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Unformatted text preview: Pa!li Pedersen E L A S T I C I T Y  A N I S O T R O P Y L A M I N A T E S "i h ma ri# form!la ion, fini e elemen s and an inde# o ma rices x 3 + z x 2 x 1 s 11 å 11 s 12 å 12 s 21 å 21 s 22 å 22 ć g SOLID MECHANICS TECHNICAL UNIVERSITY OF DENMARK II Pauli Pedersen: 1. Contents and Introduction E L A S T I C I T Y  A N I S O T R O P Y  L A M I N A T E S with matrix formulation, finite elements and an index to matrices Copyright 1998 by Pauli Pedersen, but feel free to copy for personal use. ISBN 87904160105 III Pauli Pedersen: 1. Contents and Introduction FOREWORD This book originates from notes for a course in elasticity (Pedersen (1992)) and a translation from Danish is needed because this course is now taught in English. Extensions from earlier notes on laminates (Pedersen (1988)) and from a small book on finite elements (Pedersen (1984)) are naturally included. The influence from recent research on material optimization (Pedersen (1995a)) will also be seen and finally the notation is in accordance with the 2 Ǹcontracted notation, advoĆ cated in Pedersen (1995b). It is my hope that the reader will find the efforts worthwhile, and critics, advices and information about posĆ sible errors will be most welcome (email: pauli@fam.dtu.dk). Many students and colleagues have influenced the book and without mentioning specific names, I am most thankful for their guidance. I thank Robert Zetterlund for producing many of the figures and a lot of additional help; I thank Bente Brask Andersen and Betina ChristianĆ sen for the textediting with all its revisions. Without their valuable help the book would not have been finished. In gratitude of a warm family life, I dedicate this book to my wife Hanne and my children Niels, Mads and Sine. January 1997 Pauli Pedersen IV Pauli Pedersen: 1. Contents and Introduction This second version is almost unchanged, but the known errors have been corrected. Furthermore, the book is now available at the internet: http://www.fam.dtu.dk/html/pp.html November 1998 Pauli Pedersen V Pauli Pedersen: 1. Contents and Introduction CONTENTS page Foreword III 1. Introduction 1 2. Geometry  Kinematics 9 2.1 Displacement 9 2.2 Displacement field 10 2.3 Displacement gradients 10 2.4 Stretches and strains in 1D 11 2.5 Strains from displacement gradients 15 2.6 2D Strain description 17 2.7 3D Strain transformation 20 2.8 Dilatation and distortion, deviatoric strains 26 2.9 Compatibility of a strain field 28 2.10 Examples of strain calculations 29 VI Pauli Pedersen: 1. Contents and Introduction 3. Statics  Equilibrium 35 3.1 Force 35 3.2 Moment 37 3.3 Stress concept 38 3.4 Equilibrium 39 3.5 Descriptions of stress state 43 3.6 Hydrostatic stress and deviatoric stress 45 3.7 A classic example 46 4. Physics  Constitutive Models 53 4.1 Material parameters and modelling 53 4.2 Rotational transformations 56 4.3 Classification of 2D stiffness for anisotropic elasticity 61 4.4 From 3D to 2D constitutive matrix 67 4.5 4....
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