Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.     

Effect of geometric imperfections on non-linear stability of circular cylindrical shells conveying fluid

Circular cylindrical shells conveying incompressible flow are addressed in this study; they lose stability by divergence when the flow velocity reaches a critical value. The divergence is strongly subcritical, becoming supercritical for larger amplitudes. Therefore the shell, if perturbed from the initial configuration, has severe deformations causing failure much before the critical velocity predicted by the linear threshold. Both Donnell's non-linear theory retaining in-plane displacements and the Sanders-Koiter non-linear theory are used for the shell. The fluid is modelled by potential flow theory but the effect of steady viscous forces is taken into account. Geometric imperfections are introduced and fully studied. Non-classical boundary conditions are used to simulate the conditions of experimental tests in a water tunnel. Comparison of numerical and experimental results is performed.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

A new analytical formula to compute the step length of Padé approximants in the ANM: Application to buckling structures

In this paper, we propose a new analytical formula to define the next branch in the Asymptotic Numerical Method (ANM) using the Padé approximants. The proposed formula is based on the computation of the relative error of two consecutive Padé approximants. This formula is obtained by developing the relative error with respect to the path parameter. An appropriate matrix formulation is adopted for the computation of this relative error. A comparison between the analytical formula proposed in this paper and the classical continuation Padé approximants using the step length computed numerically using dichotomy method is presented for examples of buckling structures.     

Mathematical justification of an elastic elliptic membrane obstacle problem

Starting from the 3D Signorini problem for a family of elastic elliptic shells, we justify that the obstacle problem of an elastic elliptic membrane is the right approximation posed in a 2D domain, when the thickness tends to zero. Specifically, we provide convergence results in the scaled and de-scaled formulations.     

Mathematical justification of a viscoelastic elliptic membrane problem

We consider a family of linearly viscoelastic elliptic shells, and we use asymptotic analysis to justify that what we have identified as the two-dimensional viscoelastic elliptic membrane problem is an accurate approximation when the thickness of the shell tends to zero. Most noticeable is that the limit problem includes a long-term memory that takes into account the previous history of deformations. We provide convergence results which justify our asymptotic approach.     

Chaotic vibrations of spherical and conical axially symmetric shells

Summary Chaotic vibrations of deterministic, geometrically nonlinear, elastic, spherical and conical axially summetric shells, subject to sign-changing transversal load using the variational principle, are analysed. The paper is motivated by an observation that variational equations of the hybrid type are suitableto solve many dynamical problems of the shells theory. It is assumed that the shell material is isotropic, and the Hook's principle holds. Intertial forces in directions tangent to mean shell surface and rotation inertia of a normal shell cross section are neglected. A transition form PDEs to ODEs (the Cauchy problem) is realized through the Ritz procedure. Next, the Cauchy problem is solved using the fourth-order Runge-Kutta method. Qualitative and quantitative analysis is carried out in the frame of both nonlinear dynamics and quantitative theory of differential equations. New scenarios from harmonic to chaotic dynamics are detected. Various vibration forms development versus control parameters (rise of arc; amplitude and frequency of the exciting force and number of vibrational modes accounted) are illustrated and discussed.     

Free Vibrations of Cylindrical Shells with Non-Axisymmetric Mass Distribution on Elastic Bed

The free vibrations of circular cylindrical shells partiallyloaded by a distributed mass and rested on an elastic bed are studied in this paper. Both the mass-load and the elastic bed are assumed to be applied on limited arcs and with arbitrary distributions in circumferential direction,while they are considered to be uniformly distributed in longitudinaldirection on the entire shell length. Therefore, the problem is notaxisymmetric. The solution is obtained by using the development of theflexural mode shapes in a Fourier series, whose coefficients are determinedby rendering the Rayleigh quotient stationary, so a Galerkin equation isobtained. The proposed method is independent of the boundary conditionsat the shell ends. The results are satisfactorily compared to FEM results.Finally, the influence of the mass-load and of the bed stiffness on thenatural frequencies and mode shapes of a simply supported shell is shownand discussed.     

Optimization of Axisymmetric Membrane Shells Against Brittle Fracture

The questions investigated in this paper are related to an important class of problems of optimal design of structures against brittle fracture. The primary problem of axisymmetric shell optimization under fracture mechanics constraint is formulated as the weight (volume of the shell material) minimization under stress intensity constraints. Considered problems are characterized by incomplete information concerning crack size, crack location and its orientation. Taking into account the factor of incomplete information the paper presents the formulation of optimal shell design problem based on minimax (guaranteed) approach and provides some results of analytical investigation for thin-walled shells with through cracks.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.