MECHANICS OF CURVED COMPOSITES

SOLID MECHANICS AND ITS APPLICATIONS

Volume 78

Series Editor: G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written bij authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the toundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and strucmres; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see .final pages.

Mechanics of Curved Composites

by

S.D. Akbarov Yildiz Technical University, Istanbul, Turkey and Institute of Mathematics and Mechanics of Academy of Science of Azerbaijan, Baku, Azerbaijan

and

A.N. Guz Institute of Mechanics of National Academy of Science of Ukraine, Kiev, Ukraine

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-13: 978-1-4020-0383-7 DOl: 10.1007/978-94-010-9504-4

e-ISBN-13: 978-94-010-9504-4

Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands.

Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 2000 Kluwer Academic Publishers Softcover reprint of the hardcover I st edition 2000

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TABLE OF CONTENTS

Preface ............................................................................................................................ xi Acknowledgments ....................................................................................................... xiii About the Contents ........................................................................................................ xv

Introduction ...................................................................................... 1

1.1. Types of composite materials ................................................................. 1 1.2. Specific curving of reinforcing elements .................................................... 2 1.3. Background and brief review ................................................................. .4

1.3.1. Continuum approaches ................................................................ 5 1.3.2. Local approaches ....................................................................... 6

Chapter 1. Plane-curved Composites ................................................. 7

1.1. Classical theories .............................................................................. 7 1.2. Basic equations and boundary conditions .................................................. 9 1.3. Constitutive relations ....................................................................... 10

1.3.1. Geometrical notation ................................................................ 10 1.3.2. Ideal composites ...................................................................... 11 1.3.3. Curved composites ................................................................... 13

1.4. Displacement equations; formulation and solution ...................................... 22 1.4.1. The equation of motion ............................................................. 22 1.4.2. The small parameter method ........................................................ 24

1.5. Example for exact solution .................................................................. 25 1.6. Vibration problems ........................................................................... 28

1.6.1. A general solution procedure ....................................................... 28 1.6.2. An example ........................................................................... 30

1.7. Quasi-homogeneous stress states corresponding to pure shears ....................... 32 1.7.1. Pure shear in the OxZx3 plane (Fig.1.7.1) ...................................... 33

1.7.2. Pure shear in the OXlxZ plane (Fig.1.7.2) ....................................... 35

1.7.3. Pure shear in the OXlx3 plane (Fig. 1.5.1) ...................................... 38 1.8. Quasi-homogeneous states corresponding to tension-compression ................... 39

1.8.1. Thriaxial tension-compression (Fig.1.8.1) ........................................ 39 1.8.2. Uniaxial tension-compression along OXl axis .................................. 43

1.9. Some detailed results on quasi-homogeneous states .................................... .45 1.9.1. Values of the normalized elasticity constants ................................... .45 1.9.2. Pure shear in the OXlx3 plane .................................................... .46

1.9.3. Uniaxial tension-compression along the OXl axis ............................. .47 1.10. Composites with large-scale curving .................................................... .47

1.10.1. Basic assumptions ................................................................. .47 1.10.2. The solution method ............................................................... .48 1.10.3. An application ...................................................................... 50

1.11. Bibliographical notes ....................................................................... 54

VI TABLE OF CONTENTS

Chapter 2. General curved composites ............................................ .55

2.1. Some preliminary remarks on geometry .................................................. 55 2.2. Constitutive relations ........................................................................ 55 2.3. Explicit constitutive relations for small curving ........................................... 59 2.4. Displacements equations for small curving; formulation and solution .............. 61

2.4.1. The equations of motion ........................................................... 61 2.4.2. The small parameter method ....................................................... 62

2.5. Example of the small parameter method .................................................. 63 2.6. An exact solution ............................................................................. 65 2.7. Pure shear of composite materials ......................................................... 66

2.7.1. Notation and assumptions .......................................................... 66 2.7.2. Pure shear in the OXlx3 plane ..................................................... 67

2.8. Quasi-homogeneous stress state corresponding to triaxial tension-compression .............................................................................. 72

2.9. Approximate results for layered composites .............................................. 78 2.9.1. Pure shear in the OXlx3 plane .................................................... 78

2.9.2. Uniaxial tension-compression along the OXI axis .............................. 78 2.10. The applicability of the proposed approach ............................................. 81 2.11. Bibliographical notes ........................................................................ 81

Chapter 3. Problems for curved composites .................................... 83

3.1. Bending of a strip .............................................................................. 83 3.1.1. Basic equations and formulation of the problems ................................ 83 3.1.2. FEM modelling ....................................................................... 86 3.1.3. Periodic curving: numerical results ................................................ 90 3.1.4. Analysis of the numerical results. Local curving ............................... 101

3.2. Bending of a rectangular plate ............................................................. 104 3.2.1. Formulation .......................................................................... 104 3.2.2. Semi-analytical FEM modelling ................................................. 106 3.2.3. Numerical results ................................................................... 110

3.3. Vibration problems .......................................................................... 117 3.3.1. Formulation and solution ......................................................... 117 3.3 .2. Free vibration ....................................................................... 119 3.3.3. Forced vibration ................................................................... .124

3.4. Bibliographical notes ....................................................................... 127

Chapter 4. Plane-strain state in periodically curved composites ..... 129

4.1. Formulation .................................................................................. 129 4.1.1. Plane-strain state ................................................................... 129 4.1.2. The parameterisation ................................................................ 131 4.1.3. The presentation of the stress-strain values in series form .................... 133

TABLE OF CONTENTS vii

4.1.4. Contact conditions for each approximation (4.1.15) ........................... 134 4.2. Method of solution .......................................................................... 138

4.2.1. Co-phase periodically curved composite ....................................... 138 4.2.2. Anti-phase periodically curved composite ....................................... 144

4.3. Stress distribution in composites with alternating layers .............................. 149 4.3.1. Lower filler concentration ......................................................... 150 4.3.2. Composite material with co-phase curved layers ............................... 154 4.3.3. Composite material with anti-phase curved layers ............................. 161

4.4. Stress distribution in composites with partially curved layers ........................ 165 4.4.1. Alternating straight and co-phase curved layers ............................... 165 4.4.2. Alternating straight and anti-phase curved layers ............................... 171 4.4.3. Composite material with a single periodically curved layer. .................. 175

4.5. Viscoelastic composites .................................................................... 179 4.6. Stress distribution in composites with viscoelastic layers ............................ .182 4.7. Composite materials with anisotropic layers ............................................ 188

4.7.1. Rectilinear anisotropy ............................................................. 188 4.7.2. Curvilinear anisotropy ............................................................. 191

4.8. Numerical results: rectilinear anisotropy ............................................... .206 4.8.1. Co-phase curving of the layers .................................................... 207 4.8.2. Anti-phase curving of the layers ................................................. 211

4.9. Numerical results: curvilinear anisotropy ............................................... .214 4.9.1. Co-phase curving of the layers .................................................... 214 4.9.2. Anti-phase curving of the layers ................................................. 218

4.10. Bibliographical notes ..................................................................... 219

Chapter 5. Composites with spatially periodic curved layers ........... 221

5.1. Formulation .................................................................................. 221 5.2. The equation of contact surfaces .......................................................... 222 5.3. The presentation of the governing relations in series form ............................ 227 5.4. Method of solution ......................................................................... .233 5.5. Stress distribution ........................................................................... 240

5.5.1. Uniaxial loading ..................................................................... 240 5.5.2. Two-axial loading ................................................................... 249

5.6. Bibliographical notes ....................................................................... .253

Chapter 6. Locally-curved composites ........................................... .255

6.1. Formulation ................................................................................. 255 6.2. Method of solution ......................................................................... 256 6.3. Composite with alternating layers ........................................................ 258

6.3.1. Lower filler concentration ......................................................... 258 6.3.2. Composite with co-phase curved layers ........................................ .261

·6.3.3. Anti-phase curved layers .......................................................... 263 6.4. The influence oflocal curving form .................................................. '" .267

6.4.1. Co-phase curving .................................................................. 267

V111 TABLE OF CONTENTS

6.4.2. Anti-phase curving ................................................................. .276 6.5. Bibliographical notes ........................................................................ 283

Chapter 7. Fibrous composites ..................................................... .285

7.1. Fonnulation .................................................................................. 285 7.2. Method of solution for lower fiber concentration ...................................... 287

7.2.1. Governing equations and relations ............................................... 287 7.2.2. Periodical curving fonn .......................................................... .291 7.2.3. Local curving fonn ................................................................. 298

7.3. Method of solution for higher fiber concentrations ..................................... 299 7.3.1. Two fibers ........................................................................... 299 7.3.2. Periodically located row fibers ................................................... 308 7.3.3. Doubly-periodically located fibers ............................................... 315

7.4. Numerical results ........................................................................... .320 7.4.1. Elastic composite ................................................................... 320 7.4.2. Viscoelastic composite ............................................................ 326

7.5. Screwed fibers in an elastic matrix ........................................................ 329 7.5.1. Fonnulation and method of solution ............................................. 329 7.5.2. Numerical results .................................................................. .330

7.6. Bibliographical notes ...................................................................... 333

Chapter 8. Geometrically non-linear problems .............................. 335

8.1. Fonnulation. Governing relations and equations ....................................... 335 8.2. Method of solution .......................................................................... 339

8.2.1. Plane-strain state ................................................................... 339 8.2.2. Spatial stress state .................................................................. 343

8.3. Numerical results ................................ '" ............................. , ........... 348 8.3.1. Co-phase curving ................................................................... 348 8.3.2. Anti-phase curving .................................................................. 351

8.4. Bibliographical notes ........................................................................ 352

Chapter 9. Normalized modulus elasticity ...................................... 355

9.1. Basic equations .............................................................................. 355 9.2. Nonnalized moduli .......................................................................... 360 9.3. Numerical results ............................................................................ 363

9.3.1. Plane-curved structures ............................................................ 363 9.3.2. Spatially curved structures ........................................................ 364

9.4. Bibliographical notes ....................................................................... 365

TABLE OF CONTENTS ix

Chapter 10. Fracture problems ...................................................... 367

10.1. Fiber separation ........................................................................... 367 10.1.1. Continuum approach ............................................................ .368 10.1.2. Piece-wise homogeneous model. ............................................... 370 10.1.3. Local Fiber Separation ............................................................................ 371 .

10.2. Crack problems .............................................................. · .............. 371 10.2.1. Formulation ....................................................................... .371 10.2.2. Method of solution ................................................................ 378 10.2.3. Numerical results ................................................................. 387

10.3. Fracture in compression .................................................................. 391 10.3.1. Formulation and method of solution .......................................... .392 10.3.2. Co-phase periodically curved layers ........................................... 392 10.3.3. A single periodically curved layer .............................................. 395

10.4. Bibliographical notes .................................................................... .400

Supplement 1. Viscoelastic unidirectional composites in compression ....................................................................... .401

S.l.l. Fracture of unidirectional viscoelastic composites in compression ............... .401 S.1.2. Compressive strength in compression of viscoelastic unidirectional

composites ................................................................................. .410 S.1.3. Bibliographical notes ..................................................................... 414

Supplement 2. Geometrical non-linear and stability problems .. ...... 415

S.2.1. Geometrical non-linear bending of the strip ......................................... .415 S.2. 1. 1. Formulation ....................................................................... .415 S.2.1.2. FEM modelling .................................................................. .416 S.2.1.3. Numerical results ................................................................ .418

S.2.2. Stability loss of the strip ................................................................ .421 S.2.2.1. Formulation. FEM modelling .................................................. .421 S.2.2.2. Numerical results ................................................................. 423

S.2.3. Bibliographical notes ..................................................................... .425

References ........................................................................... ......... .427

References Supplement ................................................................. .435

Index ..................................................................................................... 437

PREFACE

This book is the frrst to focus on mechanical aspects of fibrous and layered composite material with curved structure. By mechanical aspects we mean statics, vibration, stability loss, elastic and fracture problems. By curved structures we mean that the reinforcing layers or fibres are not straight: they have some initial curvature, bending or distortion. This curvature may occur as a result of design, or as a consequence of some technological process.

During the last two decades, we and our students have investigated problems relating to curved composites intensively. These investigations have allowed us to study stresses and strains in regions of a composite which are small compared to the curvature wavelength. These new, accurate, techniques were developed in the framework of continuum theories for piecewise homogeneous bodies. We use the exact equations of elasticity or viscoelasticity for anisotropic bodies, and consider linear and non-linear problems in the framework of this continuum theory as well as in the framework of the piecewise homogeneous model. For the latter the method of solution of related problems is proposed. We have focussed our attention on self-balanced stresses which arise from the curvature, but have provided sufficient information for the study of other effects.

We assume that the reader is familiar with the theory of elasticity for anisotropic bodies, with partial differential equations and integral transformations, and also with the Finite Element Method.

We have designed the book for graduate researchers, for mechanical engineers designing composite materials for automobiles, trucks, flywheels etc; for civil engineers contemplating the use of composites in infrastructure; for aerospace engineers studying advanced airframe design; and for biomedical engineers developing lightweight composites for bone replacement and repair.

We provide bibliographical notes at the end of each chapter. For the papers which originally appeared in Russian, we have tried to cite only ones which have appeared in English translation; there are, however, some which have not been translated.

ACKNOWLEDGMENTS

We must mention several individuals and organisations that were of enormous help in writing this book. First we wish to thank the editor, Prof. Graham M. L. Gladwell for his invaluable help in editing.

We also wish to thank the scientific collaborators at the Institute of Mathematics and Mechanics of Academy of Science of Azerbaijan (Baku, Azerbaijan), the Institute of Mechanics of National Academy of Science of Ukraine (Kiev, Ukraine), and the Yildiz Technical University (Istanbul, Turkey), for their assistance in the investigations described in this book.

We also thank our colleagues Ercument Akat, Nazmiye Yahnioglu, Zafer Kutug and Arzu Cilli for help in producing the text and figures.

S.D. Akbarov and A.N. Guz

ABOUT THE CONTENTS

This book consists of an introduction, ten chapters and two supplements. In the, introduction some necessary information on the classification of composite materials and the classification of the curving in their structure are given together with the mechanical properties of some promising fiber and matrix materials. Various approaches in mechanics of composite materials with curved structures are analysed, and the scope of this book is determined.

In Chapter 1 a simple new version of continuum theory for composite materials with periodic plane-curved structure is presented. It is supposed that the curves are small-scale, in other words, the period of the curvature of the reinforcing elements in the structure of the composite materials is significantly less than the size of the elements of objects fabricated from these composite materials. First, some preliminary remarks on classical continuum approaches are given. Then the three-dimensional exact equations of motion, geometrical relations and boundary conditions in both geometrically linear and non-linear statements, for deformable solid body mechanics are presented. After these preparatory procedures the basic assumptions and relations of the simple new version of continuum theory for composite materials with plane-curved structures are detailed. A solution procedure for the continuum problems is given, then some dynamic and static problems are investigated. In these cases the influence of the periodic plane curving in the structure of the constituent materials on the mechanical behaviour of the composite is studied.

In Chapter 2 a continuum theory for composite materials with general periodic or locally curved structures is discussed, generalising the analysis given in Chapter 1.

In Chapter 3 the theory developed in Chapter 2 is combined with the [mite element method (FEM) to solve some two- and three-dimensional static and dynamic problems for strips and rectangular plates. The influence of the parameters defining the curved composite on the stress distribution, and on the natural frequencies of the plates is discussed.

It is evident that any continuum approach is an approximate one and the most accurate information on the stress distribution in the curved composites can be obtained only in the framework of the piecewise-homogeneous body model with the use of the exact equations of deformable solid body mechanics. Therefore in chapters 4-10 we study the problems of curved composites in the framework of the piecewise homogeneous body model. In Chapter 4 we consider plane strain problems of periodically plane-curved composites. First, the method is developed for cases in which the materials of the layers are:l) visco- elastic; 2) rectilinearly anisotropic; 3) curvilinearly anisotropic. Many numerical results and their analyses are given for stress distribution in such composites.

In Chapter 5 the problem formulation and solution method presented in Chapter 4 are developed for three-dimensional problems, namely, for composites with spatially curved layers. The solution method is detailed for periodic curving. Many numerical results are given for the stress distribution in such composites.

In Chapter 6 the stress state in composite material with locally plane-curved layers is investigated. All investigations are carried out for plane-strain state with the

xvi

use of the relations and assumptions given in Chapter 4. For various local curving form the distribution of the self-balanced stresses on the inter-layer surfaces are studied in detail.

In Chapter 7 the problem formulation and solution methods presented in Chapters 4-6 are developed for fibrous composites with curved structures, and various concrete problems are investigated. .

In some combinations of geometric and curvature parameters of the filler layers (or fibers) and of the values of the external force intensities it is necessary to investigate the problems using the geometrically non-linear statement considered in Chapter 8. Using the results of these investigations, we can determine the limit of the intensity of the external forces for which the results obtained in the linear statement are acceptable. Furthermore, we can determine the character of the influence of the geometrical non-linearity on the mechanical behaviour of composites. In this chapter we derive which are very important for investigations of the fracture of unidirectional composites with curved structure in compression.

In Chapter 9 the approach for determination of the normalized mechanical properties of composites with periodically curved layers is developed. For this purpose the results obtained in chapters 4-5 are used. Some numerical results related to the normalized moduli of elasticity are also discussed.

In Chapter 10 fracture problems of curved composites are investigated. First, the fiber separation effect in fracture mechanics of composite materials is explained. The crack problems typical to the composites with curved structures is also investigated. Moreover, the approach is suggested for investigation of the fracture of these composites under compression along the reinforcing elements.

In Supplement 1 we propose an approach for investigation of the fracture of the viscoelastic unidirectional composites in compression. Composites with initially insignificantly curved layers are taken as a model for the study of these fracture problems, and the method discussed in the latter section of Chapter 10 is developed for viscoelastic composites.

In Supplement 2 we use the continuum theory developed in Chapter 2 and with employing FEM to investigate some two-dimensional geometrically non-linear and stability problems for strips. We discuss the influence of the geometrical non-linearity on the stress distribution and the influence of the curving parameters of strip material on its stability.