The Finite Element Method for 3D Thermomechanical Applications - Guido Dhond.pdf

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The Finite Element
Method for Three-dimensional
Thermomechanical Applications
The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt
2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8
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The Finite Element
Method for Three-dimensional
Thermomechanical Applications
Guido Dhondt
Munich, Germany
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Copyright 2004
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A catalogue record for this book is available from the British Library
ISBN 0-470-85752-8
Produced from LaTeX files supplied by the author, typeset by Laserwords Private Limited, Chennai, India
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
This book is printed on acid-free paper responsibly manufactured from sustainable forestry
in which at least two trees are planted for each one used for paper production.
To my wife Barbara and my children Jakob and Lea
Contents
Preface
xiii
Nomenclature
xv
1 Displacements, Strain, Stress and Energy 1
1.1 TheReferenceState .............................. 1
1.2 TheSpatialState................................ 4
1.3 StrainMeasures................................. 9
1.4 PrincipalStrains ................................ 13
1.5 Velocity..................................... 19
1.6 ObjectiveTensors................................ 22
1.7 BalanceLaws.................................. 25
1.7.1 Conservationofmass ......................... 25
1.7.2 Conservationofmomentum...................... 25
1.7.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . 26
1.7.4 Conservationofenergy ........................ 26
1.7.5 Entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7.6 Closure................................. 28
1.8 LocalizationoftheBalanceLaws....................... 28
1.8.1 Conservationofmass ......................... 28
1.8.2 Conservationofmomentum...................... 29
1.8.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . 31
1.8.4 Conservationofenergy ........................ 31
1.8.5 Entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.9 TheStressTensor................................ 31
1.10TheBalanceLawsinMaterialCoordinates.................. 34
1.10.1Conservationofmass ......................... 35
1.10.2Conservationofmomentum...................... 35
1.10.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . 37
1.10.4Conservationofenergy ........................ 37
1.10.5 Entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.11 The Weak Form of the Balance of Momentum . . . . . . . . . . . . . . . . 38
1.11.1 Formulation of the boundary conditions (material coordinates) . . . 38
1.11.2 Deriving the weak form from the strong form (material coordinates) 39
1.11.3 Deriving the strong form from the weak form (material coordinates) 41
1.11.4 The weak form in spatial coordinates . . . . . . . . . . . . . . . . . 41
1.12TheWeakFormoftheEnergyBalance.................... 42
1.13 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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