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An approximate shell theory for unrestricted elastic deformations
Abstract
An approximate shell theory is formulated within the framework of the general theory of finite elasticity for unrestricted deformation. A deformation field is constructed throughout the shell based on the solution of the corresponding membrane problem. An additional deformation is then superposed on this "membrane state" and the resulting equations of motion for the final state linearized in the additional displacement. These equations are then "averaged" through the thickness of the shell to yield an approximate shell theory. Details are carried through for the case in which the additional deformation is itself linearized in the thickness variable so that normals to the middle surface remain straight and uniformly extended, but not necessarily normal. The resulting theory is then applied to the problem of a uniformly twisted, extended and inflated cylindrical shell.
The stability of an inflated and extended cylindrical shell composed of a rubberlike material is studied within the framework of large elastic deformation theory. The study utilizes shell equations that are developed by superposing upon the solution to a membrane problem a linearized deformation that permits the inclusion of bending resistance and thickness effects upon stability. A comparison is made with the results published in an earlier treatment of the problem developed purely from a membrane point of view. In particular, results are plotted for neo-Hookean and Mooney-Rivlin materials showing the effects of bending resistance and thickness variations on previously found regions of stability.
This paper describes the effects of various external axisymmetric loads on pressurized hinged spherical membranes taking into account changes in internal pressure, volume, and temperature. "Exact" geometrical non-linearity along with generalized constitutive relations for a highly non-linearly clastic, isotropic, homogeneous, incompressible material are used in the analysis. The specialized case of a Hookean material is also treated. The non-linear equations of membrane equilibrium are derived in terms of additional finite displacements for the case of nonorthogonal curvilinear midsurface coordinates and are then specialized for the problem of an inflated hinged spherical membrane. The resulting two highly non-linear coupled second order differential equations are solved by means of a finite difference and Newton-Raphson iterative procedure. All results are presented in nondimensionalized graphical form.
A theory of inflatable, incompressible shells is deduced from the general theory of small deformations superposed upon finite deformations. The new aspects of this study, as compared to earlier attacks upon the problem, concern the inclusion of bending effects during inflation and a simple treatment of normal traction.RésuméOn déduit une théorie des coques gonflables et incompressibles à partir de la théorie générale des petites déformations superposées à des déformations finies. Les aspects nouveaux de cette étude, par rapport à des approches antérieures de ce problème, concernent la prise en compte des effets de flexion pendant le gonflage et un traitement simple de la traction normale.ZusammenfassungEine Theorie für auf blasbare inkompressible Schalen wird von der allgemeinen Theorie kleiner Verformungen, die endlichen Verformungen überlagert sind, hergeleitet. Verglichen mit früheren Angriffen auf das Problem besteht der neue Gesichtspunkt dieser Untersuchung in der Einschliessung von Biegeerscheinungen und in einer einfachen Behandlung von normalen Schubkräften.РефератO тeOpии нaлyвныч OбOлOЧeк. Из Oбщeй тeOpии мaлыч лeфOpмaций, нaлOжeнныч нa кOнeЧныe дeфOpмaции, вывOдитcя тeOpия нaдyвныч нecжимaeмыч OбOлOЧeк. ПO cpaвнeниy c пpeдыдyщими пOпыткaми peшeния пpOблeмы, в дaннOй qpaбOтe Oбpaщaeтcя внимaниe нa yЧeт эффeктOв изгибa пpи нaкaЧивaнии OбOлOЧки и нa пpOcтyy фOpмyлиpOвкy ycлOвий дλя нOpмaльныч нaпpяжeний.
This paper is concerned with a nonlinear theory of elastic shells with small deformations whose material response is nonlinear. The developments are carried out under the Love-Kirchhoff hypothesis. General constitutive equations are derived in which the geometrical properties (due to deformation) and the material characteristics are separable. Through this separability, it is shown how to extract constitutive equations of predetermined types. Particular examples are followed by a discussion of the membrane theory.
The analysis is concerned with an exact, complete, and fully general nonlinear theory of elastic shells founded under the Kirchhoff hypothesis (Arch. Rat'l Mech. and Anal., 1:295323, 1958). Since the fully general equations of equilibrium in terms of stress and couple resultants and the appropriate boundary conditions are well known, the main task is the determination of suitable strain measures and the derivation of constitutive equations. (Author)
C O : P R LI IN RIE AN THE GEOMETRY OF A SURFACE Relationships between space and surface tensors and their derivatives in normal coor inates Deformation and strain Stress and couple result ts; equations of equilibrium The constitutive equations of the linear theory Remarks on the general theorems of the linear heoryAD -<+< *%$N AD-272 084Div. 25 U (TISTP/TL) Cornell U., Ithaca, N. Y. PHOTON ABSORPTION BY VALENCE ELECTRONS IN MG, CR, FE AND CO, by H. Kroger. Feb 62, 103p. incl. illus. tables, 59 refs. (Technical rept. no. 4) (Grant DA-ORD-37, Proj. 2810P) (AROD rept. no. 2810:4) Unclassified report No automatic release to Foreign Nationals. DESCRIPTORS: (*Photons, *Absorption, *Valence, *Electrons, Excitation, Density.) (X rays, Vacuum systems, Ultraviolet radiation, *Plasma physics, Oscillation, Metals, Optics, Photometers.) (Magnesium, Chromium, Iron, Cobalt, Nickel, Copper.) This analysis deals with experimental studies of the absorption of soft x-ray - vacuum ultraviolet radiations by metals. Since the energies of the incident photons are just insufficient to excite x-ray levels of the atoms, the most important mechanism for the removal of energy from the photon beam is interband excitation of valence electrons. The absorption of light in this frequency range should therefore yield some information about the electronic valence and conduction states of metals. The comparison of the optical absorption spectrum with all the characteristic electron energy losses of a metal is also of interest. (Author)
A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic
in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation
of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous
extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible
material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder
with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for
a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended
and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods
are compared.
This paper is concerned with a general dynamical theory of a Cosserat surface, i.e., a deformable surface embedded in a Euclidean 3-space to every point of which a deformable vector is assigned. These deformable vectors, called directors, are not necessarily along the normals to the surface and possess the property that they remain invariant in length under rigid body motions. An elastic Cosserat surface and other special cases of the theory which bear directly on the classical theory of elastic shells are also discussed.