An Introduction to Nonlinear Finite Element Analysis

Transcription

1 An Introduction to Nonlinear Finite Element Analysis

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3 J. N. REDDY Distinguished Professor Department of Mechanical Engineering Texas A&M University, College Station Texas , USA An Introduction to Nonlinear Finite Element Analysis OXFORD UNIVERSITY PRESS

4 OXFORD UNIVERSITY PRESS

5 To My beloved teacher Professor John Tinsley Oden

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7 About the Author J. N. Reddy is Distinguished Professor and the Holder of Oscar S. Wyatt Endowed Chair in the Department of Mechanical Engineering at Texas A&M University, College Station, Texas. Prior to the current position, he was the Clifton C. Garvin Professor in the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University (VPI&SU), Blacksburg, Virginia. Dr. Reddy is internationally known for his contributions to theoretical and applied mechanics and computational mechanics. He is the author of over 300 journal papers and dozen other books, including Introduction to the Finite Element Method (Second Edition), McGraw-Hill, l993; Theory and Analysis of Elastic Plates, Taylor & Francis, 1999; Energy Principles and Variational Methods in Applied Mechanics (Second Edition), John Wiley, 2002; Mechanics of Laminated Plates and Shells: Theory and Analysis, (second Edition) CRC Press, 2004; An Introduction to the Mathematical Theory of Finite Elements (coauthored with J. T. Oden), John Wiley, l976; Variational Methods in Theoretical Mechanics (coauthored with J. T. Oden), Springer Verlag, 1976; The Finite Element Method in Heat Transfer and Fluid Dynamics (Second Edition; coauthored with D. K. Gartling), CRC Press, 2001; and Professor Reddy is the recipient of the Walter L. Huber Civil Engineering Research Prize of the American Society of Civil Engineers (ASCE), the Worcester Reed Warner Medal and the Charles Russ Richards Memorial Award of the American Society of Mechanical Engineers (ASME), the 1997 Archie Higdon Distinguished Educator Award from the American Society of Engineering Education (ASEE), the 1998 Nathan M. Newmark Medal from the American Society of Civil Engineers, the 2000 Excellence in the Field of Composites from the American Society of Composites, the 2000 Faculty Distinguished Achievement Award for Research and the 2003 Bush Excellence Award for Faculty in International Research from Texas A&M University, and the 2003 Computational Solid Mechanics Award from the U.S. Association of Computational Mechanics (USACM). Dr. Reddy is a Fellow of ASCE, ASME, the American Academy of Mechanics (AAM), the American Society of Composites, the U.S. Association of Computational Mechanics, the International Association of Computational Mechanics (IACM), and the Aeronautical Society of India (ASI). Professor Reddy is the Editor-in- Chief of Mechanics of Advanced Materials and Structures, International Journal of Computational Engineering Science and International Journal of Structural Stability and Dynamics, and he serves on the editorial boards of over two dozen other journals, including International Journal for Numerical Methods in Engineering, Computer Methods in Applied Mechanics and Engineering, and International Journal of Non- Linear Mechanics.

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9 ix Contents Preface xvii 1 Introduction Mathematical Models Numerical Simulations The Finite Element Method Nonlinear Analysis Introduction ClassiÞcation of Nonlinearities The Big Picture 11 References 12 2 The Finite Element Method: A Review Introduction One-Dimensional Problems Governing Differential Equation Finite Element Approximation Derivation of the Weak Form Interpolation Functions Finite Element Model Two-Dimensional Problems Governing Differential Equation Finite Element Approximation Weak Formulation Finite Element Model Interpolation Functions Assembly of Elements Library of Two-Dimensional Finite Elements Introduction Triangular Elements Rectangular Elements Numerical Integration Preliminary Comments Coordinate Transformations Integration Over a Master Rectangular Element Integration Over a Master Triangular Element 45

10 x 2.6 Computer Implementation General Comments One-Dimensional Problems Two-Dimensional Problems Closure 53 Problems 57 References 59 3 Heat Transfer and Other Field Problems in One Dimension Model Differential Equation Weak Formulation Finite Element Model Solution Procedures General Comments Direct Iteration Procedure Newton s Iteration Procedure Computer Implementation Introduction Preprocessor Unit Processor Unit Closing Remarks 82 Problems 82 References 85 4 Nonlinear Bending of Straight Beams Introduction Euler Bernoulli Beams Basic Assumptions Displacement Field and Strains Weak Forms Finite Element Model Iterative Solutions of Nonlinear Equations Load Increments Membrane Locking Computer Implementation Timoshenko Beams Displacement Field and Strains Weak Forms General Finite Element Model Shear and Membrane Locking Tangent Stiffness Matrix 117

11 xi Problems 124 References Heat Transfer and Other Fields Problems in Two Dimensions Model Equation Weak Form Finite Element Model Solution Procedures Direct Iteration Newton Raphson Iteration Computer Implementation Introduction Numerical Integration Element Calculations 135 Problems 139 References Nonlinear Bending of Elastic Plates Introduction Classical Plate Theory Assumptions of the Kinematics Displacement Field and Strains Variational Formulation of CPT Virtual Work Weak Forms Equilibrium Equations Boundary Conditions Stress Resultant Deßection Relations Finite Element Models of CPT General Formulation Tangent Stiffness Coefficients Some Plate Finite Elements Computer Implementation Aspects and Numerical Results of CPT Elements Computer Implementation Results of Linear Analysis Results of Nonlinear Analysis First Order Shear Deformation Plate Theory Introduction Displacement Field Weak Formulation 174

12 xii 6.7 Finite Element Models of FSDT Virtual Work Statements Finite Element Model Tangent Stiffness Coefficients Shear and Membrane Locking Computer Implementation and Numerical Results of FSDT Elements Computer Implementation Results of Linear Analysis Results of Nonlinear Analysis Theory of Doubly-Curved Shells Introduction Geometric Description Strain Displacement Relations Stress Resultants Equations of Motion Finite Element Analysis of Shells Weak Forms Finite Element Model Linear Results Nonlinear Results 217 Problems 222 References Flows of Viscous Incompressible Fluids Introduction Governing Equations Introduction Conservation of Mass Conservation of Momenta Conservation of Energy Constitutive Equations Boundary Conditions Governing Equations in Terms of Primitive Variables Vector Form Cartesian Component Form Velocity Pressure Finite Element Model Weak Form Finite Element Model Penalty Finite Element Models Introduction Penalty Function Method 242

13 xiii Reduced Integration Penalty Model Consistent Penalty Model Computational Aspects Properties of the Matrix Equations Choice of Elements Evaluation of Element Matrices in Penalty Models Post-Computation of Stresses Computer Implementation Mixed Model Penalty Model Numerical Examples Preliminary Comments Fluid Squeezed Between Parallel Plates Flow of a Viscous Lubricant in a Slider Bearing Wall-Driven Cavity Flow Backward-Facing Step Least-Squares Finite Element Models Introduction Finite Element Model Computational Aspects Numerical Examples 275 Problems 283 References Nonlinear Analysis of Time Dependent Problems Introduction Time Approximations Introduction Parabolic Equations Hyperbolic Equations Stability and Accuracy Preliminary Comments Stability Criteria Transient Analysis of Nonlinear Problems Introduction Heat Transfer Flows of Viscous Incompressible Fluids Plate Bending (FSDT) Computer Implementation 304

14 xiv 8.6 Numerical Examples Linear Problems Nonlinear Problems 314 Problems 323 References Finite Element Formulations of Solid Continua Introduction Background Descriptions of Motion Strain and Stress Measures Deformation Gradient Tensor Green and Almansi Strain Tensors Polar Decomposition Stress Tensors Energetically Conjugate Stresses and Strains Strain and Stress Measures Between ConÞgurations Notation Conservation of Mass Green Strain Tensors for Various ConÞgurations Euler Strain Tensor Relationships Between Various Stress Tensors Constitutive Equations Total Lagrangian and Updated Lagrangian Formulations of Continua Principle of Virtual Displacements Total Lagrangian Formulation Updated Lagrangian Formulation Finite Element Models of Two-Dimensional Continua Introduction Total Lagrangian Formulation Updated Lagrangian Formulation Computer Implementation Numerical Results Shell Finite Elements Introduction Incremental Equations of Motion Finite Element Models of a Continuum Shell Finite Element Numerical Examples 378 Problems 381 References 387

15 xv 10 Material Nonlinearities and Coupled Problems Introduction Nonlinear Elastic Problems Small Deformation Theory of Plasticity Introduction Ideal Plasticity Strain Hardening Plasticity Elastic Plastic Analysis of a Bar Non-Newtonian Fluids Introduction Governing Equations in Cylindrical Coordinates Power-Law Fluids White Metzner Fluids Numerical Examples Coupled Fluid Flow and Heat Transfer Finite Element Models Numerical Examples 419 References 423 Appendix A1: Solution Procedures for Linear Algebraic Equations 425 A1.1 Introduction 425 A1.2 Direct Methods 427 A1.2.1 Preliminary Comments 427 A1.2.2 Symmetric Solver 428 A1.2.3 Unsymmetric Solver 430 A1.3 Iterative Methods 430 A1.3.1 General Comments 430 A1.3.2 Solution Algorithms 432 References 433 Appendix A2: Solution Procedures for Nonlinear Algebraic Equations 439 A2.1 Introduction 439 A2.2 Picard Iteration Method 440 A2.3 Newton Raphson Iteration Method 444 A2.4 Riks and ModiÞed Riks Schemes 448 References 457 Subject Index 459

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17 Preface The objective of this book is to present the theory and computer implementation of the Þnite element method as applied to simple nonlinear problems of heat transfer and similar Þeld problems, ßuid mechanics, and solid mechanics. Both geometric as well as material nonlinearities are considered, and static and transient (i.e. time-dependent) responses are studied. The guiding principle in writing the book was to make the presentation suitable for (a) adoption as a text book for a Þrst course on nonlinear Þnite element analysis (or for a second course following an introductory course on the Þnite element method), and (b) for use by engineers and scientists from various disciplines for self study and practice. There exist a number of books on nonlinear Þnite elements. Most of these books contain a good coverage of the topics of structural mechanics, and few addresstopicsofßuid dynamics and heat transfer. While these books serve as good references to engineers or scientists who are already familiar with the subject but wish to learn advanced topics or latest developments, they are not suitable as textbooks for a Þrst course or for self study on nonlinear Þnite element analysis. The motivation and encouragement that led to the writing of the present book have come from the users of the author s book, An Introduction to the Finite Element Method (McGraw Hill, 1984; Second Edition, 1993; third edition scheduled for 2004), who have found the approach presented there to be most suitable for any one irrespective of their scientiþc background interested in learning the method, and also from the fact that there does not exist a book that is suitable as a textbook for a Þrst course on nonlinear Þnite element analysis. The author has taught a course on nonlinear Þnite element analysis many times during the last twenty years, and the present book is an outcome of the lecture notes developed during this period. The same approach as that used in the aforementioned book, namely, the differential equation approach, is adopted in the present book to introduce the theory, formulation, and computer implementation of the Þnite element method as applied to nonlinear problems of science and engineering.

18 Beginning with a model (i.e. typical) second-order, nonlinear differential equation in one dimension, the book takes the reader through increasingly complex problems of nonlinear beam bending, nonlinear Þeld problems in two dimensions, nonlinear plate bending, nonlinear formulations of solid continua, ßows of viscous incompressible ßuids in two dimensions (i.e. Navier Stokes equations), time-approximation schemes, continuum formulations of shells, and material nonlinear problems of solid mechanics. As stated earlier, the book is suitable as a textbook for a Þrst course on nonlinear Þnite elements in civil, aerospace, mechanical, and mechanics departments as well as in applied sciences. It can be used as a reference by engineers and scientists working in industry, government laboratories and academia. Introductory courses on the Þnite element method, continuum mechanics, and numerical analysis should prove to be helpful. The author has beneþted in writing the book by the encouragement and support of many colleagues around the world who have used his book, An Introduction to the Finite Element Method, and students who have challenged him to explain and implement complicated concepts and formulations in simple ways. While it is not possible to name all of them, the author expresses his sincere appreciation. In particular, it is a pleasure to acknowledge the help of the author s students Juan P. Pontaza with the least-squares Þnite element analysis of ßuid ßow problems in Chapters 7 and 8, and Goy Teck Lim with the plasticity example in Chapter 10. The author expresses his deep sense of gratitude to his teacher, Professor J. T. Oden (University of Texas at Austin), to whom this book is dedicated and without whose advice, mentorship and support it would not have been possible for the author to modestly contribute to the Þeld of applied mechanics in general and theory and application of the Þnite element method in particular, through author s teaching, research, and technical writings. J. N. Reddy College Station, Texas