CONTINUUM THEORY OF THE MECHANICS OF FIBRE-REINFORCED COMPOSITES.pdf

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INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 282 e • . CONTINUUM THEORY OF THE MECHANICS OF FIBRE-REINFORCED COMPOSITES EDITED BY A.J.M. SPENCER THE UNIVERSITY OF NOTTINGHAM SPRINGER-VERLAG WIEN GMBH

This work is subjtd to copyright. All righ ts are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 1984 by Springer-Verlag Wien Originally published by Springer-Verlag Wien-New York in 1984 ISBN 978-3-211-81842-8 ISBN 978-3-7091-4336-0 (eBook) DOI 10.1007/978-3-7091-4336-0

PREFACE This booK is made up of tne notes for tne lectures 6ze .-"ive au6zors gave at the International Centre for Mecizanica~ Sciences in Udilie in Jdy 1981. The aim of the Scizool was to present an account of rec

CONTENTS CONSTITUTIVE THEORY FOR STRONGLY ANISOTROPIC SOLIDS A. J, M. Spencer Introduction Linear Elastic Constitutive Equations for Fibre-reinforced Material 2.1 Linear elasticity - one family of fibres 2.2 Linear elasticity - two families of fibres 1 2 3 Finite Elastic Constitutive Equations for Fibre-reinforced Material 3.1 Kinematics of finite deformations 3.2 Finite elasticity for one family of fibres 3.3 Finite elasticity for two families of fibres 4 Plasticity Theory for Fibre-reinforced Material 4.1 Yield functions for one family of fibres 4.2 Yield functions for two families of fibres 4. 3 Flow rules 4.4 Hardening rules 4.5 Small elastic-plastic deformations References 1 3 10 16 18 20 22 26 27 28 30

Contents II FINITE DEFORt·1ATION AND STRESS IN IDEAL FIBRE-REINFORCED t~ATERIALS T. G. Rogers 1 Theory of Ideal Fibre-reinforced Materials (IFRM) 1.1 Introduction 1.2 Kinematics 1.3 Stress and equilibrium 1.4' Stress discontinuities and singular stresses 2 Plane Finite Deformations 2.1 Introduction 2.2 Kinematics 2.3 Stress 2.4 Hyperbolicity and singular stress layers 3 Solutions of Plane Boundary Value Problems 3.1 Force resultants 3.2 Examples involving initially straight, parallel fibres 3.3 Example- squashing of a hollow tube 3.4 Non-uniqueness and energy considerations 4 Finite Axisymmetric Deformations 4.1 Introduction 4. 2 Kinematics 4.3 Stress References II I PLANE PROBLEt4S FOR FIBRE-REINFORCED LINEARLY ELASTIC SOLIDS 1 Introduction III 33 36 37 40 43 47 51 53 56 62 64 66 67 69 70 73

IV Contents 2 Plane Strain Problems for an Ideal Fibre-reinforced Material 2.1 Displacement boundary-value problems 2.2 A stress boundary-value problem 2.3 The general stress boundary-value problem 2.4 Three-point bending of a beam 3 4 Holes and Cracks in Ideal Fibre-reinforced Materials Plane Strain and Generalised Plane Stress Problems for Fibre-reinforced Materials 4.1 Plane strain for compressible materials 4.2 Generalised plane stress 4.3 Boundary-value problems 4.4 Edge tractions on the half-plane y > 0 4.5 Edge tractions on the half-plane x > 0 4.6 Infinite strip 4. 7 Finite regions References IV STRESS CHANNELLING A1~D BOUNDARY LAYERS ANISOTROPIC SOLIDS A. C. Pipkin IN STRONGLY 1 2 3 4 5 6 7 8 9 In traduction Plane Deformations of Highly Anisotropic Materials Solution in Complex Variables The Inextensible Theory Outer and Inner Limits Boundary Stress Concentration Layers Example: Sheared Sheet Interior Stress Concentration Interior Stress Concentrations. General 10 Interior Layer through Crack Tip References 76 77 77 81 90 93 100 102 102 103 lOS 107 117 120 123 126 128 130 132 134 137 138 140 142 145

Contents V PROBLEMS FOR HELICALLY WOUND CYLINDERS T. G. Roger>s 1 Balanced Angle-ply Cylinders 1.1 Introduction l. 2 Kinematics of infinitesimal deformations 1.3 Stress and equilibrium 1.4 Constitutive equations 2 Problems with Axial Symmetry 2.1 A simple example - uniform inflation and axial extension 2.2 Singular stress layers 2.3 Example - clamped pressurised cylinder 2.4 General axial symmetry 3 Pure Bending of Hollow Cylinders 3.1 Kinematics 3.2 Boundary conditions 3.3 Special case tan¢ 3.4 General case tan¢ 12 12 4 Large Deformations of Thick-walled Cylinders 4.1 Introduction 4.2 Kinematics and stress 4.3 Finite inflation and extension References VI 1 2 FRACTURE MECHANICS OF FIBER-REINFORCED MATERIALS A. C. Pipkin Introduction The Inextensible Theory. v 147 149 151 153 156 160 164 168 169 169 170 172 174 174 177 178 179 180

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