Stability of multilayer orthotropic shells

The variation-difference method is a convenient numerical method for shells of complex forms. It is enough when only cinematic boundary conditions are satisfied because the method is based on the principle of Lagrange. Another advantage of the variation-difference method is the better opportunity to create computer programs based on it. For shell analysis in orthogonal coordinate system as well as for shell analysis in principal curvatures the system of equations describing stress-strain state can be simplified. In this paper the difference between analysis in orthogonal coordinate system and analysis in principal curvatures of the surface is considered. The main distinction of the analysis of shells in orthogonal curvilinear coordinate system is the necessity of determination of components which include curvature of torsion of coordinate lines. The addition of these components in the equations of the theory of shells for the coordinate system in principal curvatures gives possibility to analyze shells in common orthogonal coordinate system. In this article shell analysis in orthogonal coordinate system is applied to shells based on normal cyclic surfaces.

Based on the discrete-structural theory of thin plates and shells, a variant of the equations of buckling stability, containing a parameter of critical loading, is put forward for the thin-walled elements of a layered structure with a weakened interfacial contact. It is assumed that the transverse shear and compression stresses are equal on the interfaces. Elastic slippage is allowed over the interfaces between adjacent layers. The stability equations include the components of geometrically nonlinear moment subcritical buckling conditions for the compressed thin-walled elements. The buckling of two-layer transversely isotropic plates and cylinders under axial compression is investigated numerically and experimentally. It is found that variations in the kinematic and static contact conditions on the interfaces of layered thin-walled structural members greatly affect the magnitude of critical stresses. In solving test problems, a comparative analysis of the results of stability calculations for anisotropic plates and shells is performed with account of both perfect and weakened contacts between adjacent layers. It is found that the model variant suggested adequately reflects the behavior of layered thin-walled structural elements in calculating their buckling stability.

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.

Compared with 8-node and 4-node isoparametric element, the 9-node isoparametric element is more flexible and higher rate of convergence. Based on state-space techniques, the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system is established. Finally, the example shows the practicability and accuracy of the 9-node isoparametric element of piezothermoelastic material under orthogonal curvilinear coordinate system.

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This review article is devoted to the use of the Reissner Mixed Variational Theorem (RMVT) forward two-dimensional modeling of flat and curved, multilayer structures. A thorough review of the literature involving the use in the modeling of multilayered plates and shells using RMVT is also presented. In the first part, the paper overviews relevant key points that should be taken into account for an accurate description of strain and stress fields in multilayered plate and shell analysis. It is then shown that RMVT has been originated in view of the fulfillment of such key points, herein referred to as C0-Requirements (zig-zag form of the displacement fields in the thickness direction and continuity of transverse normal and shear stresses at each layer interface). Classical variational statements are used to introduce Reissner’s Theorem. In the second part, the paper presents various ways in which RMVT can be used to develop plate and shell theories in a systematic manner. The so called layer-wise and equivalent single layer variable description are considered. Both strong and weak (finite element) forms of governing equations have been derived. A Weak Form of Hooke’s Law (WFHL), is also discussed as an idea to eliminate transverse stress variables leading to standard classical models with only displacement unknowns. Two appendices display details of governing equations related to multilayered doubly curved shells and to finite element matrices of multilayered plates. A third part reviews the works that have appeared in literature which make use of RMVT. Mainly papers on multilayered plate and shell modelings have been addressed. The final part of the paper is devoted to giving an overview with selected results of numerical performance that can be acquired by RMVT applications; extensive comparison to elasticity solutions and to other significant analyses, based on classical and refined approaches, are given. It is concluded that Reissner’s Mixed Theorem should be considered as a natural tool for multilayered structure analyses; it plays a similar role to that of the Principle of Virtual Displacement in the analysis of isotropic single-layer structures. This review article includes 119 references.

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

In Part I, an analytical examination of electromagnetic boundary conditions at the interface between two different media is presented in an orthogonal curvilinear coordinate system which conforms locally and instantaneously to the interface. Subsequently, an original natural coordinate system is introduced to resolve the uncertainty in the choice of an orthogonal curvilinear coordinate system. This natural coordinate system is also appeared to reduce the number of the nonvanishing discontinuity relations, and to determine local and instantaneous natural aspects of the e.m. boundary conditions problem.

The stability problem is solved for cylindrical shells made of a laminated composite whose directions of layer reinforcement are not aligned with coordinate axes of the shell midsurface. Each layer of the composite is modeled by an anisotropic material with one plane of symmetry. The resolving functions of the mixed variant of shell theory are approximated by trigonometric series satisfying boundary conditions. The stability of the shells under axial compression, external pressure, and torsion is investigated. A comparison with calculation data obtained within the framework of an orthotropic body model is carried out. It is shown that this model leads to considerably erroneous critical loads for some structures of the composites.

Preface. Introduction. Vectors and Tensors. Kinematics of Fluid Motion. Laws of Fluid Dynamics. Dynamics of Discontinuity Surfaces. Velocity Representations and Associated Theorems. Vorticity Transport Theorems. Pressure Theorems. Two-Dimensional Potential Flows. Forces on Bodies in Two-Dimensional Flows. Two-Dimensional Flows with Vorticity. Three-Dimensional Potential Flows. Axisymmetric Vortex Flows. Vortex Tubes. Interfacial Wave Motion. Stability of Fluid Flows. Appendix A: Common Expressions in Orthogonal Curvilinear Coordinate Systems. Index.

This paper investigates the globally dynamic stabilizing effects of the geometry of the domain at which the flow locates and of the geometrical structure of the finite-energy solutions to three-dimensional (3D) incompressible Navier-Stokes and Euler systems. We have established the global existence and uniqueness of the smooth solution to the Cauchy problem for 3D incompressible Navier-Stokes and Euler equations for a class of smooth large initial data in orthogonal curvilinear coordinate systems, which has no smallness assumption on initial data for the Cartesian coordinate system. Moreover, we have also established the existence, uniqueness, and exponential decay rate of the global strong solution to the initial-boundary-value problem for 3D Navier-Stokes equations for a class of smooth large initial data and a class of special bounded domain in orthogonal curvilinear coordinate systems. Regarding application, some new classes of large-amplitude global smooth solutions with very complex geometric structures, and that are either non-axisymmetric or non-helical in $ \mathbb{R}^3 $, have been established for 3D incompressible Navier-Stokes and Euler equations. This is the first result on the global existence and uniqueness of the large smooth solution to 3D incompressible Navier-Stokes and Euler equations in general orthogonal curvilinear coordinate systems.

An incremental variational formulation is presented for large deflection analysis of elastic‐plastic continuum by considering the energy consistency of a nonlinear multivariable discrete system based upon incompatible trial functions. The functional is given in orthogonal curvilinear coordinate system and the initial stress procedure has been introduced. Equilibrium imbalance correction is included to prevent drifting of the solution during the incremental process. As an initial application of the suggested formulation, a hybrid stress laminated shell element, in which an energy constraint is introduced to improve the element performance, has been developed to analyze the through‐thickness plastic behavior and the progressive cracking in RC plates and shells. It is of quadrilateral shape with only four corner nodes and can be used for both shallow and deep laminated shells with satisfactory numerical characteristics. Example problems are also included to illustrate the element performance.

The state equations for orthotropic, doubly curved shells are established in an orthogonal curvilinear coordinate system. Simplifying hypotheses about displacement models or stress distribution that were assumed in early work are not introduced in this paper. The continuity conditions of stresses and displacements at the interfaces between the contacting layers are satisfied, and analytical solutions for statics and dynamics of the simply supported, doubly curved shells with ortho‐tropic layers are presented by means of the Cayley‐Hamilton theorem. For any number of layers considered, the problem reduces to solving a set of linear algebraic equations with three unknowns. A unified analytical solution is given for thin, moderately thick, and thick laminated shells. The solution satisfied all the equations of elasticity, and each of nine elastic constants is considered. Arbitrary precision of the desired order can be obtained. Numerical results are given to compare with solutions available in the literature.

A pedagogically pleasing and mnemonically useful method for expressing the velocity and acceleration vectors in orthogonal systems of coordinates is shown. The method relies on expressing the time derivatives of the unit vectors in matrix form. It is also noted that the spatial derivatives of the unit vectors obey a similar rule.

Post buckling analysis of smart laminated doubly curved shells