Singular Problems in Shell Theory: Computing and Asymptotics

ISBN-13: 9783642138140 / Angielski / Twarda / 2010 / 266 str.

Evariste Sanchez-Palencia; Olivier Millet; Fabien Bechet

Opis

Thin shells are three-dimensional structures with a dimension (the thickness) small with respect to the two others.Such thin structures are widely used in automobileandaviation industries, or in civil engineering, because they provide animportantsti?ness, due to theircurvature, with a small weight. Fig. 0.1. Airbus A380 Fig. 0.2. Hemispherical roof (Marseille, France) One ofthechallenges is often to reduce the weight (andconsequently the thickness)oftheshells, preservingtheirsti?ness.So that it is essential to have 1 accuratemodelsforthinandevenverythinshells, andtobeabletocomputethe displacements resultingfromagivenloading.In particular, singularities leading to fractures in some cases must be absolutely predicted a priori and ofcourse avoided (see Fig.0.3 forexample). Since the pioneeringmodels of Novozhilov-Donnell 81] and Koiter 65] 66], numerous works havebeen devoted to establish linear and non linear elastic shell model usingdirect orsurfacic approaches 18] 25] 100]. More recently, the asymptoticmethods 87] havebeen used, to try tojustify rigorously, fromthe three-dimensional equations, the shell models obtained by direct approaches - lying onapriori assumption, andto construct new models 54] 55]. This way, 1 Very thin shells are present in certain domains of industry, as plastic ?lms for pa- aging or for electronics, streched sails, or even very thin metal sheets obtained by drawing. E. Sanchez-Palencia et al.: Singular Problems in Shell Theory, LNACM 54, pp. 1 11."

Recenzje

From the reviews:

"The book under review is devoted to a mathematically rigorous study of singularities in linear elastic shell theory which appear for very small thickness. ... This well-written book is a reader-friendly and good organized research work in the field of mathematical theory of shells. It can be recommended to highly-qualified experts in this field." (Igor Andrianov, Zentralblatt MATH, Vol. 1208, 2011)

Spis treści

Geometric Formalism of Shell Theory.- Singularities and Boundary Layers in Thin Elastic Shell Theory.- Anisotropic Error Estimates in the Layers.- Numerical Simulation with Anisotropic Adaptive Mesh.- Singularities of Parabolic Inhibited Shells.- Singularities of Hyperbolic Inhibited Shells.- Singularities of Elliptic Well-Inhibited Shells.- Generalities on Boundary Conditions for Equations and Systems: Introduction to Sensitive Problems.- Numerical Simulations for Sensitive Shells.- Examples of Non-inhibited Shell Problems (Non-geometrically Rigid Problems).

Blurb

It is known that deformations of thin shells exhibit peculiarities such as propagation of singularities, edge and internal layers, piecewise quasi inextensional deformations, sensitive problems and others, leading in most cases to numerical locking phenomena under several forms, and very poor quality of computations for small relative thickness. Most of these phenomena have a local and often anisotropic character (elongated in some directions), so that efficient numerical schemes should take them in consideration. This book deals with various topics in this context: general geometric formalism, analysis of singularities, numerical computing of thin shell problems, estimates for finite element approximation (including non-uniform and anisotropic meshes), mathematical considerations on boundary value problems in connection with sensitive problems encountered for very thin shells; and others. Most of numerical computations presented here use an adaptive anisotropic mesh procedure which allows a good computation of the physical peculiarities on one hand, and the possibility to perform automatic computations (without a previous mathematical description of the singularities) on the other. The book is recommended for PhD students, postgraduates and researchers who want to improve their knowledge in shell theory and in particular in the areas addressed (analysis of singularities, numerical computing of thin and very thin shell problems, sensitive problems). The lecture of the book may not be continuous and the reader may refer directly to the chapters concerned.