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.
