Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition

We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: (a) the stretch field is found by pure algebra, (b) the rotation field is obtained by solving a system of first-order PDEs, and (c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.     

Determination of the deformed position of a thin shell from surface strains and height function

This paper focuses on the following problem: given the strain tensor of a deformed reference surface of a thin shell and the distances of the points on this surface from some arbitrarily fixed reference plane (the so-called height function) find the position of this reference surface. Two alternative procedures supplying the solution are developed. The first one follows from the ideas developed by Darboux (Leçons sur la Théorie Général des Surfaces, Troisiéme Partie, Gauthier-Villars, Paris, 1894), whereas the second one is based on the polar decomposition theorem and techniques developed in continuum mechanics. These procedures are purely kinematic, valid for arbitrary surface geometry and for unrestricted surface strains. Szwabowicz (Deformable surfaces and almost inextensional deflections of thin shells, Habilitation Thesis, 1999) proposed a relatively simple non-linear boundary-value problem (BVP) for thin elastic shells, which was expressed in three surface strains and the height function as basic independent field variables. The results of this paper suggest that this approach to the non-linear problems of thin shells may be an attractive alternative to other BVPs developed in the literature.     

On modified displacement version of the non-linear theory of thin shells

We discuss the non-linear theory of thin shells expressed in terms of displacements of the shell reference surface as the only independent field variables. The formulation is based on the principle of virtual work postulated for the reference surface. In our approach: (1) the vector equilibrium equations are represented through components in the deformed contravariant surface base, and using the compatibility conditions the resulting tangential equilibrium equations are additionally simplified, (2) at the shell boundary the new scalar function of displacement derivatives is defined and new sets of four work-conjugate static and geometric boundary conditions are derived, as well as (3) for prescribed shell geometry all non-linear shell relations are generated automatically by two packages set up in Mathematica. The displacement boundary value problem and the associated homogeneous shell buckling problem are generated exactly without using any additional approximations following from errors of the constitutive equations. Both problems are extremely complex and available only in the computer memory. Such an approach allows us to account also for those a few supposedly small terms which may be critical for finding the correct buckling load of shells sensitive to imperfections. This approach is used in the accompanying paper by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On refined analysis of bifurcation buckling for the axially compressed circular cylinder. International Journal of Solids and Structures, 46, 3111–3123.] to perform the refined numerical analysis of bifurcation buckling for the axially compressed circular cylinder.     

Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.     

Non-linear dynamic analysis of symmetric and antisymmetric cross-ply laminated orthotropic thin shells

In this paper, the governing equations for non-linear free vibration of truncated, thin, laminated, orthotropic conical shells using the theory of large deformations with the Karman-Donnell-type of kinematic nonlinearity are derived. Applying superposition principle and Galerkin’s method, these equations are reduced to a time dependent non-linear differential equation. The frequency-amplitude relationship for the laminated orthotropic thin truncated conical shell is obtained using the method of weighted residuals. In the particular case, we can obtain the similar relationships for the single-layer and laminated orthotropic cylindrical shells, also. The influence played by geometrical parameters of the conical shell and physical parameters of the laminate (i.e. material properties, staking sequences and number of layers) on the non-linear vibration behavior of the conical shell is examined. It is noticed that the non-linear vibration of shells is highly dependent on laminate characteristics and, from these observations, it is concluded that specific configurations of laminates should be designed for each kind of application. Present results are compared with available data for special cases.     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

Non-linear modal equations for thin elastic shells

In this paper we derive non-linear modal equations for thin elastic shells of arbitrary geometry. Geometric non-linearities are accounted for by utilizing the strain-displacement relations of the Sanders-Koiter non-linear shell theory. Arbitrary initial imperfections are accounted for and the shell thickness is free to vary within the limits of thin shell theory. The derivation gives the coefficients of the modal equations as integral expressions over the surface of the shell. The resulting equations are well-suited for practical applications. Weighting factors are introduced to allow for reduction of our results to the Love shell theory and to the Donnell approximation. The equations are specialized for a finite simply supported circular cylinder and numerical results are compared to those previously published in the literature.     

Solution describing the natural vibrations of rectangular shallow shells with varying thickness

Natural vibrations of shallow cylindrical shells with rectangular plan and varying thickness are studied using a spline-approximation method developed previously. Computation is carried out for different types of boundary conditions. The effect of the curvature of the midsurface on the natural frequencies is examined. The natural frequencies of shells with constant and varying thickness are compared __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 89–98, April 2007.     

On refined analysis of bifurcation buckling for the axially compressed circular cylinder

We present extensive numerical results of bifurcation buckling analysis of the axially compressed circular cylinder. The analysis is based on the modified displacement version of the non-linear theory of thin elastic shells developed by Opoka and Pietraszkiewicz [Opoka, S., Pietraszkiewicz, W., 2009. On modified displacement version of the non-linear theory of thin shells. International Journal of Solids and Structures, 46, 3103–3110.]. To solve the buckling problem we apply the separation of variables and expansion of all fields into Fourier series in circumferential direction, with subsequent accurate calculations of eigenvalues of determinants of corresponding 8 × 8 complicated matrices. The numerical analysis of the buckling load is performed for the cylinders with length-to-diameter ratio in the range (0.05, 60), with eight sets of incremental work-conjugate boundary conditions analogous to those used in the literature and partly summarized in the book by Yamaki [Yamaki, N., 1984. Elastic Stability of Circular Cylindrical Shells. Elsevier, Amsterdam], and additionally with six sets of boundary conditions not discussed in the literature yet. The results allow us to formulate several important conclusions, such as: (a) omission in the non-linear BVP small terms of the order of error introduced by the error of constitutive equations leads to overestimated buckling loads for long cylinders with clamped boundaries; (b) for some relaxed boundary conditions the buckling load decreases for short cylinders with decrease of the cylinder length; (c) the results for additional six sets of boundary conditions reveal existence of several new cases, in which by relaxing geometric boundary conditions the buckling load falls down to about one half of the classical value in a wide range of the cylinder length-to-diameter ratios.     

Nonlinear deformation and buckling of elastic inhomogeneous shells under thermomechanical loads

The paper outlines the fundamentals of the method of solving static problems of geometrically nonlinear deformation, buckling, and postbuckling behavior of thin thermoelastic inhomogeneous shells with complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure under complex thermomechanical loading. The method is based on the geometrically nonlinear equations of three-dimensional thermoelasticity and the moment finite-element scheme. The method is justified numerically. Results of practical importance are obtained in analyzing poorely studied classes of inhomogeneous shells. These results provide an insight into the nonlinear deformation and buckling of shells under various combinations of thermomechanical loads     

Vibrations of shallow noncircular membranes with initial stresses

The basic relations for a geometrically nonlinear (quadratic approximation) theory of thin elastic membranes are obtained. The relations are used to develop a variational method for studying free vibrations of initially flat membranes bounded by a stationary piecewise-smooth contour. The membrane is deformed by uniform pressure. Numerical results are given for different types of vibrations of rectangular and elliptical shells. N. V. Gogol' Pedagogical Institute, Nezhin: S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 78–86, August 1999.     

Effects of imperfections on the buckling response of compression-loaded composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling and postbuckling response of three unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells with different orthotropic and quasi-isotropic shell-wall laminates are presented. The results identify the effects of traditional and non-traditional initial imperfections on the non-linear response and buckling loads of the shells. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The non-traditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, non-uniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. A high-fidelity non-linear shell analysis procedure that accurately accounts for the effects of these traditional and non-traditional imperfections on the non-linear responses and buckling loads of the shells is described. The analysis procedure includes a non-linear static analysis that predicts stable response characteristics of the shells and a non-linear transient analysis that predicts unstable response characteristics.     

Solution of three-dimensional problems for thick elastic shells by the method of reference surfaces

A new method for solving the elasticity problem for thick and thin shells is proposed. The method is based on the concept of reference surfaces inside the shell. According to this method, N reference surfaces are introduced in the body of the shell so that they are parallel to the midsurface and located at the Chebyshev polynomial nodes, which permits taking the displacement vectors u ¹, u ², …, u ^N of these surfaces for the desired functions. This choice of the desired functions allows one to represent the resolving equations of the proposed theory of higher-order shells in a sufficiently concise form and obtain deformation relations which permit describing the shell displacements as motions of a rigid body.     

Axisymmetric static and dynamic buckling of hollow microspheres

Syntactic foams are particulate composites that are obtained by dispersing thin hollow inclusions in a matrix material. The wide spectrum of applications of these composites in naval and aerospace structures has fostered a multitude of theoretical, numerical, and experimental studies on the mechanical behavior of syntactic foams and their constituents. In this work, we study static and dynamic axisymmetric buckling of single hollow spherical particles modeled as non-linear thin shells. Specifically, we compare theoretical predictions obtained by using Donnell, Sanders–Koiter, and Teng–Hong non-linear shell theories. The equations of motion of the particle are obtained from Hamilton׳s principle, and the Galerkin method is used to formulate a tractable non-linear system of coupled ordinary differential equations. An iterative solution procedure based on the modified Newton–Raphson method is developed to estimate the critical static load of the microballoon, and alternative methodologies of reduced complexity are further discussed. For dynamic buckling analysis, a Newmark-type integration scheme is integrated with the modified Newton–Raphson method to evaluate the transient response of the shell. Results are specialized to glass particles, and a parametric study is conducted to investigate the effect of microballoon wall thickness on the predictions of the selected non-linear shell theories. Comparison with finite element predictions demonstrates that Sanders–Koiter theory provides accurate estimates of the static critical load for a wide set of particle wall thicknesses. On the other hand, Donnell and Teng–Hong theories should be considered valid only for very thin particles, with the latter theory generally providing better agreement with finite element findings due to its more complete kinematics. In this context, we also demonstrate that a full non-linear analysis is required when considering thicker shells, while simplified treatment can be utilized for thin particles. For dynamic buckling, we confirm the accuracy of Sanders–Koiter theory for all the considered particle thicknesses and of Teng–Hong and Donnell theories for very thin particles.     

Stability of Orthotropic Corrugated Cylindrical Shells

A technique for stability analysis of cylindrical shells with a corrugated midsurface is proposed. The wave crests are directed along the generatrix. The relations of shell theory include terms of higher order of smallness than those in the Mushtari–Donnell–Vlasov theory. The problem is solved using a variational equation. The Lamé parameter and curvature radius are variable and approximated by a discrete Fourier transform. The critical load and buckling mode are determined in solving an infinite system of equations for the coefficients of expansion of the resolving functions into trigonometric series. The solution accuracy increases owing to the presence of an aggregate of independent subsystems. Singularities in the buckling modes of corrugated shells corresponding to the minimum critical loads are determined. The basic, practically important conclusion is that both isotropic and orthotropic shells with sinusoidal corrugation are efficient only when their length, which depends on the waveformation parameters and the geometric and mechanical characteristics, is small     

Free vibrations of shallow nonthin shells with variable thickness and rectangular planform

The free vibrations of shallow orthotropic shells with variable thickness and rectangular planform are studied. The shear strains are taken into account. The spline approximation of unknown functions is used. The natural frequencies are calculated for different boundary conditions. The dependence of the natural frequencies on the curvature of the midsurface is examined. The natural frequencies of shells with constant and variable thickness are compared     

Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

This paper discusses the derivation of discrete low-dimensional models for the non-linear vibration analysis of thin shells. In order to understand the peculiarities inherent to this class of structural problems, the non-linear vibrations and dynamic stability of a circular cylindrical shell subjected to dynamic axial loads are analyzed. This choice is based on the fact that cylindrical shells exhibit a highly non-linear behavior under both static and dynamic axial loads. Geometric non-linearities due to finite-amplitude shell motions are considered by using Donnell’s nonlinear shallow shell theory. A perturbation procedure, validated in previous studies, is used to derive a general expression for the non-linear vibration modes and the discretized equations of motion are obtained by the Galerkin method. The responses of several low-dimensional models are compared. These are used to study the influence of the modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses of the shell. It is shown that rather low-dimensional and properly selected models can describe with good accuracy the response of the shell up to very large vibration amplitudes.     

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses

The available accurate shell theories satisfy the interlaminar transverse stress continuity conditions based on linear strain-displacement relations. Furthermore, in majority of these theories, either influence of the transverse normal stress and strain or the transverse flexibility of the shell has been ignored. These effects remarkably influence the non-linear behavior of the shells especially in the postbuckling region. Furthermore, majority of the buckling analyses performed so far for the laminated composite and sandwich shells have been restricted to linear, static analysis of the perfect shells. Moreover, almost all the available shell theories have employed the Love-Timoshenko assumption, which may lead to remarkable errors for thick and relatively thick shells. In the present paper, a novel three-dimensional high-order global-local theory that satisfies all the kinematic and the interlaminar stress continuity conditions at the layer interfaces is developed for imperfect cylindrical shells subjected to thermo-mechanical loads.In comparison with the layerwise, mixed, and available global-local theories, the present theory has the advantages of: (1) suitability for non-linear analyses, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions, considering the transverse flexibility, and releasing the Love-Timoshenko assumption, (3) less required computational time due to using the global-local technique and matrix formulations, and (4) capability of investigating the local phenomena. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.