Limit behavior of Koiter model for long cylindrical shells and Vlassov model

We propose in this article to consider the limit behavior of the Koiter shell model when one of the characteristic length of the middle surface becomes very large with respect to the other. To do this, we perform a dimensional analysis of Koiter formulation which involves dimensionless numbers characterizing the geometry and the loading. Once reduced to a one-scale problem corresponding to thin-walled beams (long cylindrical shell), using asymptotic expansion technique, we address the limit behavior of Koiter model when the aspect ratio of the shell tends to zero. We prove that at the leading order, Koiter shell model degenerates to a one dimensional thin-walled beam model corresponding to the Vlassov one. Moreover, we obtain a general analytical expression of the geometric constants involved, that improves the empirical expression given by Vlassov.     

The Space of Inextensional Displacements for a Partially Clamped Linearly Elastic Shell with an Elliptic Middle Surface

We consider a linearly elastic shell with an “elliptic” middle surface, clamped along a portion of its lateral face and subjected to body forces. Under weak regularity assumptions on the middle surface, we prove that the space of linearized inextensional displacements is reduced to zero, by using unique continuation results. Consequently, when the thickness of the shell goes to zero, the limit of the average with respect to the thickness of the three-dimensional displacement vector solves the “generalized membrane” shell model, according to the terminology introduced by P.G. Ciarlet and the first author. This revised version was published online in August 2006 with corrections to the Cover Date.     

NonLinearly Elastic Membrane Model For Heterogeneous Shells by Using a New Double Scale Variational Formulation: A Formal Asymptotic Approach

This paper is concerned with the asymptotic analysis of shells with periodically rapidly varying heterogeneities. The asymptotic analysis is performed when both the periods of changes of the material properties and the thickness of the shell are of the same orders of magnitude. We consider a shell made of Saint Venant–Kirchhoff type materials for which we justify a new two-scale variational formulation. We assume that both the data and the displacement field admit a formal asymptotic expansion with a negative order of the leading term. We prove that the lowest order term of the displacement field must be of order zero. When the space of nonlinear inextensional displacement is reduced to , this displacement field is a solution of a two-dimensional membrane model which is obtained by solving two coupled problems. The first, posed on the middle surface of the shell is two-dimensional and global and the second, posed on the periodicity cell, is three-dimensional and local.     

Nonobvious features of dynamics of circular cylindrical shells

In the framework of the nonlinear theory of flexible shallow shells, we study free bending vibrations of a thin-walled circular cylindrical shell hinged at the end faces. The finite-dimensional shell model assumes that the excitation of large-amplitude bending vibrations inevitably results in the appearance of radial vibrations of the shell. The modal equations are obtained by the Bubnov-Galerkin method. The periodic solutions are found by the Krylov-Bogolyubov method. We show that if the tangential boundary conditions are satisfied “in the mean,” then, for a shell of finite length, significant errors arise in determining its nonlinear dynamic characteristics. We prove that small initial irregularities split the bending frequency spectrum, the basic frequency being smaller than in the case of an ideal shell.     

On the Obstacle Problem for a Naghdi Shell

Starting with the Naghdi model for a shell in Cartesian coordinates, we derive a model for the contact of this shell with a rigid body. We also prove the well-posedness of the resulting system of variational inequalities.     

Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls

We study a nonlinear, unsteady, moving boundary, fluid–structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by two-dimensional incompressible Navier–Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the one-dimensional cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (two-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and the balance of contact forces at the fluid–structure interface. We prove the existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in Guidoboni et al. (J Comput Phys 228(18):6916–6937, 2009) to numerically solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk–Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.     

Homogenization of linear elastic shells

Homogenization techniques were used by Duvaut (1976,1978) in asymptotic analyse of 3-dimensional periodic continuum problems and periodic von Kármán plates.In this paper we homogenize Budiansky-Sanders linear, elastic shells with material parameters rapidly oscillating on the shell surface. We obtain a homogenized shell model which is elliptic and depends on explicitly calculated effective material parameters. We show that the solution of the periodic shell model converges weakly to the solution of the homogenized model when the period tends to zero.     

On inflating closed mylar shells

We discuss the inflating of a closed thin shell made of inextensible flexible material like mylar. The problem is to determine the extremal form of the shell, when it is inflated to the maximal possible volume. We introduce a variational problem which describes the inflating of rotationally symmetric shells. The main result presents a criteria for a rotationally symmetric shell to admit volume increasing deformations without surface stretching. Moreover explicit solutions are found for cylindrical and biconical shells.     

Shaping a thin-walled cylindrical shell on a three-roll bending machine

We use the Bernoulli-Euler kinematic hypothesis to model the steady-state process of shaping a thin-walled cylindrical shell by bending an elastoplastic strengthening parent sheet on a three-roll bending machine. We determine the curvilinear shape of the moving parent sheet in the bending area and the displacement of the central roll axis needed to obtain the prescribed curvature of the cylindrical shell when leaving the bending area. One- and multitransition shell shaping processes are considered. The computational model is in satisfactory agreement with experiments.     

Anticavitation and Differential Growth in Elastic Shells

Elastic anticavitation is the phenomenon of a void in an elastic solid collapsing on itself. Under the action of mechanical loading alone typical materials do not admit anticavitation. We study the possibility of anticavitation as a consequence of an imposed differential growth. Working in the geometry of a spherical shell, we seek radial growth functions which cause the shell to deform to a solid sphere. It is shown, surprisingly, that most material models do not admit full anticavitation, even when infinite growth or resorption is imposed at the inner surface of the shell. However, void collapse can occur in a limiting sense when radial and circumferential growth are properly balanced. Growth functions which diverge or vanish at a point arise naturally in a cumulative growth process.     

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.      

Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

Nonlinear Effects on Electrophoresis of a Soft Particle and Sustained Solute Release

A numerical study is made on the electrophoresis of a core-shell soft particle based on the first principle of electrophoresis. The soft particle consists of a charged rigid core coated with a polymer shell. Numerical computations for the electrophoretic velocity are obtained and compared with the existing analytical solution. The analytical solutions, based on the Boltzmann distribution of ions and the Debye–Huckel approximation, are valid for lower range of charge density, weak applied electric field and thin double layer. Discrepancy from the existing analytical solution is found when the Debye layer extends beyond the porous shell. This discrepancy becomes larger for higher values of the rigid core surface potential, fixed charge density of the soft shell and stronger imposed electric field. The double-layer polarization is found to have a strong impact when the shell thickness is lower than the Debye length. The electrophoretic velocity is found to vary nonlinearly with the imposed electric field when the imposed field strength is large enough to create a potential drop across the particle bigger than the thermal potential. We have also analyzed the mechanism of sustained solute release from the soft particle. Our results show that the rate of solute release is large compared to a pure diffusion dominated process.     

Exact Controllability of a Koiter Shell by a Boundary Action

We consider a thin shell whose elastic deformations are described by the system of Koiter linear partial differential equations. The shell is controlled by functions acting along its boundary. We show, using the HUM method (Hilbert Uniqueness Method) of J.L. Lions, that when the middle surface of the shell satisfies specific geometrical assumptions, it is possible to obtain exact controllability which consists in driving the system to rest in finite time. In particular we identify the function spaces where the initial Cauchy data (initial displacement and velocity fields) have to be taken and the function space where the control can be chosen in order to get the exact controllability. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Linearly Elastic Shell over an Obstacle: The Flexural Case

We study the equilibrium of a three-dimensional solid having a uniform thickness \(2 \varepsilon \) along a middle surface which satisfies the usual assumptions of shell theory. The solid is linearly elastic at small strains and is submitted to unilateral contact conditions with an obstacle on a part of its boundary. When \(\varepsilon \) tends to zero, the three-dimensional domain tends to a two-dimensional one, so that the contact conditions pass from a part of the boundary to the interior of the domain. We restrict our attention to the so-called bending case, that is when the shell undergoes only inextensional deformations. As a major difference with the case of a shallow shell, we get in general a coupling between the three components of the displacement in the contact conditions. The work is closed by explicit examples showing the corresponding variation of the non-penetrability condition along the surface of the shell and by comments about the model and the remaining difficulties.     

On the theory of elastic shells made from a material with voids

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.     

Axisymmetric equilibrium states of non-linear elastic cylindrical shells

The problem of nonuniqueness of static axisymmetric solutions for a constrained cylindrical shell under a compressive thrust is studied. Both in the elastic and hyperelastic case, we prove the existence of buckled states. For a simple choice of the elastic potential, a Hamiltonian formulation is provided.     

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.     

Modeling the Behavior of Heat-Shrinkable Thin Films

We describe an asymptotic model for the behavior of PET-like heat-shrinkable thin films that includes both membrane and bending energies when the thickness of the film is positive. We compare the model to Koiter’s shell model and to models in which a membrane energy or a bending energy are obtained by Γ-convergence techniques. We also provide computational results for various temperature distributions applied to the films.