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.     

Variational theory of thin shells and panels

Summary A comprehensive theory is developed for elastic thin shells and panels of arbitrary shape and load conditions, including the effect of large transverse displacements, non uniform temperature distributions and initial imperfections. A single variational principle is derived, that comprehends both equilibrium and compatibility conditions. In the Appendix an example of the application of such a principle is carried out.     

On shear correction factors in the non-linear theory of elastic shells

Theoretical values of two correction factors α_s = 5/6 and α_t = 7/10 are established for the respective transverse shear stress resultants and stress couples within the general, dynamically and kinematically exact, six-field theory of elastic shells. These values do not depend on the shell material symmetry, geometry of the base surface, the shell thickness, or any kind of kinematic and/or dynamic constraints. The analysis is based on the complementary energy density following from the transverse shear stresses acting only on the shell cross section. The appropriate quadratic and cubic distributions of the stresses across the thickness allow one to derive the consistent constitutive equations for the transverse shear stress resultants and stress couples with α_s and α_t as the respective correction factors. Four numerical examples of highly non-linear shell structures illustrate the influence of different values of α_s and α_t on the results. In particular, some influence of α_t is noticed on the placement of bifurcation points. In dynamic problem of flight of three intersecting plates analysed with Newmark-type temporal algorithm, the value of α_t influences the moment at which the relative error of total energy of the system begins to grow indefinitely leading to the solution failure.     

On the incremental equations in non-linear elasticity — I. Membrane theory

A new formulation of the equations of membrane theory in non-linear elasticity is described. It is based on the consistent use of certain conjugate variables averaged through the (undeformed) thickness of the thin shell which the membrane approximates. The deformation gradient is taken as the basic measure of deformation, and its average value as the membrane measure of deformation. It is shown that the average elastic strain energy can be regarded as a function of the average deformation gradient to within an error which is of the second order in a certain small parameter. Moreover, to the same order, the average strain energy is a potential function for the average nominal stress. This means that the averages of the conjugate variables (nominal stress and deformation gradient) are also conjugate.In terms of the average conjugate variables, the membrane equilibrium equations are obtained by averaging from the equilibrium equations of the full three-dimensional theory. Discussion of the order of magnitude of the errors involved in the membrane approximation is a feature of the analysis.The corresponding incremental equations are also derived as a prelude to their application in certain bifurcation problems. One such problem is examined in the companion paper (Part II) in which results for thick shells and membranes are compared.     

Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements.     

Variational principles in the geometrically non-linear theory of shells undergoing moderate rotations

Summary A general approach to the derivation of variational principles is given for the geometrically non-linear theory of thin elastic shells undergoing moderate rotations. Starting from the principle of virtual displacements, a set of sixteen basic free functionals without subsidiary conditions is constructed. From these free functionals a, number of related functionals with or without subsidiary conditions may be generated. As examples, the functionals of the total potential energy and the total complementary energy are derived.

---

Übersicht Die vorliegende Arbeit enthält eine systematische Herleitung von Variationsprinzipen für die geometrisch nichtlineare Theorie dünner elastischer Schalen, in der die Quadrate der Rotationen von gleicher Größenordnung wie die Dehnungen sein können. Ausgehend vom Prinzip der virtuellen Verschiebungen wird eine Familie von sechzehn freien Funktionalen hergeleitet, die keinen Nebenbedingungen unterliegen. Von diesen freien Funktionalen kann eine Vielzahl verwandter Funktionale mit oder ohne Nebenbedingungen abgeleitet werden. Als Beispiele werden die Prinzipe vom stationären Wert des Gesamtpotentials und der komplementären Energie angegeben.

---

---

This work was prepared under an Agreement on Scientific Cooperation between the Institute of Fluid-Flow Machinery of the Polish Academy of Sciences in Gdask and the Institut für Mechanik of the Ruhr-Universität Bochum, FRG     

A function theory for thin elastic shells

It is well known in the theory of elastic shells that a first order approximation using the shell thickness as an expansion parameter leads to the membrane theory of shells. The membrane equations have as solutions thegeneralized analytic functions. These functions have been exhaustively studied by Ilya N. Vekua [6], [7] and his students. R.P. Gilbert and J. Hile [3] introduced an extension of these systems to include elliptic systems of 2n equations in the plane and named the solutions of these systemsgeneralized hyperanalytic functions.It is shown in this paper that the next order approximation to the shell, which permits, moreover, the introduction of bending, may be described in terms of the generalized hyperanalytic functions. It is strongly suspected that the higher order approximations may also be described in terms of corresponding hypercomplex systems.     

On the energy method in the non-linear theory of thin plates

Summary The equilibrium configuration of a thin plate under normal pressure is studied. A variational non-linear treatment of the problem is considered, taking into account the bending stresses and allowing large deflections.Existence, uniqueness, and regularity of solutions are obtained.

---

Sommario Si studia il problema non lineare della configurazione di equilibrio di una piastra sottile caricata normalmente nel suo piano medio, tenendo conto delle tensioni di natura flessionale e in un regime di spostamenti moderatamente grandi.Con una tecnica variazionale si studiano l'esistenza, l'unicità, e la regolarità delle soluzioni del problema.

     

Canonical equations in the geometrically nonlinear theory of thin anisotropic shells

The functional in the principle of minimum potential energy of layered anisotropic shells with a nonlinear relationship between strains and displacements is transformed into a canonical integral that coincides with the functional in the Reissner principle. Partial forms of the functional are derived for problem formulations where the dimension can be reduced with respect to one of the coordinates. The canonical system of equations is linearized and then normalized. The boundary-value problem is solved by the numerical discrete-orthogonalization method. An anisotropic spherical shell under external compression is analyzed for stability as an example     

Geometrically non-linear analysis of arbitrary elastic supported plates and shells using an element-based Lagrangian shell element

In the present paper, the ELF (element-based Lagrangian formulation) 9-node ANS (assumed natural strain) shell element was combined with the spring element for geometrically non-linear analysis of plates and shells sustained by arbitrary elastic edge supports that are subjected to variation in loading.This particular spring element serves as tool for modeling an arbitrary elastic edge support with 6 DOF (degrees of freedom). The elastic edge support was modeled by combining different spring models. The ANS method was used to overcome shear and membrane locking problems inherent in some thin plate and shell problems. In the formulation of the ELF characteristic arrays, the expression of element strains was adopted in the framework of the element natural coordinates. The non-linear analysis results of idealized edge supports were validated against the reference solutions available in the literature. As a result of the numerical test, the combination of the ELF 9-node shell element and spring element shows an exceptional performance for non-linear analysis of plates and shells under elastic edge supports.     

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.     

An approach to the formulation of variational functionals in the theory of laminar shells

Design Institute No. 3, USSR Ministry of Construction, Odessa. Translated from Prikladnaya Mekhanika, Vol. 26, No. 5, pp. 61–69, May, 1990.     

A polar theory for vibrations of thin elastic shells

In relation to a polar continuum, this paper presents a 2-D shear deformable theory for the high frequency vibrations of a thin elastic shell. To begin with, the 3-D fundamental equations of the micropolar elastic continuum are expressed as the Euler–Lagrange equations of a unified variational principle. Next, the kinematic variables of the shell are represented by the power series expansions in its thickness coordinate, and then, they are used to establish the 2-D theory by means of the variational principle. The 2-D theory is derived in invariant variational and differential forms and governs all the types of vibrations of the functionally graded micropolar shell. Lastly, the uniqueness is investigated in solutions of the initial mixed boundary value problems defined by the 2-D theory, and some of special cases are indicated in the theory.     

On the linear and nonlinear stability analysis in the theory of thin elastic shells

Summary In the frame of the geometrically nonlinear theory of thin elastic shells with moderate rotations a set of consistent equations for the nonlinear stability analysis is derived by application of energy criteria. Some methods of functional analysis are used which enable to prove the symmetry of the stability equations and to calculate bifurcation buckling from linear and nonlinear equilibrium branches and also snap-through buckling loads by variational approximating procedures.

---

Über die lineare und nichtlineare Stabilitätsberechnung in der Theorie dünner elastischer Schalen

Übersicht Im Rahmen einer geometrisch-nichtlinearen Theorie dünner Schalen mit moderaten Rotationen werden konsistente Gleichungen zur nichtlinearen Stabilitätsberechnung hergeleitet, wobei von Energiekriterien ausgegangen wird. Die Benutzung einiger Methoden der Funktionalanalysis ermöglicht den Nachweis der Symmetrie der Stabilitätsgleichungen und die Berechnung des Verzweigungs-Beulproblems bei linearen und nichtlinearen Gleichgewichtszuständen sowie die Bestimmung der kritischen Last beim Durchschlagproblem mit Hilfe variationeller Näherungsverfahren.

---

---

Lecture: XVth Inter. Congr. Theor. Appl. Mech., Toronto/Canada, 17.–23. Aug. 1980

---

The author is indepted to Docent Dr. habil. W. Pietraszkiewicz, Institute of Fluid-Flow Machinery Gdask, for valuable remarks and the Polish Academy of Science for continuous support.