Mathematical Analysis of Thin Plate Models

ISBN-13: 9783540611677 / Angielski / Miękka / 1996 / 236 str.

Philippe Destuynder;Michel Salaun

Opis

Shells and plates have been widely studied by engineers during the last fifty years. As a matter of fact an important number of papers have been based on analytical calculations. More recently numerical simulations have been extensively used. for instance for large displacement analysis. for shape optimization or even -in linear analysis -for composite material understanding. But all these works lie on a choice of a finite element scheme which contains usually three kinds of approximations: 1. a plate or shell mndel including smnll parameters associated to the thickness, 2. an approximntion of the geometry (the medium sUrface of a shell and its boundary), 3. afinite element scheme in order to solve the mndel chosen. VI Obviously the conclusions that we can draw are very much depending on the quality of the three previous choices. For instance composite laminated plates with damage like a delamination is still an open problem even if interesting papers have already been published and based on numerical simulation using existing fmite element and even plate models. - In our opinion the understanding of plate modelling is still an area of interest. Furthermore the links between the various models have to be handled with care. The certainly best understood model is the Kirchhoff-Love model which was completely justified by P. O. Ciarlet and Ph. Destuynder in linear analysis using asymptotic method. But the conclusion is not so clear as far as large displacements are to be taken into account.

Spis treści

I — Plate models for thin structures.- I.0 — A short description of the chapter.- I.1 — The three dimensionnal elastic-model.- I.1.1 — About the kinematics.- I.1.2 — About the Principle of virtual work.- I.1.3 — About the constitutive relationship.- I.1.4 — Existence uniqueness of the solution to the elastic model.- I.2 — The Kirchhoff-Love assumption.- I.3 — The Kirchhof f-Love plate model.- I.3.1 — Existence and uniqueness of a solution to the Kirchhof f-Love model.- I.3.2 — The local equations satisfied by the Kirchhoff-Love plate model.- I.3.3 — The transverse shear stress in Kirchhoff-Love theory.- I.4 — The Naghdi model revisited using mixed variational formulation.- I.4.1 — Existence and uniqueness of a solution to the revisited Naghdi model.- I.4.2 — Local equations of the Naghdi model.- I.5 — About the rest of the book.- References of Chapter I.- II — Variational formulations for bending plates.- II.0 — A brief summary of the chapter.- II. 1 — Why a mixed formulation for plates.- II.2 — The primal variational formulation for Kirchhoff-Love model.- II.2.1 — Double Stokes formula for plates.- II.2.2 — The variational formulation.- II.2.3 — Another variational formulation.- II.2.4 — Interest of formulation (II. 12).- II.3 — The Reissner-Mindlin-Naghdi model for plates.- II.3.1 — The penalty method applied to the Kirchhoff-Love model.- II.3.2 — A correction to the penalty method.- II.4 — Natural duality techniques for the bending plate model.- II.4.1 — A mixed variational formulation for Kirchhoff-Love model.- II.4.2 — Existence and uniqueness of solution to the mixed formulation.- II.4.3 — Computation of the deflection u3.- II.4.4 — How to be sure we solved the right model (interpretation of the model).- II.4.5 — What is the meaning of ? and when is it zero?.- II.4.6 — Non-homogeneous boundary conditions.- II.4.7 — The revisited modified Reissner-Mindlin-Naghdi model.- II.4.8 — Extension to a multi-connected boundary.- II.5 — A comparison between the mixed method and the one of section II.2.4.- References of Chapter II.- III — Finite element approximations for several plate models.- III.0 — A summary of the chapter.- III. 1 — Basic results in finite element approximation.- III. 1.1 — Several useful definitions.- III. 1.2 — A brief recall concerning error estimates.- III.2 — C1 elements.- III.3 — Primal finite element methods for bending plates.- III.4 — The penalty-duality finite element method for the bending plate model.- III.4.1 — Stability with respect to the penalty parameter of the R.M.N. solution.- III.4.2 — A finite element scheme and error estimates for the R.M.N. model.- III.4.3 — Practical aspects in solving the R.M.N. finite element model.- III.4.4 — About the famous QUAD4 element.- III.5 — Numerical approximation of the mixed formulation for a bending plate.- III.5.1 — General error estimates between (0,A) and (?, ?h).- III.5.2 — Theoretical estimates on u3 — u3h.- III.5.3 — A first choice of finite elements.- HI.5.4 — A second choice of finite elements.- References of Chapter III.- IV — Numerical tests for the mixed finite element schemes.- IV.0 — A brief description of the chapter.- IV. 1 — Precision tests for the mixed formulation.- IV. 1.1 — A recall of the equations to be solved.- IV. 1.2 — Numerical tests.- IV. 1.3 — A few remarks relative to the above numerical results.- IV.2 Vectorial and parallel algorithms for mixed elements.- IV.2.1 — Three strategies for solving the system (IV. 18).- IV.2.2 — Optimization of Crout factorization.- IV.2.3 — Optimization of node renumbering.- IV.2.4 — Numerical tests.- IV.3 — Concluding remarks.- References of Chapter IV.- V — A Numerical model for delamination of composite plates.- V.O — A brief description of the chapter.- V. 1 — What is delamination of thin multilayered plates.- V.2 — The three-dimensional multilayered composite plate model with delamination.- V.3 — A plate model for large delamination.- V.4 — The three-dimensional energy release rate.- V.4.1 — The energy release rate..- V.4.2 — The energy release rate for delaminated plates.- V.5 — The mechanical example and the numerical method.- V.5.1 — The specimen studied.- V.6 — Concluding remarks.- References of Chapter V.

Blurb

Ce livre est destiné aux enseignants, chercheurs et étudiants désireux de se familiariser avec les différents modèles de plaques minces et d'en maîtriser les problèmes mathématiques et d'approximation sous-jacents. Il contient essentiellement des résultats nouveaux et des applications originales à l'étude du délaminage des structures multicouche. La démarche est guidée par un souci de mettre en avant les points délicats dans la théorie des plaques minces.