Static and dynamic responses of a circular cylindrical shell made of hyperelastic arterial material are studied. The material is modeled as a combination of Neo-Hookean and Fung materials. Two types of pressure loads are studied—distributed radial forces and deformation-dependent pressure. The static responses of the shell under these two loads differ essentially at moderate strains, while the behavior is similar for small loads. The principal difference is that the axial displacements are much larger for the shell under distributed radial forces, while for actual pressure the shell is stretched both in circumferential and axial directions. Free and forced vibrations around preloaded configurations are analyzed. In both cases, the nonlinearity of the single-mode (driven mode) response of the preloaded shell is quite weak, but a resonant regime with both driven and companion modes active has been found with more complicated nonlinear dynamics.

The techniques used by Koiter in 1968 to derive a simplified set of linear equilibrium equations for an elastically isotropic circular cylindrical shell in terms of displacements and the associated pointwise error estimate engendered in Love’s uncoupled strain-energy density are here extended to derive analogous simplified equilibrium equations and an error estimate for elastically isotropic cylindrical shells of arbitrary closed cross section.

An acceptable variant of the Koiter–Morley equations for an elastically isotropic circular cylindrical shell is replaced by a constant coefficient fourth-order partial differential equation for a complex-valued displacement-stress function. An approximate formal solution for the associated “free-space” Green’s function (i.e., the Green’s function for a closed, infinite shell) is derived using an inner and outer expansion. The point wise error in this solution is shown rigorously to be of relative order ( h ∕ a ) ( 1 + h ∕ a ∣ x ∣ ) , where h is the constant thickness of the shell, a is the radius of the mid surface, and a x is distance along a generator of the mid surface.

The aeroelastic stability of simply supported, circular cylindrical shells in supersonic flow is investigated by using both linear aerodynamics (first-order piston theory) and nonlinear aerodynamics (third-order piston theory). Geometric nonlinearities, due to finite amplitude shell deformations, are considered by using the Donnell’s nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. The system is discretized by Galerkin method and is investigated by using a model involving up to 22 degrees-of-freedom, allowing for travelling-wave flutter around the shell and axisymmetric contraction of the shell. Asymmetric and axisymmetric geometric imperfections of circular cylindrical shells are taken into account. Numerical calculations are carried out for a very thin circular shell at fixed Mach number 3 tested at the NASA Ames Research Center. Results show that the system loses stability by travelling-wave flutter around the shell through supercritical bifurcation. Nonsimple harmonic motion is observed for sufficiently high post-critical dynamic pressure. A very good agreement between theoretical and existing experimental data has been found for the onset of flutter, flutter amplitude, and frequency. Results show that onset of flutter is very sensible to small initial imperfections of the shells. The influence of pressure differential across the shell skin has also been deeply investigated. The present study gives, for the first time, results in agreement with experimental data obtained at the NASA Ames Research Center more than three decades ago.

The stability of circular cylindrical shells with supported ends in compressible, inviscid axial flow is investigated. Nonlinearities due to finite-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory; the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied at the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear potential flow theory is applied to describe the fluid-structure interaction. Both annular and unbounded external flow are considered by using two different sets of boundary conditions for the flow beyond the shell length: (i) a flexible wall of infinite extent in the longitudinal direction, and (ii) rigid extensions of the shell (baffles). The system is discretized by the Galerkin method and is investigated by using a model involving seven degrees-of-freedom, allowing for traveling-wave response of the shell and shell axisymmetric contraction. Results for both annular and unbounded external flow show that the system loses stability by divergence through strongly subcritical bifurcations. Jumps to bifurcated states can occur well before the onset of instability predicted by linear theory, showing that a linear study of shell stability is not sufficient for engineering applications.

Analytical solution for the tangential stress distribution ahead of a hole is needed for the theoretical prediction of notched strength of brittle laminate using the well-known W-N criteria. In the present study, tangential stress distribution in an orthotropic circular cylindrical shell under uniaxial loading with a circular hole is obtained intuitively with the use of a stress function. A good agreement is obtained for the stresses around and ahead of the circular hole in 0 deg 4 ± 30 deg s and 90 deg laminates with the finite element results.

The three-dimensional theory of elasticity in curvilinear coordinates is employed to investigate the dynamic buckling of an imperfect orthotropic circular cylindrical shell under mechanical and thermal loads. Accurate form of the strain expressions of imperfect cylindrical shells is established through employing the general Green's strain tensor for large deformations and the equations of motion are derived in terms of the second Piola-Kirchhoff stress tensor. Then, the governing equations are properly formulated and solved by means of an efficient and relatively accurate solution procedure proposed to solve the highly nonlinear equations resulting from the above analysis. The proposed formulation is very general as it can include the influence of the initial imperfections, temperature distribution, and temperature dependency of the mechanical properties of materials, effect of various end conditions, possibility of large-deformation occurrence and application of any combination of mechanical and thermal loadings. No simplifications are done when solving the resulting equations. Furthermore, in contrast to the displacement-based layer-wise theories and the three-dimensional approaches proposed so far, the stress, force and moment boundary conditions as well as the displacement type ones, can be incorporated accurately in these formulations. Finally, a few examples of mechanical and thermal buckling of some orthotropic cylindrical shells are considered and results of the present three-dimensional elasticity approach are compared with the buckling loads predicated by the Donnell's equations, some single-layer theories, some available results of the layer-wise theory and the recently published three-dimensional approaches and the accuracy of the later methods are discussed based on the exact method presented in this paper.

With the aid of the static-geometric duality of Goldenveizer (1961), Cartesian tensor notation, and nondimensionalization, it is shown that the equations of linear shell theory of Sanders (1959) and Koiter (1959), when specialized to a circular cylindrical shell with stress-strain relations exhibiting full anisotropy (21 elastic-geometric constants), can be reduced, with no essential loss of accuracy, to two coupled fourth-order partial differential equations for a stress function F and a curvature function G. Auxiliary formulas for the midsurface displacement components are also given. For isotropic shells with uncoupled stress-strain relations, the equations reduce to a form given by Danielson and Simmonds (1969). The reduction is achieved by adding certain negligibly small terms to the given stress-strain relations. For orthotropic shells of mean radius R and thickness h with uncoupled stress-strain relations, it is shown that the very short decay length of O( hR ) and the very long decay length of O(R R / h ) (associated with separable solutions of the form e −Rz sin nθ) depend, respectively, to within a relative error of O(h/R), only on the products of different pairs of the eight possible elastic constants.

A study on the problem of linear buckling of piezoelectric circular cylindrical shells subjected to external pressure as well as on an electric field is presented. In this paper, the structure is treated as a three-dimensional one. The results reveal that the piezoelectric effect has significant effect on the critical load, while the stress due to the uniformly applied electric field alone is not likely to cause elastic buckling. In addition, they can also be used to assess the limitation of shell theories in predicting buckling of piezoelectric smart shell structures.

The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifurcation. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Nonlinear bending deformations of generally laminated circular cylindrical shells subjected to an axisymmetric radial pressure, axial loading, and torsion are considered. Deformations of such shells are axisymmetric, i.e., while cross-sections can rotate about the shell axis, they remain circular and all derivatives with respect to the circumferential coordinate are equal to zero. In this case a nonlinear term is included into the axial strain only. Excluding this strain from the equilibrium equations, one can reduce the analysis to integration of a system of linear differential equations. In the first part of the paper the solution based on Love’s first approximation and Donnell’s theories is shown for thin shells. While out-of-surface boundary conditions can be arbitrary, several combinations of in-surface conditions are discussed. The second part of the paper outlines the approach to the solution for shear-deformable geometrically nonlinear shells. First-order shear-deformation theory is used in the analysis. The approach is based on exclusion of the nonlinear term (axial strain) and subsequent solution of linear differential equations of equilibrium by the operational method. This approach can be also used for an analysis of thin, generally laminated cylindrical shells.

The self-equilibrated end load problem for a hollow circular cylinder is considered using the Papkovitch-Neuber solution to the elastostatic displacement equations of equilibrium; both axi- and nonaxisymmetric solutions are derived. The requirement of zero traction on the surface generators of the cylinder leads to an eigenequation whose roots determine the rate of decay with axial coordinate. The locus of the smaller roots is plotted for circumferential harmonic loadings n = 0, 1, 2, and 3, for different wall thicknesses, and supplement previously known decay rates for the solid section and the circular cylindrical shell which are the extremes of diameter ratio. The loci are of considerable intricacy, and for small wall thickness, simple shell theory and two modes of decay for the semi-infinite plate are employed to identify the various modes of decay. Whereas for the solid cylinder the characteristic decay length of Saint-Venant’sprinciple is the radius (or diameter), for the hollow cylinder it becomes possible to discriminate between “wall thickness” and “ r m t ” modes of decay according to the limiting behavior as the cylinder assumes shell-like proportions; the one exception is “membrane bending” for which self-equilibrating end loading does not decay as thickness tends to zero.

The method of multiple scales is used to analyze the nonlinear response of infinitely long, circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting modes. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcations. Moreover, the frequency response curves exhibit Hopf bifurcations. For certain parameters and excitation frequencies between the Hopf values, limit-cycle solutions of the modulation equations are found. As the excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddlenode bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.

The geometrically and constitutively nonlinear response of an infinite, circular, cylindrical shell submerged in an infinite fluid medium to a transverse, transient acoustic wave is analyzed. Circumferential Fourier series solutions are obtained through the numerical integration of coupled ordinary differential equations and convolution integrals. Numerical results are presented in the form of response histories, response snapshots, and iso-damage curves for incident waves of rectangular pressure profile. Response solutions obtained with the first-order doubly asymptotic approximation are compared with their “exact” counterparts. It is found that doubly asymptotic approximations are unsuitable for two-dimensional, shock-response analysis of yielding submerged structures.

Three equilibrium equations in terms of three displacements are derived in scalar mathematics form, by linear, small-strain elasticity principles, for the case of general thick-walled shells under general loading. These reduce to well-known forms for the particular cases of flat-plates and thick circular cylindrical shells.

A combination of the Galerkin procedure and the method of multiple scales is used to analyze the nonlinear forced response of infinitely long circular cylindrical shells (or circular rings) in the presence of internal (autoparametric) resonances. If ω f and a f denote the frequency and amplitude of a flexural mode and ω b and a b denote the frequency and amplitude of the breathing mode, the steady-state response is found to exhibit a saturation phenomenon when ω b = 2ω f if the shell is excited by a harmonic load having a frequency Ω near ω b . As the amplitude f of the excitation increases from zero, a b increases linearly with f until a threshold value f c of f is reached. Beyond f c , a b remains constant and the extra energy spills over into the flexural resonant mode whose amplitude grows nonlinearly. Results of numerical investigations, guided by the perturbation analysis, show that the long-time response exhibits a Hopf bifurcation, yielding amplitude and phase-modulated motions. The amplitudes and phases experience a cascade of period-doubling bifurcations ending up with chaos. The bifurcation values are finely tuned.

Asymptotic expansions are developed for the equations governing large axisymmetric deformation of a circular cylindrical shell composed of a Mooney material. The shell equations allow large normal strains and thickness changes but ignore transverse shear deformation. For a pressurized cylinder with rigid end plugs, results are presented to illustrate the development of a primary and a secondary boundary layer as generalizations of those that occur in small-strain shell theory. The form of the WKB-type expansion divides the secondary layer into bending and stretching components, which lie within the wider primary boundary layer. While the bending component of the secondary layer can become significant when strains are still small, the stretching component emerges as a consequence of large geometry changes in the edge zone, becoming significant as strains grow large and material nonlinearity becomes important.

Harmonic vibrations of a circular, cylindrical shell of rectangular planform and with an arbitrarily located crack, are investigated. The problem is described by Donnell’s equations and solved using triple finite Fourier transformation of discontinuous functions. The unknowns of the problem are the discontinuities of the slope and of three displacement components across the crack. These last quantities are replaced, using constitutive equations, by curvatures and strain in order to improve convergence and to represent explicitly the singularities at the tips. The formulas for differentiation of discontinuous functions are derived using Green-Gauss theorem. Application of the boundary conditions at the crack leads to a homogeneous system of linear algebraic equations. The frequencies are obtained from the characteristic equation resulting from this system. Numerical results for special cases are provided.