Elastoplasticity Theory (Backmatter Pages)
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CONTINUUM MECHANICS (Lecture Notes)
CONTINUUM MECHANICS (Lecture Notes) Zdenekˇ Martinec Department of Geophysics Faculty of Mathematics and Physics Charles University in Prague V Holeˇsoviˇck´ach 2, 180 00 Prague 8 Czech Republic e-mail: [email protected]ff.cuni.cz Original: May 9, 2003 Updated: January 11, 2011 Preface This text is suitable for a two-semester course on Continuum Mechanics. It is based on notes from undergraduate courses that I have taught over the last decade. The material is intended for use by undergraduate students of physics with a year or more of college calculus behind them. I would like to thank Erik Grafarend, Ctirad Matyska, Detlef Wolf and Jiˇr´ıZahradn´ık,whose interest encouraged me to write this text. I would also like to thank my oldest son Zdenˇekwho plotted most of figures embedded in the text. I am grateful to many students for helping me to reveal typing misprints. I would like to acknowledge my indebtedness to Kevin Fleming, whose through proofreading of the entire text is very much appreciated. Readers of this text are encouraged to contact me with their comments, suggestions, and questions. I would be very happy to hear what you think I did well and I could do better. My e-mail address is [email protected]ff.cuni.cz and a full mailing address is found on the title page. ZdenˇekMartinec ii Contents (page numbering not completed yet) Preface Notation 1. GEOMETRY OF DEFORMATION 1.1 Body, configurations, and motion 1.2 Description of motion 1.3 Lagrangian and Eulerian coordinates 1.4 Lagrangian and Eulerian variables 1.5 Deformation gradient 1.6 Polar decomposition of the deformation gradient 1.7 Measures of deformation 1.8 Length and angle changes 1.9 Surface and volume changes 1.10 Strain invariants, principal strains 1.11 Displacement vector 1.12 Geometrical linearization 1.12.1 Linearized analysis of deformation 1.12.2 Length and angle changes 1.12.3 Surface and volume changes 2. -
Universit`A Di Trento Dottorato Di Ricerca in Ingegneria Dei Materiali
Universit`adi Trento Dottorato di Ricerca in Ingegneria dei Materiali e delle Strutture XV ciclo BOUNDARY ELEMENT TECHNIQUES IN FINITE ELASTICITY MICHELE BRUN Relatore: Prof. Davide Bigoni Correlatore: Dr. Domenico Capuani Trento, Febbraio 2003 Universit`adegli Studi di Trento Facolt`adi Ingegneria Dottorato di Ricerca in Ingegneria dei Materiali e delle Strutture XV ciclo Esame finale: 14 febbraio 2003 Commissione esaminatrice: Prof. Giorgio Novati, Universit`adegli Studi di Trento Prof. Francesco Genna, Universit`adegli Studi di Brescia Prof. Alberto Corigliano, Politecnico di Milano Membri esperti aggiunti: Prof. John R. Willis, University of Cambridge, England Prof. Patrick J. Prendergast, Trinity College, Dublin Acknowledgements The research reported in this thesis has been carried out in the Department of Mechanical and Structural Engineering at the University of Trento during the last three years. I would like to gratefully acknowledge my supervisor, Prof. Davide Bigoni, for all the stimulating and brilliant discussions, for his non-stop encourage- ments, remarks, corrections and suggestions. I would like to thank Dr. Domenico Capuani for his accurate, precise and fruitful help. I wish to express my sincere gratitude to Prof. Giorgio Novati, and to all members of the group of mechanics at the University of Trento: Prof. Marco Rovati, Prof. Antonio Cazzani (and Lavinia), Dr. Roberta Springhetti (and child) and little scientists Arturo di Gioia, Giulia Frances- chini, Massimilano Margonari and Andrea Piccolroaz. I am indebted to Massimiliano Gei for his suggestions about the topic of my research and to Katia Bertoldi for the help in the development of numerical applications. Last but not the least, I thank my family and the unique Elisa for their love, patience and fully support. -
Rotations in Computational Solid Mechanics
2 1 Vol. , , 49–138 (1995) Archives of Computational Methods in Engineering State of the art reviews Rotations in Computational Solid Mechanics S.N. Atluri† A.Cazzani†† Computational Modeling Center Georgia Institute of Technology Atlanta, GA 30332-0356, U.S.A. Summary A survey of variational principles, which form the basis for computational methods in both continuum me- chanics and multi-rigid body dynamics is presented: all of them have the distinguishing feature of making an explicit use of the finite rotation tensor. A coherent unified treatment is therefore given, ranging from finite elasticity to incremental updated La- grangean formulations that are suitable for accomodating mechanical nonlinearities of an almost general type, to time-finite elements for dynamic analyses. Selected numerical examples are provided to show the performances of computational techniques relying on these formulations. Throughout the paper, an attempt is made to keep the mathematical abstraction to a minimum, and to retain conceptual clarity at the expense of brevity. It is hoped that the article is self-contained and easily readable by nonspecialists. While a part of the article rediscusses some previously published work, many parts of it deal with new results, documented here for the first time∗. 1. INTRODUCTION Most of the computational methods in non-linear solid mechanics that have been developed over the last 20 years are based on the “displacement finite element”formulation. Most problems of structural mechanics, involving beams, plates and shells, are often characterized by large rotations but moderate strains. In a radical departure from the mainstream activities of that time, Atluri and his colleagues at Georgia Tech, have begun in 1974 the use of primal, complementary, and mixed variational principles, involving point-wise finite rotations as variables, in the computational analyses of non-linear behavior of solids and structures. -
Chapter 1 Introduction to Elasticity
Chapter 1 Introduction to Elasticity This introductory chapter presents some basic concepts of continuum mechan- ics, symbols and notations for future reference. 1.1 Kinematics of finite deformations We call a material body, defined to be a three-dimensional differentiable manifold, theB elements of which are called particles (or material points) P . This manifold is referred to a system of co-ordinates which establishes a one-to-one correspondence between particles and a region B (called a configuration of ) in three-dimensional Euclidean space by its position vector X(P ). As the bodyB deforms, its configuration changes with time. Let t I R denote time, and ∈ ⊂ associate a unique Bt, the configuration at time t of ; then the one-parameter family of all configurations B : t I is called a motionB of . { t ∈ } B It is convenient to identify a reference configuration, Br say, which is an arbi- trarily chosen fixed configuration at some prescribed time r. Then we label by X any particle P of in B and by x the position vector of P in the configuration B r Bt (called current configuration) at time t. Since Br and Bt are configurations of , there exists a bijection mapping χ : B B such that B r → t x = χ(X) and X = χ−1(x). (1.1) The mapping χ is called the deformation of the body from Br to Bt and since the latter depends on t, we write −1 x = χt(X) and X = χt (x), (1.2) instead of (1.1), or equivalently, x = χ(X,t) and X = χ−1(x,t), (1.3) for all t I.