PLASTIC BEHAVIOUR OF MATERIALS - Tecno-Libro

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CONTENTS
1 THE TENSILE TEST 1
1.1 Introduction 1
1.2 Engineering stress and strain 2
1.3 Analysis of the engineering stress-strain curve 3
1.4 True stress and strain 7
1.5 Maximum load 10
1.6 Empirical stress-strain relations 11
1.7 SAMPLE PROBLEM 1 14
1.8 PROPOSED PROBLEMS 15
2 UNIAXIAL STRESS STATE 17
2.1 Introduction 17
2.2 The compression test 17
2.3 The Bauschinger effect 19
2.4 SAMPLE PROBLEM 2 22
2.5 PROPOSED PROBLEMS 23
3 THE YIELD CRITERION FOR METALLIC MATERIALS 25
3.1 Introduction 25
3.2 The yield criterion 25
3.3 Isotropic material 26
3.4 Yield criterion for metals and alloys 27
3.5 Geometrical representation 28
3.6 Tresca’s and von Mises’ criteria 31
3.7 SAMPLE PROBLEM 3 37
3.8 PROPOSED PROBLEMS 38
4 YIELD CRITERION FOR NON-METALLIC MATERIALS 39
4.1 Introduction 39
4.2 Intrinsic curve 40
4.3 Coulomb’s criterion 41
4.4 Drucker-Prager’s yield criterion 43
4.5 Strain hardening 44
4.6 Anisotropic materials 51
4.7 SAMPLE PROBLEM 4 53
4.8 SAMPLE PROBLEM 5 54
4.9 PROPOSED PROBLEMS 55
5 STRESS –STRAIN RELATIONS 57
5.1 Introduction 57
5.2 Prandtl-Reuss equations 60
5.3 Levy-Mises equations 62
5.4 Total strain theory 63
5.5 Validity of Total Strain Theory 68
5.6 SAMPLE PROBLEM 6 69
5.7 SAMPLE PROBLEM 7 70
5.8 PROPOSED PROBLEMS 71
6 PURE BENDING 73
6.1 Introduction 73
6.2 Analysis of the behaviour of a slice 74
6.3 Elastic behaviour 76
6.4 Yielding 78
6.5 SAMPLE PROBLEM 8 83
6.6 SAMPLE PROBLEM 9 84
6.7 PROPOSED PROBLEMS 85
7 ELASTIC-PLASTIC BENDING OF BEAMS 87
7.1 Introduction 87
7.2 Isostatic beams 88
7.3 Hyperstatic beams 91
7.4 Continuous beams 94
7.5 Collapse load computation using the virtual work method 96
7.6 PROPOSED PROBLEMS 101
8 ELASTIC PLASTIC BEHAVIOUR OF PLATES 103
8.1 Introduction 103
8.2 The yield line theory 103
8.3 Application of the principle of virtual work 105
8.4 SAMPLE PROBLEM 10 112
8.5 PROPOSED PROBLEMS 113
9 ELASTIC PLASTIC BEHAVIOUR OF TUBES 115
9.1 Introduction 115
9.2 Thin walled tubes 115
9.3 Strains of the thin walled tube in plastic state 119
9.4 Tube with free ends 120
9.5 Tube with closed ends 121
9.6 Maximum pressure 122
9.7 Thick-walled tubes 125
9.8 SAMPLE PROBLEM 11 133
9.9 PROPOSED PROBLEMS 134
10 VISCOPLASTICITY 135
10.1 Introduction 135
10.2 Definitions 135
10.3 Creep behaviour of materials 137
10.4 Creep at low temperatures (logarithmic creep) 139
10.5 Creep at high temperatures 140
10.6 Andrade’s creep law 141
10.7 Dependence of creep strain rate on stress and temperature 141
10.8 Creep under multiaxial stress states 142
10.9 Stress relaxation 144
10.10 SAMPLE PROBLEM 12 146
10.11 PROPOSED PROBLEMS 147

Preface

This book has been prepared for the students of the Bachelor in Materials Engineering of the Technical University of Madrid. It includes most of the lessons of Plasticity of the subject Mechanics of Materials III, given on the third year of the “Grado en Ingeniería de Materiales” at the “Escuela Tecnica Superior de Ingenieros de Caminos Canales y Puertos” of the “Universidad Politécnica de Madrid”.

The text is an introduction of the plasticity of materials. It is focused to introduce the student to the basic behaviour of materials under permanent deformations and the equations that govern those mechanisms. The book starts with a description of the
tensile test, and then extended to uniaxial stress state. The two next chapters cover the yield criterion for metallic and non-metallic materials. Chapter four is dedicated to the constitutive equations of isotropic plasticity, presenting the descriptions of the models of Prandtl-Reuss and Henky. Then the book focuses on some examples and applications like pure bending and its application to plastification of beams, and
plastification of plates. Chapter nine is dedicated to the plastification of tubes focusing on both, thin and also thick wall tubes. The book ends with an introduction to viscoplasticity and creep.

The book is complemented with some sample exercises that are solved, and also some
proposed exercises for the reader.