low velocity impact in thermal environment

Based on Giannakopoulos’s 2-D functionally graded material (FGM) contact model, a modified contact model is put forward to deal with impact problem of the functionally graded shallow spherical shell in thermal environment. The FGM shallow spherical shell, having temperature dependent material property, is subjected to a temperature field uniform over the shell surface but varying along the thickness direction due to steady-state heat conduction. The displacement field and geometrical relations of the FGM shallow spherical shell are established on the basis of Timoshenko–Midlin theory. And the nonlinear motion equations of the FGM shallow spherical shell under low velocity impact in thermal environment are founded in terms of displacement variable functions. Using the orthogonal collocation point method and the Newmark method to discretize the unknown variable functions in space and in time domain, the whole problem is solved by the iterative method. In numerical examples, the contact force and nonlinear dynamic response of the FGM shallow spherical shell under low velocity impact are investigated and effects of temperature field, material and geometrical parameters on contact force and dynamic response of the FGM shallow spherical shell are discussed.     

Contact Problem for a Pneumatic Tire Interacting with a Rigid Foundation

Based on mixed finite-element approximations, a numerical algorithm is developed for solving a geometrically nonlinear contact problem for a prestressed multilayered Timoshenko-type shell undergoing arbitrarily large displacements and rotations. As unknowns, six displacements of faces of the shell are taken, which allows one to use principally new relationships for components of the Green–Lagrange strain tensor in curvilinear orthogonal coordinates, exactly representing arbitrarily large displacements of the shell as a rigid body. As an example, a tire interacting with a rigid foundation is considered.     

处理一类非线性与线性组合结构的混合边界条件方法

本文处理边界与线弹性结构连接的扁壳轴对称大挠度问题,提出了处理此类问题的混合边界条件方法,将组合问题转化为独立结构问题。给出了问题的积分方程组,用摄动法求得了解答。计算了扁球壳与柱壳组合问题的算例。     

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 1. Geometrically Linear Statement

The Hu-Washizu functional is constructed for analyzing prestressed multilayer anisotropic Timoshenko-type shells. As unknown functions, six displacements and eleven strains of the faces of the shells are chosen. Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells. The results of solving the well-known test problem on a cylindrical shell subjected to two opposite point forces and the problem on local loading of a toroidal multilayer rubber-cord shell are presented.     

The steady motion of an elastic cylinder compressed by a fixed shell

The steady mixed problem of the motion of a transversely isotropic elastic circular cylinder, compressed by a finite elastic shell, is solved by the method of piecewise-homogeneous solutions [1]. One of the relations of generalized orthogonality obtained for homogeneous solutions is used. Two special cases are considered: (1) a semi-infinite shell is placed on a movable cylinder with a specified negative allowance the edge of the shell is stress-free, and there is no preloading, and (2) a concentrated encircling load acts on the shell. The solution of the problem of a semi-infinite shell and the system of piecewise-homogeneous solutions are constructed in quadratures by the Wiener-Hopf method. (A similar problem was investigated in [2] in a static formulation. Steady mixed contact problems were investigated previously in [3–10]).     

Axisymmetric contact of anisotropic shells of rotation with rigid and elastic foundations

A class of problems are investigated on determining the stressed-strained state of anisotropic shells of rotation that are in axisymmetric one-sided contact with rigid and elastic surfaces. The shells are under the action of surface and contour loads. For some combinations of these quantities the shell may break away from the surface. To determine the contact zone, the method of successive approximations is utilized. In contrast to most investigations in which the contact zone is first determined, the method proposed makes use of a special quantity characterizing the size of the contact zone. The load on contours is determined from the solution to the problem on the stressed state of the shell and the condition specified on the boundary of the contact zone. Some examples of solving concrete problems are given. Bibliography: 5 titles. Translated fromObchyslyuval’ na ta Prykladna Matematyka, No. 76, 1992, pp 70–74.     

Analytical and numerical method of finite bodies for calculation of cylindrical orthotropic shell with rectangular hole

To solve the boundary-value problem for cylindrical orthotropic shell with sizeable rectangular hole we suggest analytical and numerical method of finite bodies. For determination of the stress state of orthotropic thin-walled cylinder we use a systemof equations that exactly satisfies the equilibrium equations of orthotropic cylindrical shell. Representation of the solutions is divided into basic and self-equilibrium state. For some loads of a shell we build the basic stress state. We obtain a countable number of resolving functions that exactly satisfy the equations of a shell and describe the self-equilibrium stress state. We develop the algorithm of the analytical and numerical solutions of boundary-value problem based on approximation of the stress state of a shell by finite sum of resolving functions and propose a universal way of reduction of all conditions of the contact parts of the enclosure and the boundary conditions to minimize the generalized quadratic forms. We establish criteria under which the construction of approximate solutions coincides with the exact one.     

Contact interaction of a slotted cylindrical shell and a deformable filler with regard for dry friction

The statement of the mixed problem of the friction interaction of a deformable filler with a slotted cylindrical shell is formulated. Using one-dimensional models of a shell and a filler, we obtain an integral equation for calculating contact stresses. On the basis of a numerical solution, the influence of geometric sizes, the number of slots in the shell, and the physical properties of the interacting bodies on the rigidity and strength of the system is investigated.     

Modeling frictional contact in shell structures using variational inequalities

A new variational inequality-based formulation is presented for the large deformation analysis of frictional contact in shell structures. This formulation is based on a seven-parameter continuum shell model which accounts for the normal stress and strain through the shell thickness and accommodates double-sided shell contact. The kinematic contact conditions are expressed accurately in terms of the physical contacting surfaces of the shell. Furthermore, Lagrange multipliers are used to ensure that the kinematic contact constraints are accurately satisfied and that the solution is free from user-defined parameters. Large deformations and rotations are accounted for by invoking the Piola–Kirchhoff stress and the Green–Lagrange strain measures. Three examples involving a strip friction test, ring contact and sheet compression tests are used to verify the developed formulations and algorithms, and test various aspects of the solution technique. Photoelastic analysis of the ring compression example is performed for experimental verification.     

Optimization of cylindrical composite shells with regard to their critical mode of imperfections

This study deals with the optimum design of composite shells under external pressure with material strength and loss of stability according to the critical mode of imperfections taken as the failure criterion. The problem of optimum design is solved and the critical mode is obtained by nonlinear optimum programming for which the geometric and initial imperfection parameters are treated as variables. Numerical results are obtained for a cylindrical composite shell supported freely at its ends. The effect of shear forces between layers on the load-carrying capacity of the shell is also investigated.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 613–620, September–October, 1998.     

Optimization of a cylindrical composite shell with a viscoelastic core in axial compression

The problem of the optimal design of a composite shell in creep is formulated. The progressive buckling of a cross-wound reinforced cylindrical shell supported on a viscoelastic core is considered as a particular case. The reinforcement structure and shell thickness corresponding to minimum weight for a given load and service life are found.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 442–446, May–June, 1975.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

Optimization of cyclindrical shells of fiber-reinforced composite materials

Problems of optimizing nonelastic circular shells are considered. The material of the shells is assumed to be a fiber-reinforced composite with fibers unidirectionally embedded in a relatively less stiff but ductile metallic matrix so that the material has the yield surface suggested by Lance and Robinson. The shell is subjected to an impulsive loading of short-time periods generating initial kinetic energy. During plastic deformation of the shell the initial kinetic energy is transformed into the plastic strain energy. The shell thickness is assumed to be piecewise constant. Various thicknesses and coordinates of the rings, where the thickness has jumps, are preliminarily unspecified. We look for a shell design for which the maximum residual deflection has a minimum value for the total weight given. The alternative problem of minimizing the shell weight for the maximum deflection given is also studied.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Octobe, 1995.Tartu University, Estonia. Published in Mekhanika Kompozitnykh Materialov, No. 1, pp. 65–71, January–February, 1996.     

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.