Finite Element Analysis: Thermomechanics of Solids, Second Edition

475 Pages·2008·2.18 MB·English

by  David W. Nicholson

Second Edition
FINITE
ELEMENT
ANALYSIS
Thermomechanics
of Solids
(cid:1)2008byTaylor&FrancisGroup,LLC.
(cid:1)2008byTaylor&FrancisGroup,LLC.
Second Edition
FINITE
ELEMENT
ANALYSIS
Thermomechanics
of Solids
David W. Nicholson
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
(cid:1)2008byTaylor&FrancisGroup,LLC.
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Library of Congress Cataloging-in-Publication Data
Nicholson, D. W.
Finite element analysis : thermomechanics of solids / by David W. Nicholson.
-- 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-1-4200-5095-0
ISBN-10: 1-4200-5095-8
1. Thermal stresses--Mathematical models. 2. Finite element method. I. Title.
TA418.58.N53 2008
620.1’121--dc22 2007042632
Visit the Taylor & Francis Web site at
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and the CRC Press Web site at
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(cid:1)2008byTaylor&FrancisGroup,LLC.
Dedication
To Linda and Mike,
with the deepest love
(cid:1)2008byTaylor&FrancisGroup,LLC.
(cid:1)2008byTaylor&FrancisGroup,LLC.
Contents
Preface totheSecond Edition
Preface totheFirst Edition
Author
Acknowledgments
Chapter1 Introduction to theFinite ElementMethod
1.1 Introduction
1.2 Overview of the Finite Element Method
1.3 Mesh Development
Chapter2 Mathematical Foundations: Vectors andMatrices
2.1 Introduction
2.1.1 Range and Summation Convention
2.1.2 SubstitutionOperator
2.2 Vectors
2.2.1 Notation:Scalar andVectorProducts
2.2.2 Gradient, Divergence, andCurlin RectilinearCoordinates
2.3 Matrices
2.4 Eigenvalues andEigenvectors
2.5 CoordinateTransformations
2.5.1 Transformations ofVectors
2.6 Orthogonal Curvilinear Coordinates
2.7 Gradient Operator in OrthogonalCoordinates
2.8 Divergence and Curl ofVectors inOrthogonalCoordinates
2.9 AppendixI:Divergenceand Curl ofVectorsin OrthogonalCurvilinear
Coordinates
Chapter3 Mathematical Foundations: Tensors
3.1 Tensors
3.2 Divergence of aTensor
3.3 Invariants
3.4 Positive Definiteness
3.5 Polar Decomposition Theorem
3.6 KroneckerProducts ofTensors
3.6.1 VEC Operator andthe Kronecker Product
3.6.2 Fundamental Relationsfor Kronecker Products
3.6.3 Eigenstructures ofKronecker Products
3.6.4 KroneckerForm ofQuadratic Products
3.6.5 KroneckerProduct Operatorsfor Fourth-Order Tensors
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3.6.6 TransformationProperties of VEC, TEN22, TEN21,and TEN12
3.6.7 Kronecker Expressions for Symmetry Classes inFourth-Order
Tensors
3.6.8 Differentials of Tensor Invariants
3.7 Examples
Chapter 4 Introduction toVariational Methods
4.1 IntroductoryNotions
4.2 Properties ofthe Variational Operator d
4.3 Example:VariationalEquation fora Cantilevered
Elastic Rod
4.4 Higher Order Variations
4.5 Examples
Chapter 5 Fundamental Notions of Linear Solid Mechanics
5.1 Displacement Vector
5.2 Linear Strain andRotation Tensors
5.3 Examplesof Linear Strain andRotation Tensors
5.4 Traction and Stress
5.5 Equilibrium
5.6 Stress and Strain Transformations
5.7 Principal Stresses andStrains
5.8 Stress–StrainRelations
5.9 Principle ofVirtualWork in Linear Elasticity
Chapter 6 Thermal andThermomechanical Response
6.1 Balance ofEnergyand Productionof Entropy
6.1.1 Balance ofEnergy
6.1.2 Entropy Production Inequality
6.1.3 Thermodynamic Potentials inReversibleProcesses
6.2 Classical Coupled Linear Thermoelasticity
6.3 Thermal and Thermomechanical Analogs ofthe Principle of Virtual
Work and Associated Finite ElementEquations
6.3.1 ConductiveHeat Transfer
6.3.2 Coupled Linear Isotropic Thermoelasticity
Chapter 7 One-Dimensional Elastic Elements
7.1 InterpolationModelsfor One-DimensionalElements
7.1.1 Rods
7.1.2 Beams
7.1.3 Beam-Columns
7.2 Strain–DisplacementRelationsin One-DimensionalElements
7.3 Stress–StrainRelationsin One-Dimensional Elements
7.3.1 General
7.3.2 One-DimensionalMembers
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7.4 ElementStiffness and Mass Matrices from the Principle
of Virtual Work
7.4.1 Single-Element Model for DynamicResponse
of aBuilt-in Beam
7.5 Integral Evaluationby Gaussian Quadrature:Natural Coordinates
7.6 Unconstrained Rod Elements
7.7 Unconstrained Elements forBeams and Beam-Columns
7.8 Assemblage andImposition ofConstraints
7.8.1 Rods
7.8.2 Beams
7.9 Damping inRodsand Beams
7.10 General Discussionof Assemblage
7.11 General Discussionon the Imposition ofConstraints
7.12 InverseVariational Method
Chapter8 Two- and Three-DimensionalElements in Linear Elasticity
andLinear Conductive Heat Transfer
8.1 Interpolation Models inTwo Dimensions
8.1.1 Membrane Plate
8.1.2 Plate withBending StressesOnly
8.1.3 Plate withStretching and Bending
8.1.4 Temperature Field inTwo Dimensions
8.1.5 Axisymmetric Elements
8.2 Interpolation Models inThree Dimensions
8.3 Strain–Displacement Relations and Thermal Analogs
8.3.1 Strain–Displacement Relations: Two Dimensions
8.3.2 Axisymmetric Element
8.3.3 Thermal Analogfor Two-Dimensional and Axisymmetric
Elements
8.3.4 Three-DimensionalElements
8.3.5 Thermal Analogin Three Dimensions
8.4 Stress–Strain Relations
8.4.1 Two-Dimensional Elements
8.4.1.1 Membrane Response
8.4.1.2 Two-DimensionalMembers: BendingResponse
ofThin Plates
8.4.1.3 Element forPlate with Membrane andBending
Response
8.4.2 Axisymmetric Element
8.4.3 Three-DimensionalElement
8.4.4 Elements for ConductiveHeat Transfer
8.5 Stiffness and Mass Matrices and Their Thermal Analogs
8.6 Thermal Counterpart ofthe Principle ofVirtual Work
8.7 Conversion toNatural Coordinates in Twoand Three Dimensions
8.8 AssemblyofTwo- andThree-Dimensional Elements
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