Koiter Model Research Articles

Anisotropic adaptive mesh procedure for computing very thin hyperbolic shells

This paper is devoted to the study of the mechanical behavior of thin elastic shells when their relative thickness e tends to zero. We focus on the case of hyperbolic shells whose middle surface has principal curvatures of opposite signs everywhere. We use the Koiter model to describe the mechanical behavior of the shell. The corresponding system, which depends on the relative thickness e, is elliptic except at the limit for e = 0 where it is hyperbolic. In a first part, we study theoretically the phenomena of internal layers appearing during the singular perturbation process, when the loading is somewhat singular. These layers have very different structures either they are along or across the asymptotic lines of the middle surface of the shell. Moreover, we may have pseudo-reflections when a part of the boundary is not along an asymptotic line of the surface. In any case, the layers become very thin when e tends to zero and the displacements inside the layers tends to infinity. In order to have a good description of the singularities of the displacements inside the layers, we propose here to use an anisotropic and adaptive mesh procedure. In a second part, we will present numerical computations performed with such meshes. They enable us to approach accurately the singularities inside the layers predicted by the theory. Finally, we test the behavior of the remeshing procedure when the shell is non-inhibited.

Numerical simulation of head–tape magnetic reading devices by a new 2-D model

In this work, a new elastohydrodynamic model governing a head–tape magnetic reading device is proposed. The coupled model is mainly based on a nonlinear compressible Reynolds equation and a Koiter model for shells. After its mathematical analysis, a set of numerical techniques is presented to solve the different difficulties associated to the problem. Finally, numerical examples illustrate the good performance of the algorithm.

Asymptotic analysis of bending-dominated shell junctions

The subject of this paper is the asymptotic analysis of a multi-structure made of two thin linearly elastic shells. It is assumed that the middle surfaces of the two shells are linked together on a part of their boundary, that at each point of their intersection, the two middle surfaces have distinct tangent planes and that the order of the applied body force density is ɛ 2 , where ɛ is the half-thickness of each shell. We then identify the limit when ɛ tends to zero of the solution of the ‘scaled’ three-dimensional linearized elasticity problem. It is shown in particular that this limit is constituted of inextensional displacement fields for each shell and such that the junction is rigid (the angle between the middle surfaces of the two shells at the junction is unchanged by the deformation of the multi-structure). Moreover this limit is the same as the limit of the solution of the classical shell model (or Koiter model) for the multi-structure with the condition of rigid junction.

Multiscale Expansions for Linear Clamped Elliptic Shells

We investigate solutions of the two-dimensional Koiter model and of the three-dimensional linear shell model in the case where the shell is clamped and its mean surface is elliptic. For smooth data, these solutions admit multiscale expansions in powers of ϵ1/2 where ϵ denotes the (half-)thickness of the shell. Both expansions contain terms independent of ϵ and boundary layer terms exponentially decreasing with respect to , with r the distance to the boundary of the mean surface. The expansion of the three-dimensional displacement contains supplementary boundary layers, exponentially decreasing with respect to r/ϵ as for plates. Using these expansions we obtain sharp estimates in various norms.

A SIMPLE PROOF OF THE ELLIPTICITY OF KOITER'S MODEL

In this paper, we give a simple proof of the ellipticity of Koiter's model for a shell, based on the three-dimensional Korn inequality applied to Kirchhoff–Love displacement fields.

THERMOELASTIC BUCKLING OF PLATES WITH IMPERFECTIONS BASED ON A HIGHER ORDER DISPLACEMENT FIELD

The nonlinear higher order strain-displacement relations for thin orthotropic plates are considered and substituted into the potential energy function for thermoelastic loadings. The Euler equations are then applied to the functional of energy and the general thermoelastic equations for a thin orthotropic plate are obtained. The stability equations are then derived through the second variation of the potential energy function. The results are extended to imperfect isotropic plates based on the Koiter model. The thermal loadings include uniform temperature rise, axial temperature difference, and gradient temperature through the thickness. The thermoelastic buckling of a thin isotropic plate with imperfections under these thermal loadings is investigated.

Mathematical analysis and numerical simulation of a Reynolds-Koiter model for the elastohydrodynamic journal-bearing device

The aim of this work is to deduce the existence of solution of a coupled problem arising in elastohydrodynamic lubrication. The lubricant pressure and concentration are modelled by Reynolds equation, jointly with the free-boundary Elrod-Adams model in order to take into account cavitation phenomena. The bearing deformation is solution of Koiter model for thin shells. The existence of solution to the variational problem presents some difficulties: the coupled character of the equations, the nonlinear multivalued operator associated to cavitation and the fact of writing the elastic and hydrodynamic equations on two different domains. In a first step, we regularize the Heaviside operator. Additional difficulty related to the different domains is circumvented by means of prolongation and restriction operators, arriving to a regularized coupled problem. This one is decoupled into elastic and hydrodynamic parts, and we prove the existence of a fixed point for the global operator. Estimations obtained for the regularized problem allow us to prove the existence of solution to the original one. Finally, a numerical method is proposed in order to simulate a real journal-bearing device and illustrate the qualitative and quantitative properties of the solution.

Numerical implementation of composite Koiter shells including membrane/bending coupling coefficients

We propose a way for determining the generalized coefficients of rigidity – some of which are membrane/bending coupling coefficients – which appear in the deformation energy of the Koiter model of thin shells. This is concerned with a heterogeneous material in the thickness direction. A new program to compute these coefficients is implemented in the finite element code Modulef, in order to simulate problems of thin multilayered shells with linearly elastic anisotropic layers. We propose an example of an inhibited multilayered thin shell, with hyperbolic middle surface, involving a composite material with unidirectional fibres. To cite this article: H. Ranarivelo, C. R. Mecanique 330 (2002) 273–278.

Existence of solution of an elastohydrodynamic Reynolds–Koiter model

In this paper the existence of solution for a new elastohydrodynamic model for journal bearing devices is proved. The lubricant pressure and the concentration are governed by Reynolds equation combined with Elrod–Adams model for cavitation, while the bearing displacement follows a Koiter model for shells. A regularization procedure is proposed to overcome the Heaviside concentration–pressure relation. A fixed point theorem leads to the solution of the regularized problem. The introduction of prolongation and restriction operators copes with the additional difficulty of elastic and hydrodynamic subproblems posed on different domains. The estimates for the regularized problem allow to pass to the limit and state the existence result.

Justification des modèles linéaires de Koiter et de Naghdi pour des coques totalement encastrées soumises à des forces «non admissibles »

We consider a shell, i.e., a three dimensional body with a small thickness (denoted by 2ε), which is clamped along its entire lateral boundary and subjected to regular loads. In the linear case, one can use the two-dimensional models of Koiter or Naghdi to calculate the displacement vector field of the shell. Some error estimates have been obtained in [3,4] between these models and the three-dimensional displacement for flexural or membrane shells. Here, we do not make any assumptions on the geometry of the shell. In particular, the space of linearized inextensional displacements can be reduced to zero. The assumptions on the loads are weak: for instance, one can consider regular loads (in H 1 ) which do not depend on the transverse variable. We then establish a relative error estimate for the scaled linearized deformation tensor between Koiter's model (or Naghdi's model) and the three-dimensional model. These estimates hold though the limit model is not known. In addition, further assumptions on the data allow to recover the error estimates concerning the displacements which have been proved in [3,4].

Développements asymptotiques dans les coques elliptiques : Modèle de Koiter

We study the asymptotic expansion of the solution of the two-dimensional Koiter model (see [7]) in linear elasticity, for an elliptic shell clamped along its whole lateral face. We show that for smooth data, this solution admits an asymptotic expansion in powers of ε 1/2 where ε represents the (half-)thickness of the shell. This expansion contains terms independent on ε and boundary layer terms exponentially decaying with respect to r/ ε where r denotes the geodesic distance to the boundary of the mean surface.

An Up-to-the Boundary Version of Friedrichs's Lemma and Applications to the Linear Koiter Shell Model

In this work, we introduce a variant of the standard mollifier technique that is valid up to the boundary of a Lipschitz domain in $\mathbb{R}^n$. A version of Friedrichs's lemma is derived that gives an estimate up to the boundary for the commutator of the multiplication by a Lipschitz function and the modified mollification. We use this version of Friedrichs's lemma to prove the density of smooth functions in the new function space introduced in our earlierwork concerning the linear Koiter shell model for shells with little regularity. The density of smooth functions is in turn used to prove continuous dependence of the solution of Koiter's model on the midsurface. This provides a complete justification of our new formulation of the Koiter model.

Finite element solution of a Reynolds–Koiter coupled problem for the elastic journal–bearing

The aim of this work is to present a numerical algorithm for solving a new elastohydrodynamic free-boundary problem which models the lubricant pressure behaviour by means of Reynolds equation, jointly with the free-boundary model of Elrod–Adams for cavitation and the elastic deformation of the bearing with the Koiter shell model; a non-local constraint on the pressure is imposed. The proposed algorithm uncouples the hydrodynamic and elastic parts to achieve a fixed point. For the two-dimensional problem, a combination of finite elements, transport-diffusion and duality methods is performed. For the three-dimensional elastic problem, a finite element discretization is applied to a mixed formulation for the Koiter model.

Asymptotic Justification of the Kirchhoff-Love Assumptions for a Linearly Elastic Clamped Shell ?

The displacement vector of a linearly elastic shell can be computed by using the two- dimensional Koiter's model, based on the a priori Kirchhoff-Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x3. The term independent ofx3 of this approximation is equal to the displacement vector of the two- dimensional Koiter's model. The term linear in x3 depends on the infinitesimal rotation vector of the normal. After an appropriate scaling, we estimate here the difference between the three-dimensional displacement and this affine vector field in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal allows to improve the asymptotic estimate between the three- dimensionnal displacement and Koiter's bidimensional displacement. Mathematics Subject Classifications ( 2000): 73C15, 73C99, 35F15.

Formulation bidimensionnelle exacte du modèle de coque 3D de Kirchhoff-Love

An exact two-dimensional formulation of the 3D Kirchhoff-Love shell model is given by an integration over the thickness, without making the thickness assumption h ≪ R of the Koiter's model. Based on h/2 ≤ inf (¦R 1(x)¦, ¦R 2(x)¦, x ε gw) — δ, δ > 0 assumption, the proposed two-dimensional shell model permits to consider small radii of curvature of the middle surface of the shell, of h order. This model exhibits a coupling between membrane and bending effects and displays the same asymptotic behavior as the Koiter shell model when the thickness h tends to 0. An existence and uniqueness result is given.

Existence and uniqueness for the linear Koiter model for shells with little regularity

We give a simple proof of existence and uniqueness of the solution of the Koiter model for linearly elastic thin shells whose midsurfaces can have charts with discontinuous second derivatives. The proof is based on new expressions for the linearized strain and change of curvature tensors. It also makes use of a new version of the rigid displacement lemma under hypotheses of regularity for the displacement and the midsurface of the shell that are weaker than those required by earlier proofs.

Une justification du modèle de coques de Naghdi

We consider a thin linearly elastic loaded shell allowing non-zero inextensional displacements. Under some assumptions on the loads, we prove that the tangential and normal parts of the stress tensor are small compared with the transverse pan, when the thickness of the shell goes to zero. Besides, the displacement vector and the transverse pan of the stress tensor are of the same order of magnitude with respect to the thickness when the material constituting the shell is Isotropic and homogeneous. The limit model, which is a flexural model, can also be obtained from Naghdi's model but not from Koiter's model. In some cases of anisotropic materials, the displacement vector is of a larger order of magnitude than the stress tensor, when the thickness goes to zero.

Two-Dimensional Models of Linearly Elastic Shells: Error Estimates between Their Solutions

The author considers a family of linearly elastic shells, all having the same middle surface that is assumed to be elliptic. Under some regularity assumptions on the data, the author gives an error estimate between the solution of Koiter's model and the solution of a two-dimensional membrane shell problem. The proof uses a corrector method.

Ellipticity of koiter's and naghdi's models for nonhomogeneous anisotropic shells

We show that the bilinear forms associated to the linear thin shell models of Koiter and Naghdi, for a nonhomogeneous and anisotropic material are elliptic. Essentially we use the fact that these bilinear forms are deduced from the three dimensional elasticity tensor, which is positive defined and also the techniques introduced by M. Bernadou, P.G. Ciarlet and B. Miara for homogeneous and isotropic shells (cf. [1], [2], [3], [4]). The ellipticity assures the existence and uniqueness of solution for these models. For Koiter's model, a similar problem was considered by D. Caillerie and E. Sanchez-Palencia [5]

Justification asymptotique des hypothèses de Kirchhoff-Love pour une coque encastrée linéairement élastique

The displacement vector of a linearly elastic shell can be computed by using the twodimensional Koiter's model, based on the a priori Kirchhoff-Love assumptions. These hypotheses imply that the displacement of any point of the shell is an affine function of the transverse variable x 3. The term independent of x 3 of this approximation is equal to the displacement vector of the two-dimensional Koiter's model. The term linear in x 3 depends on the rotation vector of the normal. After an appropriate scaling, we here estimate the difference between the three-dimensional displacement and the affine function in the case of shells clamped along their entire lateral face. Besides, in the case of shells with uniformly elliptic middle surface, taking into account the term depending of the rotation of the normal, allows to improve the asymptotic estimate between the three-dimensionnal displacement and Koiter's bidimensional displacement.