Metal and Ceramic Based Composites (Composite Materials Series, Volume 12)

METAL AND CERAMIC BASED COMPOSITES COMPOSITE MATERIALS SERIES Series Editor: R. Byron Pipes, Center for Composite Mat...

Author: S.T. Mileiko

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METAL AND CERAMIC BASED COMPOSITES

COMPOSITE MATERIALS SERIES Series Editor: R. Byron Pipes, Center for Composite Materials, University of Delaware, Newark, Delaware, USA Friction and Wear of Polymer Composites (K. Friedrich, Editor) Fibre Reinforcements for Composite Materials (A.R. Bunsell, Editor) Textile Structural Composites (T.-W. Chou and F.K. KO, Editors) Fatigue of Composite Materials (K.L. Reifsnider, Editor) Interlaminar Response of Composite Materials (N.J. Pagano, Editor) Application of Fracture Mechanics to Composite Materials (K. Fiedrich, Editor) Vol. 7 Thermoplastic Composite Materials (L.A. Carlsson, Editor) Vol. 8 Advances in Composite Tribology (K. Friedrich, Editor) Vol. 9 Damage Mechanics of Composite Materials (R. Talreja, Editor) Vol. 10 Flow and Rheology in Polymer Composites Manufacturing (S.G. Advani, Editor) Vol. 1 1 Composite Sheet Forming (D. Bhattacharyya, Editor)

Vol. Vol. Vol. Vol. Vol. Vol.

1 2 3 4 5 6

Cover illustration - The fracture surface of a sapphire fibre. The pore, certainly initiating the failure, can be seen. For further details see section 14.2.2.

Composite Materials Series, 12 METAL AND CERAMIC BASED COMPOSITES

SOT.Mileiko Solid State Physics Institute, Russian Academy of Sciences Chernogolovka Moscow district, 142432, Russia

1997 ELSEVIER Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam. The Netherlands

ISBN 0-444-82814- 1 (Vol. 12) ISBN 0-444-42525-X (Series)

0 1997 Elsevier

Science B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright and Permissions Department, P.O. Box 52 1 , 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper Printed in The Netherlands

PREFACE Modern scientific and technological fields are frequently of an interdisciplinary (composite) nature. The field of fibrous composites should be no exception. It has therefore long been the author's intention to combine the various aspects of composites into one composite field. Unlike fibre-reinforced plastics, the family of metal- and ceramic-based composites is still quite a new group of materials with a large variety of mechanical and physical properties. It is still rather tricky in producing such materials to track down the necessary technical information as this is not yet always well documented. To be an expert in the field of metal- and ceramic-matrix composites one has to be able to stroll freely over the whole field of composite materials. The main idea of the present book is to link together.fabrication - s t r u c t u r e properties chains, in order to make clear for composite makers what kind of structure provides the necessary properties and how to get the proper composite structure. People who are going to analyze material properties or use the materials should know about modern fabrication routes and understand what can be expected of these in the future. They have to be aware of the permanent and temporary limitations for making the things they would like to see. For a book to help reach goals such as these, it should contain, among topics of a purely technical nature, a description of the failure mechanics of metal- and ceramic-matrix composites because this is the key to understanding the s t r u c t u r e - properties segment of the chain mentioned. As it is expected that not all readers will be experts in theories of elasticity and fracture mechanics, the description of these is given in simple terms with references to the original papers for the more detailed mathematics. The general composition of the book is as follows. Part I Towards Composites presents a general view of composites with the accent on metal- and ceramic-matrix composites. The reader should come to understand that the occurrence of composites was an inevitable result of both the evolution of materials and the demands of modern technology. This part also contains a brief description of modern fibres and composites and can be considered, at least for beginners, as a base from which to proceed with further reading. Part II Failure Mechanics of Composites throws light on the composite microstructures considered to be either optimal or reasonable to resist a particular loading. A variety of mechanical, physical, and chemical potentialities for organizing such microstructures are described in Part III Technological Processes and Materials. Systematic information on some composites in connection with the corresponding fabrication processes is also presented here. It should be noted that experimental data on technologies, material structures, and material properties are used throughout the book to support theoretical conclusions or to obtain important physical parameters.

vi

Preface

The Russian author of this book set out to make a composite product of the book itself, in the sense that complementary Western and Russian results are an essential feature of it. The author understands the international character of science and technology but he still sees some important peculiarities of the recent Soviet (Russian) situation that are relevant to look at here. The R&D activity in the composite materials field in the Soviet Union was very high during the last two decades of the existence of the Union. There were several reasons for this. First, there was the obvious technical demand of the aerospace and some other high technology industries. The Government, directed by the Party Central Committee, was always very cautious about the technical standards of corresponding products. This brought a special result, namely that prices were fixed by the State. So society appeared not to need the mass production of composites as parts of sporting goods and similar things to stimulate a price reduction and a proper influence of the consumer goods market on aerospace and similar fields of production. This caused the situation to differ drastically from that in the West.. A second reason has obviously been the natural scientific interest of people who came into composites from related fields such as, for example, elasticity theory and physical m e t a l l u r g y - fields which were traditionally of very high professional level in this country. There appears to be a third reason which is partly psychological in nature. Indeed so many people here worked enthusiastically on composites not only because they had the necessary background but also because they had a subconscious hope to wake up one day and find themselves in a country whose economy was returning to a civilized state from the 'topsy-turvy' world in which ideological dogmas ruled instead of economical laws. Such situations lead to the differences between the present activity in composites in this country and in the West. For example, the levels of scientific results in mechanics and fabrication of fibre-reinforced polymers are about the same, but the quantity of materials used differs by an order of magnitude. On the other hand, the usage of metal matrix composites on both sides of the former curtains and walls is similarly small, but the quantity of research results obtained in those days on the Eastern side seems to be larger. A relevant point here is that Western researchers used to know little about Soviet results in this field because, firstly, Russian authors were publishing only a very small part of their results in international journals and, secondly, quite a large part of the results were classified - without any real justification. Since the Soviet Union transformed into a number of independent states (with Russia taking on a new role) and well-known events such as demilitarisation and the restructuring of the economy (with the enormous problems that accompany this), research in the composites field has continued (sometimes only because of i n e r t i a ) b u t has been changing its priorities and standards. Obviously, the situation will stabilize at some point, but at what point in the priorities/time space, one may ask. Nevertheless, history has set a challenge to use the Western and Eastern developments as complementary parts of a future world economy. Because a similar

Preface

vii

situation is certainly observed in various fields of science and technology, the challenge seems to be an impelling one. If this book contributes even a little to future international cooperation, the author will consider his mission as having been fulfilled. S.T. Mileiko May 1997

ACKNOWLEDGEMENTS I was writing this book in a rather hard time for this country, so the friendly support of my family and colleagues was especially stimulating. I am thankful to them. I am also pleased to acknowledge the participation of my lab and people of other groups in obtaining results, discussing them and presenting them in publications. I should particularly mention with many thanks V.M. Anishshenkov, I.L. Aptekar, T.A. Chernova, M.V. Gelachov, V.I. Glushko, S.I. Gvozdeva, V.I. Kazmin, A.A. Khvostunkov, V.M. Kiiko, L.S. Kozhevnikov, Nelly Prokopenko, A.M. Rudnev, Natalie Sarkissyan, O.A. Sarkissyan, D.B. Skvortsov, S.I. Trifonov, and V.V. Tvardovsky. Regrettably two of them, V.I. Kazmin and M.V. Gelachov, have passed away in the last decade. The patience of the staff of Elsevier involved in this project, which has gone on longer than originally planned, is gratefully acknowledged. Chernogolovka, Moscow district October 1997

viii

CONTENTS Preface v Acknowledgements

viii

Part I: Towards composites

Chapter I

Structural materials 3

1.1. 1.2. 1.3. 1.4.

Effectiveness of materials and structures Stiffness and strength of materials 20 Structural materials 26 Concluding remarks 34

3

Chapter I I

Fibres and fibrous composites 37 2.1. Fibre strength 37 2.2. Some structural fibres 46 2.3. Composites 63

Part 11: Failure mechanics of composites

Chapter III

Deformation and failure of composites 77 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

Elastic and plastic behaviour of anisotropic materials Elastic behaviour of composites 85 Non-elastic behaviour 11 1 Failure criteria 122 Fibre/matrix stress transfer 123 Shear-lag analysis 134 ix

77

C‘hupter I V

Macro- and microcracks in non-homogeneous materials 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7.

147

Cracks in homogeneous solids 148 Energy dissipation 164 Fibre cracking 169 Matrix cracking 177 Interface cracking 189 Cracks in dually non-homogeneous solids 204 Cracking in joints 217

Chupter V

Strength and fracture toughness

5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.

233

Strength of a fibre bundle 233 Brittle-fibre/ductile-matrix: strength 235 Composite strength and fibre properties 263 Brittle-fibre/ductile-matrix: fracture toughness 280 Ductile-fibre/brittle-matrix 289 Ductile-fibre/ductile-matrix: strength 291 Ductile-fibre/ductile-matrix: fracture toughness 295 Brittle-fibre/brittle-matrix 297

Chupter V I

Creep and creep rupture 307 6. I . Short fibre composites 307 6.2. Continuous fibre composites 323 6.3. Concluding remarks 33 1

Chupter V I I

Fatigue and ballistic impact 7. I . 7.2. 7.3. 7.4. 7.5.

333

Fatigue of a metal laminate 333 Ballistic impact 337 Fatigue of metal matrix composites 346 Fatigue: further experiments 363 Concluding remarks 371

xi

Contents

Chapter VIII

Compressive strength 373 8.1. 8.2. 8.3. 8.4.

Rods under compression 373 Tubes 381 Shells 392 Technological impact 408

Part 111: Technological processes and materials Chapter I X

Interfaces and wetting 415 9.1. 9.2. 9.3. 9.4.

Structures of the interface 416 Interface layers 419 Surface energy and wetting 424 Interface properties 432

Chapter X

Diffusion through fibre/matrix interface 441 10.1. Kinetics - a simple case 441 10.2. Kinetics in the case of chemical reactions

442 10.3. Effects of component interaction on composite properties 454 10.4. Diffusion barriers 468 10.5. Sintering 470

Chapter X I

Hot pressing 475 1 1.1, 1 1.2. 11.3. 11.4. 1 1.5. 1 1.6. 1 1.7. 11.8. 11.9.

Fabrication of composite precursors 475 Processing parameters 480 Techniques 495 Rolling and drawing 500 Explosive welding 502 Boron/aluminium composites 503 Silicon-carbide/titanium composites 508 Glass- and glass-ceramic matrix composites Graphite-aluminium composites 515

512

xii

c o n 11'11 I J

cll~lpttv.X I 1

Powder metallurgy methods 517 12.1. 12.2. 12.3. 12.4. 12.5.

Basic scheme 5 17 Variations of the basic scheme 521 Pyrolysis of matrix precursor 523 Short-fibre composites 526 Continuous fibre composites 543

C ' h q m , . XI11 Liquid infiltration 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7.

547

Infiltration mechanics 547 Techniques 561 Aluminium-matrix composites 571 Magnesium niatrix composites 579 Titanium-matrix composites 580 Intermetallic matrix composites 58 1 Ceramic-matrix composites 582

C'hrrpttv X I

v

Internal crystallization 587 14.1. 14.2. 14.3. 14.4. 14.5.

Technique 587 Fibres and composites obtained by ICM ICM with pre-made fibres 622 Blotting paper technology 630 Fibres produced by ICM 636

Bibliography 639 Author index 669 Subject index 683

591

Part I TOWARD COMPOSITES

Fibrous composites occurred as a result of the previous history of development of materials for structures at a point where metal alloys were approaching their limiting properties and, at the same time, some technological fields of vital importance, such as transportation of all kinds, energy transformation, etc. called for better material properties to balance society's demands and natural resources. At this point, a basis for the development of composites had been formed within both fracture mechanics and physical metallurgy fields. Therefore, the challenge mentioned was adequately met by materials science. The two chapters included in this part briefly illustrate the reasons that determined the occurrence of composite technology and give an illustrative review of it.

This Page Intentionally Left Blank

Chapter I STRUCTURAL MATERIALS

Metal and ceramic based fibrous composites are supposed to replace metals and ceramics in some engineering applications. Enhancing efficiency of transportation and energy production machines, production equipment and tools is an aim we are going to reach by such a replacement 1. Therefore, we start with a very brief discussion of effectiveness of structures and materials and then proceed with limitations to the efficiency imposed by traditional materials. We need materials to make load bearing structures of them. Materials and structures are to be considered together. Discussing the problem briefly, we should finally point at the answers to a number of questions such as 9 What is the effectiveness of a material? 9 What is the effectiveness of a structure made of a particular material? 9 Why do we need light structures? 9 Why do we need heat resistant structures? 9 Do we need non-metallic materials? Discussing these points very briefly, as a reader would expect in a book on composite materials, we should restrict ourselves just with some of these questions. 1.1. Effectiveness of materials and structures

What is the criterion to say what is good and what is bad in evaluating the effectiveness? Perhaps, different users of materials will give different answers. But we need a universal criterion that could be accepted by a majority. Certainly, energy saving (or wasting) by using a particular material in a particular structure or machine is a universal criterion. We can follow here an elegant discussion on materials, fuel and energy presented by G o r d o n [202]. He writes ... coal and oil store a great deal of energy packed into a small volume. Engines process a great deal of this energy very quickly and within a small space. They then deliver the energy as electricity or mechanical work in concentrated forms. On this concentration of energy our whole contemporary technology rests. The

1In the present context, we do not include concrete into the ceramic field.

4

Ch. I, w1.1

Structural materials

materials of this technology, steel, aluminium and concrete, themselves require a great deal of energy to manufacture them; how much energy is indicated in Table 1.1... We are not only investing money capital in a technical device; we are also investing energy capital, and in both cases it is necessary to secure a fair return on the investment... Advanced engines, such as gas turbines, process more and more energy, more and more hectically, within less and less space. Advanced devices require advanced materials, and the newer materials, such as high-temperature alloys and carbon-fibre plastics, consume more and more energy in their manufacture. Nothing to add, except perhaps to remind that money capital is actually a result of previous labour accompanied with energy expense. Why do we need light structures? The answer of a qualitative nature is obvious. In the case of transportation machines, the smaller the weight of a machine, the larger the payload that can be delivered at the same expense: the lighter the aeroplane structure, the smaller is fuel spending per passenger or per weapon unit, etc. In the case of a product to be delivered from a point of its fabrication to a store point and further on to a consumer, the lighter an equipment producing a given quantity of a useful performance, the cheaper is its delivery: the less heavy the tubes for gas or oil transportation, the less fuel is to be spent for its transportation to a service point; the lighter a satellite structure, the cheaper is its launching. And so on. Mixed examples just presented illustrate a complicity of a very non-linear problem of optimal design of engineering systems that normally include a number of structures and are subjected to a variety of constraints. Therefore, it is illuminating to evaluate the answer of a quantitative nature, at least for specified problems and to some approximation. Let us consider transportation by marine, ground, atmospheric, and transatmospheric vehicle. TABLE 1.1 Approximate energies required to produce various materials. After Gordon [202]. Material

Energy to manufacture 103 kJ/kg

Relative energy to manufacture

Steel (mild) Titanium Aluminium Glass Brick Concrete Carbon-reinforced plastic Wood (spruce) Polyethylene

60 800 250 24 6 4 4000 1 45

1 13 4.2 0.4 0.1 0.066 66 0.016 0.75

Note that a unit in the third column corresponds to 1.5 kg of oil equivalent per kilogram of a material.

Effectiveness of materials and structures

Ch. I, w

5

For a ship (no hydroplane!) of a linear characteristic size L, the resistance to its movement with velocity v is [590] Sv 2

F-

cp

2

(1.1)

where c-

c f ( R e ) + C~w(0, Fr).

Here S is the wetted surface of the ship, S e( L 2, Re = Reynolds and Froude numbers, respectively, r / a n d water, respectively, and ~ a value depending on the Energy necessary to carry a unit of the mass over Fv L2/3 v 2 e -- ~ = ~ c p ~ . my 2m

(1.2) p v L / q and Fr = v / v / ~ - s are the p the viscosity and density of shape of the ship. a unit of the way is

(1.3)

where ~ is a constant. Assuming the total mass of the ship, m, is proportional to L, we obtain v2 e -- ~c(v)El~3

(1.4)

where ~ is a constant. The dependence of c on the ship velocity is due to both a relatively weak dependence of cf on the velocity and a very strong non-linear ! dependence of c w on the Froude number [590] that makes dependence e ( v ) stronger than e o( v2. The accountir~g for a dependence of the propulsion coefficient, which shows the efficiency of the propulsor, will yield even stronger dependence of e on the velocity. Therefore, we see, first, that the energy spending for the transportation on the water increases strongly with increase in the velocity. This means that the importance of using light materials grows as the necessary velocity increases. Secondly, large-size ships are more effective from the point of view of the energy efficiency than small ones. Assuming that the ratio of the payload to total weight of a ship decreases with decreasing ship size, we conclude that making a light construction is especially important for small ships. The resistance to a motor car motion is a sum of those arising due to friction between the tires and the road and aerodynamic resistance, so that F = fmg

+ cSv 2

(1.5)

where m and S are mass and middle-section of the vehicle, respectively, f the friction coefficient, v the vehicle speed, and c a coefficient depending on the car shape. (Normally, f ~ 0.015, c ~ 0.25 kg/m 3 for cars and c ~ 0.7 kg/m 3 for trucks.)

6


Ch. I, w

Therefore we have S v2 .

e -- fg + c--

m

(1.6)

The second term is approximately equal to the first one at speeds about 150 km/h for cars and 100 km/h for medium-size trucks, so that from about these speeds and further on, e increases very strongly with an increase in the speed. This sets limits for reasonable transportation speeds on the ground and shows the ways of improving the efficiency. It is important to point out that one way is to decrease the total mass of a motor car by decreasing mass of the bearing structure. To give a feeling of the corresponding value, we remind that there exists an empirical relationship between the fuel consumption and total automobile weight. For vehicles of equal performance, the relationship is a straight line. The slope obviously depends on the performance (velocity, engine, tires, etc), an order of magnitude of the slope is between l0 -5 and l0 -4 1/(km 9kg) [124]. Therefore, weight saving in the automotive technique yields a predictable economy in fuel consumption. Equation (1.4) can also be used in the case of the transportation in atmosphere by subsonic aeroplanes at sufficiently high speeds, although the sources of the resistance to the motion is different for the two cases. Correspondingly, values of c and ~ are different. We can get a feeling of the quantities just reminding that the ratio, K, of the drag to the aircraft weight changes from about 2 for a small and fast aircraft of a fighter type to about 10-15 for large subsonic aeroplanes. The efficiency of jet engines can be characterized by fuel consumption per unit thrust per unit time (see below, in this section). For modern turbofan engines in cruising flight, this value is about 0.05 kg/(N 9h). Hence, the proportions between airframe and fuel mass and payload, which clearly determine the effectiveness of the transportation, are strongly dependent on airframe weight. To compare the energy efficiency of various transportation methods just considered, we note that it is affected by the size of a vehicle which, in turn, depends on technical constraints (roads, aerodromes, navigation, etc). Therefore, it should be perhaps better to use the value of e L 1/3 as a measure of the efficiency of a method but such a kind of value looks rather ugly. So we plot a schematic e ( v ) dependence on the log-log plane (fig. 1.1). Actually, the dependence for a whole family of a transportation means is a band that just follows the plotted line and changes with increase in technological level. Figure 1.1 illustrates the universal validity of principle "the higher is the transportation speed the more profitable is the weight saving". This means that the more expensive is a material, the more rapid vehicle is an appropriate point for its usage. Finally, consider transportation of a payload by a ballistic rocket. Neglecting the atmospheric drag and other secondary forces applied to the rocket, we can use a classical formula to estimate the final velocity of a single stage rocket, that is the velocity reached by the rocket when fuel has been burnt out. We have Vf -- --re ln(lAf).

(1.7)

Ch. I, w


7

Fig. 1.1. Schematic of a dependence of the energy efficiency on the transportation speed for various methods of payload carrying.

Here mf mo

Ve is exhaust velocity, mo and mf the initial and final values of the rocket mass, mf = m o - mfo where mfo is the mass of fuel and oxidizer. The total distance of the rocket flight, L, consists [166] of (i) active part A'A where the engine works and supplies final velocity VA to the rocket, (ii) ballistic flight along an elliptic trajectory ABC, and (iii) re-entry path CC' (fig. 1.2). Therefore, L - ll + 2(R + rA)fl + 12 with the notations shown in fig. 1.2. It can be seen [166] that

L ~ L' ~ 2Rfl and fl is determined by tan

fl -

%- VA t a n

0A

1 - VA + tan 2 0A

The flight distance reaches a m a x i m u m when the angle between the velocity vector at the end of the active part of the trajectory, 0A, and the local horizon is such that

t a n 0 ~ -- V/1 - VA =

1--

VA

8


Ch. I, w

Z B A

u

Fig. 1.2. A trajectory of a rocket launched at point A' of the Earth surface.

where Vcr is the first critical velocity. Hence,

pf-exp

/--4(' / x/2vcrtan~" Ve

1 + ~ - 1 tan 2

.

(1.8)

Here ~" = (L/2R) where R is the radius of the Earth. Figure 1.3 presents the dependence given by eq. (1.8). We see that the relative mass of fuel reaches one-half of the total rocket mass very quickly and then goes up to about 0.8 even for a most effective fuel. Therefore, the efficiency of the rocket flight is greatly dependent on the weight of load-carrying structures. It should be also emphasized that in the remaining mass of the rocket containing an engine, fuel and oxidizer tanks, other structural parts, and a payload, the latter is just a small part. Because all the elements of such rockets, which are mainly of military purposes, are not reusable, it is of no use to consider the efficiency of such method of transportation in the energy terms as we did in the previous analysis: the cost of the whole structure is too high as compared with the cost of the full tanks. The delivery of a payload to a low Earth orbit and from the orbit to the surface, as well as sending a payload to far-out space can be sufficiently effective by using rocket systems, a part of which are reusable, and all are optimized to reach weight perfectness. In the case of a rocket with n stages which work serially, the final ideal velocity is V f - - --Vel ln(/Afl ) -- re2

ln(pf2) . . . . .

Uen l n ( P f n ) .

(1.9)


Ch. I, w

1.0

i

!

i

!

!

i

i

i

i

1

0.8

1

!

i

i

!

v~ v~-

o}0.004

i

!

i

,

3.5 4.4

I

!

!

i

i

!

!

!

i

9

!

km~s km/s

02

O

I

I

1

i

i

0

I

I

I

I

I

[

2

i

I

I ,I

I

L.IO -s /

I

I

I

I

km

]

I

I

1

I

I

I

[

l

I

4,

Fig. 1.3. The maximum distance of the rocket flight versus the ratio of final mass of the rocket to the initial one for two values of exhaust velocity and vcr - 8 km/h. Note that ve - 4.4 km/s is close to an ultimate value for chemical fuels [166].

where tt k (1 _< k _< n) is the ratio of the mass of k , . . . n stages without fuel mass of the kth stage to the total mass of k , . . . n stages. So the total mass of the k + 1,... n stages is a payload for the kth stage. We see clearly that the higher a rocket stage, the more effective is the usage of light materials in it. It is also clear that the efficiency of using light materials can easily be evaluated in this case. To conclude, we show a schematic of the dependence of a commercial profit from weight saving on velocity of the transportation. Figure 1.4 is dimensionless on the profit-axis since the absolute value depends on many factors that are far from physical and technical nature of the problem. However, it can be useful for the purposes of a comparison of output gained from input of various materials into various types of the transportation systems. Why do we need heat resistant structures?

Energy saving strategy makes us use higher and higher temperatures in engines and energy transforming machines. The efficiency of the ideal thermal cycle is tit---1

T1 T2

(1.10)

where 7'1 and T2 are the lowest and highest temperatures of the cycle, respectively.


l0

1000-

,

,

,

I

i

,l

I

Ch. I, w

'

100-

o 0

10

i

I

I

i

I

I

I

I

I

i

i

i

!

i

i

10000 Fig. 1.4. The profit obtained from saving a unit mass of a structure versus velocity of the transportation. Profit is normalized by that for v = 100 km/h.

For relatively simple heat engines, like piston ones, eq. (1.10) can be used straightforward to estimate the thermal performance coefficient. For very good diesel engines, it is about 38% . If we have a material on ceramic base with higher temperature resistance than that of metal alloys, we will expect an increase of thermal efficiency to about 65% [401]. When we estimated the fuel efficiency of aeroplane transportation we considered the propulsive energy supplied by the propulsor. To obtain this energy, thermal energy has to be input into the engine. Transformation of thermal energy into mechanical thrust is necessarily accompanied by losses according to the ideal process (given, in total, by thermal performance coefficient, r/t, eq. (1.10)) increased by deviation from the ideal cycle and those arising due to a particular interaction of gaseous currents and the engine (given by propulsive coefficient, r/p). For straight jet engine, we have [591] 7

~t

-

1 ~22-2~

T,* ~+ 1 ~

1

2 ,

r/p

(1.11)

where 7"1" and T~ are the stagnation temperatures at the inlet section and in the combustion chamber, respectively, 21 and 22 the inlet and outlet gas velocities normalized by the corresponding sound velocities in the gas, 7 the adiabatic exponent. We see that the total performance coefficient, q - r/tr/p, of the straight

Ch. I, w


11

engine depends strongly on the ratio of the maximum temperature in the combustion chamber and the ambient temperature. One sees that v = 0 at 21 = 0. Moreover, it can be shown that such type of the jet engine is effective at very high supersonic flight speeds only. An aircraft engine for lower flight speeds is effective when it is designed according to a more complicated scheme. Normally it includes a compressor to compress the air, a turbine to drive the compressor and a combustion chamber to supply the turbine with mechanical energy resulting from burning fuel. A part of mechanical energy stored in the working gas (air plus products of burning) can be used in a propelling nozzle to produce thrust, as well as a part of mechanical energy produced by the turbine can be used to drive a propeller or fan to make the propulsion more effective. Therefore, the fuel efficiency of such engines depends on the effectiveness of the number of parts. Equation (1.11) remains to be a basic relationship to determine the performance coefficient, just as 22 depends on the pressure ratio in the compressor, that is [591] 22 = ~/~(1_~)/~ (2 ~

7+1) 7+1 7 1 + 7~- 1 -

For modern aircraft engines, the pressure ratio in the compressor may be about 20 that yields a temperature rise by about 600 K. This means that to input a sufficient amount of energy into a limited amount of air passing through the combustion chamber in a time unit, we need to increase its temperature by as large an amount as possible. The limit is set by fuel but a much lower limit is determined by the high temperature strength of turbine details, the blades being most critically loaded, then vans, rotating disk and stationary elements which shape the air flow. The fuel efficiency is usually expressed by specific fuel consumption (sfc) given in mass of fuel per unit thrust per unit time. A detailed analysis [387] shows that sfc is mainly determined by the pressure ratio and, correspondingly, by a gas temperature at high pressure stages of the compressor. The turbine inlet temperature determines the specific power, which is the power output per unit mass of air passing through the turbine in a unit time. This value does directly determine the weight of the engine and the weight of an airframe structure supporting it as well as its cross-sectional area that effects the drag and, correspondingly, a necessary value of the thrust. Therefore, the first motivation for enhancing working temperature of a material is a need to increase the efficiency of engines. A corresponding benefit can be evaluated. Actually, the benefit is to be compared with an amount of energy necessary to produce the material (see Table 1. l) but a result of the comparison can be of indirect practical importance because of either a demand to decrease mass or cross-sectional area of the engine, or another technical, military, and environmental requirement. The second motivation relevant to a transportation machine is heating of its skin due to the stagnation of the gaseous flow. Kinematic energy of a gas flowing with velocity v converting fully to thermal energy gives a temperature rise equal to

12


AT-~

Ch. I, w

v2 = 7-1M2 T 2Cp 2

Here M is the Mach number, Cp is the specific heat; for air, Cp ~,~ 103 J/kg 9grad. So loss of the air velocity due to interaction with wing and fuselage of an aeroplane yields a rise in the temperature of the airframe. For instance, during the re-entry of space planes like American Shuttle or Russian Buran the maximum temperature can reach about 1800~ Similar heating will occur in space planes with horizontal takeoff as well as in transatmospheric planes. The temperature of outer surfaces of atmospheric supersonic plane for Mach 2.4 reaches 150-200~ The next motivation is an increase in the efficiency of some technological processes which operates more effectively if they are performed at higher temperatures. W h a t is the effectiveness of a material?

A structural element under service loading can fail to resist the loads because of the tensile fracture (fig. 1.5a), of a change in its form due to non-sufficient rigidity (fig. 1.5b), or as a result of the buckling (fig. 1.5c). These types of failure can actually be caused by a variety of fracture or deformation processes, such as creep, fatigue, environmental effects (oxidation, corrosion, radiation, etc), or coupled effect such as aeroelasticity. Nevertheless, the first failure mode depends mainly on the material strength, two other modes depend on the material stiffness. The example shown in fig. 1.5a is very instructive. Consider a body rotating around axis O with angular velocity o9 (fig. 1.6). The stress at radius r will be -

( )p o2R2 -

2

(1.12)

where p is the material density, and ~(r) is determined also by dependence of crosssectional area of the rod on the radius at ~ > r. Therefore, the maximum linear speed at the external radius of the rod is determined by the square root of material strength, a, to density ratio

Vmax (X V ~ .

(1.13)

Ratio a,/p is called the specific strength and has a dimension of the square of a linear speed, (m/s) 2. If a shape of the rod along the radius is predetermined (say, by thermo- and aerodynamical considerations as in the case of turbine blade) there are no means to provide the failure resistance except for an appropriate choice of the material. If no material is available to satisfy the requirement and all possibilities to decrease' a temperature of the material (ceramic coating or gas cooling) are exhausted, one has to either decrease the temperature of the gas with a corresponding loss of fuel efficiency or decrease the speed to lower power. And vice versa, with increasing specific strength at higher temperature the efficiency of gas turbine can be made to increase.

Effectivenessof materialsandstructures

Ch. I, w

13

( Xo, and finally there should be

fo

~ a(x)dx - 27,

(1.29)

where 7 is the surface energy of the solid. There might be set up a requirement about the behaviour of the function in the vicinity of Xo, namely da/de = E, where E is the Young's modulus, and e = ln(x/xo) the deformation. No doubt there can be written a number of function satisfying the requirement mentioned. For example we can define the function as (see fig. 1.12) a-

Ax, a* (x, /x) ~

at at

x <x,, x>x,,

(1.30)

where a* is the maximum stress, x, the separation between the planes corresponding to the maximum stress, ~ a constant to define a shape of the falling part of the curve, A =a*/(x,-Xo). We have 27 -

Axdx +

a* (x,/x) ~dx = 2a'x,,

(1.31)

22

Ch. I, w


X 0

7C,

X

Fig. 1.12. A possible dependence of the stress (normalized force) on the separation between the atomic planes.

where2= 1/2+l/(aSubstituting

1).

x, = Xo exp(e,) and exp(e,) = a*/E + 1 into eq. (1.31) yields 27 - 2 (a .2/E + a*)Xo,

(1.32)

and finally we obtain a* -- E / 2 [(1 + 87/AEXo) 1/2 - 1] - fiE ,~ 2~,/2Xo,

(1.33)

because obviously 87/2Exo < ~

< x x x x

x

x

x

x

x

x

x x

x x

~ ~" ~

5"

x

x

x x

x x

o o

3

.m "u

, ~ ~

x x x x x

x

x

.

.

.

•

[x:xxx>

x x x x x

X X X X >

x" x

x"x" x

.

.

x ' W Y Y

x

x- x

•

x

x

x

x x

• x x

~
~YxJvx/vx~ ~

X X X X > X) x x x x x

x x x- x x

•

x

x

x x

x x

x

x

x

x

•

-

y

y

x

x

x

x

x

•

x x

"x" x

< x ,< x

~

.~ x

(xXyX~

~ x

•

x

x

x

x

x

x

x

x

3

_m -11

XPk.x.x~ x " Y W "x'"~ y x • • 2 1 5 x x x x ) ~

XXXX/X x x x x • x x x x x

>~xxxx •

x x x x x x x x x x • x x x x x •

< x x x x ........

x x

x

x

< X X X X

K X X X X < x x x x < x x x x ........

< x x x x ,( *

y

y

< x x •

......... L/L/..-..-.

, ~ x x x x

.........

km, ]Af > ]Am

the 'plus' and 'minus' superscripts denote the upper and lower bounds, respectively: -- )~ , "-1- flm/Af Of--I) ]A~3)--]Am 0~(X_

(+) 4k]A23

(+)

E22 :

4_~'~ '

k+]A~3 ) 1 + e,, ]

v~3)= E~2) 2]A(+) 23

1

]A~3 ) -- ]Am -~-

1

~- /Afkm+2Pm ,

Pr- Pm

2/A m (kin q- ~tm )

(3.54)

(3.55)

(3 56) 9

Elastic behaviour of composites

Ch. III, w Here C~- (l/g + f l m ) / ( ] 4 p -

//It

,uf ]2m , tim

1), ),-

1 3--4Vm'

.2 o2 3 VfVmp m --/.tp flf 1 + timl+l,A~f v~

1 3 - Vf

flf

Here and further on, the asterisk superscript for denoting effective values is dropped. It is clear that moduli Ell and v12 can be well approximated by simple averaging, that is Ell --

E,

V12 -- f12.

These moduli are not sensitive to a particular reinforcement geometry. On the other hand, moduli E22 and P23 are very sensitive to the reinforcement geometry 2. The upper and lower bounds for values of these moduli provide a rather big gap, so numerical methods for known geometry are preferable if one needs sufficiently exact values. Direct calculations of the elastic characteristics of composites based on the homogenization of eigenstrains have been carried out by a number of authors starting with Nemat-Nasser et al. [487]. In particular, Luciano and Barbero [374] obtained closed-form expressions for components of the stiffness tensor of the transversely isotropic composite: ell-

(), + 2/t)m--Vf [A~-2m

2BA

aA

lt2 g

lam c

t

B 2 -- C 2 ~ + l~2mg2

aB + bC a 2 _ b 2] ~ ltm g C + 4c 2 j / D , (3.57)

s-c

C 1 2 - J~m + vfbI'2Al4m

C23

I aC

a+bl/ -4--~2]

2Cltm9 ba c --

L b2]/

4C2

.]

(3.59)

D,

aB aA C22 -- (2 + 2/~)m-Vf -2-~fc + 2#m9C +

C44 --/2 m -+- of

[2r

+ #f-

1

(3.58)

D,

4c2 j

D,

(3.60)

4C ]-1 ]Am

~

'

(3.61)

2Numerical calculations revealing the influence of irregularities in the square and hexagonal fibre arrays on the effective transverse stiffness of composites were published long ago [7].

92

Deformation and failure of composites

C66 -- ]'/m + of

k

+

1 11

Ch. III, w

(3.62)

f l f - tim

where

aA 2 O

_

.

_

2#2c +

aBA

a ( 9 2 - C 2)

p2 gc

2#292c

B(a 2 - b 2 ) + C ( a b + b

2)

2ltmgC 2

+ +

A (b 2 - a 2)

2/.tm c2 a 3 _ 2b 3 _ 3ab 2 8c 3

and a =/.if (1 - 2Vm) - / 2 m (1 - 2vf), b - - / t f v f ( 1 - 2Vm) - ].tm Vm ( 1 -- 2Vf), C ----- - - ( / . t f - / . t m ) ( a q- b),

9=2(1-Vm). The series, A, B, C, given by N e m a t - N a s s e r et al. [487] in tabular form for some values of the fibre volume fraction are a p p r o x i m a t e d [374] with parabolas: A - 0.49247 - 0.47603vf - 0.02748v~, B - 0.36844 - 0.14944vf - 0.27152v~, C - 0.12346 - 0.32035vf + 0.23517v 2. Effective elastic characteristics can be defined by considering d y n a m i c response of a n o n - h o m o g e n e o u s m e d i u m (see for example [3, 477]). This yields both effective moduli and dispersive characteristics of the medium. In particular, for a unidirectional composite M u r a k a m i and Hegemier [477] obtained: Cll -- (2 + 2p) -- (2f-/].m)2/Al, C12 - ~ . - (/].f- ~m)[(/]. + /-/)f-- (/]. -+- fl)m]/A1,

C22 - (R -+- 2 1 , ) - [(R -+- p ) f - ( A -+- fl)mJ2/Al - (]./f- flm)2/A2, C33 - ~ . -

[(~ + fl)f-(/]. + fl)m]2/A1 - + - ( f l f - ,/./m)2/A2,

C44 - (C22 - C 2 3 ) / 2 ,

where

C55 - C66 - ~ -

( f l f - flm)2/A2

(3.63)


Ch. III, w

93

a~ = (2 + p) + Urn Vf Vm

A2 --- ~ --F-

(,~ + ~,)

m

2 v f Vm

and the double bar means a sum of the form a - - af

am

Vf

Vm

-- + - -

(3.64)

When we are using rather rough models of composite behaviour, to use exact models of elastic behaviour. Actually in such cases, approximations for the engineering elastic constants, an example homogenization approach [54]. The geometry of the model is shown components are isotropic, then Ell -- E,

E22 =

tJf

(3.65) Vm -+- ~ - / ] ,

(3.66) (3.67)

V21 - - 1,'31 - - ~, -1

//

Fibre

we do not need we need simple being a simple in fig. 3.3. If the

/Matrix

Fig. 3.3. The plane model of a composite.

Deformationandfailure of composites

94

Ch. III, w

where 2--1

--

E~f v f ( VEm_ ~- - fVf )

2

Structural fibres exhibit usually anisotropic elastic behaviour (Section 2.2). Independent of a real type of the elastic symmetry, the fibres can be assumed to be transversely isotropic since the orientation of their x2, x3 axes are random in a composite. Therefore, just five effective constants are necessary to characterize the (f) E22, (f) P12, (f) P23, (f) and v12 (f). Hashin [234, 235] elastic properties of any fibres, namely, Ell, derived the necessary formulae by using analogies between isotropic and transversely isotropic elasticity equations. T , ,-,(f,m) f,m f,m f,m . ,-,f,m/2 (1 f,m f,m 2 ) Let/511

, v12 , ]212 , ]223 , Kf,m -- 1522

-- Y23 -- (Y23)

be the elastic c o n s t a n t s

of the fibre and matrix, respectively. Then the expressions analogous to those given by eqs. (3.50) to (3.54) can be written -

Ell -- Ell nt- 4 v12 -- V

o9,

Vl2_~,12+(v(f)_vl2))(1

(3.69) l)

(3.70)

, (f) ]2(m) t'12 ~- ~12 ]212--12 17 )

(3.71)

p~3 ) k - - ~ kx~+ k ) '

(3.72)

P

+Pl2

Vf , 1 km+2/t~7) (f) . (m) + 2//~7)(kmq_./z23 (m)) 1123-/z23

p~3) - p~3 ) +

(+) -- fl~7) I1 + fl23

(1-[-flm)Vf / 2 2 .1 ow~+l ]

p - vf 1 + 3flmVm'~

where fO - - Vf Vm

Vm

Vf 1 -~- ~m -+-

C~= tim -- ]2pflf

1 + ]2uflf

p---

]2p -[- tim

]2~- 1 '

(3.73)

(3.74)

Ch. III, w

kf

km tim

95


km -+- 2//(m) ' flf 23

#(f)

23 ,., (f) , ] A p - (m)" kf + z/t23 1123

3.2.3. Two-dimensional fibre packing We shall consider briefly two cases. First, the laminated structure well studied in mechanics of fibre reinforced polymers, and second, in-plane arbitrary arranged fibres. Laminates

Comprehensive reviews of the elastic behaviour of laminates are presented in recent books dealing mainly with polymer matrix composites [85, 265]. A general scheme of theories of thin laminates is obvious (see also [88, 234, 289, 360]). First, the stress/strain relationships for a unidirectional lamina of a particular elastic symmetry are written down for the plane stress conditions. In the case of orthotropic symmetry, corresponding formulae are given by eq. (3.9). Second, aiming at obtaining stress/strain relationships of a laminate in its main axes which differ from the main axes of the lamina, the a/e relationship for the lamina is transformed to the general form written in the x - y axes rotated in the lamina plane on angle 0:

(xx) ~yy ]"xy

--

812 S16

822 826

16)(xx) 826 866

O-yy '~xy

(3.75)

or the inverse relationship

( ) ()() O'xx Oyy Zxy

---

C_ll C12 C16

C_12 C16 C22 C26 C26 C66

s 6yy ~xy

9

(3.76)

The transformation of components of a fourth-order tensor, eq. (3.5), yields Sll - Sll cos 4 0 -~- (2812 + $66) sin 2 0 c o s 0 + 822 sin 4 0, 822 - Sll sin 4 0 + (2812 + 866)sin 2 0 c o s 2 0 -~- 822 cos 4 0,

S12 - $12 (sin 4 0 + cos 4 0) + (&l + $22 - $66)sin 2 0 cos 2 0 816 = (2311 - 2812 - $66)sin 0cos 3 0 -

(2822 - 2812 - $66)sin 3 0cos 0,

826 -- (2811 - 2812 - $66)sin 3 0 c o s 0 -

(2822 - 2812 - 866)sin 0cos 3 0,

$66 - 2(2&, + 2822 - 4812 - $66)sin 2 0cos 2 0 + $66 (sin 4 0 + cos 4 0),

(3.77)

96

Ch. III, w


and C'll = Qll COS4 0 + 2(Q12 + 2Q66) sin 2 0cos 2 0 + Q22 sin 4 0, C'~2 = (Qll + Q22 - 4066)sin 2 0cos 2 0 + 012 (sin 4 0 4- COS4 0),

C'22 - Qll sin 4 0 + 2(Q12 + 2Q66)sin 2 0 cos 2 0 + Q22 COS4 0, C'16 -- (Qll - Q12 - 2Q66) sin 0cos 3 0 + (Qll - Q22 - 2Q66) sin 3 0cos 0, C-'26 --" (Qll - Q12 - 2Q66)sin 3 0 c o s 0 4- (Qll - Q22 - 2Q66)sin 0cos 3 0, C-'66 -- (Qll 4- Q22 - 2Q12 - 2Q66) sin 2 0cos 2 0

4- Q66 (s in4 0 4- COS4 0),

(3.78)

The engineering constants of the lamina referring to the x - y axes can be expressed as functions of the off-axis angle, 0, by using eqs. (3.24) and (3.77). For example, the Young's moduli are Exx-[Elll

cos4 0 4 - / / l l ( 1 - 2 v , 2 ( l ~ 1 2 / E , l ) ) s i n 2 O c o s 2 0 + E 2 ~ s i n 4 0 ]

Eyy-[Elllsin40+kt121(1-2v12(la12/Ell))sin20cos20+E221cos40]

-1,

(3.79)

-1 .

(3.80)

Third, assuming the Kirchhoff hypothesis for a laminated plate (fig. 3.4) being valid yields the force and moment resultants of the laminate as

,y k,-Tz

k=n-1

MID -PLANE

k

0

k=2 k=l Fig. 3.4. A laminate composed of n layers.

Ch. III, w


(Nx, Ny, Nxy)

_ fh/2

97

(Oxx,O'yy,~xy)dz,

(3.81)

fh/2(ffxx , O'yy,Txy)zdz

(3.82)

J-h~2

( M x , M y , M x y ) -- J-h~2

where h is the laminate thickness. Substituting eq. (3.76) into eqs. (3.81) and (3.82) leads to the following relationships connecting N and M components to the laminate mid-plane strains, exx~ s Zxy0,and the mid-plane curvature components, Xxx, Kyy Xxy, namely

()( ()( Nx Ny

Nxy

Mx My mxy

=

All A12 A16 A12 A22 A26 A 16 A26 A66

--

Bll B12 916

B12 B16 B22 B26 B26 B66

)(o) ( )( ) )(o) ()() exx 0 eyy 0 7xy

+

Bll B12 B16

B12 B16 B22 B26 B26 B66

exx 0 Eyy 0 7xy

+

Dll D12 D16

D12 D22 D26

D16 D26 D66

Kxx Kyy 2Kxy

Kxx Kyy 2Kxy

(3.83)

,

.

(3.84)

where n

Aij - Z

e~/(hk - hk_,),

k=l 1 n Bij -- 5 Z e!'k) ( h2 -- h2-1 ) k=l 1 n -(k)

Oij - - s Z C i j k=l

(3.85)

(h~c-h3_l)

where hk are shown in fig. 3.4. Note that the mid-plane strain and curvature determine strains at any point of the laminate according to

()(o) () s eyy

--

7xy

s 0 tYyy 0 ]~xy

+ Z

Kxx Kyy 2Kxy

.

(3.86)

Fibres in-plane There are two known approaches to the evaluation of elastic moduli of a composite with fibres randomly distributed in-plane. The first one was originated by Cox [106] and based on the averaging of values of moduli of a corresponding

Deformation andfailure of composites

98

Ch. III, w

unidirectional structure. For example, to derive the value of the in-plane Young's modulus, the following averaging procedure is nearly obvious: Ex --

1f'~x(0)

rc

Cx(0)

dO -

if"

-

E(0)d0

(3.87)

where E(O) is given by eq. (3.79). The second approach is based on a laminate analogy suggested by Halpin and Pagano [226]. It is assumed either stacking up of unidirectional laminae of all possible orientations or just three ones, (0, +2rt/3), which yield to a quasi-isotropic laminate. Corresponding formulae are simple in the final form, they can be found, for example, in [85]. Both approaches to evaluating the elastic moduli of in-plane-fibre composites can be modified to account for non-homogeneous angular fibre distribution. In the first approach, the value of E(O) in eq. (3.87) should be multiplied by distribution density function qg(0); in the second approach function q)(0) is to be used to approximate the laminate structure. In the case of a short-fibre composite, the effective axial Young's modulus of a fibre is to be introduced to account for partially unloaded fibre ends. If the Cox's approximation, (see below, eq. (3.136)), is taken for the normal stress distribution along the fibre of length l, then the average fibre stress is

1

(1

-

tanh(/~//2)) ~l/2

(3.88)

where c is the composite strain. Therefore, the effective Young's modulus of the fibre, that is Eff f - (1

-

tanh(fll/2)) ~1/2 Ef,

(3.89)

should be used to obtain the elastic moduli of the composite.

3.2.4. Three-dimensional fibrous structures Such a type of structure is characteristic to either whisker or short-fibre reinforcement as well as to carbon/carbon composites, the latter being a typical structure for composites produced by filling a fibrous carcass with the matrix through the vapour or liquid phase.

Random packing of continuous fibre This is a case of the isotropic elastic behaviour of a composite, which can be considered as a limiting case for two structural types, namely (i) a regular structure of carbon/carbon composites formed by arranging fibre bundles in three or more


Ch. Ill, w

99

directions, (ii) reinforcement of a matrix by randomly packed short fibres or whiskers. In the first case, increasing the number of directions can lead to increasing a degree of the isotropy; in the second case, the elastic moduli of a composite approach those calculated on the basis of a model with continuous fibres as the fibre length becomes much larger than the critical length. Again, as in the twodimensional case, the elastic response of the composites can be evaluated by two averaging procedures described in the previous section. Assuming the stress/strain relationships given by the Hook's law with the stiffness matrix given by eq. (3.10), let us write following [88], instead of eq. (3.87), ratios

1/0 /0

Crxx - m -~xx 7 - - 2rc

~rxxsin 0 dO d o _-7-s

(3.90)

and

yy,2 1/0 /0 Yysin~176

-!

!

6xx

s

where axx and 0"yy a s well as a fixed value, exx are obtained according to the general transformation rule for tensor components, eq. (3.5). Integrating in eqs. (3.90) and (3.91) yields finally to the random fibre distribution K -- (1/9)(Cll + 2(C22 + C23) nt- 4C12) -- (1/9) (Ell + 4(1 + V12)2k)

(3.92)

and p = (1/30)(2Cll + 7C22 - 5C23 -4C12 + 12C66)

= ( 1 / 1 5 ) ( E\ l l

+ (1 - 2v12)2k + 6(#12 + P23)~/

(3.93)

To deduce the Young's modulus and Poisson's ratio, we can use eq. (3.17). The remarks in the previous section about the ways of taking into account a length of the fibre and the evaluation of a material texture due to 'preferred fibre orientation are relevant to the case under consideration as well. It should be noted that the upper and lower bounds, derived from energy considerations and the application of a variational principle, are of importance since a real geometry of the composite is normally unknown in detail. In particular, the bounds derived by Hashin and Shtrikman are especially useful for a threedimensional random fibre distribution. We have [234]: Vf 1 3Vm < -t- ~ Kf-Km 3Km-l-4#m

Km -Jrand

K < Kf-

Vm 1 Kf-Km

3.v.f 3Kf+4#f

(3.94)

1O0


]Am -[-

~ _ 1 ]Af--]Am

Vf 6(Km+2/~m)Vm [_ 5/-tm(3Km +4Pm)

~ ~ ~

]Af-

Vm 6(Kf+2pf)vf 1 ~5pf(3Kf+4/~f) ]Af-~ m

Ch. Ill, w (3.95) "

Still, regularizing a three-dimensional structure and using a numerical procedure allows to analyze the effect of such parameters as thickness and stiffness of coating, which can be either applied on the fibre predominantly to prevent unwanted fibre/ matrix interaction during fabrication and service or arise as a result of the interaction mentioned [521,522], fibre breakage and the existence of two or more kinds of the fibre in the composite [68].

A regular structure To evaluate the elastic constants of a regular three-dimensional structure, Halpin et al. [225] suggested to use the laminate analogy as was used for two-dimensional composites. Again, to calculate the stiffness matrix in a reference coordinate system, it is necessary to perform a transformation of the stiffness matrix in the coordinate system associated with the reinforcement direction to the reference system by using eq. (3.5), and then to average the contributions according to the volume fractions of a particular reinforcement direction. The procedure yields rather complicated formulae which can be found in [644].

3.2.5. Experimental data Elastic properties of the material determine rigidity of a specimen, its vibration behaviour, and ultrasonic speed in it. Therefore, corresponding measurements are used to evaluate elastic characteristics of the material. We shall present some experimental data revealing both peculiarities of the elastic behaviour of fibrous composites and comparison of experimental data with calculated ones. Because of a number of reasons, elastic characteristics obtained by different methods can be different to some extent. Without going into full details of the reasons we shall just illustrate the difference in the elastic constants measured by different methods. We start, however, with a brief discussion of the method based on measuring ultrasonic wave velocity.

Ultrasonic technique Ultrasonic technique is now widely used in various applications, such as nondestructive testing, radioacoustics, acousto-optics, microscopy, etc. This technique has also been employed to measure the elastic characteristics of fibrous composites [313, 477, 636, 645, 659]. This particular application of the technique has some feature to be pointed out. Relationships between constants Cij and group velocities V of the ultrasonic longitudinal a n d transverse waves are [645]

Elastic behaviourof composites

Ch. III, w Cll -- pV,21,11

C22 -- pV,1,22 2

~

C44 - pV~s,23 2 ~ C23-

~

C33 - pV,1,33 2

C55 - t9V~s,213 ~

C66 - / 9 V _ s,212

__[(C22-'~-C44-2pV2n)(C33-t-C44-2p __

C13 -- [ ( C l l - k - C55 - 2pV-q2, m ) ( C 3 3 - ~ - C , 5 --

--

101

q,k)(C22--t-C66

V'2q,n)]1/2- C44~ _ 2pVq2m , ) ] 1/2 , )]

-

(3.96)

C55, C66.

Here p is the material density, subscript 1,s, and q refer the velocity to a wave type, those being longitudinal, transverse and quasi-longitudinal (quasi-transverse), respectively. The first digit in a subscript to 'V' points at the direction of elastic displacement, the second one relates the velocity to a wave propagation direction. Subscripts m, n, k show directions of the quasi-longitudinal (quasi-transverse) wave propagation. Note that for the case under consideration:

m

,0,

,

n

0,

2

,

,

'

2

'

"

Interpreting results of measuring the wave velocities, it is necessary to take into account the wave dispersion, that is a dependence of the velocity, V, on frequency f . When testing non-homogeneous specimens, it is preferable to choose such a frequency as to obtain a wave length, 2 = V / f , satisfying the inequalities: s .~

"> n 8

r~ o~ o0

II !

!

~>~>~->

!

!

r~

"a ~"

By using eqs. (3.108) and (3.110), the stress and strain components with superscript A F and AM are eliminated from eqs. (3.104) and (3.106) and we obtain

~o

,.,_,

+

+

>.

~:

I ~ >

"~

II

~

II

II

--

II

II

---S n S j

II

-3

where

>

,'rj

II I

+

-..,o

~._.., .~-...

I

,~

I

II

...,,

~

>

.-~

>-~

II

I

>

II

l

II

and

o

@

IxO

,

rh

r~ II

9

~. ~>

Inverting eq. (3.1 12), we obtain

I

~>

II

~>

where

o

go

.~

,

c~

~..~o

o

go

o

st are obtained according to eq. (3.1 1 I), just a t Mare replaced o

~-~

o

0 9

~.:~ ~

where components with at.

~%

~

~

~

0

~

~

~ o~""

For region B containing only the matrix, we have


118

Ch. III, w

The inversion of eq. (3.114) is {da B} - - [ c a ] {de B}

(3.115)

where C B - (sB) -1. Now eq. (3.107) together with eqs. (3.113) and (3.115) yields the effective stress/ strain relationship for the composite in the incremental form {da} = [C]{de}

(3.116)

[c] - [c A]

(3.117)

with

+ [C"]

The inversion of eq. (3.116) is (de} = [S]{da}

(3.118)

where

S = C -1. The relationships given by eqs. (3.116) and (3.118) are non-linear since a0B, a AM and Sij depend on the current stress-strain state in the matrix, i.e. a AM and a AM. Therefore, Sun and Chen [631] used a numerical procedure to obtain C and S for a particular loading path. The first load increment is obviously assumed to be purely elastic. Robertson and Mall [568] modified the Aboudi's model used by Sun and Chen by, first, decoupling normal and shear effects which simplifies the situation, and second, introducing separation and/or slip on the AF/B1 and AF/AM interfaces (see fig. 3.18) which produces additional effects. The separation and slip are controlled by interfacial compliance. Another simplification of a real composite geometry is a model based on thin slices which do not interact mechanically. Such model was adopted by Dymkov [146] (fig. 3.19a). Stress/strain relationship for a layer is da = Bde

(3.119)

where a and c are the stress and strain in the layer, respectively. That for the representative element shown in fig. 3.19a can be written as da =/~dc

(3.120)

Ch. III, w

Non-elastic behaviour

119

Fig. 3.19. (a) A representative element. (b) An elementary layer in which the fibre and matrix are interacted through a spring.

where

B = -~

B(z)dz

(3.121)

Here 0- - [033 , f i l l , 0"22, T12]T and e - [s E l l , c22, ~12]T are the vectors of average stresses and strains. It is convenient to decompose vectors 0- and e into the blocs:

~z

=

[~33, ~1,] T

s

_

[s

ell IT

, ~r -

, s

_

[~22,~12] ~,

[E22,~12]T

and to rewrite eq. (3.119)as daz

rr]"I Bzr

dcr]"

(3.122)

The equilibrium equations and the compatibility of deformations within a layer are expressed as dalr -- da~l -- do-r, vfd0-/z 4- vmda~ -- daz,

ddr - de~t - dcr, vfdetz 4- vmde" - dez.

(3.123)

Here prime or double prime refers a value to the fibre and matrix, respectively. Equation (3.122) and similarly written relationships for the fibre and matrix together with eq. (3.123) yield

~rr (/~rr/ 1/) -1 (vf/~) l+vm/~rm~)-1) -1


120

Ch. III, w

Brz -- B r r ( ( O r r ) - l O r z ) ,

Bzr

-

-(Azr/Azz)Brr ,

-

Bzz -- ( l / A z z ) + Ozr(Orr)-lOrz 9 The matrix plastic yielding is described by a more realistic theory in which

de~j -

H Of Of d~k,

(3.124)

OO'ij ~O'kl

where H depends on the loading path. Adding the plastic strain to the elastic components modifies B(pq) components that can be evaluated numerically. Comparing the calculation results with the stress/strain curves of boron/ aluminium composites loaded in the transverse direction (fig. 3.20), Dymkov revealed that the material stiffness was much higher than that predicted by the model. This made him to adjust the model to the real behaviour by introducing an interaction of neighbouring layers in the representative element via a spring as shown schematically in fig. 3.19b. The compliance, ~', of this spring was determined by fitting theoretical and experimental stress/strain curves in the transverse direction. Then the value of ~ obtained in such a way was used to describe offaxis stress/strain curves (see fig. 3.21). Sometimes an extreme simplification of a real geometry of a composite based on the plane model shown in fig. 3.3 can be useful. Certainly, it can be applied to an

300

--

'

I

"

I

'

I

'

~=0 Experiment

200 -

/

""--,,x/

.

_.

~;=10

100

0

0.000

!

0.001

0.002

S

0.003

0.004

Fig. 3.20. Experimental and calculated stress/strain curves in the transverse direction of a unidirectional boron/aluminium composite. After Dymkov [146].

Ch. III, w

121

Non-elastic behaviour

500

'

I

I

"

I

"

'

I

Calculated

400

.~ ] "

l

I

9ssJssSSSisjsssssssss SSISS1

300 200

//

100 0

O.000

,

s

I

O.O01

,

I

,

O.002

I

,

O.003

I

O.004

E;

O.005

Fig. 3.21. Experimental and calculated stress/strain curves of a l a m i n a t e d b o r o n / a l u m i n i u m [+45o/02]. After D y m k o v [146].

composite

approximate analysis of simple structural elements like tubes, cylindrical shells, etc, simply loaded, for example, by the axial force or hydrostatic pressure. Normally, the fibre directions in a laminate coincide with those of main stresses. In such cases the loading is proportional and a micromechanical model to describe the stress/strain response of the laminate may be very simple [325]. Consider a laminate of the geometry shown in fig. 3.3 under the proportional loading, K = o-1/o- 2 - const. We may obviously write O'1

I

-

II

O'lVf--[-O" lvm~

-

0"2

l

-

0"2

-

II

-

-

0"2~

~1

-

s

-

I -

-

s

l/

(3.125)

Assuming the deformation theory and incompressibility for the matrix we have , _ 1 _ 1 (o.,1

Cl - - ~--~1- - E---f

_

Vf/s

,, e 1 --

'

1

(

E m (0"~)

a'1

tc

al

)

9

(3.126)

Introducing x p

~_

1 gm (o'g) _ql_~f Ef ( l -- gVf/)f) Emvrn 1 + Ef Vf

yields t

(71 - - p a l ,

t/

O"2 - - KO'I,

t/

0"0 - - Or] V / P 2 +

K2

-- p K .

(3.127)

122


Ch. III, w

Here ag is the equivalent strain in the matrix, Em(a~) is the secant modulus. Therefore, we have s =

K Em (o'g)

and s

=

O'1 Vm(K- p / 2 ) + Vf K - Vf vl Em (~fl~) ~ 0"I,/

(3.129)

where t ~ 0"1 = - ~ f ( l - - p V m )

Experiments performed on thin-walled tubes of a steel-wire/aluminium-matrix composites loaded by internal pressure and axial tension-compression give the elastic-plastic response of the material described by the model outlined fairly good [325]. Time-dependent non-elastic deformation of the matrix can be taken into account in both Aboudi's [568] and Dymkov's [10] types of the geometry.

3.4. Failure criteria

Evaluation of the ultimate strength of a unidirectionally reinforced composite loaded at some angle to the fibre axis is usually performed by assuming a procedure aimed at finding plastic limiting state and using strength and/or plastic characteristics of the components and fibre/matrix interface [301]. Numerous failure criteria for anisotropic solids, which are, in fact, generalizations of yield criteria, have been proposed to describe limiting states of composite laminates (an excellent review see in [573]). A most widely used is the so called Tsai-Hill criterion [29]: al

_ ala____~2_

a2

= 1

(3.130)

where al, and 0"2, are the stresses in the fibre direction and the transverse one, respectively, zl2 is the corresponding shear stress, and the values marked by asterisks are the related ultimate stresses. To apply criterion given by eq. (3.130) to estimation of the strength of a unidirectional lamina loaded in its plane at angle 0 to the fibre direction by stress o, we use the general rule of the transformation of tensor component, eq. (3.5), to find stress components involved in eq. (3.130), as

Fibre~matrix stress transfer

Ch. III, w 0"1 :

0" COS2 0 0"2 - - 0" sin 2 0 1712 :

0" s i n 0 c o s 0

123

(3.131)

Substituting eq. (3.131) into eq. (3.130) yields the ultimate stress for the lamina:

(7*

[COS4 0 sin 4 0 -[/ a*2i + "2 + sin2 0 COS2 0 0"2

( 1 . 1 )1 -lj2 -,2 1712

_~2

(3. 132)

A direct use of the Tsai-Hill criterion in evaluating the limiting state of a laminate is doubtful. Actually, the procedure should be based on consideration of the fracture kinetics in the layers. Corresponding approaches are discussed by Chou [85].

3.5. Fibre/matrix stress transfer Normally a composite performs its functions as far as the fibre is loaded via the fibre/matrix interface. Hence, it is very often necessary to know how the stress transfer is carried out. The problem of stress transfer involves a number of cases because a variety of factors, such as elastic/plastic behaviour of the components, debonding and friction on the interface, non-homogeneous fibre packing, etc, can be responsible for various features of the stress/strain state in a composite structure. In this section we consider some problems of the stress transfer as well as some associated problems. McCartney [397] lists problems of composite mechanics which can be analyzed provided the fibre/matrix stress-transfer problem is solved. These are some of them: 9 loading the fibre in a short fibre composite; 9 stress state of fibre and the matrix in a brittle fibre composite in the vicinity of the matrix crack; 9 analysis of possible suppression of the matrix cracking in a brittle fibre composite; 9 analysis of pull-out and push-out experiments to assess the characteristics of the fibre/matrix interfaces; 9 analysis of the cracking of fibre coating. So the problem of the interface mechanics looks as a key problem in many fields of composite mechanics. Actually, interface behaviour influences composite behaviour essentially, in some cases, like fracture behaviour of brittle matrix composites, processes that are going on at the interface determine most important properties of a composite providing the very possibility of using~such composites as structural materials (see the next two chapters). That is why we need to consider this problem in detail. Also we shall come back to various aspects of the interface problems further on.

3.5.1. Elastic interface When formulating a corresponding problem of the elasticity theory, the forces applied on the interface, unknown beforehand, are used to write integral equation of

124

Ch. Ill, w


the problem. The solution of the equation gives stress fields in the fibre and surrounding matrix as well as traction and displacement on the interface. The essence of the problem can be elucidated by a model problem on a rigid cylindrical fibre of a finite length in the infinite elastic matrix loaded in the fibre direction so that at infinity the matrix is homogeneously strained [649]. It was shown that the shear forces f(z) on the interface should satisfy the following equation Uz(Z) -- c0z -

fl 1 ~02rtUz(~, z) f~( ~ )

(3.133)

~d~dO = 0

where co is the matrix strain at infinity, and

x+ '

8rt/t(2 + Z/z)

/

z)2 ( ( ~ _ z)2 + e2)3/2

2+3#

1

t

/], nL// ((~_ Z)2 _+_0~2)1/2

Here - 1 _< z _< 1 and ~ is dimensionless fibre radius, ~ al the corresponding relations are obtained by replacing q by iq so that

cosh(qz/Ri) ---+cos(qz/Ri),

sinh(qz/Ri) ~ sin(qz/Ri)

The formulation of the problem can be further simplified [25, 385, 396]. Namely, at the end of the debonding zone, it is assumed that the stress states in the fibre and the matrix are the same as at z ~ 0o. The stress fields are unidirectional (vf = Vm = v), so that the stress components in both the fibre and matrix do not depend on radius; the problem to satisfy the continuity conditions for the displacements are eliminated; a constant value of the interfacial shear stress, v, becomes a free parameter. In this case a'(z) . . . . Vf

2v Ri

z,

0 s at y - 0 we can obtain

a~) _ a 1 --~I(s, 1 t)f

r) 1

(3.216)

138

Deformation and failure o f composites

Ch. III, w

Here I(s, r) -

a0 +

q=l

aq cos rO cos rO sin ~ dO

(3.217)

where aq (q = 0, 1,... s) are obtained as a solution of a linear algebraic system. That is the simplest case which has been improved many times by including various factors into consideration. The improvement has been in five directions, namely (i) more rigorous consideration of the elastic stress state in the matrix; (ii) consideration of a fibre array in space instead of the plane array; (iii) accounting for non-elastic effects in the matrix concentration on non-homogeneous packing of fibres; (iv) consideration of dynamic effects. 3.6.2. Amendment o f the stress state

In the first direction, Eringen and Kim [157] dropped the assumption of the transverse stress in the matrix being zero. Equation (3.201) is then supplemented by the equation n, --O" x/n l, q'--~-

-'l- 17(n- l, ) --0.

(3.218)

The system was solved by using the same technique based on the Fourier transformations. The accounting for ax yields to a slightly higher load concentration factor than in the pure Hedgepeth's case; for example, for a single broken fibre, k = 1.3724 instead of 4/3. Ochiai et al. [509] attempted to avoid a most important weakness of shear-lag analysis describing the stress state in a composite. Actually, they prescribed the same type of behaviour to both the fibre and the matrix that resulted in shear stresses in the fibre and so in non-constant axial stress on a fibre cross-section as well as in nonzero axial stresses in the matrix. In particular, this allowed to distinguish between the cases of the fibre cracking and fibre + m a t r i x cracking. In this model, only a finite number ( 2 N - 1) of the elements in a composite is considered. The problem formulation for a two-component composite is clear from fig. 3.25. Here index at displacement refers the value to the centerline of a component, two indexes, a s /)n/n+l o r Zn-1/n refer the value of displacement or shear stress to the corresponding interface. Therefore, we can write

"Cn-1/n---]~'(Un--On-1/n)/(~)--]2it(On_l/n--Un-1)/(~)

(3.219)

"On_l/n -- (2/~'/z"/(h'/z" + httlAt))(v n - Vn-I )

(3.220)

and

Shear-lag analysis

Ch. III, w

2in_ 2

'Un_ 1

13 ,n,+ l

1.)

139

Vn+2

... . . ... ...

[ ..., ... ...

Thin+ 1

[

..... ... .I. . 9

99

'. . .ii.!.il

n-1

n,+l

12. ...

9

component 1 component 2 Fig. 3.25. Ochiai-Schulte-Peters model. See text for details.

Here prime relates the value to the first component, two primes relate the value to the second component. The equilibrium equations are d2vn

@2

1

h'E' (~'n/n+l -- ~'n-1/n) -- 0

(3.221)

for the first component, and d2vn

1

dy 2 = h,,E,----7(17n/n+l -- ~n-1/n) -- 0

for the second component. Substituting eq. (3.220)into dimensionalization, we obtain

eqs. (3.221) and (3.222) and m~tking non-

d2~n 2--7-Sy~9 + 2(Vn+l -- Vn --[- Vn-1) -- 0

clr

(3.222)

(3.223)

140


Ch. III, w

where

Vn -- Vn ~

O'e(~

r

-

(k,

h'(h'p" + h"#')

'

~, "~" /At]At' -t- hit]At)) 1/2 ,

(3.224)

(3.225)

and

2-

1 E"h"/E'h'

if eq. (3.223) is applied to component 1, if eq. (3.223) is applied to component 2

(3.226)

Note that letting 2 = 1 transforms the problem to the original Hedgepeth's formulation. Making now the problem to be symmetrical with respect to no element with index 1 and assuming element N + 1 to be unaffected by the cut fibres, the equations given by eq. (3.223)can be rewritten as

d2vl Pl - d ~ + 4(v2 - vl) - 0 d2vn ]An-~ - --[-2(Vn+l -- 2On + Vn-l) -- 0

(3.227)

d2VN

Py-d~2 + 2(~ -- 2VN + VN-I) -- 0 The bar over v was here dropped. A general solution of the system given by eq. (3.227) is expressed as

N 2N Vn -- ~ -}- Z A m B n ' m e x p ( - k m ~ ) + Z AmBn,m exp(km~) m=l m=N+l

(3.228)

where Am are constant to be obtained from boundary conditions, (km) 2 are eigen values of a matrix coming from eq. (3.227) which is written down in [509] together with expressions for constants Bn,m. The most important qualitative conclusion drawn from the results of calculation for some particular configurations is that the difference in the stress concentration factors for the cases shown in fig. 3.26 can be essential. For example, for 2 = 0.2, the stress concentration factors in the 2F fibre are 1.14 for case (a), 1.44 for case (b) and 1.09 for case (b), being 1.17 for the 1F-fibre in the latter case. For case (a), as the value of 2 increases, the stress concentration in the 2F-fibre decreases starting from value 4/3 corresponding to Hedgepeth's case. On the contrary, for the case (b), the 2F-fibre experiences increasing overload with increasing 2.

Ch. III, w

141

Shear-lag analysis

Component I (Fibre)

,+i+ li. 12M

IF

2F

Component 2 (Matrix)

23M

3F

il ] (b)

(c) Fig. 3.26. Some fibre-matrix-crack configurations considered numerically by Ochiai et al. [509].

3.6.3. Three-dimensional configuration

Three-dimensional geometry of a composite (the second direction of improving the simple model) was analyzed first by Hedgepeth and Van Dyke [242]. The model remains to be uni-dimensional in the sense that only the axial component of the fibre displacement is considered. Instead of eq. (3.201) the following equilibrium equations are written: d2vn,m a2 dy2 ~ (Vn+l,m -t- Vn,m+l + Vn-l,m --[- Vn,m-1 - 4Vn,m) - 0

(3.229)

142


Ch. III, w

for the square fibre array, and d2vn,m a 2 ~y2 ]'- (Vn+l,m +/)n,m+l + Vn-l,m -+- Vn,m-1 -4- Vn+l,m-1 -4- Vn-l,m+l -- 6Vn,m) -- 0 (3.230) for the hexagonal array. Here n,m neighboring a (n, m)-fibre,

are corresponding indexes of the fibres

a 2 =/Am h / E--TAfh '------S"

(3.231)

Here Af is the cross-sectional area of the fibre, h' is the characteristic diameter of the fibre, and h" is the characteristic interfibre distance. The numerical solution of the problem obtained by using the standard techniques of Fourier transformations gives the stress concentration factors dependent on the number of broken fibres and configuration (n,m) of the broken-fibre area. In particular, the maximum stress concentration factor for the case of the only broken fibre appears to be 1.146 for the square array and 1.104 for the hexagonal array.

3.6.4. Non-elastic effects An important direction of the investigation is taking into account non-elastic effects in the matrix and on the fibre/matrix interface. Hedgepeth and Van Dyke originated these studies in [242]. They considered the case of a single broken fibre (n = 0) at y -- 0 loaded along length + a by constant shear stress tau. The equilibrium equation for this fibre is d2vo dy 2

2 ~r ~ - 0 Efh'

at

[Y 1< a

(3.232)

where .cO _ (EfAfh"/Pmh') l/2,rmh'/a'.

(3.233)

Here "cm is the yield shear stress of the ideally plastic matrix. Fibre n = +1 are loaded as the zeroth fibre on one side, and as in the simplest Hedgepeth's model on the other side. The corresponding equilibrium equation is d2vl

@2 +c2(v2-vl)-~

1 'c0

-0

at

[Y l g, and purely imaginary at [Xl [< g. Hence, Re W ( X l ) " l s zero outside of the crack interval, and equals to • we obtain

~ / l 2 - x 2 on the crack surfaces. Differentiating w(z) with respect to z v

w'(z) - ~

A/z

V/Z2 __ g2

(4.11)

Values of w' will be real on the crack plane at I Xl 1< g; therefore, the crack surfaces are free from loads, z2=0. At z ~ ~ , w' ~ Ai. Hence, according to eq. (4.5), there should be 0 - izo - Ai, and A - -z0.

(4.12)

Equations (4.10) to (4.12) present the solution of the problem. We will not write down complete expressions for the stresses and strain, but just look at the stress state at the crack tip. To do it, put the local coordinate system (, q at the crack tip (fig. 4.2) and take ~" - p e i~ Then z - g + ~ and w -- - z0iv/2g(

+ ~2.

Ch. IV, w

Cracks in homogeneous solids

x2

151

z7

y

l

v-

Fig. 4.2. A crack of a finite length.

Expanding the expression for w in a power series with respect to ~, w--r0ix~

1+~-~+-..

,

and differentiating the series, we see that only the first term contains ( in a negative degree, ~-1/2, all other terms contain ~ in positive degrees. Hence, all the terms of the expansion go to zero when ~ ~ 0, except the first one that goes to infinity. So at the vicinity of the crack tip, we need to consider the first term only, i.e. w(~) - - i z 0 x ~ .

(4.13)

Equations (4.7) and (4.13) are identical if K -- z0v/~.

(4.14)

The stresses at the vicinity of the crack tip, q71 and "C2, are now given by eq. (4.9). Parameter K appearing in the solution is called the stress intensity factor, its dimension is stress • length 1/2, its value depends on a particular geometry of a body and load application. The stress intensity factor, K, is introduced to write the expressions for the stress state in a convenient form. To connect this parameter to values of direct physical meaning, let us consider a change in the elastic energy of a body when the crack advances by a length Al. If we cut the material ahead of the crack tip by the distance Al we can restore the configuration by applying shear stresses to the edges of the cut. The values of the stresses are given by eq. (4.9) and (4.14) at 0 = 0, that is K

Now let the stresses be decreasing. When they reach zero we obtain the original crack with the length increased by Al. The displacement u3 will be determined by eq. (4.9), just the origin of the local coordinate system is transferred by a distance Al, so

152

Macro-

U3 - - - -

and microcracks

in n o n - h o m o g e n e o u s

materials

l

AI.

Ch. IV, w

(A/-x)

~t

The change in the elastic energy will be

AU

-- -

/o

At zu3

dx --

K 2 gfl

a;

x dx - -

(4.15)

Replacing the finite increments with differentials yields a definition of the energy release rate as dU K2 G. . . . . dl 2#

(4.16)

The corresponding p l a n e p r o b l e m s for elastic isotropic body, that for the normal crack belongs to such a class of problems, are being solved by using a method developed by Kolosov and Muskhelishvili. In this method, a solution of a problem of the theory of elasticity is found via arbitrary analytical functions of complex arguments by choosing their particular form through satisfying boundary conditions. Because many formulae are to be used further on are expressed via such solutions, we give the fundamental expressions for the plane strain situation (s

=0): 0"11 + 0"22 4Re qg'(z), 0"22 -- 0"11 -k- 2i0"12 = 2(2~0"__(z)+ -

-

(4.17)

ff'(z)),

2#w - xq~(z) - z~' (~) - r Here q~ and ff are arbitrary functions and w - - Ul -k- i u 2 ,

z - - x 1 .qt_ i u 2 ,

K

=

3

-

4v.

The bar over a quantity means that ~ - a - ib if v - a + ib. The plane strain problem corresponding to a crack occupying interval - g _< Xl _< g, x 2 - 0 in a body loaded at infinity by 0 " 2 2 - 0", 0 " 1 1 - 0 " 1 2 - - - 0 is constructed by taking tp(z) -- C v / z 2 - g2,

The second expression in eq. (4.18) provides expression for ff into eq. (4.17) yields 0-11 -- 2(Re q~' + x2Im qg"), qg' - x 2 I m 99"), Re q)".

0"22 - - 2(Re 0"12 - - --2X2

(4.18)

~ ( z ) = ztp'(z) + qg(z). 0"12 - - 0

at

X2 - - 0 .

Substituting the

(4.19)

Cracks in homogeneous solids

Ch. IV, w

153

At X2 -- 0, 0"12 -- 0 and 0"11 = 2Re qg' = 0"22, hence, /tUl -- (1 - 2v)Re q~,

pu2 -- (1 - v)Im q~.

(4.20)

At infinity, eq. (4.19) gives 0"11 = 0"22 ---

2C,

0"12 - - 0.

Let 2C = 0", then function qg(z) defines the solution of a problem which differs from that formulated initially, that is a problem of homogeneous tension of a body. However, the crack oriented in the direction of the applied stress, 0"11, does not disturb stresses 0"22. Therefore to convert the solution obtained into a solution we are looking for, we need just to extract from the stress, 0"11, given by eq. (4.19), the value of 0". However, because we are interested in the stress state at the crack tip only, where the stresses go to infinity, this correction is not important and may be neglected. For the sake of simplicity, we write the expressions for stress 0"22 and displacement u2 on the XlX3-plane. Equations (4.19) and (4.20) yield -

o,

-

u2 -

P

0"22 - - ,~/Xa2x l

U2 - - 0,

v) / _

y-

,

/g2 _ xff,

Ix l < t

Ixll>t.

g2

At the vicinity of the crack tip Ix1 - g I = r 2d (see fig. 4.10) the series can be presented, to a good approximation, by the first term for all the microcracks except that nearest to the macrocrack tip, for the latter one linear term is to be preserved. It brings the whole problem to the solution of a system of linear equations, the coefficients of the series being unknown values. Averaged stress values a~ obtained are used then to calculate the stresses in the layers, a~ and ai~. Now coming back to the criterion given by eq. (4.101), take it as

(Axa x + Ayay) 2 + Axy(-Cxy)2 - 16yf/~zd

(4.102)

where constants A are expressed via elastic characteristics of the components and their volume fractions. The critical values of the stress intensity factor is taken as

xf

_

! ,/2Em(TmVm

V

1 -

ntv2

~fvf)

(4.103)

One can see that eq. (4.103) provides a possibility for the macrocrack to propagate also in the absence of the microcracks. The stress intensity factor is determined from the solution of the problem of the crack with microcracks in a homogeneous anisotropic solid (see Section 4.1.1). The computer simulation procedure is as follows. At a particular load, we have a particular number, N, of the microcracks and a length of the macrocrack, 2Ln - - d + 2nd/vf, where d is shown in fig. 4.10 and n an odd integer. So we start with a particular configuration of the crack system. Now increment APs of the applied load necessary to transform sth defect into the microcrack is being calculated. For the same system configuration the value of the stress intensity factor,

176

Ch. IV, w

M a c r o - a n d m i c r o c r a c k s in n o n - h o m o g e n e o u s m a t e r i a l s

Kl, necessary to make the macrocrack to propagate, is also calculated as well as corresponding value AP0 necessary to push ahead the macrocrack. If at least one value of AP found is negative, the configuration is assumed to be unstable and the defects with AP < 0 are replaced by the microcracks. Then new values of all load increments are calculated, again under the initial applied load. if AP0 < 0, the macrocrack increases its length by a characteristic size of the structure. Again the system goes to a new configuration without an increase in the applied load. If all values AP >_ 0, a lowest value of APs, say ~kPk, is to be chosen, the applied load is increased by this value, a corresponding defect is transformed into the microcrack, and the system goes to a new configuration to be analyzed at the next step. The procedure is being performed until the stability of the system configuration is impossible to reach. The point of instability corresponds to the ultimate load. A typical sequence of the system configurations is shown in fig. 4.13. One can see the growth of a number of microcracks when the applied load increases and an increase in the stress intensity factor at the macrocrack tip due to the formation of a microcracking zone (steps 1 through 7). At step 8 the macrocrack jumps by a characteristic structure size, the microcrack number does not change, but their configuration becomes such that they shield the crack tip (AK < 0). The shielding remains now up to the final step. Still, the stress intensity factor at the microcrack tip

1 n=37

2

3

N= 2

-

n=37

N= 4

AP= 40.83 A K / K = O. 0 2 9

-

AP= /5.88 AK / K =

re= 3 7

5

."

".

AP=5.85 ~ K / K = O. 0 4 /

-.

-

re= 3 7

O. 0 3 6

4 n=37

-"

5 N= 7

"

n=37

6 N= 9

.

A P = 3. 7 0 K/K=

O. O S S

AP= 0

/3

~P= e.sz aK/K= O. O S a

--

aXlE=

n

=

39

.

O. o e a

-

39

N = 2 0

. _ - -

~ P = e. 7 s _-.aK/K= - 0/0 /S - -

- O. 0 8

N=

/3

-

-

re= 3 9

n=39

-

N=22

N=

/8

_

.

.

-

aP: o .-.-. ~ : / K - - -o. o ~ / - -.

II .

-

9

n=

-

_ .

a x / z : = o. o s z

" =

I0 Ap= 0 K/K=

//

AP= 0 . 3 7 5

8 N=

N=

.

"

7 n=37

N=

12 -

-

n=41

N= 21

-

-

-

"

-

Ap= /.20 aI(/K= -0. 009

.'.'. - " "

A P = O. 1 0 5 AK~K-0. 035

- _ - _"

Fig. 4.13. Growth of the microcracking zone and stable macrocrack growth with applied load increasing. Value n characterizes the crack length, N is the number of microcracks, AP is an increment of the applied load transforming the systcm from the previous state to the present one, AK is the change in the stress intensity factor at the tip of the macrocrack due to the presence of microcracks. The parameters taken for the computer simulation experiment: E m = 70GPa, V m = 0.29, Ef = 400 GPa, vf = 0.24, ~m = 79.5 kJ/m 2, 7F = 0 . 5 5 kJ/m 2, vf = 0 . 5 , h i d = 5, d = 5 mm. (Calculations by Suleimanov and Tvardovsky [629].)

Ch. IV, w

Matrix crack&g

177

reaches a critical value between steps 11 and 12 that makes the macrocrack to advance. Hence, we observe a complicated interaction of the crack and growing microcracks born by the crack. At some stages of the loading, the crack tip is shielded by the microcracks; at another stage, it is 'sucked' by the microcracks. The process as a whole is developing in a stable manner until the stress a* when the macrocrack becomes unstable. Note that non-linear process of crack propagation through a changing material at the process zone is analyzed here by linearization of the problem at each step of crack advance.

4.4. Matrix cracking Matrix cracking in brittle matrix composites is an inevitable process that, in the absence of fibres properly included into the composite structure, leads to the quick fracture of the material. The fibres and properly designed fibre/matrix interface prevents the material from premature fracture. A sufficiently large number of observations of matrix cracking in ceramic matrix composites under tensile loading is known. Because of the obvious practical importance of ceramic-matrix composites to be used in high-temperature structures, modelling for this phenomenon has led to a large number of results ranging from those of qualitative nature to sufficiently rigorous solutions to rather idealized problems. All the approaches are useful if corresponding results are properly applied. We will discuss here some of the models and results located somewhere between the two extremes mentioned.

4.4.1. Aveston-Cooper-Kelly theory Aveston, Cooper and Kelly proposed a theory of crACKs in brittle matrix composites [25, 26, 229] based on the assumption of the matrix cracking without fibre breaking between the crack surfaces (ACK theory). For such a process to be really observed, two conditions should be obviously fulfilled. First, the fibre/matrix interface should deviate the crack in its plane leaving the fibre intact. Second, the fibres in a specimen cross-section can carry the additional load initially taken by the matrix, there should just be 0"~Vf ~ O"m Vm nt- O't,Vf

(4.104)

where a., is the fibre stress when the matrix cracks. A simple force balance yields the equation for the average matrix stress, a it, arisen due to the load transfer from the fibre of the circular cross-section to the matrix, as da" dx

Vf "C(X)

= 2--~

Vm r

where r is the fibre radius, and the x-axis directed along the fibre.

(4.105)

178

Macro- and microcracks in non-homogeneous materials

Ch. IV, w

If z - Z}m where Z}m is a constant, then the stress in the matrix rises from zero at :r the crack surface to the ultimate value, am, at distance (4.106)

Xt = vm(Tm vf "c $

Obviously, if the first matrix crack occurs at O'm, no cracks can be observed within Ix I< x ~. If value a m has no scatter, the largest distance between two neighbouring cracks will be 2x~. In such a case, the matrix cracking goes on at constant stress on the composite a** provided m o n o t o n o u s increasing load is applied to the specimen. When both components are fully elastic, the composite strain for the cracking onset is e** = 6m/Em and a** = O'mV m -[- e**Efvf = O'm(V m -[- (Ef/Em)vf). The overall strain increment during matrix cracking varies between l e**(Emvm/ Efvf) and 3e**(gmvm/Efvf) with crack spacing varying between 2x' and x'. After cracking is complete the effective modulus of the composite is equal Efvf. This is depicted in fig. 4.14a. Obviously, if a composite of the type just considered is stretching at infinity, a stress drop occurs when strain e** is reached (fig. 4.14b). If the composite with a matrix characterized by a scatter of value a m is loaded as in the former case, the cracking process proceeds under growing stress as shown schematically in fig. 4.14c. Consider now the energy balance in the matrix cracking process. We do it by following McCartney [396] mainly. As in the above analysis, Section 3.5.3, a singlefibre model (fig. 4.15) will be considered. The composite cell with the fibre diameter Ri and the external diameter Re, such that vf - (R/Re) 2, is stretched at the infinity so that the average strain is e. Hence the axial stress in the fibre in the crack plane is

0"(0) --

(Efvf -+- gmvm) (1 - v2)vf

"

(4.107)

G**

~**

(~)

~**

(b)

~**

(c)

Fig. 4.14. Schematic stress/strain curves of a brittle matrix composite. Point (a**,e**)corresponds to the onset of matrix multiply cracking. (a) The specimen is loaded at infinity (the Aveston-Cooper-Kelly case). (b) The specimen is stretched at infinity. (c) The matrix is characterized by a scatter of the cracking stress, the specimen is loaded at infinity.

Ch. IV, w

M a t r i x cracking

179

TTTTTTTTTTTTTTTTTTTTTTTT' q'

r rrrrT T 'll'II'[I"J'J'IT rrrTrrT r.~

AoA /Y

-

-

-

R .---J R e

Fig. 4.15. Schematic presentation of a single-fibre model of a composite with the matrix crack and debonding at the fibre/matrix interface.

The additional displacement between planes A and B, A1, is given by eq. (3.196), the crack opening, A0, is given by eq. (3.196). The work, AWe, done by the external load on the additional displacement A1 becomes available to an increment of both elastic energy in the system, A Wc, and the energy of newly formed surfaces. All these values relate to the unit area of the composite. We have the additional work Ag{ -- 2aA l

(4.108)

where a is the average stress in the composite cell. The strain energy stored by the composite between planes A and B before matrix cracking takes place is Wc -

I

vfEf vmEm "~ e2" l _ v2 + l _ v2 J

(4.109)

The strain energy stored by the fibre and matrix in the region of the interface debonding after matrix cracking is

180

M a c r o - a n d m i c r o c r a c k s in n o n - h o m o g e n e o u s materials

W~ - 2v~

fo' 1 1 - v2 (a(~)(z)) 2 dz 2 E~

Ch. IV, w (4.110)

where subscript ~ is either f or m and superscript (cr is either single or double prime. Hence, the elastic energy increment is AWe = a +

(4.111)

Wm -- Wc

The energy dissipated due to frictional slipping along the interface is

Rif0l

AWF -- 4Z-~ee

(u'(z) -- u " ( z ) ) dz .

(4.112)

Neglecting the debonding energy we write the energy balance as AWl = AWe + AWF + AG

(4.113)

where AG = 2~/mVm is the energy of newly formed surfaces in the matrix, 7m the effective surface energy of the matrix material. Making use of eqs. (3.188)-(3.194), (3.195), and (3.196), it follows from eq. (4.113)

** 6-

**

12~/mZ Vf2 ( 1 - v2)2Ef

- - em - -

Ri

l)m

E2mE

) 1/3 '

(4.114)

where E

= Efvf + Emvm.

The expression for the critical composite strain, that is also the strain at which the matrix cracking occurs, is the main result of the ACK theory. One can see that e _ * * o ( ( T m ' c / R i ) 1/3 so that the matrix cracking strain depends on the composite parameter and may be larger than that for the unreinforced matrix. Actually, in the A C K theory, the matrix cracking process is not analyzed, just available elastic energy in the initial state and energy that has been used, i.e. either stored or absorbed, in the final state of a composite are compared assuming that there is no energy barrier to be overcome in the process, such an assumption being physically reasonable. Nevertheless, McCartney [396] considered an equivalent homogeneous model of the crack of a finite length 2L in a composite with continuously distributed traction on the crack surfaces that replaces the fibre bridging action to prove the validity of Irwin's relationship, eq. (4.16), coupled with the fracture criterion given by eq. (4.35), that is g 2 _

2"YmvmE/(1

-

v2).

(4.115)

Ch. IV, w

Matrix cracking

181

The continuous distribution, q(x), of the traction on the crack surface is assumed to be

q(x) -- 2 x/-U~

(4.116)

where 2 is a constant, and

u(x) -- (u+(x) - u-(x))E/4rc(1

- v2).

(4.117)

Here u + - u- is the displacement jump at the crack, the value of 2 is chosen such that to equate work done by the external load resulting from the crack formation to the effective surface energy, accounting also for the stress state and frictional dissipation in the region of the interface debonding considered above. This yields -

2

vf /2rc'cErE

(4.118)

Analyzing under these conditions the energy balance formulated via the rates associated with the rate of the crack tip,/2, and making use of the dependence of u(x) on both a y and q(x) given by eq. (4.29), McCartney finally obtained 27mvmE 9 -x2)

L-~

(4.119)

from which eq. (4.115) follows. The expression obtained has been proved by the experiments with carbon fibre reinforced glass [533], fibre reinforced cements [229] and other brittle-matrix composites. On the other hand, accurate observations by using acoustic emission, replication and optical microscopy in conjunction with the stress/strain curves, of the cracking behaviour of glass and glass-ceramic matrix composites unidirectionally reinforced with high-modulus carbon fibres and Nicalon silicon carbide fibres have shown [316] that most of matrix microcracking takes place at strain much below those predicted by eq. (4.114). This can be a result of a scatter of the parameters in the equation mentioned. Two indirect confirmations of this can be seen. First, the arrest of transverse cracks observed in [316], and second, the extremely wide intervals of the stress and strain within which the matrix microcracking occurs in the same type of composites [296]. Finally we should point out that despite the McCartney's refinement of the ACK theory, it is not an objective of the Aveston-Cooper-Kelly to estimate the crack resistance of a composite as a macroscopic result of the theory. The theory explains in general terms a possibility of the stable matrix cracking in brittle matrix composites as well as predicts the stress/strain curve and the ultimate stress, the latter being predetermined by the fibre strength and fibre volume fraction. The

182


Ch. IV, {}4.4

problem of estimation of the fracture toughness will be considered in the next sections.

4.4.2. Crack propagation To estimate the conditions for crack propagation in a composite when the matrix crack leaves intact fibres, at least in some regions, one needs (i) to evaluate either the stress state at the vicinity of the crack tip via calculation of stress intensity factor or a value characterizing elastic energy concentration at the crack tip, those being strain density function or J-integral; (ii) to formulate fracture criterion; (iii) to apply the criterion in an appropriate manner. In the case under consideration, meeting these problems, the time to rely on a routine mathematical formalism has not perhaps come. This is true even when we are dealing with the first problem. To solve it, a number of assumptions to be held with is usually quite large, and a final result depends on their physical 'quality'. FormuJating the fracture criterion involves inevitably some physical ideas; experimental observations are normally necessary to provide them. Also, if the criterion is to be formulated in terms of 'effective' characteristics of the composite, a correlation between effective values and local ones is needed. Again, the scale of the locality is an important issue. This means, in particular, that constructing a complete theory of the process needs a great deal of consideration of various aspects of the problem. A large contribution to developing such a theory has been made by Evans, Marshall, Cox with co-authors, in a series of papers published during the last decade [159, 161,385, 386]. Their results shall be mainly discussed in this section in which we consider the cracking of brittle matrix composites when either fully or partial fibre bridging of crack surfaces occurs. Suppose the crack in a fibrous composite carries a fibre bridging zone at its tip (fig. 4.16), similar to the crack in an elastic-plastic metal that pushes ahead of its tip the plastic zone. The analogy can be more radical as we can picture such zone by a linear image like the Dugdale or Panasyuk-Leonov model (Section 4.1.2). Pragmatically, we can go on with this analogy by calculating J-integral around the zone although the material behaviour in this case has hardly anything in common with the deformation theory of plasticity. The corresponding technique was used when a phenomenological equation for the material behaviour in a bridged part of the crack was assumed (Section 4.1.2). Another way to consider the crack with the fibre bridging zone is to calculate directly the effective stress intensity factor for a crack with a part of its surface loaded by tractions that model forces imposed by the fibres. The corresponding mathematics is also known (Section 4.1.1). In both cases, to elaborate the result executing either eqs. (4.26)-(4.27) or eq. (4.45), the dependence of the traction, p(x) or trq(X) on crack opening u(x) is required. Obviously, the fibres that remain intact at the vicinity of the crack tip (fig. 4.16), change the stress state ahead of the tip. The problem can be solved evaluating the stress intensity factor via the tractions applied on a part of crack surfaces in accordance to eq. (4.26) and (4.27). Again, as in the above consideration,

Ch. IV, w

Matrix cracking

183

ttttttttttttttttttttttttttt

x...~

Fig. 4.16. A schematic view of a bridged crack.

a continuous distribution of the tractions, q(x), is assumed on the bridged part of the crack surface, so that instead of eq. (4.116) we write

q--

{ 2X/~(x) 0

at x > L - c, at x < _ L - c

,

(4.120)

where u(x) is the displacement of the surface relatively to the middle plane, that is a half of the value in eq. (4.116). To connect q(x) to the crack opening displacement, u, eq. (3.195) is used, that yields q -- 2(uzv2EfE/Emvmr) 1/2,

(4.121)

where E -- g f v f q-- g m Vm,

r is the fibre radius, and z the sliding frictional stress on the fibre/matrix interface. Equation (4.121) is to be solved together with that describing the crack profile (see above, eq. (4.29)). Marshall and Evans [386] in an attempt to get an approximate analytical solution, assumed a crack subject to uniform traction, with the crack opening determined by the net stress intensity factor K ~ given by eq. (4.27). Then

184


u(~) - 2(1 - v2)KZl/2(1 -~2)1/2/~1/2E.

Ch. IV, w (4.122)

Combining eqs. (4.122) and (4.121) yields

q(~) _ (~KOL1/2(1 _ ~,2)1/2)1/2

(4.123)

where - 8(1 - v2)~v2Er(1 + q ) / g r ~ 1/2. Now the net stress intensity factor can be written as

K=K

~ -Kq

(4.124)

where K ~176 = ~o "~176 1/2 Kq - (16~/91t) l'/Z (K~ l/2c3/4 (2 - c/L) 3/4, and f* = 2/v/~. The term Kq represents the decrease of the stress intensity factor due to the fibre bridging a part of the crack.

4.4.3. Pseudo-macrocracks Let us consider again a geometrically plane model shown in fig. 4.10, but unlike the situation analyzed above, we have no original macrocrack in it. Suppose one kind of the layers is brittle, the second kind is relatively ductile. The strength of the interface is assumed to be large enough to prevent delamination during crack propagation. At the same time the interface strength is sufficiently small to prevent the direct penetration of a crack into a ductile layer. When such a solid is put under increasing tensile load, microcracks present in the brittle layer can propagate leaving the other layers intact. Corresponding mechanical model have been developing during the last decade, we will follow [299, 460]. We call the pseudo-macrocrack a set of finite number of identical cracks regularly spaced along one axis, say x on fig. 4.17.

Qualitative analysis of the pseudo-maerocrack Consider the problem of pseudo-macrocrack using the energy balance approach of the Griffith type. The change in the elastic energy, A W, of an isotropic body under the plane strain condition as a result of the crack formation with length 2L is

p2 Aw

-

5 2 (1 -

(4.125)

Ch. IV, w

185

Matrix cracking

p~

t

t

t

t t

t t

t

t

Fig. 4.17. Pseudo-macrocrack in a plane geometry.

in which p - ~ry is the applied stress at infinity, E and v the Young's modulus and Poisson's ratio, respectively. Since the quantity p 2 ( 1 - vZ)/2E is the elastic energy density per unit area without the crack, the quantity ~rtL2 can be treated as the effective area unloaded by the crack. Coefficient ~ defines a particular geometry. In the case of the Griffith's crack, ~ = 2 and the configuration is that of an ellipse with semiaxes L and L x/2 in x- and y-direction, respectively. If the crack surfaces are bound by unbroken layers, the unloaded zone is different. The dimension in the ydirection should decrease down to le which will be shown below to equal approximately to the load transfer length along the interface. The corresponding change in the elastic energy now becomes p2 A W -- ~7~Zl e ~

(1 - v2)(1 - vf)

(4.126)

where uLle is the area of the ellipse with semiaxes L and le. The factor (1 - vr) is volume fraction of the phase susceptible to cracking (say, matrix). The effect of fibre overloading can be accounted for in the determination of 1r which is proportional to the period, T, of the structure and depends on yr. Equation (4.126) is valid for L >> lr only. The energy balance

o(Aw) ~L

= 47m (1 -- vf)

(4.127)

can be used to obtain the critical load 2

(4.128)


186

Ch. IV, w

Here (4.129)

K m - V,/2E~,m/(1 - v2)

is the critical stress intensity factor for the matrix material and ])m is the specific surface energy of the matrix. Equation (4.128) is derived by assuming both phases have the same elastic modulus, i.e. Em = Ef = E. If Em -7(=Ef then the ratio Em/Ef and vf will enter into eq. (4.127) while le will also depend on Em/Ef. For a penny-shaped crack we have p* = K m V/3/21e.

(4.130)

An important conclusion that follows from eqs. (4.128) and (4.130) is that the critical load p* does not depend of the length of the pseudomacrocrack.

Modelling a pseudo-macrocrack Let us consider the same problem of the plane pseudo-macrocrack in more detail. An anisotropic elastic body characterized by the effective engineering constants, Ey = Ez, Ex, Vxz, Vyz, Vxy, with the xy-plane being that of the elastic symmetry, contains a regular system of cracks located in one plane, the length of each crack being 21 and the period being T (fig. 4.17). The analytical solution for finite number M of such microcracks can be found in a paper by Sih and Liebowitz [610], but here we will follow a simpler but approximate solution of the problem [460] based on well known presentation of stress and strain components via complex potentials, eq. (4.31). The critical state of a pseudo-macrocrack is determined if the variation of the elastic energy 6AW corresponding to a unit extension can be calculated. We can write 1

A W _ -~ ~

fXk+l Jxk-l

o (O'~(X)(Uy+ (X) -- (U;(X)) + ~Txy(X)(U+(X) -- Ux(X))

)dx (4 .1 31)

where u~+ ( x ) - u-~(x) are the displacement jumps at a cut, ~ - x,y. For constant 0 = p, axy(X)= 0, eq. (4.131) can be integrated by parts and external loads, ay~ written as

AW -

P

--2 k

[xk+l

+ (x - X k ) ~ x (Uy (x) -- u ; ( x ) ) d x

(4.132)

A standard procedure using the complex potentials yields

AW -- pxl 2 1 - Vy i 2z Re 1/21+/22 ] Z ak 2Ey

/21/22 J k

(4.133)

Matrix cracking

Ch. IV, w

187

where /~l and ~t2 are the non-equal roots of the corresponding characteristic equation, ak the unknown real numbers determined by satisfying the boundary conditions on each cut. For sufficiently large number of the cracks in the array, L >> T, the summation Y~k ak is substituted by integration of function a(x) which is approximated as

ao(x)

- - 6 y~

-

412/r2

Finally, using the energy balance, eq. (4.127), we obtain

p * ~ 2Km l

EY v/ T2 - 412 rcTlEm Re [i ~lnt-/22] ~-~S~j

(4.134)

Comparing now eqs. (4.134) and (4.128), a characteristic length le is found

4TIEm Ev/T 2 _ 412 ,

le

(4.135)

where E is the effective Young's modulus of the solid in the y-direction. The stress intensity factors at the right (superscript 'plus') and left (superscript 'minus') tips of the jth cut is determined by

K - K~ =

1 f, , V It+, ~-~a(xj

~--~7

+ t)dt.

(4.136)

If the function a(xj + t) alters just slightly at It I< l, then K~ -- a(xj)~/~.

(4.137)

In the case of an infinite periodic array of cracks engulfed by homogeneous stress field a y - p at infinity, all constants ak are identical and equal to a0. In such a case we have

ao

-

-

pl V/1 - 412/T 2.

(4.138)

Therefore, the crack tip stress intensity factor is 2

2 G.C. Sih has pointed out the exact solution for small ratios l/T obtained by Koiter in 1959 which is K~ = p T T t a n ( ~ )

Macro-andmicrocracksin non-homogeneousmaterials

188

KI = px/~ V/1 - 412/T 2.

Ch. IV, w (4.139)

For large values of the microcracks (L >> T), using an approximation for a(x) yields Ki--p~--1

1+

T - v/T 2 - l 2) v/T2_4/2 .

(4.140)

Again, it is important to note that the stress intensity factor at the margin microcrack does not depend on the number on microcracks or the length of the pseudo-macrocrack. Substituting eq. (4.134) into eq. (4.140) gives the critical stress intensity factor 2 - 12~ K* - 2K~ q Ey v/ Ti - 412 ( 1 + T -~T--v/T ~ _- 4"12 /" TEmRe i"1+"2]

(4.141)

~1/'12 J

Here K m is given by eq. (4.129).

Application to fibrous composites The results obtained in the previous section can be applied for estimating the ultimate properties of the fibrous composites. We simply need to express T and l as functions of fibre diameter and fibre volume fraction. For the hexagonal fibre arrangement

d T--~V/-3/vf,

2l-

(1 -

vf)T- vmT.

(4.142)

Substituting eq. (4.142) into eq. (4.141), the critical stress intensity factor Ki~ at the tip of the pseudo-macrocrack is obtained

K*-2flKm~/Eyv/1-V2m( l+Em

1 - V/1 - v2/4~ @--i ----VTm j

(4.143)

In the same way, eqs. (4.128), (4.130) and (4.134) can be applied to obtain the ultimate load for a body containing a penny-shaped pseudo-macrocrack. The result is

~/~Evf V/3 ( 1 + Vm) . , - flK v

-UdGJ

(4.144)

The dimensionless factor fl in eqs. (4.143) and (4.144) depends on the ratio of elastic moduli of the matrix and the fibre, which is given by

Interface cracking

Ch. IV, w

-

]21~2 J

.

189

(4.145)

If Em/Ef ~ 1, then/3 ~ 1/2. It should be noted that quantity 0-* in eq. (4.144) can be regarded as the composite strength if vf is sufficiently large. This gives 0-* > 0-~ where 0-~ is the effective strength of the fibres in a bundle.

4.5. Interface cracking Interface cracking can arrest a microcrack and enhance fracture toughness of a composite. In this section we present a simple model suggested by Cook and Gordon in 1964, then we proceed with a modern analysis of the problem.

4.5.1. Cook-Cordon's model Cook and Cordon [97] were the first to show clearly a possibility to stop a crack by a weak interface. 3 They were analyzing numerically the stress field in the plane stress situation illustrated in fig. 4.18(a). Under the stress, a ~ , applied at infinity, a complex stress state ahead of the continuation of the long axis of the ellipse is generated. The normal stress, all, reaches a maximum value at the hole nose, Xl = x2 = 0 (fig. 4.18(b)). The transversal normal stress, 0-22, reaches a maximum value at some distance away from the point mentioned, say at point C. The numerical results by Cook and Cordon show that decreasing the semiaxes ratio, that is making the hole to approach a crack shape, (max) (max) one can observe the ratio of corresponding maxima 0-22 /0-11 , approaching a constant value, ~ 1/5. Therefore, if the values of the strength of the bulk body, 0-~1, and the cleavage strength of the interface, 0-~2, are such that

then the general configuration may change drastically (fig. 4.18(c)) with the appearance of longitudinal crack at the vicinity of point C. Hence, the main crack gives birth to a secondary one at the interface that does obviously decrease the stress ahead of the primary crack essentially. The behaviour of C o o k - G o r d o n ' s crack in an orthotropic material was treated analytically in [542]. Limits for ratios of the characteristic stresses were found when

3Actually, Stepanov [624] discussed similar ideas in 1949; perhaps, he did it in not so clear form.

190 t

,.

t

Macro- and microcracks in non-homogeneous materials t

t

t

t

t

t

t

t

t

t

t

t

Ch. IV, w

ott

b)

Xl

X2

MODRL CRACK

a)

0 Fig. 4.18. The Gordon's crack in a body with a weak interface.

ratio e of the semiaxis of the ellipse with axis coinciding with those of elastic symmetry of the material is going to zero, e ~ 0. There occurred to be { (max)

2--1im~0~,Crll

lim t' ool e---~o \

Vmax --

11/

1/4

3,/3 1

z~2 --+ o-~ 2,

with the anisotropy increasing, for an isotropic material,

2max -- 1 / 3 V ~

v

)

/ ~r~2 ~ 0

o '~ ~.z-/

with the anisotropy increasing, for an isotropic material, for an isotropic material, with the anisotropy increasing.

Ch. IV, w

Interface cracking

191

00 is Here 0-00, a02 and '6o2 are the largest values of the corresponding stresses; 0-11 reached at a point on the ellipse contour, x2 ~ 0, and 0"11 00 f... 0"11 (max).,'612 0 is also reached at a point on the ellipse contour, x2 :/: 0. The most important finding in [542] is that for a sufficiently high anisotropy, the ratio '602/0"02 does not depend on the shape of the stress concentrator and is not going to zero unlike the values of 2 and v. Hence, this ratio is a characteristic parameter of the structure which is determined only by ratios of elastic characteristic. An expected change in the geometry of the crack tip that leads to the crack arrest is delamination by shear to originate at a point on the crack contour where a largest value of '602 is reached, say point S in fig. 4.18d. Actually, whether delamination occurs and the crack stops or fibres at the crack front break and the crack propagates, depends on which of the following inequalities is satisfied:

"602 > '6~2 0"22

"602 O'02

'6~2 O'22

(4.146)

where '6~2 is the shear strength of an interface in the composite. If the first inequality is satisfied then the crack is to lose its sharpness as a result of the delamination. Otherwise, it propagates by breaking fibres. An optimal structure is certainly such that the equality of the two ratios is observed. If one tends to enhance the crack resistance together with tensile strength and rigidity of the composite, one has to increase the shear strength when increasing the tensile strength by increasing, for example, the fibre volume fraction. Metal- and ceramic-matrix composites, in this respect, are much more versatile materials than fibre-reinforced plastics with their inherently low values of the shear strength.

4.5.2. Cracks near the &terface

Suppose, there appears a crack with its tip placed at the interface between two media, A and B, with different elastic properties (fig. 4.19). Without looking at the pre-history of this event which could be non-real if the interface strength is sufficiently low, let us analyze possible after-effects. The problem of the stress state at the crack tip in such a situation was first solved by Zak and Williams [727] who did the work at about the same time as Cook and Gordon were looking at the consequences of meeting a weak interface by a crack. An important finding by Zak and Williams is that singularity of the stress state is different from that of a homogeneous medium. The stress field for the case of a crack perpendicular to the plane interface is given by aij cx/Cr ~

(4.147)

where, unlike the case of a crack in a homogeneous solid, 2 is a real root of the equation [239], that is

192

Macro-and microcracks in non-homogeneous materials

Ch. IV, w

:re

B

A

Fig. 4.19. A crack in the Zak and Williams' problem.

cos 2~r -

2 ( / / - a) 1+/3

a

q_ f12

(1 _/~)2q 1 - - f l 2

and obviously, 2 -r 1/2, so the dimension o f / s is M P a - m ~. Here - (EB - E A ) / ( E B + E A ) ,

f l - (/~B(1 -- 2VA) --/~A(1 -- 2VB))/(/~B(1 -- VA)+/~A(1 -- VB))

(4.148) (4.149)

where m

E

E D ~

1 -- V2"

Note that the singularity is stronger than - 1 / 2 when the crack is moving from more rigid to less rigid material and vice versa. He and Hutchinson [239] considered conditions for a Zak-Williams crack to penetrate through the interface or debond the interface (fig. 4.20). If the crack penetrates the interface, its stress intensity factor follows from dimensional considerations as KI -- F(~, fl)Kc 1/2-~

(4.150)

Interface cracking

Ch. IV, w

A

193

/7

xt

C

Fig. 4.20. If the crack reaches the interface it can either penetrate through the interface or debond the interface.

where F is dimensionless. The energy release rate is Gp =

1-

(4.151)

VB F 2 k 2 c l _ 2 2

2/~B

In the case of debonding, the stress state is obtained as a solution of a system of the integral equations, the final result that is the ratio of energy release rates Gd and Gp for debonding and penetration, respectively, is G d = 1 -- f l 2 1 d Gp

1 - ~

12 + l e [2 +2Re(de)

(4.152)

F 2

where F, d(a, fl) and e(a, fl) are obtained from the solution of the system mentioned. Note that Gd/Gp is independent of c and K. The dependence given by eq. (4.152) is schematically illustrated in fig. 4.21 which can be called the debonding diagram. The qualitative applicability of the debonding diagram to metal matrix composites was proved by Chan [74] who plotted experimental points for B-B4C/Ti, SiC/Ti and A1203/Mg composites on the diagram. The corresponding values of a lie between 0.53 (for B-B4C/Ti) and 0.78

194

Macro--and microcracks & non-homogeneous materials

Ch. IV, w

FIBRE / FRACTURE /

DEBONDING .

.

.

.

.

.

.

.

.

o~

.

4 ,,

1

Fig. 4.21. Schematic representation of the calculated dependence of the critical ratio of values of energy release rates for interface debonding, Gd, and the crack penetration, Gp, on Dundurs' parameter ~. Gfm and Gf are the critical energy release rates of the fibre/matrix interface and the fibre. The diagram suggested by He and Hutchinson [239] can be called debonding diagram.

(for AI203/Mg) with the values of Gfm/Cf being between 0.03-0.10 (for SiC/Ti) and 0.95 (for AI203/Mg). The experimentally observed regions of the interface debonding and fibre cracking (crack penetration) appear to correspond to the HeHutchinson diagram fairly well, despite in this approach, finite sizes of the components are not considered. An attempt to account for finite sizes of the crack and the matrix layer is performed by Popejoy and Dharani [544]. They used the shear-lag analysis accounting for the matrix stresses to calculate the stresses and displacements in a model shown in fig. 4.22a. To exclude a possibility of oscillatory behaviour of the stress field at the crack tip occurring within a bimaterial interface, an interlayer at the interface is introduced as shown in fig. 4.22b. The elastic properties of the interlayer are chosen in such a way as to replace jumps of the properties with continuous changes in them. In general, this gives reasonably good results as long as the interlayer is thin relative to the other regions. Two important situations were modelled. First, coating a fibre with materials of various Young's modulus values. Results of the calculation presented schematically in figs. 4.23 and 4.24 show that (i) the debonding/penetration behaviour depends on the coating thickness not strongly; (ii) coating with a stiffer material enhances a critical value of Gd/Gp, that means that debonding occurs within larger interval of the ratio Gi/Gs where Gi and Gs are critical energy release rates for either the matrix/coating interface and coating, respectively, or the coating/fibre interface and fibre, respectively. Secondly, the matrix crack approaching a debond part of the fibre/matrix interface (fig. 4.22d). A rather unexpected result shown schematically in fig. 4.25 reveals a non-monotonic dependence. A method to analyze the behaviour of cracks in ductile-matrix composites in terms of strength criteria for the fibre and fibre/matrix interface coupled with the yield criterion for the matrix was suggested by Chan [73]. The analysis is based on

Interface cracking

Ch. IV, w

195

TTTTTTTTTTTTTTTTTTT ~ ~

ii

2a

---~

(a) MATERIAL 2

INTERLAYER

INTERFACE

i

(b) ---t

COATING

/_

Z2 ~

--- t

1,9

-(d)

-L-Fig. 4.22. A crack in a plane composite model of Popejoy and Dharani [544].

calculation of the stress/strain fields in the matrix in the vicinity of a crack. To calculate the fields, plastic zone in the matrix is represented by a continuous dislocation distribution on slip planes normal to the crack plane. Then a constant shear stress equal to the yield stress, Zm, of the matrix material equates, within the plastic zone, to a sum of the stresses determined by the crack in the elastic medium and a total action of the dislocation dipoles. This yields to a formulation of the


196

Ch. IV, w

..................................

/

T

E:<E~

,

E,.,, < E~ < E I

E i < Ern

0

....

I

o

I

~ (t/~).

t o -~

2

Fig. 4.23. Critical value of the Gd/Gp ratio for crack impinging on matrix/coating interface (crack 1 in fig. 4.22c) versus normalized thickness of the coating. Schematic representation of the data by Popejoy and Dharani [544] for E f - 200 GPa, Em = 85 GPa, vf = Vm = 0.25, v i - 0.30. Index i stays for the coating.

_

T El
~ . (7~ g ( l / x ) \ O ' m J gl

(4.159)

Finally, Chan used his expression [74] for the fibre stress in the bridged fibre (fig. 4.26c) * "Cfm

(4.160)

r 1/4

which yields, for the single bridging fibre, the condition for crack bridging prior to fibre fracture as

7s < C( "c~~ 1/2(,c~t~ 1/2(O.mV/~~ 1/2

ff~

\T~m]

\amJ

\

Ki

J

(4.161)

with C being a constant.

4 It is important to point out that only effective fibre strength may be introduced here because (i) stress o-' is calculated for the fibre center in the crack plane and (ii) the fibre break is determined by an appropriate defect at the vicinity of the point under consideration.

Interface cracking

Ch. IV, w

199

Dislocation dipole

(~)

(b)

(o)

Fig. 4.26. Schematics showing (a) localizing plastic zone, (b) interface debonding, and (c) crack bridging in Chan's consideration [74].

Plotting the conditions given by eqs. (4.156), (4.160) and (4.161) as the dependencies of ~ / a ~ on CrmX/~f/Ki, we obtain the fracture map shown in fig. 4.27.

4.5.3. A crack on interface A direct stress analysis for the interface crack leads to oscillatory singularities and corresponding query when applying fracture criteria (see [141] for a review of the

~ABTRRIXFY IAEL?JN~/ d b'~

ITEDDEBONDING// ~

L ]

/

/

EXTENSIVE DEBONDING/

~

,-~ ~ / / 2

Fig. 4.27. A map of failure mechanisms suggested by Chan [73]. Four regions correspond to (i) matrix yielding followed by fibre fracture, (ii) limited interface debonding (l < 2d0 and fibre fracture, (iii) extensive interface debonding (l > 2d0 and fibre fracture, (iv) crack bridging.

200


Ch. IV, w

7"

T

Fig. 4.28. An interface crack under longitudinal shear. The maximum applied shear stress is shown, so Z13 = Z C O S ~ and 1523 --" T sin ~ .

problems and possible ways to overcome them). So we restrict ourselves with consideration of a relatively simple problem of the interface crack under longitudinal shear (fig. 4.28) which, being uniaxial, does not include a possibility of the oscillatory behavior of the stresses near the crack tip and clearly illustrates the interface behaviour. Interface crack under longitudinal shear We consider here formulation of the problem and the results obtained [461,664]. The method of solution of the problem was described in general terms in Section 4.1.1. Let a circular inclusion be placed in an infinite solid. The shear moduli of the inclusion (fibre) and the matrix are/if a n d / t m. The crack is located along lo(t) while /l(t) denotes the bonded portion of the interface. Tractions f + ( t ) and f - ( t ) are presented on lo(t) and continuity of tractions and displacements are satisfied on ll(t). Constant stresses r ~ and r ~ are applied at infinity. Again, the arbitrary function w' given by eq. (4.5) supplies solution to the problem. The stress at an element with normal n: (1, m) can be written as

Zn = lZl + mz2.

(4.162)

The condition at the interface is 2Zn -- tw'(t) + tw'(t).

(4.163)

The quantity

l + im = cos0 + isin0 = t has been used to obtain eq. (4.163).

(4.164)

Interface cracking

Ch. IV, w

The b o u n d a r y conditions on "Cn+ ( t ) - -

f + (t) - 0,

lo(t)

201

are

r~ (t) - f - (t) - 0.

(4.165)

Continuity on Ii (t) implies that OU +

~7n+ ( t )

~U-

~8 0 -- - - 8- 0

"/7n ( t ) ,

(4.166)

"

At infinity, it can be written w'(t) -

(4.167)

T + N . z -1

where T - -r~ - i-r~ and 2rcN is the total force applied to the crack surfaces. The solution to the b o u n d a r y problem just stated is k n o w n [181], that for the crack situated along an arc of radius R symmetrically with respect to the x-axis is ,

w (z) - Wo(Z) +

Wtm(Z) -- Wo(Z ) +

l,f

/Am +/Af /Am

/Am +/Af

v--

T

Z2 / Z2 /

(4.168)

with

r(z-RcosOo)z 2 -

!

Wo(Z) /Am + /Af

Z2 V / z 2 - - 2 R z

R2(R-zcos00) cos 00 + R 2

It can be shown that the change in elastic energy is given by AU-

7tR 2

/Am

/Am --/AfCOS 00 TT q 1 #f sin 2 0o Re(T2)). /Am + / i f 2 /Am +/Af

(4.169)

F o r symmetrically loading ( r ~ - 0), the Griffith's condition OAU 800

- 4RTfm

(4.170)

yields the critical stress / 47fm/Am(1 +/Am//Af) Z, -- V-x--~n~oi 1 5r ~osOoo)"

(4.171)

F o r non-symmetrical loading (r~ -r 0) fracture would start at one of the two crack tips. The symmetry of the system changes as one crack tip moves. With r ~ - r cos and r ~ - v sin ~, eq. (4.171) takes the form

202


z, -

V/

4~'fmttm (1 + ttm/,Uf) ~R sin Ooc o s 2 ( 0 o / 2 - ~)

Ch. IV, w

(4.172)

if cos2(00/2 - 0t) > cos2(00/2 + e). Otherwise, /

4)'fmttm(1 + ~m/~f)

(4.173)

- VThe dependencies obtained are shown in fig. 4.29. Figure 4.30 presents the dependence schematically to point out that a crack with angular length 20o < 20, propagates in an unstable manner until it reaches the condition 20o > 20,. Then it propagates in a stable quasistatic manner as the load increases. If the initial angular 20o _> 20,, then the crack length grows with increasing load in a stable quasistatic fashion. The situation depends on a particular combination of the values 00 and ~. Note, in particular, that the value of 0, is a solution of the equation cos 0 + cos(2(0 + ~)) = 0

(4.174)

which lies within the interval (0, r~). It should be noted that eq. (4.168) can be rewritten [664] in the local coordinate system (~, r/) as defined in fig. 4.28. For crack tips a and b they are

,~10

|

~

w

I

,

,

|

!

a=TT/2

i

1

a=O/

~4

C2 0

!

1

2

3

Fig. 4.29. Dependencies of the normalized critical stress on the angular length of the interface crack for different symmetries of the crack given by ~.

Ch. IV, w

203

Interlace cracking

r

"01

I

I

~'0,

1"02

I

"0

I

J

Fig. 4.30. Schematic dependence of the curve family shown in fig. 4.29. If initial angular length, 201, of the crack is less then that corresponding to a minimum on the curve, 0,, then the crack jumps to the length 202 to propagate further under increasing load along the path marked by stars. a ]Af ~/2R sin 00 ( 00 ~) "c~cos + ' c ~ s i n - "C~3 ]Af -t- ]Am r -2-

"Ca __ __

2R sin 0o -c~ cos

]Af

-

r

"C~3-- ]Af-[-]Af]Am

+ r~ sin

cos

(re qo) 4 2

sin Tc q~

5-

( 00 2R Sinr00 "c~ cos-~- -

(4.175) ~ sin--

'c~3 -- ]Af-+-]Af]Am 2R Sinr00 ~ cos-~- - "c~ sin--

cos ~ + ~sin ~ +

The shear stresses reach maximum values at tangential planes 'ca=

]Af

~/2R sin 00 cos (O0

]Af nt- ]Am

~)'c

r

~ +~/2R ] ~A rm sin0~c~ ( -- ~)'c.

"cb-- nt-]A~f ]Af

(4.176)

These formulas can be obviously written in usual form using the stress intensity factor. The energy release rate and the stress intensity factor can be related to each other by using well known Irwin's procedure already used in obtaining eq. (4.16). This yields _

/2Gfm]Am]Af

KI*II V ~m ~ ~ "

(4.177)

204

Macro- and microcracks & non-homogeneous materials

Ch. IV, w

The interface crack will propagate along the interface according to the above consideration provided K, 0321~

216

Macro- and microcracks & non-homogeneous mater&&

C,,O1o

6,)x

~l x

~ 60

Ch. IV, w

6de0' 6de Q}21

Fig. 4.36. A bi-modal distribution function for the interface energy.

7-

._.To/

....

1

Fig. 4.37. Stress/strain curve for a material with uni-modal (dotted line) and bi-modal distribution function for the interfaces energy.

(0 2

CXz (iim -~- iif)(ii0 -- i l l )

accounting for eq. (4.209). One can see that the values of CO 1 and O3 2 characterize the dispersion while the areas Sl and s2 determine the third moment, which is related to S = S1/(S1 + s 2 ) .

In the calculations the mean value of o~, (o~), is taken to be constant, the reference material is characterized by the (co) and the stress/strain curve with the "yield stress" z0. Some results of the calculations are presented in fig. 4.38. It should be pointed out that the dispersion does strongly influence the critical stress; the symmetry of the distribution function (values s in fig. 4.38) has also the influence, although that effect is smaller. In a material with bi-modal distribution function for the interface energy, sufficiently long cracks can be completely arrested. An explanation in physical terms of the effect of enhancement of the crack resistance of a composite due to non-homogeneous interface energy as compared with that of the reference material is enlargement of the fracture process zone in which energy dissipates.

Cracking in joints

Ch. IV, w '

1"01~ FI" 1

d.rve

,,/,o

0.8b I

I

]-

'

2

]

s__.._s._ - -_

-3

I

"

4-

5

-6

I

7-

217

"

8

-9

0.8 o.s 0.6 0.8 0.8 0.6 o.s 0.6

0.70 0.85 0.85 0.55 0.40 0.70 0.25 0.55

2

0.6

67

0.4

8

0.2 0

,

I

10

,

I

20

,

,

I

30

"

I

40

Fig. 4.38. Normalized ultimate stress versus normalized crack length for a material with bi-modal distribution function for the interface energy. Constant parameters are Vf = 0.55, V0m -- 0.44, ]Af/#m = 50. After Mileiko and Tvardovsky [458].

It should be noted that in the above consideration, values of the interface energy may be divided by the fibre radius R without changing the results. This means that the same enhancement of the effective surface energy of a composite can be achieved by introducing distribution function for the fibre radius. 4.7. Cracking in joints

This section is an introduction to an important subject of the load transfer to a composite element. Perhaps, designing structural parts to load metal-matrixcomposite elements reveals advantages of such composites as comparing with fibrereinforced polymer. That procedure applied to a ceramic-matrix-composite elements is a special subject since many precautions have to be observed in a corresponding design due to relatively high brittleness of fibre-reinforced ceramics as compared with metal-matrix composites and, usually, quite a high mismatch in coefficients of thermal expansion of ceramics and metals or composites to be joint to ceramics. Hence, we shall describe here mechanical models and support the conclusions by experimental data obtained for metal matrix composites only. We shall consider just two types of load-transfer designs, namely those via pin-loaded holes and overlapping. In strict formulations, these two problems are complicated contact problems of the elasticity theory of anisotropic non-homogeneous solids. 4.7.1. Plate with a hole

In addition to a contact problem arisen there is a question of formulation of failure criterion. So we start with a simpler situation which does not include the contact problem, that is the behaviour of a composite containing a hole.

218

Macro- and microcraeks in non-homogeneous materials

Ch. IV, w

A hole in a composite plate produces the stress concentration well studied for elastic solids. For composites, which are non-elastic and non-homogeneous materials, at least two problems arise. First, non-homogeneity may lead to the fracture starting not at a point of a maximum stress, A in fig. 4.39a, but at some point away, where the stress reaches a critical value. Hence, the situation calls for a statistical analysis, for example, by applying the strength distribution function of the Weibull type which can be written for the plane stress state as

/ ds}

(4.221)

Here we take the threshold stress equal to zero. Secondly, when a most dangerous volume in the vicinity of point A has given out, the corresponding stress is taken by neighbouring volumes and the applied load can increase. This is especially true for composite laminates and that type of behaviour is similar to plasticity in metals. So three approaches are known to the problem: 9 Calculation of the stress/strain state around the hole by using analytical [17] or numerical [357] methods and applying either a set of fracture criteria to account for an elastic moduli reduction due to the damage or just a single criterion to assume the stiffness of a damaged volume becoming zero. 9 Calculation of the average, through the thickness of the plate, stress components and applying a Weibull type statistics to estimate the failure probability under the applied load [709].

u

~ f

A

.....

x

i

D

Fig. 4.39. Schematic illustration of failure criteria of a plate with a hole.

Cracking in jo&ts

Ch. IV, w

219

9 Assuming a fracture criterion for a particular case of loading and applying it to the elastic stress state calculated. The last approach, which is very simple, has been rather popular since Whitney and Nuismer [699] suggested two simple criteria. Both include a linear dimension explicitly. According to the first criterion, failure occurs when the stress at a distance aw from the hole contour reaches a critical value a, (fig. 4.39a), quantities a, and aw being two material parameters to be measured in an experiment. According to the second criterion, failure occurs when the integral of ay over length aN (fig. 4.39b) reaches a critical value, that is 1 jR+aN 0", - - - aN dR

O'y(X, 0)dx.

(4.222 /

Here again a, and aN are the material parameters to be determined in an experiment. The latter criterion corresponds to the experimental results sufficiently well. Yet, two lines of the attempts to go on with the developing new criteria are known. The first line is directed towards improving the phenomenological criteria. This line originates, perhaps, in paper by Waddoups et al. [677] published before [699]. The authors assumed that the regions of a heavy damage of a length a (fig. 4.39c) can be treated as cracks originated from the hole surface. The stress intensity factor at the crack tip is

KI = ax/~-af (a/R)

(4.223)

where f(a/R) is a known function [56]. 6 A recent version of an improved failure criterion [697] is written as fRD - - ' ~1 s

(4.224)

where aij is the stress tensor at point Xk, f(k) is the failure surface, which can be, for example, of a polynomial type, and ~0(Xk) a weight function which needs to be approximated. The integration is performed over region D (fig. 4.39d) which should be specified. Assuming a fracture criterion of such a type and aiming at the estimation of strength of a plate with an opening, we have, actually, to perform a procedure comparable with that necessity to calculate, step-by-step, the damage around the opening. So we are losing the simplicity of a quick, approximate estimation of a necessary value. If we really need a means for a rough estimation, we would rather simplify the criterion leaving sufficiently exact strength evaluation for numerical calculations. 6Note that in [17], the Bowie's problem is solved for an anisotropic plate via potential functions satisfying boundary conditions strictly.

220


Ch. IV, w

To give an example of a very simple procedure [455], let us consider a sharp crack of length 21 in a composite body with no external load. During loading, a fracture process zone arises at the crack tip. Assume that the influence of this zone on the crack results in the transformation of the crack into an elliptical hole and the radius of curvature of this hole at the "crack" tip reaches a characteristic size for a given material, say r*, at fracture. Then assume that at the same time the local strength ~r* of the material also is reached at the same point. Neglecting a change in elastic moduli of the material in the fracture process zone and using the well known solution of the problem of an orthotropic plate in tension with an elliptical hole [361], we obtain for the limit load (or stress ~)

a*/a-l+

p+q~, pq

(4.225)

where p and q are the roots of a characteristic equation which depend on the elastic moduli of a material as:

P' q =

Ex 2Gxy

Vyx

-t-

Ex 2Gxy

__

2

Vyx

Ex - ~yy

(4.226)

Here, Ex, Ey,Gxy,Vyx a r e the engineering elastic moduli. Equation (4.225) contains two parameters a* and r* to be determined in an experiment. Note that the dependence of 1/~ on ~ is given by a straight line. The intersection of line with the 1/#-axis gives the value of or*, and the slope of this line determines the value of r*. Therefore, the first check of the validity of the assumptions made above is the linearity of the experimental dependence of 1/# on v/7. The bulk of such experimental data showing this linearity is given in [457]. We just present here fig. 4.40 to illustrate the behaviour of boron/aluminium composites. It is interesting to note that the value of a* obtained in such a way is always larger than the mean strength of a specimen without a notch. This is a result of the scale effect. Now it is possible to give a more general sense to the assumption made above. Assume now that a defect of arbitrary shape exists with the radius of curvature at a dangerous point less than r*. Also, assume that the defect transforms under the load into an elliptical hole with the radius of curvature r* and at failure the stress at the dangerous point reaches the value of or*. Thus, plates with defects of equal lengths in the direction normal to the applied tensile load and with radii of curvature less than r* will fail at the same load. The limiting load is given by eq. (4.225). In the case of a plate with a central hole of radius r, the dependence of 1/~ on vQ can be easily obtained (schematically drawn in fig. 4.41). Here, interval B C corresponds to a straight line given by eq. (4.225) and the slope of this line is related to the value of r*. Point A corresponds to the strength, or0, of a specimen without a notch, such that

221

Cracking in joints

Ch. IV, w

1.4 O

T

1.2

\

1.0

.

~0.8 0.6

0.4-

0

~/2 /

m2 m

1/2

3

4

Fig. 4.40. The dependence of 1/~ upon x/1 for unidirectional boron/aluminum plates with a central notch. The values of ~ are obtained taking into account a final width of a plate, = # / v / ( w / n l ) t a n h ( n l / w ) where # is the experimental value. (Experimental data by Awerbuch and Hahn [28].)

/

/

O"

p+q

A ! / (7*

/

/i .

4B

~/r o

[

i

i!

i i i i

/

i .

I

I ~/'r*

~/r

Fig. 4.41. Schematic dependence of the strength of a composite plate with a central hole on the hole radius.

Macro- and microcracks & non-homogeneousmaterials

222

2.5

7 a. q~

w

l

w

I

w

i

i

I

w

l

w f_ I

I

I

i

Ch. IV, w

2.0

\ 1.5

1.0

.5

o o o o o Sl~t * * * * * Ho~e

0

I

I

I

I

/

w(th I

I

m.,

s~its I

I

Fig. 4.42. The dependencies of 1/# upon vQ for boron/aluminum plates ([+45/02]s) with various discontinuities. The correction procedure for experimental data accounting for a final width of a plate is the same as in fig. 4.40. (Experimental data by Mar and Lin [383].)

[pq(a~ r0-

P+q

~

)]2 -1

r*.

(4.227)

The horizontal part of the curve at r > r* corresponds to the elastic behaviour of the plate and the fracture occurs when the stress at the tip of the hole reaches value a*. Part AB needs also some comments. Here (at r < r0), the rupture does not necessarily occur in the vicinity of the hole. The local strength here may be higher than the local stress when rupture condition is fulfilled elsewhere in the specimen. Corresponding experimental facts obtained in testing specimens of fibre-reinforced plastics are presented in [457]. Value a0 (point A) depends on specimen size. If the elastic moduli of the material are known, then the slope of part BC gives the value of r* and a point on the line BC corresponding to r = r* presents the value of a~, that is the location of point C. Fig. 4.42 shows the corresponding experimental data. The model discussed does not give a sufficient description of all possible cases of fracture of a composite element with macrodefects, but it corresponds well enough to real fracture processes which occur on the micromechanical level. Therefore, the model can be used for an approximate evaluation of the strength of composite plates containing openings.

4.7.2. Pin-loaded holes The stress/strain field in an anisotropic plate with a pin-loaded hole can be found either by an analytical (see, for example, [17, 510]) or numerical (for example, [96]) method.

Ch. IV, w

223

Cracking in join ts y

t Contact

IIllIIl d-~~

-

area

o

21.)

0

Tensile failure

Shear-out

?

Bearing

Q

Fig. 4.43. Schematics of loading and failure modes of pin-loaded plates.

Analytical methods are based on exploiting of two Lekhnitsky functions 9 and q~ following eq. (4.31). Annin and Maximenko [17] did it in a most general form for an arbitrary plate containing arbitrary openings. Some results obtained for a particular case of a pin-loaded hole shown in fig. 4.43 are presented in fig. 4.44. In the calculations, the normal pressure was applied as p(O) = ( 4 ~ / n ) c o s ( r e / 2 - 0) where 0 < 0 < ~z and # = Q/td with t being thickness of the plate. Two instances are exemplified in fig. 4.44, that is of a plate of infinite width and w = 1.5d. One can immediately see the difference in the stress concentration for those two instances. As the stress state of the laminate around a hole is obtained, the next step is to adopt a failure criterion. Continuing the work presented in [17] Maximenko and Ravikovich [393] calculate the normal longitudinal and shear stresses, ~r(n) and z (n), in kth lamina along a curve 7

r(O) - d/2 +

rt

-t- (rb

--

rt)COS(n/2 -- 0),

I0 I< ~z/2

(4.228)

where rt and rb are the characteristic radii for tensile and bearing fracture, respectively, the former being, in fact, similar to the characteristic length introduced in Section 4.7.1. Failure is assumed to occur when the stresses in at least one lamina satisfy the condition

or* / + \ - - ~ - / = 1.

(4.229)

Here a* and z* are the longitudinal and shear strength of the lamina, respectively. The polar coordinate 0 --- 0* of the critical point determines the failure mode. It is

7 The authors refer to [77] for the idea of such a characteristic curve.

Macro--and microcracks in non-homogeneous materials

224

0.0

........................

2.0

,

,

,

,

!

,

,

,

,

,

,

,

,

,

Ch. IV, w

!

,

,

,

,

!

,

,

,

,

,

,

,

I

,

,

~

,

b

_,

1.5

(~) -0.5

b

1.0 -1.0

0.5

\,":',,, . "

=

-.

1.5d

" ~",~,._ "'--~

oo

1.0

1.5

2.0

x~ (~) 0 . 8

I

I

"}../.)

. . . . .

'

'

'

'

'

~

O0

2.5

~/(a/2)

'

'

'

'

'

'

,,

3.0

-1.5

3.5

w = 1.5a,

'

'

'

'

'

'

'

1.0

1.5

. . . .

2.0

v/(a/e)

2.5

,

,

3.0

'

F0.4

0.2

0.0

i/,/

V; 1''

0.0

"tO

:

co

. . . . . . . . . . . . . . . . .

0.5

1.0

u/(a/e)

1.5

I 2.0

Fig. 4.44. Stress components, normalized by the applied stress a = Q / w t , versus coordinates normalized by the hole radius. Elastic moduli of the material are El -- 53.84 GPa, E2 = El~3, Gl2 -- 8.63 GPa, vl = 0.25. Solid/dotted lines correspond to the case when axis 1 of the elastic characteristics coincides with the x-axis/y-axis. Calculations are performed for the plate of infinite width and w = 1.5d, in both cases l = 3d. (After Annin and Maximenko [17].) a s s u m e d that the tensile failure occurs when 0* < 15 ~ bearing takes place when 0* > 75 ~ and the intermediate angles c o r r e s p o n d to shearing out. One can see that in such procedure, the values of rt and rb should be determined in an experiment with pin-loaded holes, and this is to be done, strictly speaking, for each particular stalking sequence of the laminae. Then three values of the strength, if(n) (for tension and compression) and "C(n), should be m e a s u r e d in testing unidirectional composites. This was d o n e by the a u t h o r s and still the experimental d a t a o b t a i n e d for b o r o n / a l u m i n i u m laminates with various stalking sequences did not seem to fit the calculated predictions sufficiently well. The main reason for a r a t h e r large discrepancy between two sets of the data is certainly the algorithm for failure evaluation a d o p t e d . A m o r e reliable algorithm would definitely call for m o r e reliable d e t e r m i n a t i o n of material p a r a m e t e r s . On the other hand, in this situation, a simple empirical a p p r o a c h to the p r o b l e m can be justified.

Cracking in joints

Ch. IV, w

225

To mark possibilities, consider first the results of a systematical series of the experiments carried out on boron/aluminium composites, mainly of unidirectional structure [444]. Four butches of the materials were used, the material characteristics necessary for the present purpose are given in Table 4.1. The peculiarities of the three modes of failure are as follows.

Shearing-out The dependencies of ultimate load Q/t on shearing length l ~ (fig. 4.45) is linear up to about 10 mm. The average shear stresses, ~, are rather high, 100-110 MPa for unidirectional plates and ~ 160 MPa for [0 90]s plates. This certainly means that plastic deformation of the matrix is smoothing shear stress concentration. A definite dependence of the shear strength on the pin diameter revealed in fig. 4.45a can be caused by a dependence of normal stresses acting on the shear surfaces upon the pin diameter. Increasing the shearing length would enhance the effect of stress inhomogeneity if the bearing mode of failure did not occur. Bear&g First, there is observed a dependence of the bearing strength, #, on the plate thickness (fig. 4.46). Second, both values of # and the ultimate compressive stress of short composite rods (Section 8.1) depend on the same material parameters [444]. Therefore, the bearing-mode failure can be assumed to be a kind of shear-mode buckling occurring in a volume of the composite. Third, the average bearing stress depends strongly on the pin diameter (fig. 4.47).

Interpretation Let us plot average failure stress a0 = Q/tw in the plate away from the pin, versus ratio d/w (fig. 4.48a). Points corresponding to the bearing-mode failure are located along a line going from the origin of coordinates with the slope equal to bearing strength. With the ratio d/w increasing, a point A is reached when the tensile-failure-mode replaces the bearing-mode. Realizing curve AD requires TABLE 4.1 Characteristics of boron/aluminium composites used in testing specimens with pin-loaded holes. Code

vf

Tensile strength MPa

Compressive strength MPa

1

0.36

500

1300

2

0.5

-

-

3

0.5

800

2350

4

0.57

-

-

Bearing strength, mean value for the hole diameter shown in brackets MPa (mm)

900 (4) 600 (8) ~350

1000

1 Pin diameter is 4 mm.

(3)

1000 (4) 1400 (4) 750 (6) 1400 (3) 1500 (4) (6)

Average shear stress during shearing out 1 MPa

100 11 -

226

Macro-and microcracks in non-homogeneous materials '

400

I

'

I

Material c o d e - 2 Uni-directional

~300

A

d=3mm 9 d=3mm o d=4mm 9 d=4mm 9 d=6mm [] d=6mm

. A

200

O [] A

100

~

?J

Ch. IV, w

I

I

I

5

10

15

,

I

25

20 I"1 m m

600

t

Material code -3 d=4mm

500 400

A o

300

[]

[0~176176176176176

s

9 [0~176176176176

200 100

Uni-directiona/ 9 Unidirectional

o A

[0~176176176176176 s 9 [0~176176176176176

i

I

5

10

=

i

15

,

I

20

I'1 mm

25

Fig. 4.45. Ultimate load of a pin-loaded boron/aluminium plate normalized by plate thickness versus shearing length. The slope of linear part of the dependence is equal to the average shear stress. Open points stand for the shearing-out-mode of failure, solid points stand for the bearing-mode. Material codes are in Table 4.1. After Mileiko et al. [444].

9

i

Solid points stand for tensile fracture

1000

% 9 o%

800

o

o v

O0o o

o

600

0.0

'

015

Code 1 Code 4

o

'

1.'0

t / m' m

1 15

'

2.0

Fig. 4.46. Ultimate bearing stress of a pin-loaded unidirectional boron/aluminium plate versus plate thickness. Material codes are in Table 4.1. After Mileiko et al. [444].

Crackinginjoints

Ch. IV, w .5

,

,

,

,

,

,

'

',

,

,

,

9

v

t

'-

|

a~

\

,

2 2 7

1.o

Ib

o 9 9

0.5

9 9

9 O

9

| .0

a

1

,

I

2

I

4

i

1

d/mm

l

I.

6

8

Fig. 4.47. Ultimate bearing stress of a pin-loaded unidirectional boron/aluminium plate versus pin diameter. Material 1 in Table 4.1. After Mileiko et al. [444].

(b)

(a)

/ i ~\\..

II

~,~

II

v~ ~ c

0

Beaching i

Zo

l"

Fig, 4.48. Schematic illustration of the evaluation of ultimate loads of pin-loaded plates. calculation o f the stress state a n d a p p l y i n g an a p p r o p r i a t e failure criterion. As we h a v e seen, sufficiently reliable results in this d i r e c t i o n r e m a i n to be o b t a i n e d . So a m e t h o d o f picking up j u s t a s c h e m a t i c s h a p e o f the curve can be used. E q u a t i o n (4.225), w h i c h is valid if 2r0 O a n d ~ 1 at d/w~O. If this r e l a t i o n s h i p is a s s u m e d to be valid for the case u n d e r c o n s i d e r a t i o n , t h e n the curve

228

3OO

0200 go o o

100 8

0.~

I/,/' 9 d=~mm

o

I o 80 ~ ~ o ~o

200

%

Ch. IV, w


014 ~

0.6

-

400

mm

:

8ram

0.0

0.2

0.4

~

0.6

0 0.0

9

0 8

o

,P

oo oo

v

o

d,,4mm d=6mm d.4mm d-6mm 9 d=Smm

9

012

01, a,

0~

Fig. 4.49. Representation of experimental data obtained by testing pin-loaded boron/aluminium plates with unidirectional reinforcement on the plane according to the scheme shown in fig. 4.48. Open points stand for bearing, solid ones are for tensile-mode failure. (a) material 1 (see the code in Table 4.1), d = 4 mm; (b) material 1, the lines are the best fit linear dependencies for the pin diameters shown at the lines; (c) material 3. After Mileiko et al. [444].

AD shown schematically in fig. 4.48a is obtained. Here also a horizontal part of the dependence shown in fig. 4.41 (part AB) is taken into account as well as a requirement for the tangent, at point D, to curve AD to go via point 0, 0*. Experimental data are plotted on the plane o o - d/w in fig. 4.49. The scatter expands point A in a domain.

Experimental algorithm To build up a scheme presented in fig. 4.48, one can follow, in an experiment or, perhaps, in numerical calculations, the following algorithm. First, shear strength ~ is to be determined by testing specimens with small enough shearing length l t (point C in fig. 4.48b). It should be noted that the value of ~ may be dependent on the pin diameter. Second, the location of point B is determined by testing specimens with d/w < (d/w)A and sufficiently large l (figs. 4.48a, b). Note that the location of point M is now determined which gives a minimal length l~. Third, the location of point A is determined by testing specimens with sufficiently small widths. Note that point A' suggests an approximate location of point A.

4.7.3. Overlapping joints The single- and multi-laps are common patterns of the design for load transfer to metal-matrix-composite elements, an example being a tube under tension or compression (see Section 11.3.2). The stress analysis of the single-lap joints was certainly started by Goland and Reissner [198] half a century ago and since then it has been undergoing numerous refinements with a rather little change in the philosophy of the model. Note that the problem is similar to that of the load transfer to a fibre in a strained matrix, except for a necessity to account for bending, peeling, etc. We shall not go into the details of the stress analysis and refer a reader to a recent paper [661] for a brief review of the corresponding developments. We will remain here with attempts to evaluate the ultimate load.

Cracking in joints

Ch. IV, w

229

In this section, we shall use the results obtained by Mikhailov [408, 409] for the stress field asymptotics in the vicinity of an angle edge and those by Kryssan and Nikitin [344] who have applied the above results for a metal/composite joint. In [408,409], D' (fig. 4.50) is an isotropic wedge adhered to a transversally isotropic semi-infinite plate, D", in such a manner that at the interface q9 = 0 (0 _< q9o _< re) the continuity of tractions and displacement is satisfied. Tensile load in the direction of the x-axis is applied at infinity. It is shown that in the vicinity of point r = 0, singular stresses akl 0( r -sn dominate where Sn are the roots of transcendental equation A(s, qg0,E , v, Ex,Ey, Vxy, Gxy) -- 0.

(4.231)

Here E and v are the elastic characteristics of the isotropic wedge and Ex, Ey, Vxy, and Gxy are those of the anisotropic plate. In [344], the load transfer to a composite plate of a finite thickness from metal tabs as shown in fig. 4.51 is modelled in configuration presented in fig. 4.50 by special boundary conditions at x > 0, y = 0: 0"y=0,

"/Txy--0, - - ~ < ~ < 0;

ay--Kl~-S

rxy=K2~ -s,

0 < ~< ~

(4.232)

where point (~, 0) belongs to the interface, K1 and K2 are u n k n o w n constants, s a root of eq. (4.231) with a maximal real part in the interval 0 < Re s< 1. The stress components are given by eq. (4.31) with changing the axis notations, xl ~ x and x2 ~ y, and potential functions for conditions expressed by eq. (4.232) as [361]

1 Klq+K2fo~176 ~r d~-s ~, (I)'(Zl) -- 2rci p -- q -- Z1 1 KIp + K2 f o e

W'(z2) - 2rti

~

p - q

J0

(4.233)

~-s

~~ d- { .z2

(4.234)

g

X

Fig. 4.50. Isotropic wedge adherent to anisotropic semi-infinite plate.

230


Ch. IV, w

By using known methods of calculation of Cauchy-type integrals, eq (4.234) reduces to: (I),(Z1)

__

2Kaq + K2 1 p-q z]'

~'(z2) = 2 K'p + K2 1 p-q z~

(4.235)

where 2-

exp(ircs) 2i sin(ns) "

Since at infinity the load is applied in the x-direction only, there should be

f

(4.236)

oe o ' y d x - - 0. OO

Equation (4.236) is satisfied only if Kl -- 0. Therefore eq. (4.235) can be rewritten in the form: (I)'(z,)-2 K2 1 p - qz]'

W'(Zl)--2 K2 1 p - qz~

(4.237)

Hence, eq. (4.3 l) yields O'x- 2Re [K(~I2 - ~ ) ] , (4.238)

O'y- 2Re [K ( ~ - li-)l , ~xy- -2Re [K (zP~-i- zq-) where K

.~_

, K2 p-q

In the case Rep = Re q - 0, eq. (4.18) yields for y - 0: 1 O'y - - (ill -[- f 1 2 ) t a n ( ~ z s ~

where fll - Imp and 9xy(X, 0) x(X,0)

~

1 +

f12 --

K2 x S'

K2 O'y - - 0,

37xy - -

Xs

(4.239)

Im q. Therefore,

tan(ns(q~o) ) .

(4.240)

Cracking in joints

Ch. IV, w

Me t a l

Composite

231

7' i

Fig. 4.51. Specimen for testing load transfer to composite plate via metal tabs.

We see now that constant K2 plays a role of the stress intensity factor and its dimension is M P a . m s. The value of s depends on the ratios of elastic constants of two materials as well as on angle ~00. So the Irwin's type criteria are practically fruitless. The result obtained can be used to interpret experimental data and, perhaps, to choose a desired value of angle q~0 in the design configurations of a type shown in fig. 4.51. In particular, the ratio given by eq. (4.240), can be used to estimate the value of q~0 that corresponds to the transition from shear delamination in vicinity of the edge to tensile fracture. The authors [344] conducted an experimental study of failure of boron/aluminium plates with titanium tabs (fig. 4.51). For boron/aluminium composite with vf = 0.5: fll = 1.730, f12 = 0.796, ~* ~ 100 MPa, ~, ~ 1000 MPa. This corresponds to the dependence of s on q~0 given in fig. 4.52 and the transition mentioned at q~0 ~ 10~ The experimental data presented in fig. 4.53 show sufficiently good correspondence to the theoretical evaluation and, at the same time, reveal a decreasing of the ultimate load with the angle increasing when delamination occurs. A microscopical observation of the failure region a jump-like fracture process: after initial

0.5 J

0.4 0.3 0.2 0.1 .0

0.0

~

I

0.4

~

~

I

0.8

1.2

Fig. 4.52. The dependence of degree of singularity on the tab angle in the boron/aluminium-titanium specimens shown in fig. 4.51. After Kryssan and Nikitin [344].

232

Macro- and microcracks in non-homogeneous materials 1.1

Ch. IV, w

ii

1.0

J

x ~ 0.9 b

0

0

0.8

o

0.7

8

0.6 9 9 9 9a T e n s i l e

0.5

failure

ooooo Delarnination

04.

i

0

~

I

15

L

,

I

30

I

a

initiates I

45

i

i

failure I

L

i

60 ~0o /

I

75

i

L

90

grad

Fig. 4.53. Ultimate stress in boron/aluminium part of boron/aluminium-titanium specimens shown in fig. 4.51. normalized by the average strength of boron/aluminium composite versus the tab angle. After Kryssan and Nikitin [344]. delamination, the outer layer of b o r o n / a l u m i n i u m fails, then a n o t h e r delamination stage occurs followed by the failure of the next layer, etc. Hence, tensile stress c o n c e n t r a t i o n remains to be a m o s t i m p o r t a n t factor determining the ultimate load. The whole p r o b l e m remains to be solved in the future.

Chapter V STRENGTH AND FRACTURE TOUGHNESS

In this chapter, we present micromechanical models of failure and fracture of fibrous composites. We shall consider mainly continuous-fibres composites because corresponding models are clear and provide useful technological recommendations. It is important to note that pure mechanical models of metal and ceramic based composites, especially those concerned with non-elastic properties, have limitations due to the variety of non-mechanical ways the components influence each other. Therefore, we have to deviate at some points from pure micromechanics towards structural considerations of composites.

5.1. Strength of a fibre bundle

The strength of a fibre bundle is an old subject of research because it has been always necessary to understand the failure behaviour of textile, ropes and similar things. So the matter is now quite clear and well known. The full account of the fibres-bundle behaviour can be found in classical papers by Daniels [115] and Coleman [94], as well as in [85, 538]. Suppose the fibre bundle is composed of a sufficiently large number N of elastic fibres of equal length, l, and cross-sectional area. All fibres have the same Young's modulus. Let P ( a ) be cumulative fibre strength distribution function. If the stress on unbroken fibres in the bundle is a then the true stress carried by the bundle is ab - - a ( 1 - - P ( a ) ) .

(5.1)

Because stress a depends linearly on strain c, eq. (5.1) can be considered as the stress/strain curve of the bundle (fig. 5.1). Maximum value of the bundle stress, a~, can be obtained by searching for the maximum of function ab(a), i.e. d a(1 - P ( a ) ) = 1 - P ( a ) - a p ( a ) - O.

da

where p(a) is the probability density function. If P(a) is the Weibull distribution given by eq. (2.5), then eq. (5.2) yields

233

(5.2)


234

Ch. V, w

%

Fig. 5.1. A v e r a g e stress in a fibre b u n d l e versus stress in a u n b r o k e n fibre.

a b = o0

13

e x p ( - 1/fl).

(5.3)

Comparing eq. (5.3) with eq. (2.7) shows that the strength of a loose bundle of length l is always lower than the mean strength of single fibres of length l by following ratio k - [fl'/~ exp( - 1/fl)F(1 + 1/fl)]

.

(5.4)

Factor k, which is called as Coleman factor, decreases with the Weibull parameter/~ decreasing (or the coefficient of variation of the fibre strength increasing). Note that eq. (5.4) does not contain the fibre length. It was shown by Daniels [115] that the distribution function of bundle strength a~, tends for large N toward the normal distribution with the distribution density

p(x) -- Db ~

exp

2D~

and standard deviation Db -- a~ v/P(a)[1 - P(a)]N -'/2

(5.6)

where O'f - - O"0

The loose bundle may be a simplest model of a fibrous composite, in which (i) the load carried by a broken fibre is distributed equally among the remaining unbroken

Ch. V, w

Brittle-fibre~ductile-matrix:

strength

235

fibres and (ii) stress distribution along the fibre is homogeneous. The second assumption was actually dropped by Gfiser and Gurland [214] who considered a model composed of a series of layers of height 6, which is a minimum height to assimilate all the fracture events. The layers do not interact with each other, and the failure of the weakest one means the failure of the model as a whole. Rosen [570] defined the ineffective length of the fibre which in fact is a half of the critical length, took li to be equal to the critical length and finally applied the Daniels-Coleman analysis to the calculation of the statistical characteristics of the layer strength. The final result, that is the mean strength of the weakest layer, is a R - - VfaO

fl

exp(-- 1/fl).

(5.7)

The expression obtained differs from eq. (5.3), which gives the strength of the loose bundle by coefficient l/li. Obviously, if we knew real stress distribution around a broken fibre it would have been a routine statistical problem to calculate the composite strength. However, the stress distribution, even for the case of a regular composite geometry, can be influenced by a number of factors those being inelastic properties of the matrix, interface failure, dynamic effects, etc. So in numerous models of the composite strength various assumptions have been made to overcome natural complexities to consider main features of a failure process. For contribution of various factors to a final result varies from one kind of the composite to another, a general model of the composite strength can hardly be expected.

5.2. Brittle-fibre/ductile-matrix: strength A metal matrix composite has usually a ductile materials for fibres are graphite, boron, carbides, and First we shall present a failure model proposed some et al. [452] which has been checked many times in developed since then [419, 448, 456].

matrix. The most efficient oxides which are brittle. twenty years ago by Mileiko experiments and essentially

5.2.1. A model At sufficiently small values of the fibre volume fraction, a fibre accumulates breaks without interactions with neighbouring fibres. The dependence of the mean strength of a composite on the fibre volume fraction will be

d is assumed to be unstable in a bundle if it arises in a composite under load. It grows up to size nd that is to be called the characteristic microcrack size. A first fibre break leading to a microcrack with a length equal to nd happens when the mean composite stress reaches, on average, the value @[email protected] [email protected]@[email protected]

#b)

Fig. 5.15. The geometry of the Kopiev-Ovchinsky's model of a composite (a) Fibre distribution in a cross-section of a composite specimen and the necessary notation. (b) Linear and angle dimensions involved in shear-lag analysis of the stress state around a broken fibre.

The fibre centers are arranged hexagonally in a cross-section of a specimen, as shown in fig. 5.15a. Suppose now that the fibre labelled by "O" is broken. Then shear strain in the corresponding sectors 0 (see fig. 5.15b) is expressed via differences of the displacements of pairs of the fibres and corresponding distances as 7 (pq) -- (Up --

(5.14)

Uq)/b(O)

where p , q determine the fibres position, q = 0 corresponds to the broken fibre, p , q = 1 correspond to the closest fibre neighbours surrounding the broken fibre, p, q = 2 correspond to the second band and so on. Only such pairs of fibre points are considered in eq. (5.14) which "see" each other through the matrix directly. Shear stresses in the matrix and on the fibre/matrix interface are written assuming a bi-linear approximation of the elastic-plastic strain/stress curve for the matrix material. Hence, we have

z.(pq) __ {

~l,m)~(Pq) TO q- ~1t ('Y(Pq) --

]1'0)

at 7 -< 7o at 7 > 7o

(5.15)

where gm and g~m are the matrix shear modulus and shear secant modulus, respectively, ?0 and r0 the strain and stress corresponding to alteration of the elastic behaviour of the matrix to plastic one.

Brittle-fibre~ductile-matrix." strength

Ch. V, w

249

A unidimensional nature of the model under consideration does not leave choice for the equilibrium equation, this should be as

dNp dz

= Tp

(5.16)

Here z-axis is directed along the fibre, Np and Tp are normal and shear forces applied to the fibre cross-section and its surface, respectively. Because of the symmetry, we have q=n

(1t/6

Tp -- 2rf Z j ~

"c(Pq)d0

(5.17)

q=O

where n is the largest number of p, q which is taken into account. By using a common shear lag analysis procedure, a system of differential equations for the fibre displacements, u, can be written. For example, for n - 3 we write -

2(uo -

tt

0~2

tt

0~2

U1 --g

U2 -- g

ul)

-

0

(5.18)

(4Ul -- UO -- 2U2 -- U3) - - 0

(5.19)

(6U2 -- 2Ul -- 2U3) - - 0

(5.20)

O~2 II U3 - - - g (6U3 -- Ul -- 2U2) - - 0

(5.21)

where (X2

E

-

[~_

1

8 (5.22)

Ef/lam,

6

fr~/6 dO

rtrf jO I

b(O)

The analysis of solutions of the system shows that overloading of the fibres in the third fibre cycle on the plane of the fibre break is too small, normally less than 1%, so there is no reason to keep considering more than two fibre cycles. The statistical procedure of Monte Carlo type is used to construct each individual specimen to be tested. The total number of fibres in a specimen is some hundredths. All defects in the fibres are located on ten planes, and the distance between two neighbouring planes is sufficiently large to neglect the interaction between neighbouring defect along a fibre.

250


Ch. V, w

Then a strength value is assigned to each defect point by making a sequence of random values to operate on the strength argument in a Weibull type distribution density function, eq. (2.5). In this procedure, the average strength of a fibre is taken as to characterize a fibre of the critical length in a particular matrix. The next step is the loading of a specimen, this means that a process of fibre breaking is being monitored when the load increases taking into account a possible load concentration according to the solution of the equation system, eqs. (5.18) to (5.21). Actually, each plane containing the fibre defects are analyzed independently, and at some value of the applied load, the fibre breaks continue without a further load increment, just due to overloading of intact fibres. The load giving the start to such a process in a weakest plane determines the strength of a specimen. In fact, Kopiev and Ovchinsky [329, 519] analyzed the experimental data on boron/aluminium composites obtained by Mileiko et al. [452]. They attempted to explain some features of failure behaviour observed and succeeded in this. It is interesting to note that the authors introduced non-homogeneous fibre packing in their specimens to check the importance of this structure parameter which had been discovered in [452]. The result obtained is similar to that shown above in fig. 5.8. The simulation process described has weak points in two components. In the mechanical model, the procedure of calculation of fibre overloading when sufficiently large number of fibre in some area are broken becomes doubtful. This is important because at this stage a non-stop process of fibre breaking can occur, and this determines the specimen strength. In the procedure of the transformation of a random set into a set of definite specimens, the distribution of fibre defects onto the predetermined planes can also idealize a real configuration drastically. In particular, such an ordering does not permit to consider a crack in a composite. These assumptions were dropped out by Zaitsev and Malarenko [726]. Consider a composite specimen of length L in the fibre direction (z-axis) containing N fibres. Let ith fibre occupy the position xi,yi (i c [1,N]). Suppose, at some value of external load, each fibre carries Ki breaks with coordinates zil~ (/~ c [1, Ki]) and Ni defects with coordinates z~ (~ c [l,Ni]). A defect transforms into the break when the stress at the defects reaches random value o'ia defined below. Assuming Ef >> Era, consider the tensile stresses in fibres only and determine the composite stress at the final stage of the calculation by adding the matrix stress. The stress distribution along a fibre is taken as

f(s)- { rls/e*l a

at Isl _< e* at Isl > e*

(5.23)

Now consider a moment of the break of ith fibre at Zil3 point. The increment of the stress in jth fibre can be written as Aaij (z) - A j ( a - f (sij)) / q9(rij)

no summation!

(5.24)

Brittle-fibre~ductile-matrix: strength

Ch. V, w

251

where sij - [z - zip] and rij = V/(xi - xj) 2 + (yi - )~)2. We introduced here a function q~(rij) to be defined later. Coefficient Aj will also be determined later. Let O(z) be the fibre ensemble in a cross-section with coordinate z for which P[(z) - min [zi -

z,jl _< ~*.

Then the total stress increment in j t h fibre resulting from all previous fibre breaks is

A m (z) -

E

iEO(z)

Aoij (z).

(5.25)

The total fibre stress in a cross-section is N~. Equations (5.23) and (5.25) yield

N~- E f (Vj(z)) + Z (a + Aaj(z)). jEO(z)

(5.26)

jEIl(z)

Here l l ( z ) = ( I ) - O(z), where (I) is the total ensemble of fibres in the specimen. Therefore, we can rewrite eq. (5.26) in the form

jEO(z)

jen(z)

jE~(z)

jEn(z)

and combining summation over FI(z) obtain

Z s(~(z)) + ~ (~-. + A.j(z))

jE O(z)

jE II (z)

~

..

(5.27)

jE fl(z)

Substituting eq. (5.25) into eq. (5.27) and combining summation over ~(z) yield

Z A.~j(z)- ~ (--f(~)).

jEll(z) iE~(z)

jE~(z)

Changing the summation order in the above equation we can write

=0. iEn(z)

(5.28)

)

The condition expressed by eq. (5.28) is fulfilled when E

Aaij (z) - a - f (~).

jEll(z)

Substituting eq. (5.24) in eq. (5.29) we obtain after rearranging the terms

(5.29)

252


E

( Aj

Ch. V, w

m0"

i61-I( z ) ( p (rij)

This equation gives the value of coefficient A in eq. (5.24) via function q~(rij) as -1

Aj-

(5.30)

) q0(rij)

i

Function q~(rij) can be chosen assuming the material be a homogeneous anisotropic solid at a distances r much larger than the characteristic size of the structure. Therefore, the perturbation of the stress state may be assumed to arise due to a force dipole, F = • applied at the fibre break point. All components of the stress tensor are proportional to 1/r 3. Hence, as a first approximation we assume q~(rij) - r~ for the whole interval of r. To make the defects supply to a fibre of length L with the Weibull distribution, eq. (2.5), we have to accept F(a)-

~

1

a t a _ < (~)'//~ at a > (~) '//~

(5.31)

where No is the total number of defects on the length L. The comparison of the results of calculation performed according to the procedure just described with the experimental data on boron/aluminium composites obtained by Mileiko et al. [452] have shown [726] that to fit the experimental data into the results obtained by computer simulation, it is necessary to make the dependence on r given by function q~(r) stronger than r 3 in a vicinity of a broken fibre. The authors have chosen

q)(r) -

rk//(R,) k-3

at r _< R,

r3

at r > R,

(5.32)

and discovered that the value of R, controls the value of fibre volume fraction which corresponds to the beginning of strength decreasing (point A in fig. 5.2). The value of k determines the strength decrease, but at vf > vA the latter tends to be limited. The experimental data mentioned are described sufficiently well if R, - 4d, k -- 14 and the interface strength r* = 5 MPa. A systematical computer simulation experiment with the plane composite model was performed by Ochiai and Osamura [504, 506, 507] to study influence of the matrix effective surface energy, fibre/matrix interface strength, fibre volume fraction and non-uniformity of fibre packing on the composite tensile strength. The composite model is divided into a number of layers, the fibre within each element (a fibre in a layer) is characterized by the strength value as to the whole set

Ch. V, w

Brittle-fibre~ductile-matrix: strength

253

of the fibres obeys the Weibu|l distribution. The Monte Carlo technique is used to supply each element with a particular value of the strength. At the first stage of the procedure, evaluating stress/strain state of a model under loading, the shear-lag analysis in the version accounting for elastic/plastic behaviour of the matrix and fibre/matrix delamination as well as for non-uniform fibre packing (see Sections 3.6.4 and 3.6.5), is used. Two failure criteria are applied. The first criterion gives the composite strength as the maximum value of stress reached in the procedure of applying strain on the model specimen step-by-step and observing fibre breakage followed by the overloading of intact elements. The second criterion is applied when a crack of a size 2c = Ndf + (N - 1)dm occurs as a result of breaking of N fibres in row (Here dm is the interfibre distance in the plane model of the composite). Then the composite stress is given by eq. (5.10) with the effective surface energy of the composite replaced with a value of "ductility of matrix" U. If a strength value given by the second criterion occurs to be lower than the value of the stress at a corresponding point of the stress/strain curve on its loading part, then the second criterion is assumed to yield the composite strength, otherwise the loading is continuing. Most interesting conclusions that can be drawn on the basis of computer simulation performed by Ochiai and Osamura on the model with parameters pertinent to boron/aluminium composites are as follows: (i) The strength of composites increases linearly with increasing fibre volume fraction at low values of yr. Some combinations of the "ductility of matrix" and interface strength produce a maximum on the strength/fibre-volume-fraction dependence. The lower the "ductility of matrix", the lower the fibre volume fraction at which corresponding maximum is observed. (ii) The strength of composites with non-uniform fibre packing is lower than that for uniform packing. The larger scatter of the fibre strength, the larger is the effect. (iii) For composites with the matrix of high "ductility", the strength increases with the interface strength increasing. When "ductility of matrix" is low, the strength increases with the interface strength increasing, then reaches a maximum and starts to decrease. The first two conclusions are in harmony with the model considered above (Section 5.2.1).

5.2.4. Discontinuous fibres composites A discontinuous fibre composite usually contains either very thin whiskers, typically 1 ~tm or less in diameter, or thin fibres like graphite, silicon carbide, sapphire, etc. with diameters just one order of magnitude larger. Therefore, in this case, a characteristic size of a composite structure may be of the same order as that of a matrix structure, and an essential influence of the fibre on the matrix properties can be expected.

254


Ch. V, w

I n t r o d u c i n g thin fibres into a metal matrix produces [21, 39, 266, 488] 9 an increase in dislocation density as a result of differential thermal contraction of the components; 9 a decrease in subgrain size in the matrix due to the same reason; 9 formation of a work hardening zone around the fibre as a result of creating of the Orowan loops by dislocation moving via the fibre array. All these can change mechanical behaviour of the matrix drastically. Therefore, a reliable micromechanical model of such a composite should either quantify properly results of the above cited papers or introduce parameters to be determined from a particular experiment. Moreover, known fabrication processes of discontinuous fibre composites do not normally provide homogeneous spatial fibre distribution, so non-homogeneity of the fibre volume fraction is to be taken into account by a failure model of short-fibre composites. Otherwise, it would be impossible to evaluate quantitatively important features of the fracture behaviour of the composites observed in experiments. In particular, all the experimental data obtained at room temperature show either no increase in the composite strength with fibre volume fraction increasing or an increase at low fibre volume fractions only. It can be seen in fig. 5.14 shown before as well as in figs. 5.16 and 5.17. However, the most complete treatment of the problem is that of limiting state. For example, Fukuda and Chou (see [85]) considered limiting state of a discontinuous fibre composite with variable fibre length and orientation. They used a concept

325

,

,

320 "

\

315

*b 310

305 300 0.00

l

0.05

I

0. I0

,

VI

0.15

,

0.20

0.25

Fig. 5.16. Tensile strength versus fibre volume fraction of Al203(SAFFIL)-fibre/aluminium-alloy-matrix composites. Experimental data by Hayashi et al. (cited after Baxter [43]).

Ch. V, w

Brittle-fibre~ductile-matrix." strength

1600

g

n

!

255

I

ooo o as o b t a i n e d ar~Doo a g i n g I200~ - / h ~ ~ aging 1200~

r

1400

[]

1200

\

o

b

lO0O

800

600 0.00

'

'

0.05

0.I0

'

'

0.15

0.20

0.25

Fig. 5.17. Room temperature bending strength of graphite-fibre/titanium-matrix composites. After Mileiko et al. [425].

of 'critical zone' which is of the same order as the fibre ineffective length (see Section 3.5.3). If in the case of aligned fibres of a constant length I the length of the critical zone is lc - fll where 0 < fl < 1, then in the case of a variable fibre length there can be assumed lc - fl[ where { is the average fibre length, that is

[-

f0 Q

lh(l)dl

(5.33)

where h(l) is the probability density function of fibre length distribution. A fibre can either bridge a critical zone (bridging fibre) or have its end within the critical zone (ending fibre). Let the applied stress, ~0, be directed along the z-axis. Then the critical zone is a layer normal to the z-axis. Let the angle between a fibre of a length l and the z-axis be 0, then the z-direction component of the fibre length is I cos 0 and all the fibres of such an orientation with l < fll are the ending fibres. Therefore, for fibres randomly distributed with respect to the z-axis the probability Pe that a fibre of length l is an ending fibre is

_

fl[ = Pe

fll/lcosO

0 _< 0 _< Oo and fll _< l

1

Oo < 0 _< z~/2 or fl[ >_ 1

= 1

(5.34)

where 00

-

-

COS -1

(fl[/l)

(5.35)

Strength and.fracture toughness

256

Ch. V, w

The average length of the projection of fibres on the z-axis can be written as lz -

l cos Oh(l)g(O)dldO - [

30

(5.36)

g(O)dO

where g(0) is the probability density function of the fibre orientation. The total length of projection of all fibres on the z-axis is N[z where N is the total number of fibres in a rectangular-shaped specimen with the lengths of the edges being a, b and c. If r is chosen to be parallel to the z-axis, then the average number of fibres which cross an arbitrary section normal to the z-axis can be expressed as Nc .

Nlz . .

.

abvf f z r / 2 l g(O)cosOdO Af aO

(5.37)

where Af is the fibre cross-sectional area. The average probabilities of finding an arbitrary fibre being either an ending or bridging fibre are qe --

JO0x/2 fO e~ p e h ( l ) g ( O ) d l d O ,

qb --

(5.38)

p b h ( l ) g ( O ) d l dO -

(5.39)

1 - qe.

Here Pb = 1 -- Pc. Substituting eq. (5.34) into eqs. (5.38) and (5.39) yields

qe -

~O0,1(~Ofl-] dO

h(l)g(O)dl +

~floefl-[

1COS 0

h(l)g(O)d

l)frc/2~o~ [ +

d Oo

h ( l ) g ( O ) d l dO,

(5.40) qb --

dl

1

Z c-d-s O

h(l)g(O)dO.

(5.41)

The normal strength in the continuous fibre is a~ - ao cos 2 0

(5.42)

and the average normal stress on a short fibre is given by eq. (3.200). 2 Therefore, the average force on the fibre is Af# cos 2 0 and the z-direction component of the fibre force becomes

2 This is an assumption of the limiting state which is to be found without considering fracture processes in the composite.

Ch. V, w

Brittle-fibre~ductile-matrix." ,strength

Fz - A f # ' cos 3 0

257 (5.43)

and the average value among the bridging fibres is F'z -

/0 0 2 ram.

K-calibration The direct calculation by Bowie and Freese [55] as well as similar results of other authors (see [704]) showed that the K-calibration obtained for specimens of isotropic materials, eqs. (5.102) and (5.103), are valid for those of anisotropic composites. On the other hand, the difference in values of the critical stress intensity factor obtained by using different specimen configurations (see fig. 5.32) can be accounted for errors in K-calibration. Anyway, it is obviously preferable to use one type of the specimens for a comparative study, for example in investigating dependencies of K* on microstructural parameters. Notch h,ngth Despite some authors' (for example, in [8, 257]) claim a dependence of fracture toughness values on the notch length, this is in general not so important, and sometimes the dependence arises due to an improper application of Kcalibration, for instance, outside of an interval of the notch-length/specimen-width ratios in which the calibration is valid. Necessity to sharpen the notch by.fatigue loading It can be suggested to avoid the preliminary sharpening of the notch by fatigue loading when dealing with composites in which crack propagation is accompanied with the development of a diffused fracture process zone [228, 453]. In such cases the notch is blunted either during fracture toughness test or even by cycling loading. On the hand, in the case of a catastrophic type of crack propagation the degree of the notch sharpness can effect the critical value of the load. 3 Data in this and other figures referred to Mileiko et al. [453] are obtained by using the original experimental data.

Brittle-fibre~ductile-matrix." fracture toughness

Ch. V, w

/ COD

285

COD

COD

Fig. 5.34. Schematic of typical shapes of the load/crack-opening-displacement curves obtained in experiments.

Critical load Possible shapes of the load/crack-opening-displacement curves for brittle-fibre/metal-matrix-composites are shown schematically in fig. 5.34. Irregularities on the curves are caused by fracture processes going on under the load. So if we are able to relate a particular irregularity to a particular damage accumulation process, we can evaluate the dependence of the process rate on the stress intensity factor. Otherwise, and this is a usual case, it is recommended to use a maximum value of the load as a characteristic load for evaluation of the critical stress intensity factor. Finally, it should be noted that a diffused fracture process zone interacts with the residuum of the specimen in a complicated manner and many discontent points in the interpretation of experimental data mentioned have the origin in the extrinsic factors. The situation is somewhat similar to that analyzed by Cox [105] for the case of bridged cracks. 5.4.2. Experimental data It should be mentioned that in unidirectional composites, splitting at the very tip of the precrack can occur according to diagram in fig. 4.27 due to achieving the shear strength of the composite. So if the intention is to obtain inherent fracture characteristics of a unidirectional fibrous structure, a providence is to be conformed to make the crack go through the fibres by cutting them. For instance, a small volume fraction of thin steel wires can be added in the transverse direction of a boron/aluminium specimen to guide the crack at the wanted angle, although such a thing influences the value of the apparent fracture toughness. We shall deal here with the situations when the crack propagates in its plane and avoid those when the crack deviates from its plane. Systematical experimental data to reveal dependencies of fracture toughness of various composite systems upon structural parameters of the materials are unknown. So we present just some pieces of the experimental data to be used to draw conclusions.


286

50

I

I 9

Ch. V, w

I

40

\20

-

lO

0

i

0.0

I

0.1

:

I

0.2 V$

.

i

f

0.3

0.4

Fig. 5.35. Energy release rate versus fibre volume fraction for boron/aluminium composites with 2024-T6 aluminium alloy as a matrix. The values of K* were measured on SEN-specimens of a thickness between 1.2 and 2.5 mm and then corrected to a constant thickness 2.5 mm. Experimental data after Mileiko et al. [453].

Figure 5.35 shows the dependence of the energy release rate of a set of boron/ aluminium composites obtained via measuring the value of K* and then applying eq. (4.33). Another set of the experimental data for boron/aluminium composites was obtained by Hoover [257] (Table 5.4). One can see a quantitative difference between the values of fracture toughness of boron/aluminium composites presented in fig. 5.35 and Table 5.4. However, if one scans over a broader field of the experimental data (see, for example, Table 5.5), one can see an extremely large spectrum of the fracture toughness values. The main reason for such a large scatter of the data is not the various techniques used in the measurements, but various fibre and matrix properties and composite fabrication routes used for specimen preparation. The following qualitative conclusions can be drawn at the present stage: 9 Fracture toughness of a unidirectional brittle-fibre/metal-matrix composite goes up with the fibre volume fraction, fibre diameter, matrix fracture toughness increasing and fibre/matrix-interface decreasing. 9 The fibre strength characteristics effect the fracture toughness of the composite. As mentioned above, three main mechanisms of the interaction of the macrocrack with fibre system in the composites under consideration are possible, those being 9 Generating a diffused zone of microcracking in front of the crack tip and the energy dissipation at each point of the fibre break. 9 Crack bridging accompanied usually with fibre pull-out yielding a decrease in the effective stress intensity factor at the crack tip and additional energy dissipation due to sliding on the fibre/matrix interface.

Brittle-fibre/ductile-matrix:fracture toughness

Ch. V, w

287

T A B L E 5.4 Strength and fracture toughness of boron/aluminium composites. After H o o v e r [257]. Aluminium alloy as a matrix

I 100 1100 1100 1100 1100 1100 1100 1100 1100 6061-O 6061-O 6061-O 606 l-T6 606 l-T6 606 l-T6

Fibre diameter mm

vf

lib

0.14 0.14 0.14 0.14 0.2 0.14 0.14 0.2 0.2 0.14 0.14 0.14 0.14 0.14 0.14

0.2 0.2 0.2 0.48 0.49 0.48 0.48 0.49 0.49 0.48 0.48 0.48 0.48 0.48 0.48

0 0.498 0.618 0 0 0.495 0.595 0.488 0.605 0 0.478 0.593 0 0.48 0.565

Strength (av. value) MPa

K*

G

M P a . m I/2

kJ/m 2

609 1524 1376 1571 1683 -

19.76 19.2 32.19 31.08 31.75 33.63 30.64 31.64 48.4 49.06

4.29 4.05 8.18 7.63 7.85 8.81 7.41 7.91 18.5 19.0

T A B L E 5.5 Strength and fracture toughness of some unidirectional metal matrix composites prepared under different fabrication conditions. Matrix material

Fibre

Fibre diameter mm

vf

Strength (av. value) MPa

K* (av. value) M P a . m 1/2

G kJ/m 2

Source

6061-F 1 6061 6061 6061 6061 6061-F l 1100 2024-T6 2024-T6 AI-Li alloy

Boron Boron Boron Boron Boron Boron Boron Boron 3 Boron 4 FP-AI203

0.142 0.142 0.142 0.142 0.142 0.142 0.142 0.1 0.1 0.02

0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.3 0.3 0.55

2004 1290 1432 1526 1526 9075 6145 -

112.7 107.3 91.8 91.9 91.9 64.5 57.7 -

97.63 88.5 64.78 64.92 64.92 912 1482 41.22 32.82 152

[28] [28] [28] [28] [28]

1 2 3 4 5

[614] [614] [453] [453] [614]

As fabricated. Measured as work-to-fracture by using Tattersall-Taplin's specimens. Fibre: I kind (see fig. 5.11.) Fibre: II kind (see fig. 5.11.) See fig. 5.11

9 Cutting

fibres by the macrocrack

"perfect",

with low strength

fibre breaks tungsten

outside

fibres

in

in its plane, which

scatter and occasional

of the macrocrack Cooper-Kelly's

plane,

is a u s u a l c a s e w h e n

fibres are

effective flaws which

can cause

an example

experiments

on

being sufficiently brittle

fibre-reinforced

of

W/Cu

288


Ch. V, w

composites [99]. Energy dissipation due to plastic deformation of the matrix occurs in a narrow band around the crack plane. Obviously, various combinations of the main mechanisms are also possible. Consider, following Sarkissyan and Mileiko [584] the first mechanism obviously yields largest values of composite fracture toughness. Combining the results of model considerations (Sections 4.3 and 4.2.1) and the experimental data we can assume for the increase in the energy release rate of a composite, AG, over that of the matrix, Gm:

where (a~) is the effective fibre strength on the characteristic length. Strictly speaking, it would be necessary to introduce flaw distribution into the dependence written. However, we have actually replaced it with the assumption of a particular mechanism of fibre cracking to occur. Moreover, we are to introduce the ratio (a~)/T~m that determines a distance between neighbour fibre breaks as an argument of function q~(-). In general, dimensional considerations [590] yield

G--m=q9 vf, "rfm , , (~Fd 3

(5 104)

and, therefore,

Gm

\ ~Cfm/

Om)q

(a;>d3

(5.105)

If results of either systematical computer simulation or experimental study had been available it would be easy to evaluate constants k, m, n, and q. However, there are known just experimental data presented above which allow only to make estimations. Using the data for boron/aluminium composites with two kinds of fibres of the same diameter (Table 5.5) we estimate the value of ( n - q). Then using data for boron/aluminium composites with 1100 matrix and fibres of two diameters we estimate value of q. Finally, the data presented in fig. 5.35 yield the value of m. Approximate values of the exponents can be taken as follows: m ~ 2 n ~ 3 q ~-1.5.

5.4.3. Technological recommendations Usually, the requirements for a particular interval of the fracture toughness values are not included in a list of necessary mechanical properties of a composite to be developed. A real importance of that characteristic becomes clear when testing a particular structural element under static or fatigue conditions. Hence, when either

Ductile-fibre~brittle-matrix

Ch. V, w

289

choosing or determining composite parameters which are present in eq. (5.105) and are known to determine other strength characteristics of the composite, one should look at their effect on fracture toughness.

5.5. Ductile-fibre/brittle-matrix The ACK-model and its modifications, including those developed by Evans et al. considered above (Section 4.4.2) as well as the model of a pseudo-macrocrack may be applied to the analysis of strength and fracture toughness of a composite with brittle matrix. However we shall start the discussion with the phenomenon of a ringlike microcrack which may be characteristic for fracture process of composites with fibres which have the ultimate strain higher than that of the matrix. Actually, to observe this phenomenon it is necessary to have just a small difference in the ultimate strain of the components. A microphotograph of localized cracking of the matrix in carbon fibre reinforced soda-lime glass presented by Phillips in fig. 2 in [533] shows clearly the physical appearance of the crack we are to model now. We assume the existence of inherent microdefects in the matrix. Let a defect with radius a/2 occur at a distance h from the center of a fibre. It is clear that if the defect is to propagate, it will transform into a ring-like microcrack. Its internal diameter is equal to the fibre diameter, d, and the external diameter will be D = 2h + a (see fig. 5.36(a)). The stress intensity factor, K, for the external front of such a crack can be obtained from an asymptotic solution of a corresponding system of integral equations of the problem given, for example, in [14]: KI -- q~r~tc(e).

O{

@ @ ct

(5.106)

< b

Fig. 5.36. (a) A defect with diameter equal to a and a ring-like microcrack with external diameter D. (b) A possible stable configuration of the crack (position 2) after jumping from position 1.

Strength andfracture toughness

290

Ch. V, w

Here q is the stress in the matrix when there is no microcrack in it. Function K(c), = d / D ~ x / ~ , can be expressed as follows: x(e)=x/1-e

4(

2+~-~-2e

4 162- 0 . 7 7 2 e 3) +

l+~-2e+~-~

and n

v/l_

C

for small and large (e ~ 1) values of e, respectively. The condition for the crack to propagate can be taken as KI = K m. If we take into account the thermal stress in the matrix and stress redistribution between the matrix and the fibre, the critical stress for the composite will be

O'1 -- EM

V -~

KV/-V-'f O'm "

Here a mTis the residual stress in the matrix defined by eq. (13.47). As the crack advances, the effective value of the stress intensity factor, K, can go down as a result of its interaction with the neighbouring fibres. Hence, it is necessary to check a possibility of stable configurations of the kind shown in fig. 5.36(b). A plane analogue of such a configuration, called pseudo-macrocrack, was considered in Section 4.4.3. Therefore, there may be observed two possible ways of composite failure. First, a composite can fail when an inherent microdefect in the matrix starts to propagate, going through a configuration of a ring-like crack (fig. 5.36(a)) and never meeting a situation with the stress intensity factor at its front decreases. That is the unstable failure process and the composite strength in this case is given by eq. (5.107). Secondly, a microcrack of a ring-like type can grow via stable configurations of the type shown in fig. 5.36(b) characterized by decreasing the stress intensity factor at its front when the external loads are constant. The microcrack becomes a pseudo-macrocrack and its effective stress intensity factor does not depend of its length. The composite strength in this case is given by eq. (4.144) corrected by the value of residual stress, that is

O'* --

flgm]/7tEvfV/3(1 / -1- V m ) vfEf(~xf- ~zm)AT Emvmd V

(5.108)

Obviously, at a low fibre volume fractions, a ring-like microcrack propagates in an unstable fashion: the curves corresponding to an asymptotic expression for composite strength in the presence of pseudo-macrocracks rise above the curves corresponding to a single microcrack. Strictly speaking, it would be instructive to

Ch. V, w

Ductile-fibre/ductile-matrix." strength

291

analyze the ultimate stress dependence on the pseudo-macrocrack length to check the possibility of arising a maximum on this dependence. But one can see no real physical reasons for the maximum. At high fibre volume fractions, a single ring-like crack propagates in a stable fashion until the composite stress reaches the value given by eq. (5.108), that is a critical state of a pseudo-macrocrack. The transition from one type of behaviour to the other takes place at the point of intersection of the two curves mentioned above. Finally, it should be mentioned that if the hypotheses on the microdefect behaviour are wrong then the strength of a composite containing a microdefect in the matrix is 0"0(Vf)--Emm

-~

~"~ E m (0"m - 0"T)"

(5.109)

We now compare the experimental data on the dependence of the composite strength on the fibre volume fraction with the dependencies predicted by the theory. This is done in fig. 5.37(a) for composites with Mo-fibre and A1203 + ZrOz-matrix (see Table 13.1). Here the curves corresponding to eq. (5.109) are also shown. Note that since we do not know the real values of T which determine thermal stresses, we have calculated the values of the composite strength for a number of values of AT. Assuming AT = - 1 4 0 0 ~ gives the best fit of the experimental data to the theoretical curves. This value of AT can be considered as effective, which provides the appropriate level of thermal stresses. It can be used in calculations of composite properties after small changes in the composite structure, the examples being changes in fibre diameter, fibre volume fraction, etc. 1 The assumption made on the behaviour of microdefects in the matrix describes the real behaviour very well. Otherwise, the composite strength would be much lower than that observed in the experiment. (See fig. 5.37a in particular, curves corresponding to eq. (5.109)). Fracture toughness of ductile-fibre/brittle-matrix composites is directly given by expressions in Section 4.4.3. If we deal with really ductile fibres which make an essential contribution to the fracture toughness of the composite, then to values determined by eq. (4.143) a corresponding term is to be added.

5.6. Ductile-fibre/ductile-matrix: Strength This is a case when both the matrix and fibre are metallic. In such composites, we do not expect to celebrate synergism in strength properties. However, the analysis shows that fracture toughness of such composites can be determined by non-linear contribution of the constituents. 1Such an approach, which measures a parameter of clear physical meaning by testing a composite specimen and comparing the results with a micromechanical model, can be useful in other cases when direct measurement of the parameter is possible in a rather complicated experiment (i.e. strength of the fibre/matrix interface). On the other hand, the approach cannot be considered to be a strict one.


292

~1000

,

.

,

.

Ch. V, w

,/o. 1

800 600

400 200

a

"*"x"x...,

0

,

0.0

I

0.2

,

i

0.4

,

I

0.6

=

0.8

mlO00

0.

~. ,g

t:)

800 600 400 2 0 0 ..

b

"x ,

0.0

I

0.2

~

I

0.4

,

I

0.6Vf

0.8

Fig. 5.37. The composite strength versus fibre volume fracture. The experimental points are for specimens 89 to 92 (Table 13.1). (a) The solid lines correspond to eq. (5.108), dash-dotted ones to eq. (5.107), dashed lines to eq. (5.109). Curves 1, 2, 3 correspond to -AT = 1600~ 1400~ 1200~ respectively. (b) The solid line was obtained using eq. (5.107) for vf < 0.34 and eq. (5.108) for vf > 0.34 and AT = - 1400~ F o l l o w i n g [412] we c o n s i d e r first a b e h a v i o u r o f a d u c t i l e r o d l o a d e d by tensile l o a d Q. T h e n o m i n a l stress a = Q/Ao r e a c h e s a m a x i m u m v a l u e a* w h e n n e c k i n g at s o m e section o f the r o d begins. (Ao is the initial value o f c r o s s - s e c t i o n a l a r e a A.) If the s t r e s s / s t r a i n curve, e x p r e s s e d in t r u e c o o r d i n a t e s ,

s-

Q/A,

e - In g/go,

is a p p r o x i m a t e d by a p o w e r f u n c t i o n

= (S/S*) n

(5.1 10)

w h e r e s* a n d n are c o n s t a n t s , t h e n a s s u m i n g i n c o m p r e s s i b i l i t y o f the m a t e r i a l , one o b t a i n s the c o n v e n t i o n a l - s t r e s s / t r u e - s t r a i n d e p e n d e n c e in the f o r m

cr = s*el/n e x p ( - e ) .

(5.111)

Ductile-fibre~ductile-matrix." strength

Ch. V, w

293

The maximum nominal stress is reached at e - e*, which is determined by the condition da/de = 0. Hence, eq. (5.111) yields e, - 1In. Therefore, all the constants in approximation given by eq. (5.110) are expressed by values obtainable in a tensile test: n-

l/e*,

s* - a*(e*)-~*expe *.

Now we can rewrite eq. (5.111) in the form cr - o-*(e/eo)~~

(5.112)

- e).

Let the stress/strain curve of ductile components of the composite be expressed by eq. (5.112) with values of or* and e* obtained in an experiment. We assume that the bond between the fibre and matrix is an ideal one, it means that the necking of any one component is impossible without necking of the composite as a whole. If the stress and strain of the composite are a and e, then we have o" -- E e-

ef-

o" - E

v~o'~,

(5.113)

em,

(5.114)

v~~ (e/e:)~;exp(e: - e).

(5.115)

0{

Here subscript ~ is either f or m. Differentiating eq. (5.115) with respect to e and using the instability condition da/de- 0 we obtain the dependencies of ultimate strength a* and ultimate strain e* on fibre volume fraction as

[I + B

,

0{

vf--

- e l (e,) (~ -%) * em

]1

(5.116) (5.117)

e*

Here 13- a~ (em) ~m expe~ * ( e~ ) ~ exp em *' O'm /~f

-

(5.118)

(e * /e~) , e*e x p ( e,~ - e,),

E 0{

It can be shown that e~ < e* < em, if ef < em, i.e. the critical deformation of the composite is larger than that for separate fibres. As we have assumed above that the strength of the fibre/matrix interface is sufficient to prevent the fibre necking without necking of the composite as a whole, the result obtained implies that achievement of the maximum on the stress/strain

Strength andfracture toughness

294

Ch. V, w

curve of the fibre at e - c~ is not accompanied by the beginning of composite necking. In short, more stable matrix restrains the less stable fibre. The stress/strain curve of the fibre follows eq. (5.112) up to the moment of necking of the composite, i.e. the homogeneous stable strain of the fibre reaches a value c* > e~. This situation is illustrated by fig. 5.38. It is worth noting that at the critical composite strain, e*, the composite stress achieves a maximum value ~*, but the fibre stress has passed beyond the maximum point. It is obvious that the values of 2 in eq. (5.116) are less than unity. A theory for the case of a non-ideal interface has not been evaluated but experiments conducted by Ochiai and Murakami [502] show that eqs. (5.116) and (5.117) describe the actual behaviour of a composite with a weak interface sufficiently well. In a metal-matrix composite the interaction between components does usually occur (see Chapter 10) which can produce an interface layer with its own properties. It is important to understand the influence of such layers on the behaviour of composites. First, it should be noted that the plastic properties of the matrix in the vicinity of the more rigid fibre can be different from those of the bulk matrix. This was discovered by Kelly and Lilholt [302] who tested tungsten-copper composites. In their experiments the effective tangent modulus d~/de of the matrix at plastic yielding region (0.05 < e < 0.4%) appeared to be one or two orders of magnitude larger than the value of d~/de for the unreinforced matrix material. The effect is certainly determined by a dislocation pile-up at the fibre-matrix interface [488] and so it should increase with decreasing average distance between the fibres [183].

~ Fltt R E Cr

f

-'~COMPOSI TE

MATRIX 7

~*

?s

Fig. 5.38. Schematic of the stress/strain behaviour of a ductile/ductile composite and its components.

Ch. V, w

Ductile-fibre~ductile-matrix." fracture toughness

295

Secondly, an intermetallic compound formed at the interface can give rise to an increase in the strength of a composite to some extent, if this third component is sufficiently strong [176]. Thirdly, the cracking of a brittle interfacial layer can give rise to an apparent increase in the ultimate stress of a fibre similar to the increase of the limiting stress of a rigid-plastic specimen with a notch [501].

5.7. Ductile-fibre/ductile-matrix: Fracture toughness Let us prescribe the values Gf and Gm of the critical energy release rate to the components of a composite. We assume also that the value of G of an arbitrary homogeneous material depends linearly on the ultimate strain e, of the material if we change the value of e* and do not change any other characteristic of the material. Then we can write (~

G Gf e* e* . . . . -Z Vf --[- ~ Vm Gm Gm ef ~m

(5.119)

where the value of e* is to be found from eq. (5.117). Some possibilities to control fracture toughness of metal-fibre/metal-matrix composites are suggested by the curves 0(vf) shown in fig. 5.39. Obviously such composites can have a very high fracture toughness with respect to cracks normal to a fibre direction. To proceed with the illustration of the crack resistance behaviour of ductile-fibre/ ductile-matrix composites, we present the effect of the ratio of critical values of the energy release rate for the fibre and matrix for various values of parameter/3 given by eq. (5.118) (fig. 5.40). The effective surface energy of a composite occurs to be very sensitive to small changes of its structural parameters. The comparison of the behaviour of a composite described by eq. (5.119) with that observed in experiments was carried out in [20]. Some results are presented in fig. 5.41. The large scatter of the data is obvious. It is certainly due to a variety of heat treatments the specimens have undergone. The thickness of the specimens was also changing. Even the macrostructure of the composites was not constant. For example, the only specimen with 50% fibres contained traversal reinforcement (vf = 5%) to prevent delamination ahead of the crack. Nevertheless, the general behaviour of the composites follows eq. (5.119) assuming Gf/Gm--1.5-2.5. It should be noted that such a presentation of the experimental data can be used to estimate a value of fracture toughness of a metal wire. Perhaps there are no other ways to obtain this value. The data obtained in the experiments [20] show also that the situation at the interface in a steel-aluminum composite has no essential influence on fracture toughness of the composite. It does not mean that this is always true because in these experiments such situation occurs, perhaps accidentally, that decreasing the interface strength leads to a decrease of the plastic energy dissipation nearly equal to an increase in the energy dissipation at the interface.


296

Ch. V, w

5

4

i 0.0

0.2

0.4 V~ 0.6

0.8

1.0

0.8

1.0

5

4

i o 0.0

0.2

0.4

0.6

Fig. 5.39. Critical energy release rate for ductile-fibre/ductile-matrix composites normalized by that for the matrix versus fibre volume fraction. The mechanical properties of the individual components are taken as follows: (a) a~ = 5GPa, e~ = 0.01, ~m = 0.2, a m = 750 MPa (solid lines), 500 MPa (long dotted lines), 250 MPa (short dotted lines); (b) a~ = 5 GPa, e~ = 0.01, a m = 550 MPa, em= 0.3, (solid lines), 0.2 (long dotted lines), 0.1 (short dotted lines).

T h i s effect h a s b e e n effectively u s e d in a c o m p o s i t e w i t h b r i t t l e fibres a n d a t o u g h c o m p o s i t e m a t r i x (see S e c t i o n 5.2.5). F i g u r e 5.39 s h o w s , h o w e v e r , t h a t s u c h a n effect c a n be o b t a i n e d o n l y if c o m p o n e n t s w i t h a p p r o p r i a t e c h a r a c t e r i s t i c s h a v e b e e n accurately chosen.

5.8. B r i t t l e - f i b r e / b r i t t l e - m a t r i x C e r a m i c m a t r i x c o m p o s i t e s a r e t h e m o s t i m p o r t a n t m a t e r i a l s in this class, w h i c h also i n c l u d e s c a r b o n / c a r b o n c o m p o s i t e s a n d f i b r e - r e i n f o r c e d c e m e n t . W e shall

Ch. V, w

Brittle-fibre~brittle-matrix

297

1.00

0.75

~

0.50

0.25

0.00 0.0

0.2

0.4

0.6

v!

0.8

1.0

2.0 9

b)

1.5

~ 1.0 0.5

0.0

i

0.0

l

l

l

l

0.2

l

l

l

l

0.4

l

l

l

l

l

0.6

l

l

l

l

0.8

l

1.0

Fig. 5.40. Critical energy release rate for ductile-fibre/ductile-matrix composites normalized by that for the matrix versus fibre volume fraction. The mechanical properties of the individual components are characterized by e~ = 0 . 0 1 , cm = 0 . 2 , and various values of /3. (a) Gf/Gm -- 0.1, (b) Gf/Gm = 0 . 5 , Gf/Gm -- 1.

consider behaviour of composites containing initially continuous fibres, then just make some remarks on the behaviour of those with short fibres. Strength and fracture toughness of such composites are much more connected to each other than those of composite of other types, so a description of the crack behaviour in ceramic-matrix composites is naturally included in the discussion of the strength behaviour.

5.8.1. Continuous fibre composites." strength Let a unidirectional composite with brittle matrix be loaded by stress ~ in the fibre direction as illustrated in fig. 5.42(1). Normally the characteristic fibre strength is much higher than that for the matrix, so the matrix cracking occurs (fig. 5.42(2)) at a


298

Ch. V, w

8

6

~J4

2

0 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 5.41. Critical energy release rate for steel wire/aluminum matrix composites normalized by that for the matrix versus fibre volume fraction. The lines are plotted after calculation in accordance with eq. (5.119) The mechanical properties of the individual components are taken as follows: a m - 460 MPa, em = 0.2, a~ = 315 MPa, c~ = 0.01.

stress a** given by the ACK model via matrix critical strain, e**, calculated according to eq. (4.114). At a matrix crack plane, the fibre stress is a/yr. With the applied load increasing the fibres start to break at some defect points (fig. 5.42(3)), the process proceeds in a stable fashion (fig. 5.42(4)) until the load carried by unbroken fibres plus the axial fibre load imposed to the fibre by friction on the fibre/ matrix interface ceases to balance the applied load (fig 5.42(5)). In the process seen in such a way, some key parameters of the composite can be detected. These are (i)the stress at which the matrix cracking begins, (ii)the characteristic distance between neighbouring matrix cracks, (iii) that between fibre breaks, (iv) the shear stress on the fibre/matrix interface and corresponding stress recovery length lf(a ~) where a t is the regular fibre stress. The first two parameters were analyzed above (Sections 4.4.1 and 4.4.2); we will here refer to the results already discussed. The third one is common for many problems considered earlier (see, for example, Sections 2.3.1 and 5.2.1) and are to be considered further on. The fourth parameter is of a physical nature. In mechanical models, a usual assumption is that friction imposed by normal compressive stress on the fibre/matrix interface causes a constant interface shear stress ~. Hence, the stress recovery length is of exactly the same nature as the critical length defined above, Section 3.5.3. It should be also emphasized that in mechanical models known at present, the assumption on the uniform stress redistribution of the original matrix stress as well as the stresses from broken fibres onto intact fibres is usual. Two cases are to be distinguished in this problem [536]. The first case applies to composites for which the spacing of matrix cracks is much smaller than the recovery

Ch.


V , {}5.8

299

45

~TTTTT

I 0** and until failure follows directly from eq. (5.133) as [113]

a - vfEfe 1 - g

e < e*

(5.136)

with e0 _ aO/Ef and failure strain e* - e~ + 1)) 1/(/~+l). In the case of a single matrix crack (lm 2>>/f), the probability of a fibre to fail under stress s is given by eq. (4.84), which was rewritten by Phoenix [536] in terms of a ~ that is

P(s)-1-

e x p [ - ( f l + 1)-1 (s/o~

(5.137)

Considering Vb in eq. (5.130) to be equal P(s) and replacing eq. (4.86) with the equivalent expression


Ch. V, w

(lp(s)) -

6~ foSz 1/(/3+1)exp(-z)dz 2P(s)(fl + 1) -/3/(/3+1)

303

(5.138)

where s -

+ 1),

Phoenix arrived to the mean stress in the fibre system: (a) = s[1 - P(s)] + (3 -+- 1) -/3/(/3+1)

fos

Z1/(/3q-I) exp(-z)dz

(5.139)

and replaces the equation obtained with an approximate one, namely (a) ~ s{ 1 - [(/3 + 1)/(fl + 2)]P(s) }.

(5.140)

The maximum value of (a) determines the strength, so that (O'*)Vf --

(O.max)~ S*{ l

/3+2

+ [(fl + 1)/(fl + 2)] e x p ( - 1 / f l ) }

(5.141)

where s* - a~

+

1)/fl] 1/(/3+1).

(5.142)

5.8.2. Remarks on discontinuous fibre composites Toughening and strengthening of ceramic materials through discontinuous fibre reinforcement is perhaps a most promising way to make future ceramics a heavily loaded structural material. Despite during the previous time all attempts to enhance the fracture toughness of ceramics relying on qualitative understanding of what is really governing the fracture resistance of discontinuous-fibre/ceramic-matrix composites have led to just a relative success: the values of fracture toughness of the composite materials under consideration reported in open literature, are within the range of ~ 8-12 M P a . m 1/2 (Table 2.4) as compared to these values for ceramic matrices that are less than 5 M P a . m 1/2. Therefore, to enhance fracture toughness of short-fibre-reinforced ceramics to make such materials to serve in severe environments of jet engines, diesel motors, etc. further studies are to be performed to reveal all possibilities of crack arrest. It is recognized [44, 48, 711] that such toughening mechanisms as fibre pull-out and crack bridging considered in the previous chapter as well as crack deflection and branching (fig. 5.43), and possibly some others contribute to fracture toughness of short-fibre/brittle-matrix composites of random, three-dimensional geometry. To exploit advantages of using ceramic materials (see Sections 1.3.2 and 2.3.2) we need


304

Ch. V, w

first to evaluate contributions of all toughening mechanisms to fracture toughness of the material. The appraisal of the state-of-the-art in this field ranges from optimistic [48] to rather pessimistic, the latter being expressed, for example, in a review paper [711]. The optimistic view is based on the assumption of validity of linear summation of the separate increments of the energy release rates due to separate mechanisms mentioned to obtain the total increment. Pull-out and crack bridging contributions can be evaluated (see Sections 4.1.2, 4.2.1 and 4.4.2). An increase in the effective stress intensity factor due to deflection of a crack meeting a weak fibre/matrix interface can also be calculated [163]. However, the linear summation procedure can hardly be justified as the fracture mechanisms mentioned are related processes and in practice some of them may operate simultaneously. If we compare fracture behaviour of brittle matrix composite with that of ductile matrix composite (Section 5.2.4) we see that the difficulties related to changes in plastic behaviour of the matrix due to the presence of thin fibres are not observed

CB PO

CD

Fibre pull-out

Crack deflection

Crack bridging

Fig. 5.43. Revealed mechanisms of the interaction of a macrocrack with short fibres in a brittle-matrix composite.

Ch. V, w


305

here, other factors, like changing fibre properties at the blending stage of a powder metallurgy fabrication process, are of the same importance. As in the case of discontinuous-fibre/metal-matrix composites, a complete theory of composite fracture remains to be built up. We have just seen that the basis for such a theory, that is the understanding of main mechanisms of fracture of continuous-fibre composites, exists. We shall demonstrate the applicability of this basis to quantitative interpretation of experimental data below, in Section 12.4.1.


Chapter VI CREEP AND CREEP RUPTURE

Metal and ceramic based composites are mainly designed for use at elevated and high temperatures. Therefore, creep and creep-rupture properties of such composites are of primary importance.

6.1. Short fibre composites We start with the creep and creep-rupture behaviour of composites with short aligned fibres and creeping matrix. This analysis being worth by itself due to its applicability to high temperature behaviour of metal matrix composites, will also be a base for the examination of creep and creep-rupture of continuous fibre composites.

6.1.1. Creep Let us consider first a model of the shear-lag type following the author's papers [413, 415]. Being simple, the model can easily be used in the analysis of a continuous fibre composite which experiences fibre breaking under the load, becoming a composite with short fibres in the creep process.

A simple model Let the creep behaviour of the matrix be described by the power law ~ : g]m(O'U/O'm)m

(6.1)

where qm, O'm, and m are constants. The creep of the fibres may occur under sufficiently high stresses and be described in the same manner -- ~f(O't/O'f)n .

(6.2)

The ultimate tensile strength of the fibre is a~. We consider a model of the composite with rigid fibres of hexagonal form in a transverse section. The fibres are situated at the nodes of the hexagonal plane lattice 307

308

Creep and creep rupture

..4

Ch. VI, w

9

A

o

1/- /

Fig. 6.1. Array of fibres in a transverse section.

as shown in fig. 6.1. The fibre aspect ratio is p = L'/h'. The distribution of fibres in a longitudinal section is such that their axes form continuous lines along the whole composite specimen. The axial distance between two neighbour fibres is assumed to be negligible. Since the fibres are assumed to be rigid, we may neglect the tensile stress in the matrix. Let us cut off an elemental cell shown in fig. 6.2. The way of cutting off is traced in fig. 6.1 by bold lines. The fibre overlap, l', is assumed to have a value ranging from 0 to L' with equal probability in each transverse section. The cell is loaded by constant tensile stress ~r' applied to the free ends of the fibres. Relative motion of the fibres occurs because of shear within the volume of the matrix. Under stationary conditions, the rate v of the relative motion can be found as follows. The matrix shear stress is t"-t~y

hI

1 hi 1 h" 5 -< y - < 2 ( h ' + )

at

(6.3)

h I O.t where t - 89 is the interface stress . Using creep law, eq . (6.1), and the TrescaSt. Venant condition, we write the dependence of shear strain rate r/" on the shear stress as

O -I

I

i i iI o"

Fig. 6.2. Elemental cell.

Short fibre composites

Ch. VI, w

rft -- 2m rlm ('r,U/ 6m ) m .

309 (6.4)

Therefore, the shear rate within the volume of the matrix will be

q"(y) - 2-mtlm(crtl~rm)m(ht/lt)m(ht/y) m

(6.5)

and the rate of relative motion of the two fibres will be

f~(h'+h")"" (Y)dy.

(6.6)

/)=2j 89

Here a is the average stress in the cell, and v f - (h'/(h' +htt)) 2. Let us consider now a chain of cells described by eq. (6.6). The chain after stretching is shown in fig. 6.3. The matrix has free surfaces in the vicinities of fibre ends, which is tolerable because the tensile stress in the matrix is assumed to be zero. The strain rate of a chain will be - (1 - 1 / q ) 2 v / U .

(6.7)

Here q is the number of fibres within a given length, and q is usually quite large. The rate of relative displacement of two fibres will be 2v

-/)1

-q-/)2

that is m-I

2 v - - m -r/m -,

(u-~) m h 1 '--vr~ ~VT

[ ( ~ ) _ km_ ( L t h

--t l') mj.

(6.8)

where ~ is the stress in a chain. Equations (6.7) and (6.8) may be rewritten in the form: = ~m (8"/O'm)m

(6.9)

V 2vt -~

llllllli /////llllllll // 12

L"

j-.

Fig. 6.3. A chain of the cells.

310


Ch. VI, w

where m-I

qm 1 1 - v~~-~m -- m - 1 pl+m v~n O-m

'z]

~m --

1/m ~

lt Z

Lt .

~-~ -[- (1 )m

If a parallel set of the chains without transverse interaction is deformed with a fixed rate, ~, then the stress in a chain will be proportional to ffm for a given chain. In fact it can be shown [410] that if the average stress, a, is applied to a parallel set of filaments, and creep of an individual filament is governed by eq. (6.9) with the value em distributed over the interval (0, era) in such a manner that the distribution function ~b(em) exists and f o am r

-- 1,

then the creep rate of the set will be ~ -- qm(O'/(~m)) m

where (O'm) -- fo am 6"m(~ (5"m) d6"m

is the expectation of ~m. In the case of the set of the chains considered above we have (O'm) -- O"m I ( m )

where

I(m)- fO1(Z -m

+ (1 -- z)-m) -1/m dz.

(6.10)

Finally we find the expression for the creep rate of the model ~___

r/m O" l [/(m)]_ m m - 1 (~mm) m pl+m V

(6.11)

where 1 -

V

(m-l)/2 vr

(6.12)

Ch. VI, w


311

Equation (6.11) can be rewritten in a compact form:

( )m o

-

-

(6.13)

~m Ko'm

where K(p, vf) has a meaning of the ratio of the composite stress to the stress for a non-reinforced matrix which would produce the same creep rates. We have

K(p, v f ) - ( m - 1)l/mpl+l/mI(m)V -1/m .

(6.14)

Note that K is the factor of efficiency of the reinforcement of a creeping matrix with short rigid fibres. Obviously, I ( m ) ~ 1/4 as m ~ oc; at m = 4 we have I(m)= 0.239. Because normally m > 3 [558], we can assume I(m) = 1/4. The result given by eq. (6.11) should be treated as the lower bound of composite stress corresponding to a particular value of ~, since we have assumed the existence of free surfaces within the volume of a composite and also neglected transverse stress interaction between individual chains of the elemental cells. The upper bound for a composite with the same free surfaces can be found as a result of consideration of a homogeneous (in the longitudinal direction) field of shear strain rate in the matrix, such that the rates of the relative motion of fibres are the same for a whole specimen. It has been shown [413, 415] that the ratio of the upper bound to the lower bound of the stress corresponding to a given value of the creep rate is 2 l-l/m, and that is a rather high value. However, more exact and more sophisticated analysis of this highly simplified model can hardly be justified. The conception of chains with fixed parameters is an assumption which simplifies the analysis drastically, but its contribution to the final error is uncertain. Bearing this in mind we shall consider further on only the lower bound. Now we are getting off the assumption of the fibres being rigid. Real fibres loaded by high enough stress will either break or begin to creep so that the strain rate of the fibres becomes too high to be neglected. We introduce the first critical value, p*, of the aspect ratio as such a value of p that corresponds to the beginning of fibre breaking. If we assume the fibres to have constant strength along their length, then the fibres to start to break are those which are loaded by highest stresses. The most stressed fibres are surrounded by the six neighbours in such a manner, that z = 0 for each couple. Using eq. (6.11) we obtain after simple calculations

(

p,__ 2(l~,m/~:)l/(l+m) [1 ~m-1)/2]m--1 J 1/(l+m) 0"~ m/(l+m) .

(6.15)

am/

The above analysis shows that the hardening of the matrix depends strongly on the aspect ratio, the creep rate being proportional to the value p-(l+m). However,

312

Ch. VI, w


/ /

"CO

/ / //

/ /

// ~

log

o-

Fig. 6.4. Schematic diagram for the creep behaviour of a composite with p > p* at e > e*. this strong dependence is valid only if p _< p*. When p > p* this is no longer true. The situation is illustrated in fig. 6.4. The second critical value p** of the aspect ratio is such a value of p at a prescribed level of external load (or resulting strain rate) that the creep rate of the fibres and the matrix are the same. For sufficiently large volume fraction we have

[( o')m(o')-n p**--4 ~mm

1

1/(l+m)

v(n-m)/(m+l) f

(6.16)

It should be noted that the critical values of the aspect ratio of the fibre are functions of the applied load. At sufficiently small values of the fibre volume fraction, the carrying capacity of a matrix should not be neglected. It is possible to find approximately such a minimum volume fraction /)fmin, so that at vr =/)fmin the creep rate of the matrix without reinforcement by fibres, eq. (6.1), and that of the model, eq. (6.11), are equal. The comparison of these values when vf is small and m is sufficiently large gives: /)fmin

1 1 (m--1.1/mpl+l/ml/m

1 ~, )~

(6.17)

Ch. VI, w


313

To deal with a composite with fibre volume fractions comparable with /)fmin, we can sum the stress carrying by the composite due to the presence of rigid fibres, which is given by eq. (6.13) as a-

ffm~Tml/mK(p,

vf)~ 1/m ,

and that carrying by the matrix alone, that is ACt -- Vmamqml/mr 1/m .

Therefore, for small values of the fibre volume fraction, eq. (6.13) is replaced by

~--r/m

(6.18)

with the efficiency factor or hardening coefficient being

/~ = g -[- Vm.

(6.19)

Now we shall find the distribution of tensile stress along a fibre. The creep rate of a chain with the dimensionless overlap zi is given by eq. (6.9). The maximum stress in the triangular portion of a transverse section of a fibre will be

~-t(zi)- O-m(~/~]m)1/m

(Z? m -[- ( 1 - zi)-m) 1/m

(6.20)

if the influence of the other portions of the same fibre is not taken into account. Here

I//--['l s

1/mpl+l/m .

(6.21)

When m > 3, eq. (6.20) can be written in the approximate form

O't(Zi) --

~Tm(~/qm)l/m~Zi(l

--

Zi).

(6.22)

The average tensile stress in a hexagonal fibre is a result of the interaction of a given fibre with the six neighbours fibres. At the point Zk (k is the number of a surrounding fibre, k = 1 , 2 , . . . , 6 ) , the fibre stress will be the sum of six components [415]

1+g

.

(6.23)

314

Ch. VI, w


Here -- I]/O'm(~/qm) 1/m

Z1 < Z2
Em, the loops are repelled from the fibre and they find an equilibrium distance from the interface creating a work hardened boundary zone. The zone extends to about 1 gm, so the effect of it will increase with fibre diameter decreasing. If Ef < Em the loops are attracted to the boundary and they provide a slipping mechanism without disturbing the coherency.

316

Ch. VI, w


Goto and McLean [206] analyzed the effect of the interface layer of relative thickness 6 / d on the creep properties of the composite. To simplify calculation they considered a fibre divided into a central zone that is fully loaded and end zones that carry no load. The average creep rate ~ is taken as = fec + (1 - f)/:e

(6.28)

where ~c and ~e are characteristic values of the creep rate of the central and end zones, respectively, and Bl/m m

f - 2m + 1

(6.29)

Here B = 1 for ideal interface bonding, B = 0 for weak bonding, B = h/(h6) for work hardening at the interface layer, h is the characteristic size of the structure in the direction normal to the fibre direction, i.e. h-

5

L~

vf

--1

(6.30)

where A is the distance between the two neighbouring fibre ends. Considering the equilibrium conditions and strain compatibility in the model yields the following system of differential equations -- f ( f l l ~ m ( 1

-- S1 - $2) m) + fl2~i(S2) q -+- (1 - - f ) f l 3 ~ m ( 1

-- $3) m

$1 = Hl~ R 2 ( $ 2 ) q where ill, 132, and /~3 are constants determined by the volume fractions of the components including the interface, ~m and ~i are the creep rates that the matrix and interface zone would experience if they each carried all of the applied stress a, i.e. ~m -- r/m(O'/O'm) m and ~i = r/i(O'/O'i) q where r/i, ~i, q are constants in the creep law for the interface. Also $2 = H 2 ~ -

O'ttt Ui

O't l)f S1 :

o"

,

vfEf H1 = ~ , O"

32

- - ~

o"

viEi H2 - - ~ O" ~

R2 -

H2~i 1)q

where f14 is a constant similar to other fls, Vi is the interface volume fraction and d" is the interface tensile stress.

Ch. VI, w


317

A numerical solution of the problem obtained in [206] accepting constants appropriate for SiCw/A1 composites, shows that weak interfaces (small values of B) can produce a pronounced effect on the creep behaviour of discontinuous fibre composites. At the same time, the calculations for the case of work hardening of the matrix in the interface zone show a rather negligible effect. This is a reasonable result because in the model, a relative interface thickness is a characteristic parameter. Let us consider zone thickness as the dimensional parameter and estimate an effect of the work hardened zone on the creep rate of a composite in an approximate fashion. If the thickness is 6, then assuming the creep resistance of this zone is much higher than that of the matrix, we just enlarge the fibre volume fraction according to

(

V f - V~ 1 + - ~

(6.31)

where v~' is the pure fibre volume fraction. Substituting eq. (6.31) into eqs. (6.11) and (6.12) yields the ratio of creep rates ~ and ~o of composites with fibre diameters d and do, respectively, as

~=

V(v~,3,d)

v(q, 6, do)

(6.32)

where function V is given by eq. (6.12). Applying eq. (6.32) to the interpretation of experimental data provides a means to estimate effective thickness of the interface zone. The results obtained by Bullock et al. [60] in creep experiments on the unidirectionally crystallized Ni - Ni3A1/Cr3C2 eutectic are shown in fig. 6.5 together with the curve corresponding to eq. (6.32). To provide best fit to the experimental data one has to assume 6 = 0.142 ~tm. It seems to be a reasonable estimation. All the models discussed are of semi-quantitative nature in the sense that they simplify stress/strain state in a composite to produce results in an explicit form (except that by Goto and McLean [206]). Such a form shows clearly the dependencies of composite properties on those of components and composite geometry. Since the very beginning i.e. late 60s, it has always been a desire to validate the results of the simple theories by strict solution of the problem formulated without drastic simplifications. One such attempt [167] was already mentioned, this had been a strict solution for a simplest material constitutive equation. Another one [133] was performed by using a finite-element computer procedure with a realistic description of materials behaviour. However, the author stayed with a regular composite geometry. Still their finding of the essentially triaxial stress state in the matrix that has a strong effect on reducing the creep rate of the composite, calls for accounting for this effect in further studies. With regard to the experimental verification of the simple theories, the model experiments performed by Kelly and Tyson [306] and Kelly and Street [304] long


318

1

,

i

'

I

i

i

,

I

i

i

i

I

i

off i

I

( % l j ~ '

Ch. VI, w i _

0 _

0.I

'

0.2

~

~0

1

0.4

,

,

~

|

0.6

I

I

i

I

0.8

I

I

d/~m

I

l

1.0

I

I

I

1.2

Fig. 6.5. Minimum creep rate of N i - Ni3AI/Cr3C2 composite versus fibre diameter. The experimental data have been obtained by Bullock et al. [60] at 980~ they are approximated by eq. (6.32), v~ = 0.11, m = 8.2, 6 = 0.142 ~tm. The creep rate is normalized by that, ~o, for do = 1.08 ~tm.

time ago, seem to be most systematic and best documented. Their results are presented in fig. 6.6 in comparison with two simple theories. The deviations of the experimental data from the theoretical predictions are clearly induced by some random reasons rather than defects of both theories. Systematic experimental study of compression creep of aluminium-alloy-matrix composites reinforced with either SiC (p ~ 10) or AIBO (p ~ 8) whiskers (vf - 0.15) was performed by Peng et al. [528]. The fibres were aligned by a hot-extrusion procedure. A rather low fibre volume fraction calls for using eq. (6.18) to take into account a matrix contribution to the composite creep resistance. A result of the comparison of the hardening coefficient obtained by calculation (for m - 16) and those revealed in the experiments is presented in fig. 6.7. We see, first, that small and irregular variations in the experimental data correspond to those of the exponent in the power law for the matrix (it varies between 15 and 18, such high values of n are certainly explained by introducing intermetallic particles into the matrix). Secondly, bearing in mind a real value of p mentioned and a possible reduction in the initial value of p during the composite fabrication process, we conclude that the calculation yields a lower bound for K. This is in accord with the way eq. (6.18) was derived.

20

319


Ch. VI, w

,

400bC,

,

i

t0

600~

p -

Doooop

3oooo 4 0 0 ~

=

50

100/~

3O

1~ ,,'~Y

e.-"

I-_~,@/

t

0 0.2

'

' 0.3

I

;!

'

J 0.4

20

a'

t

_

0 0.2

0.5

,

i

I

0.3

I

0.4

"of

i

0.5

v s-

Fig. 6.6. The hardening effect predicted by Mileiko's and Kelly-Street's models (rigid fibres, ideal interface) compared with the effect measured by (a) Kelly and Tyson [306] on W/Ag composites (p = 30%), and (b) Kelly and Street [304] on phosphor-bronze-wire/lead-matrix composites at room temperature.

1.75 !

!

i

!

t

1.50

j

/

/

450~ 300~

1.25

350Oc

1 . 0 0

,

5

,

,

,

t

10

,

,

P

,

-

,

15

Fig. 6.7. Comparison of the calculated value of the hardening coefficient and that obtained in a creep testing of unidirectionally reinforced SiC-whisker/aluminium-alloy-matrix composites (horizontal lines). Experimental data by Peng et al. [528].

320


Ch. VI, w

6.1.2. Creep rupture To evaluate the creep rupture properties of a composite with short fibres, we need to assume a fracture criterion. Suppose there is a critical value of the shear strain at the interface, 7,, such that the interface transfers the shear stresses until the interface shear strain, ~, is less than ~,,. At ~ = ~, the stress transfer ceases and the tensile load on the corresponding chain becomes zero [414]. The m a x i m u m shear rate follows from eq. (6.5) at y - 89 "" dTmax (--~m) m 1 1 )'max -dt = r/m (vfp)m ~,m

(6.33)

where ~, = z at z 1/2. Equations (6.9) and (6.33) lead to

,m1,,2[ ( )m]

1--vf (m-- 1)p

~max

1+

.

1-~

(6.34)

We see that the strongest chain is that with z = 1/2. Therefore, its failure, that is 6 - 0, means the composite failure. The corresponding ultimate strain of the composite is e*

=

27* 1 -- /)~m-1)/2 m-]

p

.

(6.35)

Comparing eqs. (6.34) and (6.35) yields a relationship between the composite strain, e, and the value z, _< 1/2 such that all the chains with z < z, have failed, that is e

1

[ ()m] z,

l+

l-z,

"

(6.36)

Note that at z, - 0 e/e, ~ 1/2, this means that the chains start to fail when the composite strain, accumulated on the steady-state stage of creep, reaches a half of the ultimate strain. Also it should be noted that the value of e, corresponds to the creep life during the secondary and tertiary stages of a creep curve. If we repeat the arithmetic that leads to eq. (6.11) we will see that I(m) transforms to I(m,z,) and

I(m,z,)--

fz,

1 --Z,

(z-m + ( 1 - z ) - m ) l / m d z .

(6.37)

Here the upper limit of the integration follows from eq. (6.33) at z > 1/2. The creep curve including the tertiary stage, which is a result of chains failure process, can be written as

Ch. VI, w

321


de _ e-o [ I(m) ]m dt [I(m,z,) '

(6.38)

where Go is the stationary creep rate given by eq. (6.11). 2 hence At large enough values of m (m > 3), I ( m ) ,,~ 1/4 and I ( m , z , ) ,~ 1 / 4 - z,, de eo at = (1 - 4z,2)m "

(6.39)

Substituting z, obtained from eq. (6.36) into eq. (6.39) and integrating from e = 0 to e = e, we obtain the rupture time as e,

(6.40)

t, -- ~ o ( 1 + mY(m)), where tP(m) --

js 1

[ (2X)2] x m-1 1 - 1 + x

m

dx.

(6.41)

The first member in the brackets of eq. (6.40) determines the time of the secondary stage of the creep which transfers into the tertiary stage at e = e,/2. The second member determines the tertiary stage. 6.1.3. A general approach

A most general approach based on the Eshelby's equivalent inclusion method (see Section 3.2) was proposed by Taya et al. [647]. Let the elastic matrix, characterized by elastic moduli tensor C m, contain both perfectly bonded and debonded ellipsoidal inclusions (fibres) that aligned in the x3-direction. The elastic moduli of the bonded and debonded inclusions are C b and d and C ~ respectively. It is assumed t h a t c ~ j d l - 0 with the exception of Cla111 - C2222 cd122 -- cd211 that reflects a special behaviour of the debonded interface. Under the external stress a ~ applied in the x3-direction, disturbance stresses ~i (i = b , d ) arisen due to the introduction of the inclusions of two kinds, are to be added to the stress a ~ = cme ~ and eq. (3.38), with ~ replaced by g + ~ (where g is the volume average of the strain g), is to be rewritten for two stress fields ~i as

0.o _.[_~.i __ C i " [co _[_ ~- _jr_gi] _ C m " [~o _+_g __[_g i

s

(6.42)

where e*i are the eigenstrains in the inclusions. Repeating now the procedures given by eqs. (3.38) to (3.43) we obtain the following expression

-(s-

i).

+

(6.43)

322

Ch. VI, w


which replaces eq. (3.44) in the problem of the inclusions of one kind. Here/)b and vd are the volume fractions of the bonded and debonded fibres. Finally, the average stress in each phase is written as (6.44)

(O') i -- R i . o "~

where i = m, b, d, and R i is given explicitly in terms of elastic constants C m, C b, C d, S-tensor, introduced above by eq. (3.40), and volume fractions Vb and yd. Suppose now that because of creep in the matrix, the matrix strain get an increment de c = ~dt. Due to linearity of the problem, eqs. (3.38)-(3.43), and (6.42) m a y be rewritten in terms of the increments, and the result will be (do"i) = U i . de c ,

(6.45)

(dei) = ffi. deC

(6.46)

where again i = m,b,d, and explicit expressions for the coefficients U i and j i a r e given by the same parameter R. The overall stress increment of the composite, deo, can be obtained as the weighted average of those for the fibre and matrix, i.e. deo = t~b(de b) q- t~d<de d) -q- (1 - lPb -- t~d)<dem).

(6.47)

Now we are to specify the nature of the matrix strain increment assuming the creep of the matrix to be described by the exponential law: ~c = A e x p ( # m / a ~

(6.48)

3 C lJC 0 and Crrn - = V~2-o-i.o where e~c = . v / -e..e.. 3 J i.J are the effective creep rates and effective stress in the matrix; A and a ~ are constants. During creep of the matrix, the average stress in the matrix can be determined by integration of eq. (6.45) with the initial condition set by eq. (6.44) to yield

(6.49)

(O') m = R m . o"~ -+- U m . e c .

For the case under consideration, a ~ - a~3, ec - e~3 and e~l - e~2 = lec, eq. (6.49) can then be written as m =Rlo.o a~ - a22

+ U1 ec ,

o'3n~ -- R3 0"~ + U3 ec

(6.50)

m where R1 - R~33, R3 -- R3333 and

u, -

1

(U l 1 -+- U~122) + U~133,

U3 -

1

m

- ~ (U~311 -k- U ~ 2 2 ) %- U~333.

Taking the deviatoric components of eq. (6.50) we obtain the effective matrix stress as

Continuous fibre composites

Ch. VI, w O'm :

( e 3 - R 1 ) o "~ -~- ( U 3 -

U1)s

_

323

eaO + scC.

Now eq. (6.48) gives

~c _ A exp ( ea~ +--seC). 2

(6.51)

~o

The solution of the non-linear first order differential equation obtained, with the initial condition ec(0) - 0, yields the creep strain of the matrix: o

ec(t)---e--+amln S

S

(

exp

-e

-A~t

o O"m

.

(6.52)

Now eq. (6.47) is to be used to express the overall creep strain of the composite via that of the matrix. In this model, there is a possibility of predicting the tertiary part of the creep curve and creep rupture by introducing a kinetic of fibre debonding, in which case Vb and vd become functions of time, providing vf = Vb + Vd = const. Taya et al. did it assuming the bond strength, a~m, to follow the Weibull distribution. Hence, when the interface stress of a bonded fibre, ab3, increases as a result of stress redistribution from the creeping matrix to the elastic fibre, and finally reaches critical value a~m, which is a random value, the volume fraction Vd increases with time. Equation (6.51) remains to be valid, however e and s are now functions of time -and a numerical procedure is needed to integrate this equation. As was demonstrated by the authors of the model considered [647], this approach can be effectively used for constructing composite models and analyzing their creep behaviour numerically. Perhaps it would be instructive to relate the particular model of such a type to a range of the response of corresponding simple models. Certainly it remains to be done.

6.2. Continuous fibre composites 6.2.1. Creeping fibres When the fibres are continuous, the axial strain rates of the fibres, the matrix, and the composite as a whole will be equal, if a specimen is large enough to neglect end effects. Thus a linear relationship between the average composite stress a and the volume fraction vf will be valid for a fixed strain rate ~ as a result of the uniaxial consideration. This means ~7 -- fft Vf -'}- r ( l -- V f ) .

(6.53)

Suppose the constituents creep according to power creep law, eqs. (6.1) and (6.2). Then the stresses in the fibre and matrix for creep rate of the composite will be

324


fit __ ff f

(~ff) l/n ,

0 "tt "--

Ch. VI, w

(6.54)

0"m (-Cm)l/m .

Equation (6.53) with the stress distribution given by eq. (6.54) has always been employed since McDanels et al. used it for the first time [398]. The hardening coefficient, which is the ratio of the stresses caused equal to strain rates in the composite and the matrix, is

O"

k. - - ~ - -

O'M

O'f (~m)1/n-1/m Vm -+- V f ~ . O'm

(6.55)

As usual, here qm = qf are chosen. Obviously, k~ depends on the creep rate unless n--m.

To evaluate the rupture time of the composite [443], we assume, for the sake of simplicity, that the creep-rupture behaviour of the matrix follows Hoff's model [558] and the fibre ruptures as a result of damage accumulation with a simplest kinetic [559]. This means that the corresponding rupture times are t' -- tm

(ff~f)-n

*

,

t t' -

*

tm

(fftt~-m ~

(6.56)

k,O'm/

where

ef tf -- (n + 1)r/f

tm

mr/m

and e~ is a constant that is supposed to be the ultimate creep strain. Obviously, the fibre breaks before the matrix fails. Two scenarios can be expected for subsequent events: either the first fibre break will lead to the composite failure immediately or the break will be localized and the composite will be waiting for the next fibre break. Evidently, the more homogeneous creep-rupture properties of the fibre and the higher fibre volume fraction, the more probable is the first scenario. In this case, neglecting stress redistribution between the fibre and matrix during the tertiary stage of creep process in the fibre, we will have the strengthening coefficient, which is the ratio of the stresses to cause equal rupture times for the composite and the matrix, as

kt

-

O" crM

= Vm

(~m) tf 1/m

+ vf

O'f t,.l/m-l/n ~rm t l / m t f l / n

.

(6.57)

This is obviously the lower bound of kt. We can obtain the upper bound by postulating statically admissible stress field such that creep-rupture times of the constituents are the same [411]. This yields

Ch. VI, w

325

Continuous f i b r e composites

~4

J

I

0.1

~

I

i

!

i

i

i

i

I

4 681

I

~

i

i

4

i

i

i

i

i

[

6810

i

2

i

I

i

i

i

i

i

4 6~0 0

Fig. 6.8. The strengthening coefficient for molybdenum/titanium composite (vf = 0.089) versus creeprupture time. Testing temperature is 600~ The line corresponds to eq. (6.56) with the appropriate values of the constants. (After Mileiko et al. [443].)

.1/m-1/n

O'f t,

k t+ - - v m nt- v f a m tffal/mtfl/n .

(6.58)

Again, kt appears to depend on the creep life. Mileiko et al. [443] also considered the second scenario and obtained explicit expressions for the creep-rupture time. However, in their experiments with metallic fibres (it was molybdenum wire in [324]) and metal/metal composites (molybdenum/titanium tubes in [443]), it was found that, first, creep-strength scatter of the wire is low, the Weibull parameter fl ..~ 50, and secondly, even at fibre volume fractions as low as 10% , eq. (6.56) predicts the creep rupture time sufficiently well (see fig. 6.8). Thus more elaborate approaches to the evaluation of creep-rupture time can hardly be justified, at least at the present time.

6.2.2. Non-creeping fibres Suppose now that a composite contains continuous fibres with Young's modulus EU. Let the fibre strength follow the Weibull distribution, (a~(L)) being the mean fibre strength at length L, and/~ being the Weibull parameter. To start the model [417], we assume for a moment that fibres do not break. Then neglecting a difference in values of the Poisson's ratio of the fibre and matrix we write the initial fibre stress after the composite is loaded by stress Cro as at _ Ef O'o t=0 - - E m v

Here

326

Ch. VI, w


1) - - tam --~ v f g f / g m

An elemental solution of the corresponding non-steady creep problem gives the fibre stress as a function of time

o"(t)

tro

1--Vm

I

1)m -if- r l m ( m -

v

()m-1t

(6.59)

1) vrEf _% O"m O'rn

The creep curve of the composite is determined by the expression

,r

e = ~

Ef

'

~ ~

,~o Efvf

at

t~c~.

N o w we turn back to the case of fibre breaking. The first fibre break occurs, on the average, when

m-1 t - tl =

qm (m - 1) Ef

m vf

(

1 - vf

O-o

- 1

(6.60)

where L1 is the total fibre length in the composite road. If the composite can sustain fibre breaks and it fails by the accumulation of the fibre breaks, that is a case vf < /)A in fig. 5.2, then new fibre breaks follows the first one and the composite is transforming into that with fibres of the aspect ratio which are equal, on the average, to p, = L,/d, L, being the mean fibre length in the composite with broken fibres. To estimate the value of L, we write the m a x i m u m value of fibre stress as , _ 21_l/m tro O'max /)f

(6.61)

E q u a t i o n (6.61) follows from eq. (6.9) at z = 1/2. According to the definition of p*, the value of O'xm'a may be taking as the average strength of a fibre of length L, = p*d. Hence,

L,

=

(21_l/m

o"o l ) -/~ (L)----~ (tr~ ~f "

(6.62)

Suppose that the failure time t, - t! 1) + t!2)

(6.63)

Ch. VI, w

327


where t! 1) is the time for continuous fibres to transform into short fibres with the average length equal to L,, and t!2) is the rupture time of the composite containing fibres with p = p*. We determine time t! 1) taking the average fibre stress at t - t! 1) being equal to a half of that given by eq. (6.61). Then eq. (6.59) yields

t! 1) -- gF(vf)t7 o1-m

(6.64)

where

1 ffm m-1 Y]m(m - 1) Ef ffm ,

K

vm

F(vf)----~f

(6.65)

{f ]m--1} --I

Vm

(1 - 2 - ~ ) / )

.

(6.66)

It is obvious that eq. (6.64) gives a rough estimation of the time because eq. (6.59) has been obtained from an analysis of the normal stress redistribution between the fibre and matrix resulting from the creep of the latter; and the process has been assumed not to be effected by fibre breaking process. On the other hand, eq. (6.61) follows from the model with no normal stresses in the matrix. This contradiction reflects an essential feature of the creep process of a metal matrix composite with continuous brittle fibre: at the beginning, the normal stresses in the matrix are decreasing, the fibres are breaking, the matrix shear stresses are developing; then the composite structure stabilizes, the matrix are creeping around the fibres. Note that at vf > v ~ -

1+

eq. (6.64) yields t! 1) < 0. Bearing in mind the above

21/m_ l

remark on the approximate nature of eq. (6.64) we shall take t', - 0 at vf > v}. At t > t! ~) creep and creep rupture of the composite is described by the model of a composite with short fibres considered above. In particular, the stationary creep rate is given by eq. (6.11), that is m+l

m- 1

1 - vf(m-l)/2

(~r~(L))

(6.67)

where L is the base length of a fibre in the tensile testing. Time t!2) is given by eq. (6.40), that is t!2)_ e, c, 2~o [1 + mY(m)] ~ 2~o " -

because actually mY(m) 0, the realization of the ideal b o n d takes place and vice versa. Then for a cantilever vibrating b e a m of c o n s t a n t cross-section, the m i c r o c r a c k is chosen for which the stress intensity factor, Ki, has a m a x i m u m value. It is k n o w n [526] t h a t the fatigue crack rate depends on the stress intensity factor

Fatigue and ballistic impact

336

/

Ch. VII, w

DeFect

\~F - - - - -

u

d

X/

D m _.e

Free

I

/

surface

0

c

k (9 .t.a • I,I

DeFect Fig. 7.3. The model of a laminated material with imperfect bond between the lamina. Fatigue cracks start at defects located both on the external surface and interfaces.

amplitude non-linearly. So we may observe the growth of the only microcrack just chosen. In fact, for this case of symmetrical loading the relationship for the fatigue crack growth is taken as

dl _ f(K,) _ flf KZ +ln( 1 K~))

dN

~,KiZc

- KlZc,]

(7.1)

where Kic is the critical stress intensity factor for the material and coefficient/3 is taken such that to make the calculated fatigue strength of a monolithic material be equal to an experimental value. The number of cycles, N, necessary for the microcrack to reach the interface is

Nh

_ [h

dl

J]~l f(KI)

(7.2)

At the interface the crack can meet either a free surface or ideally bonded part (fig. 7.3). In the latter case, the crack does not "feel" any peculiarity and goes further on to meet the next interface. However, if the crack meets a non-bonded portion of the interface, it is arrested, but a possibility exists to change the configuration of the interface. If the length of the non-bonded portion of the interface is much smaller than the layer thickness, the stress intensity factor at the tip of the interface crack may be assumed to be

where ~ - 1 (a' + a"), and a', a" are the normal components of the stress at the tips of the interface crack caused by the crack propagating through the layer. Free surfaces at the interface including those formed during the delamination just

Ch. VII, w

Ballistic impact

337

described, carry the same population of the defects as the external surface. Hence, when a crack is arrested at the interface and perhaps new free surfaces are formed, the computer program comes back to the start moving to the computer memory a number of cycles accumulated. The result of "testing" one specimen by computer is the stress amplitude corresponding to the fatigue life equal to 106 cycles. An example of the predicted dependence of the fatigue strength on the value of s is shown in fig. 7.4. The comparison of the result with the experimental data presented in fig. 7.1 yields s ,,~ 0.4 for a laminate made at 525~ - 30 M P a - 60 min. The dependence of the fatigue strength on the number of the layers is given in fig. 7.5. The quantitative agreement between the theory and experiment is not achieved, certainly due to poor estimation of the value of s. However, qualitatively the behaviour of the model is in good correspondence with the physical experiment. Finally, it should be noted that laminated metal specimens are characterized by higher damping value comparatively to corresponding monolithic specimens. Figure 7.6 illustrates this property of laminates. It may be assumed that the enhanced damping of a laminate with the imperfect bonding is a result of alternative shears at the non-bonded interfaces.

7.2. Ballistic impact

The penetration of a projectile into a target has always been a subject of intensive interdisciplinary investigations. Highly complicated interactions of processes 1.6

!

|

!

I

|

|

!

I

l

,

!

I

!

,

,

!

|

|

|

O

b

~,, 1.4 b

1.2

1 . 0

i

0.0

,

,

I

0.2

i

i

,

I

0.4

,

,

8

,

l

0.6

,

i

i

I

0.8

i

,

,

1.0

Fig. 7.4. The theoretical fatigue strength of the laminated aluminium composite normalized by that of a homogeneous metal versus the portion of the ideal bond at the interface. The number of the layers is 10, the fatigue base is 10 6 cycles. (After Egin [152].)


338

Ch. VII, w

2.0

o 1.8

b b

1.5

o

1.3

1.0

I

0

I

I

20

I

I

40

I

60

,

80

n

Fig. 7.5. The theoretical and experimental fatigue strength of the laminated aluminium composite normalized by that of a homogeneous metal versus the number of the layers. In the physical experiment, specimens were made at 525~ - 30 MPa - 60 min, the portion of the ideal bond at the interface in computer experiments, s = 0.4. The fatigue base is 10 6 cycles. (After Mileiko et al. [424].)

1.5 :ooooot :u~o t -***** t

= 10 rr~ir~ = 30 m~n - ?20 train

.

/~

1.0

0.5

0.0

0

50

cr /

I00 MPa

150

Fig. 7.6. The decrement of vibration of laminated titanium versus the amplitude of the maximum stress in a cantilever specimen. The temperature-time condition of the diffusion bonding is 6 0 0 ~ 30 MPa, the time is shown on the figure field. The decrement was measured by half-power bandwidth method. (After Mileiko et al. [424], the data for homogeneous specimens are after [541].)

Ch. VII, w

Ballistic impact

339

involved, the adiabatic shear band formation, plugging, spalling, projectile fracture being some of them, as well as the obvious practical reasons have brought numerous results although not always available. So we do not intend to discuss the problem in detail, but we shall present some experimental data to make clear potentiality arisen as a result of the rational use of a composite idea in the applications mentioned. We shall discuss here neither well known idea of making ceramic/metal armour plates (see for example [287]), nor results of the intensive studies of the penetration of projectiles into polymer matrix laminates and the energy dissipation due to delamination around the impact area (see for example [604]). The behaviour of Kevlar type of the fibres under dynamic loading [729] shall also not be discussed. We restrict ourselves to the description of the ballistic-impact behaviour of aluminium laminates identical to those used in model fatigue experiments (see Section 7.1). But first we present a simple method to interpret and analyze the experimental data. 7.2.1. Perforation o f thin metal targets by projectiles

We consider a rather rare occasion when the real situation can be modelled by a single process, that is, perforation of a metal alloy target by a projectile flying with the velocity of an order of hundredths ms -1 and impacting a target normally to its surface. The target thickness is of an order of the projectile diameter. During the projectile penetration into a metal target, adiabatic shear bands in the target develop [626, 705], then a plug may be formed and pushed out of the target ahead of the projectile [442]. An essential part of the energy dissipates in a cylindrical layer just outside a hole formed by the plug [442, 449]. This energy dissipation occurs as a result of either plastic deformation of the layer or its brittle fracturing. In the case of plastic deformation, the force F acting on a projectile away from the free surfaces of the target can be assumed to depend on the projectile velocity, v, as [449] F(v) - - K v n

(7.3)

where K and n are constants. Assuming eq. (7.3) be valid from the front to rear surface of a plate, we obtain a simple result of the integration, that is ~2-.

_

1 -

v-2-n o

(7.4)

where ~o and ~1 are the initial and terminate values of the projectile velocity, respectively, normalized by the ballistic limit, v*, that is such value of ~o for which ~1 - 0 . The detailed experimental procedure to obtain the value of v* is described in [449]. This is similar to that used to define the fatigue limit in fig. 7.1. Curve ~1 (~o) does not depend on the target thickness, the shape of this curve is determined by the value of n only, the latter is a characteristic of the target material. Assuming the penetration resistance is determined by plastic deformation of the


340

Ch. VII, w

target material, we should expect the value on n to coincide with the exponent in a power approximation of the dependence of the yield strength cr* on strain rate ~. The characteristic size to determine strain rate during penetration is the thickness of the cylindrical layer where the energy dissipation takes place. The deformation is localized here and for the case under consideration this thickness was evaluated to have an order of 1 mm [442]. Therefore, at the projectile velocities between l02 and 103 ms -1 the strain rate values occur to be between 105 and 106 s -1. Equation (7.4) was used in [449] to interpret experimental results obtained in testing A1-6% Mg alloy, titanium alloy, and a pure copper. An example of dependence ~1(~o) for the A1-6% Mg alloy targets is presented in fig. 7.7; the experimental conditions are described in Table 7.1. Note that the value of n

0

0

0

2 0~

1 A

0

f i

I

i

i

l

2

i

i

VJ'O"

i

I

3

Fig. 7.7. Terminal velocity normalized by ballistic limit versus normalized initial velocity. The target material is the AI-6% Mg alloy. The legends are defined in Table 7.1. (After Mileiko and Sarkissyan [449].)

T A B L E 7.1 The experimental conditions for data presented in fig. 7.7 the target material is the Al-6%Mg alloy. Legend

Lamina thickness mm

Projectile diameter mm

(3

1.97

6.35

Ballistic limit ms -I 250

/~

3.00

6.35

350

[]

4.00

6.35

405

9

3.37

10.3

228

Ch. VII, w

Ball&tic impact

341

determined using these experimental data is equal to 0.35; that agrees fairly well with n = 0.3 in dependence ~r* e~ ~n obtained earlier [125] in the direct measurements. Another feature of the perforation of a metal target accompanied with plugging is decreasing the plug mass with increasing the initial projectile velocity [450].

7.2.2. Perforation of laminated metal targets Now we consider the ballistic-impact behaviour of laminated metals, and we use as targets the same aluminium alloy laminates as in the fatigue experiments (Section 7.1). Temperature and pressure of the diffusion bonding are 485-525~ and 30 MPa. Again, changing the process parameters changes the interlayer strength. A typical set of the original experimental data is presented in fig. 7.8 as the dependencies of the ballistic limit normalized by the ballistic limit, v~, of a homogeneous target of an equal thickness on the average thickness of the lamina. In the figure, solid points correspond to perforated targets and open points stand for velocities below the ballistic limit. In some cases the projectile leaves the target with the velocity nearly equal to zero, the corresponding points are half solid-half open. The experimental data yield the dependence of the ballistic limit on diffusion bonding time presented in fig. 7.9. At least two conclusions may be drawn immediately. First, under some conditions of the projectile-target interaction, a laminated target can absorb much higher projectile energy, at least twice as high as a homogeneous target made of the same material. Second, assuming the interlamina strength depends on diffusion bonding time monotonously, the ballistic limit depends upon the interlamina strength nonmonotonously reaching a maximum as well as a minimum at some values of the interlamina strength. This means that a number of factors effecting the penetration resistance of the laminated metal target is greater than two. If we compare pictures of the projectile penetrating into homogeneous (fig. 7.10) and laminated (fig. 7.11) targets, qualitative difference will be clear. First, shears on the interfaces in the laminated target appear. The corresponding energy dissipation due to the friction enhances the penetration resistance. Secondly, unlike the case of a homogeneous metal target, when adiabatic shear bands are known to cause a cylindrical crack which forms the plug, in the case of a laminated target, the interfaces prevent the formation of such a crack. So in this case the perforation is not accompanied by the plug pushing out. Let us validate the assumption about an essential effect of the friction between neighbouring lamina on the energy dissipation during the penetration of a projectile into a laminated target. Figure 7.12 presents the experimental data for laminated targets in the form of ~1 (~0) dependencies. The approximation of these dependencies according to eq. (7.4) yields the values of n presented in Table 7.2. One can see a qualitative difference in the values of n for homogeneous target and laminated one. The value of n is positive in the former case and negative in the latter case. Positive values of n correspond to a normal behaviour of ductile metals. The change in the sign means certainly that another way of energy dissipation is opened. The most

342

Ch. VII, w

F a t i g u e a n d ball&tic i m p a c t

1.50

.

.

.

.

,

525~

-

.

30

.

.

.

,

MP a

-

5

.

.

.

.

,

.

.

.

.

rn,'i.'n,

BALL 1 0 . 3 ~,~r~ 9

o

/

1.25-

0

9

1.00

.

.

.

BALL 6 . 3 5 rn,m

.

'

0.I

1 . 5 0

.

.

.

.

i

0.2

,

,

,

525*C

,

,

-

,

30

,

,

,

,

,

,

MPa

,

-

,

I0

.

,

0.3

.

.

.

0.4

,

,

,

,

0.5

,

,

,

,

rrui, n

9

BALL 10.3 rrLrr~

o

1.25

-

/o BALL 6 . 3 5 rrLrn,

8-----'-'~

1.00

. . . . 0.1

0

9

' . . . . 0.2

o

' , , , 0.3 h/m'm

e

i

"--'--

0

0

, . . . . 0.4

0.5

Fig. 7.8. The ballistic limit of the laminated targets normalized by that of a homogeneous target made of the same aluminum alloy (AI-6%Mg) versus average thickness of the lamina. (After Mileiko et al. [450].)

p r o b a b l e w a y is the i n t e r l a m i n a friction. If the friction is i n v o l v e d essentially, the n e g a t i v e value o f n seems to be explainable. It s h o u l d be n o t e d t h a t in the case of l a m i n a t e s , the c o n t r i b u t i o n of plastic d e f o r m a t i o n o f the m e t a l to the p e n e t r a t i o n resistance m a y be less t h a n that in the case o f h o m o g e n e o u s targets. M o r e o v e r , a strict m e c h a n i c a l m o d e l o f the penetra-

Ch. VII, w

1.50

Ballistic' impact

_

J

i

I

'

l

I

'

I

I

'

'

343

~

ol.25 La~rti~

~Melcrtess

* * * * * O. t 6 ~rt~'rt

~oo_0_~ O. 18 ~,rt~'rt z x ~ 4 s 0 . 2 0 ~rt~,t ~ 0 . 2 2 ~rtrrt

-

0.24 AAAAA 0.26

*****

1.00

J

0

,

t

15

,

,

i

/

30

,

mi, n

,

~rt~rt ~rt

l

45

,

,

60

Fig. 7.9. The ballistic limit of the l a m i n a t e d targets n o r m a l i z e d by t h a t of a h o m o g e n e o u s target versus diffusion b o n d i n g time. T h e target material is the A1-6% M g alloy. (After M i l e i k o et al. [450].)

Fig. 7.10. P e n e t r a t i o n of the ball projectile into h o m o g e n e o u s m e t a l target. (After M i l e i k o et al. [442].)

tion, that remains to be evaluated, will perhaps reveal a decrease of the volume surrounding the projectile passage, in which the friction is localized, with the interlamina strength increase. A reason for this may be the necessity to rupture the interface to initiate shear friction. The thickness of a corresponding process zone

344

Ch. VII, w


Fig. 7.11. Penetration of the ball projectile into laminated target. (After Mileiko et al. [450].) TABLE 7.2 The ballistic limit, z'*, and the exponent, n, in eq (7.3) characterized the targets, and the conditions of the experiment yielding the results presented in fig. 7.12. Diffusion bonding time, min

0

15

30

30

30

30

60

90

Homogeneous

Number of layers v*, m/sec n

15 194 0.42

15 315 -1.07

15 323 -0.53

l0 332 -0.56

5 316 -0.62

2 324 -0.76

15 321 -0.19

15 310 -0.40

259 0.38

of a cylindrical shape should perhaps depend on a dimensionless combination of the lamina thickness and a characteristic size of the projectile. Coming back to the approximation given by eqs. (7.3) and (7.4) with the constants obtained in the experiments described, we see that different structures reveal best performance at different projectile velocities, an example is presented in fig. 7.13. Obviously there exists a particular structure of the laminated target to correspond to a maximum force applied to a projectile moving with a particular velocity. Therefore an optimal project of the target for a definite ballistic limit may be obtained if to take into account this dependence. In fact, to get such a project one needs also a knowledge of the interaction between the layers which are to be passed by a projectile moving in a decelerating manner. It is also important to account for a peculiarity of the projectile/target interaction in the vicinity of free surfaces of the target.

Ch. VII, w

345

Ballistic impact 9 9 9 9 9 Nor,- bonded t = 15 m i ~

t = 30 mi, r~ oo0oo t = 90 rrvi,~ * * * * * t = 60 r r ~ n ooooo l t o m o g e n e o ' t c s

ooooo

O0000

O O

/

r

/~'"

I

/*

/

//

o

Cd

,

IX~ o~"

=47 0

i

i

I

1

I

I

2

|

!

I

3 1

I

'

! 0 layers

(

5

'

'

I

I

2) o / V / *

I

layers

I

i

2

/

,

,

//o # ]#:,,,-'-

" z T,,, ~

/o

1 2

,

layers

+

I

I

2

2

o

o

o~ o

r~

0

=

I

1

- _ - o.~= , i o '

0.56

-

I

:o.~

I

2 1

V o~'U *

1 2

2

0

Fig. 7.12. Terminating normalized projectile velocity versus initial normalized velocity for laminated targets. The target material is the A1-6% Mg alloy. The thickness of specimens is 2.10 + 10 mm. The ball diameter is 6.35 mm. (a) Changing the diffusion bonding parameters, the number of the layers is 15. (b) Changing number of the layers, the diffusion parameters are constant, 485~ - 30 M P a - 30 min. (After Mileiko et al. [450].)

346

Ch. VII, w


5o

"""'""'""'"t

40 La~lered plate (~t = - 0 . 4 0 )

3O

\ r~2o

lo

Ho~rtogeneo~ts plate ( n = 0.38) O

|

0

|

|

|

!

!

|

!

!

200

|

t

|

|

|

|

I

|

i

|

t

400

|

|

i

I

|

600

|

|

!

|

|

|

!

|

800

|

|

i

|

|

I

1000

Fig. 7.13. The calculated dependence of the force acting at the projectile penetrating the homogeneous and laminated targets. (After Mileiko et al. [450].)

Finally, looking at figs. 7.1 and 7.9 we see an analogy between the fatigue and ballistic-impact behaviour of metal laminates with non-ideal interlamina bond. The analogy is physically based on the influence of the non-ideal interface on the crack propagation in a laminate.

7.3. Fatigue of metal matrix composites

First we shall discuss the main results of experimental investigations to elucidate the features of composite fatigue failure. Then we consider a micromechanical model to be used in computer simulation.

7.3.1. Experimental observations We will use experimental observations presented in [423] as a basis. Either boron fibres or steel wires were used as reinforcement in these experiments, and two aluminium alloys, D16 and AI-Zn-Mg alloy were taken for matrices. Specimens were prepared by a hot-pressing procedure. The D16 alloy matrix was brought into a composite by using a foil and the A1-Zn-Mg alloy was plasma-spayed (see Chapter 11). A batch of boron/steel/aluminium composite specimens prepared according to the instruction outlined in Section 5.2.5 was also tested. This led to a variety of the microstructures of composites that provided a basis for the observation of the fatigue mechanisms under various conditions. All combinations of the fibres and matrices as well as corresponding fabrication conditions are given in Table 7.3.

Fatigue of metal matrix composites

Ch. VII, w

347

TABLE 7.3 Composites used in the basic fatigue experiments. Material label in the text

Matrix

Fibre

B/AI(F)

D 16T, foil

Boron

St/AI(F)

D16T, foil

B/St/AI(F)

D16T, foil

B/AI(P1)

A 1 - M g - Zn

1 3 C r - 1 3 N i - 2Mo steel Boron + 1 3 C r - 1 3 N i - 2Mo steel Boron

Fabrication route

Hot pressing in vacuum, 485~ 25 M P a - 1.5 h, then aging Hot pressing in vacuum, 485~ 30 M P a - 1.5 h, then aging Hot pressing in vacuum, 485~ 30 M P a - 1.5 h, then aging Hot pressing in gaseous isostat, 500~ - 40 MPa - 0.5 h,

All the fatigue tests were conducted in cyclic bending of a cantilever specimen at a constant deflection amplitude of the free end, a, and at the natural frequency of a specimen. Because of the progressive damage during the test the effective modulus of the specimen was decreasing, so the stress distribution was changing and the maximum stress and the natural frequency were decreasing. Still, each test was labelled by a maximum stress in the specimen at the test start, that is Eh (kl) 2 a E h 3.50 ~r -- 2 l 2 U ( k l ) v2(kt) "~ -~---i-~ - a s(kt)

(7.5)

where h and I are the thickness and length of a specimen, U, V, and S are the Krylov functions, k l - 1.875 for the main oscillation form. During a test, the frequency was recorded and typical dependencies of the natural frequency on cycle number are shown in fig. 7.14. In what follows, the fatigue life is defined as the cycle number corresponding to the 5% decrease of the natural frequency. Original results of fatigue testing are given in fig. 7.15 in the form of SN curves. The dependencies of the fatigue strength, determined on the 106 cycles base, upon fibre volume fraction are shown in fig. 7.16. Note that the data presented in figs. 7.14 and 7.15 evince that for both boron/aluminium composites with high values of the fibre volume fraction and the unreinforced matrix alloy, the difference between the fatigue life defined by criterion f i f o = 0.95 and the entire failure is minor. On the other hand, in the case of boron/aluminium composites with low fibre volume fractions, the difference is rather large. Figure 7.17 reveals microstructural features of the fatigue fracture in the composites. Considering the experimental observations we may draft the following peculiarities of the fatigue behaviour of metal matrix composites. First, in a composite with a low fibre volume fraction (vf < 0.2) the fatigue process is mainly influenced by crack propagation in the matrix. The fast increase in the fatigue strength is observed in composites with a foiled matrix, this resulting from fatigue crack arrest by weak interfaces within the matrix volume as illustrated in Fig. 7.17a, b, c.


348

Ch. VII, w

1.00

0.95

0.90

0.85

0.80

I *****'V t = 0.26, r = 6 f 7 MPa *****v! = 0 . 1 0 , g = 338 MPa

0.75

,

,

0.0

,

,

0.2

,

,

,

I

0.4

,

,

N/N.

,

i 0.6

,

,

t ,

I

0.8

,

,

,

1.0

Fig. 7.14. Relative change in the natural frequency of B/AI(F) specimens versus cycle number normalized by the ultimate cycle number. (After Mileiko and Anishshenkov [423].)

The fibre strength is also of no importance (compare the behaviour of two batches of the boron/aluminium composite in fig. 7.16). The fibre elastic modulus is of most importance since the value of the ratio of elastic moduli of the components determines stress values in the matrix. Assuming the elastic behaviour of the matrix we have the stress amplitude in the matrix

(

O"m -- O" Vm -~- Of

(7.6)

So the fatigue strength of boron/aluminium composite, vf = 0.09, N, = 106, that is = 300 MPa (fig. 7.15a) corresponds to the matrix stress O"m - - - - 2 1 4 MPa. This is about 30% higher than that of a monolithic matrix alloy, which is in good accordance with the behaviour of laminated metal specimens discussed above (Section 7.1). If this is true, then the enhancement of the fatigue strength for a composite with a plasma-sprayed matrix should be not so pronounced as in the case of a foiled matrix. In fact, in the former case, ~r = 220 MPa (fig. 7.15a) and Om---- 130 MPa; this is only about 8% higher than that for the unreinforced matrix. This means that cracks in the composites with low fibre volume fractions are arrested mainly by interfaces in the matrix volume but not by the fibre/matrix interfaces. The fibre type (brittle or ductile) is of no significance as well. The cracking is also going on in the matrix (fig. 7.17e.) and the cracks are arrested by the matrix/matrix interfaces. Secondly, at large fibre volume fractions (40-50%), the dependence of the fatigue strength of boron/aluminium composites on fibre volume fraction is rather weak,

Ch. VII, w 500

349

Fatigue o f m e t a l m a t r i x composites

,

i

,

0

o

~400 b

3O0

"~ ~ 200

=

o. o s .

.

..._.._

~ _

_ ,

100

~

_..~~

2"~J. -

~

fRI

~ = o.o~i r~

,'-~---~u

5.0

; I.+

6.0

1200

~

"~ ,,,

,

7.0

,

1400

i

i

va/"u,~,,,= = 0 . 4 4 / 0 . 0 ?

0

1000

1200

0

\

800

vy = 0.30

.

9

600

0

0

'

9

9

i 6.0

log(O)

0

9 re=

600 =

5.0

1000

800

400

200

!

~

0.51

c

"u! = 0.45

0.29:

,

400 7.0

5.0

'

~ 6.0

' '7.0

Fig. 7.15. The initial maximum stress amplitude in a cantilever specimen versus number of cycles corresponding to f i f o = 0.95. Open circles stand for boron/aluminium composites with a foiled matrix, dark circles stand for boron/aluminium composites with a plasma-sprayed matrix, dark triangles stand for steel/aluminium composites with a foiled matrix, and open triangles stand for boron/steel/aluminium composites with a foiled matrix. The couples of points connected by horizontal lines correspond to f i f o = 0.95 and the final failure. They were obtained by testing boron/aluminium specimens from the second batch of the composite. The values of vf for the latter couples are shown in fig. 7.16. (After Mileiko and Anishshenkov [423].)

the fatigue strength is determined by the strength characteristics of the fibre. In this case, the fibre stress occurs to be sufficiently high to trigger fibre breaking process as is observed under monotonous tensile loading (see Section 5.2). The brittle fibre breaks at a certain point; the break may be of the "fatigue" or "instant" type and may initiate a chain breaking process in the vicinity of its own plane as a result of stress redistribution between the fibres and the matrix. Under favorable conditions the process might be slow or just delayed. This may be due to either a decrease in the stress intensity factor (because the crack propagates in a macro-nonhomogeneous

350


Ch. VII, w

B/AL(foiO, Batch '1 BLAb(foil), Batch "2 . 9 9 9 9 9B / A l,(:p~op'm,o,-,s.pra'ye~) 9m a r e . S t e e l / A ~ f o ' i , l , )

ooooo

ooooo

1200

'

'

'

i

i

.

.

I

o,

\

.

|

!

!

A

800

400

|

i

|

0.0

I

,

I

I

0.2

2

,

|

I

1

0.4 .

i

w

i

i

i

0.6

|

b

a--" f

J

I

.

0.0

i

I

0.2

vl

I

0.4

i

|

0.6

Fig. 7.16. (a) The fatigue strength (fifo = 0.95, N, = 10 6) of composites versus the fibre volume fraction. (b) The specific fatigue strength of the steel/aluminium composite, normalized by that of the matrix, versus fibre volume fraction. (After Mileiko and Anishshenkov [423].) stress field), or encountering an obstacle like the interface. Figure 7.17d illustrates the situation. Let us estimate the stresses in the components. Using eq. (7.6) yields, for vf - 0.5, 0"m < 0.31cr and ~rf _> 1.7~r (inequality appears if the matrix deforms plastically). Hence, if permanent stresses are taken into account (see Section 5.2.2), the matrix stress in boron/aluminium composites at 105 _< N, _< 107 (fig. 7.15c) occur to be higher than the yield stress, so the fibre stresses in the composites with the foiled matrix are higher than 1700 M P a (for N , - 107) and 2500 M P a (for N , - 105). Therefore, fibre m a y really start to break under such conditions. In the case of ductile fibre (steel/aluminium composites), a crack intersects both the matrix and fibres approximately in the same cross-section; as a rule, multiple cracking takes place here (fig. 7.17f, g). However, the matrix stress calculated by using the experimental data and eq. (7.6) appears to be about twice the fatigue

Ch. VII, w


351

Fig. 7.17. Microphotographs of composite specimens after fatigue testing. The surfaces presented are either parallel to the plate surface (P) or normal to that surface and parallel to the fibre direction (PP). See Table 7.4 for the description of the specimen. (After Mileiko and Anishshenkov [423].)


352

Ch. VII, w

TABLE 7.4 Description of the specimens presented in fig. 7.17. Figure label

Orientation

Material label

vf

~ MPa

N, 910-5

a b c d e f g

P PP PP P P P P

B/AI(F) B/AI(F) B/AI(F) B/AI(F) St/AI(F) St/AI(F) St/AI(F)

h

P

B/St/AI(F)

0.09 0.09 0.09 0.53 0.09 0.45 0.45 0.44/0.07

255 245 255 1100 265 450 450 1080

48.00 23.50 48.00 3.20 2.54 5.60 5.60 5.30

strength of the matrix. It may not be explained by any single reason like crack arrest at the interfaces. Perhaps, high fracture toughness of ductile-fibre/ductile-matrix composites (see Section 5.7) may be related to the enhancement of the fatigue strength of such composites. Third, at intermediate values of the fibre volume fraction (20-40%), both fatigue mechanisms inherent to the extreme intervals of the fibre volume fractions are observed together. Fourth, introducing a small quantity of steel wire into a boron/aluminium composite with high fibre volume fractions, as was suggested to improve strength characteristics of the composite (Section 5.2.5), yields an essential decrease in the fatigue strength scatter and an increase in the mean values of the fatigue limit. The comparison of the crack configurations in a pure boron/aluminium specimen (fig. 7.17d) and that containing a small addition of steel wire (fig. 7.17h) shows that unlike the former case, when the crack cuts the whole (or nearly the whole) layer, in the latter case the crack length does not exceed some fibre diameters. The fatigue mechanisms observed are the basic experiments presented that have been confirmed in a number of studies performed with specific composites and aimed at special problems. For example, Rosenkranz and Gerold [572] testing steel/silver composites with medium fibre volume fractions (35%), found that two failure mechanisms dominated, i.e. failure by a single fatigue crack and that by gradual accumulation of fatigue damage in the matrix and eventually in the fibre/matrix interface. The latter was observed mainly in composites which had a weaker matrix (after recrystallization) and weaker interface (as a result oxidation of it). An important finding was reported by Nayeb-Hashemi and Seyyedi [486] who tested graphite/aluminium composites with various thicknesses of the interface zone. The dependence of fatigue strength on the interface is shown in fig. 7.18. The result is explained by the authors' assumption of crack initiation in the interface layer, which is confirmed by microscopical observation. Direct observation of a decreasing of the fatigue crack rate with weakening fibre/ matrix interface in a B/Ti composite are also known [72].


Ch. VII, w

353

600

\

500 b

400

300

,

0

,

,

t

200

~

,

,

i

400

,

,

~

h/rim

t

600

,

,

800

Fig. 7.18. The fatigue strength on the 106 cycles base of graphite/aluminium composite versus interface zone thickness. Experimental date by Nayeb-Hashemi and Sayyedi [486].

Therefore, experimental studies of the fatigue in metal-matrix composites may be briefly summarized as has been done by Johnson [286]. The possible failure modes can be grouped into four categories: (1) matrix dominated, (2) fibre dominated, (3) self-similar crack propagation, and (4) fibre/matrix interfacial failures.

7.3.2. Computer simulation of fatigue failure It is our intention to present here a computer simulation procedure in detail to illustrate potentialities of the technique. This occurs to be possible and certainly instructive since in this particular case the computer simulation appears to be firmly confined to the physical experiment. A model

Let a characteristic composite element containing the fibre and matrix may be damaged as a result of the fatigue process. This may be either matrix cracking, or fibre break, or trough-cracking (fig. 7.19). The height of an element, h, is the thickness of the composite layer and the length, l, is equal to the average distance between the microcracks (fig. 7.20). The latter is to be either chosen from experimental data or calculated by using a micromechanical model. In the initial state, the element has the effective Young's modulus Eo - Efvf -[- g m Vm In a limiting state the Young's modulus is


354

(

Ch. VII, w

(a)

C (b)

m m Fig. 7.19. Possible types of the element damage. (a) Matrix cracking, 2f = 1, 2m -- 1/2. (b) Fibre break, 2f = 1/2, '~.m -- 1. (C) Through-cracking, 2f = 1/2, ~,m = 1/2. Er -

2fEfvf-4- 2mEmvm;

~,f, 2m _< 1

If the limiting state is reached because of matrix cracking, then Er - E l ;

2f = 1;

2m = 2rn;

If the limiting state occurs as a result of fibre breaking, then

Y /

2

3

9

(

j

9

9

T2 3

1-z

Fig. 7.20. A longitudinal section of the cantilever rod divided into the elements.

.z"


Ch. VII, w

Er - E2;

2f -- ,~;

355

2m -- 1;

If the crack cuts both the fibre and matrix, then in the limiting state: E r -- E3;

/~f-/~?;

2m - / I ' m

Let us introduce damage value o9 such that in the initial state c o - 0 and at the limiting state (D -- (_D* z

Eo - Er E0

that is ~

,

=

1 -+- K/~m. 1 --~-K '

,

/~f nt- K

602 = 1 + K ;

, __

CO3

+

I+K

where ~c- Emvm/Efvf. Let the damage be accumulating according to the following equation do)

dN = fl

(~nn) n

(7.7)

where fl, ~rn and n are the phenomenological constants, N is the cycle number, and is the maximum normal stress amplitude in an element. Equation (7.7) will be utilized to describe the behaviour of large volumes composed of a sufficiently large number of the elements. Moreover, the stress state of the composite is non-homogenous. So it is certainly reasonable to accept a jumpwise transition of an element from the state with modulus E0 to that with modulus Er. The transition occurs when o9 reaches the critical value, o)*. It is assumed that in a particular fatigue process just one damage mechanism takes place.

Calculation procedure Consider a cantilever rod of a rectangular cross-section which remains constant along its length (fig. 7.20). At an arbitrary cycle number, Nq, i.e. at the q-step of the damage process, contour Fq in a longitudinal section of the specimen divides the area with the initial value of the effective modulus, E0, from that with modulus Er (fig. 7.21). This means that we deal with the vibrating rod of stiffness E1 changing along the axis in a step-wise manner. For interval (x~, c~+ 1) of the constant stiffness, (El)a, the vibration is described by equation [654]

(EI)~ ~4y

(7.8)

356

Ch. VII, w

Fatigue and ballistic impact bt +

+

+

+ ++

+ +

"~T" +

+ +

+ +

+ +

+ i

+ +

+ i

+ i

+ + I

/ I 1

x~

+

+ |

+

+

+

+ + + +

+

+ + ! i

i

1i 1

Eo

1 i I

I i

f

:ca+ 1

Z

.Z"

Fig. 7.21. The contour of the damaged area at the q-step of the damage process. where fq is the natural frequency of the rod and # is the mass of a unit length, I is the moment of inertia, and (El)a-- fF E ( y ) y Z d y -

Eol ( ~3a -[-'~0 Er (1 - ~a) 3)

-- EoI~a,

Here ~a - 2Y(a) (x~)/H, y(a)(x~) - yr(xa + 0). Since contour Fq changes with time and so the distribution of the effective stiffness, (El)a, along the length does also change with the number of cycles, we cannot exploit usual approximate approaches, like the Rayleigh-Ritz method [654] directly. Hence we intend to apply a numerical procedure and write down the expressions for deflection Y~, rotation angle Oa, bending moment Ma, and shearing force Qa at x = x~ belong to interval x~ < x _< x~+l, as follows Ya - a Y a - a ( C l a ) V l ( a ) + C~a) Vz(a) + C~a) V3(a) + C~a) V4(a)) |

-- aka ~'~ - akoOa

= ajl/2ko~;1/4(Cla) V4(a)-[- C~a) Vl(a)-k- C~a) V2(a)_[_ C~a) V3(a)) 2 Ma - akaEaIY itt - akZEollf/la " 2 a1/2Eol(Cla) V3(a)-k- C~a) V4(a)-k- C~Ot)Vl(a) + C~a)V2(a)) =afqko~ 3 tit - ak3EolQ~ Qa - akaEalY~

"-" uj'~3/2/'3rl/4 L - ' t~0 q 'a ~01"(CI a) V2(a) -[ C~a) V3(a) -[- C~a) V4(a) -[- Ci ~) VI(~ ']]

(7.9)

where k~ - f l / 2

#

1/4

-- fl/2ko~-~l/4

1/4

Ch. VII, w

357


Vi (~) - Vi(k~x~) are the well known Krylov functions, and a is the amplitude of the free end. Denoting r/l~) - I?~, r/~~) -1~(~), r/~~) -j~7/~, ~/~)_ Q~, we rewrite eq. (7.9)in the form

4 r](ne) --/7~!R(n)-'a'~-t) Z C(~)gS(n,m)(kexc~) m=l

(7.10)

where R ( n ) - (1 - sgn(5 - 2n))/2 S(n, m) - ( 2 - 2sgn(m - n)) + m Solving eq. (7.10) with respect to C, we obtain

Cn(~

n + 1.

4

Z/~~m~'!R(m)-!-~)A S(n,m) (k~x~)rl (~) m=l

Substituting eq. (7.1 l) into eq. (7.10) for x -

(7.11) X~+l, yields the recurrence

4 ?/(c~+l) O) -- Z R(~+I) 1 = r/l(Xe+l + ~"l,m g/(e) m=l

(7.12)

where B(,~r+ 1)

~-~y (R(p)-R(r)-P@)

-- Jq ~

Finally, for x -

4

Z As0,r)(k~x~) gS(p,1)(k~xc~+l). 1=1

L, we have 4

r/n(L) __ q(~,+l) = ~ Dn,mr/(m1) m=l

(7.13)

where (2) On,m-- Z ... Z Z O(p,~m +1) .B(~s) r,p "" . . . B n,l' 1

r

p

as is the number of the intervals of a constant bending stiffness, l, p, r, n, m = 1,2,3,4. Hence, the problem is n o w reduced to solving four simultaneous equations (7.13), the results being values of /I(1). Note that ql 1) =/72(1) _ , t13(as+l) _ 0 and r/~~s+r) - 1; hence, the number of equations (7.13) is reduced to 2. Havhag obtained (1) r/~1) and/14 from remaining two equations, we obtain, from eq. (7.12), values q(n~) (n = 1,2,3,4; ~ = 1,2,...,C~s + 1). Then eq. (7.11) yields values Ck (k = 1,2,3,4) for all the intervals of a constant stiffness. The natural frequency is given by the frequency determinant

358


det IDf(fq)[ - D33D44 - D43D34 -- 0

Ch. VII, w

(7.14)

The number of cycles at the q-step of the damage process is

l~/'q--

(_O*-- (9!q~l)

/ O)* (.o(q-1) /

~--~ff(q_-i'---~-j-- - - m i n ~ ~-~~_ k'~a-) ~, k,l 1)/O'n) n k,1 ~ ~ n

(7.15)

where elements (k, l) are located within the area with E = E0. Element (i', f ) changes its state to the limiting one, i.e. o~i,,j, - o~*, at the q-step. The increment of o~ in the area with E = E0 will be Aoj(q) {'o.(q_ 1) )n k,1 -- ~ , k,l / an Z~g'q

(7.16)

Following the experimental observations outlined above, we may introduce probability s for element (iI+ 1,f) to change its state to the limiting one immediately after element ({,f) reaches the limiting state. Obviously, s depends on an interface structure. The damage accumulation process may lead to formation of a macrocrack that propagates with the rate given by the Paris law [526]

dN where c is the crack length, K is the stress intensity factor amplitude, k, K0 and m are constants. Assuming that the propagation time of a macrocrack is much shorter than the damage accumulation time, which corresponds to the experimental data (the shape of the natural-frequency/cycle-number curves, fig. 7.14), we introduce the other probability, p, of the transition of an element damage into the macrocrack. The macrocrack cuts the whole cross-section if only the condition of the damage-tomacrocrack transition is fulfilled for all mc elements of column j' that have not reached the limiting state (mc = i', i' + 1 , . . . , My). Again, probability p relates mainly to the interface structure. In terms of statistics, the formation of the macrocrack is a stepwise damage process in a series of tests, in each test one of two possible events occurs, either the crack jumps or not. The probability of macrocrack formation at the q-step of such a process, provided it has not been created at a previous step, is Pq - 1 - (1 - Pq_,)(1 _p)(M~ ,+,) Simulation procedure

For a given value of the free end amplitude, a, at the q-step of the damage, the stresses in all the elements are computed. Then number of the cycles, ANq, at this

Ch. VII, w

359


damage stage is calculated according to eq. (7.15), as well as the increment of the damage, Ao~, is determined by using eq. (7.16) for all elements with E - E0. The natural frequency is calculated by solving eq. (7.14). Then, still staying at the q-step, two series of the random values, ~)p and 4~s, distributed homogeneously over interval (0,1), are generated. The specimen is assumed to be fractured at this step if for mc members of the ~)p series ( m e - M y - i ' + 1), inequality ~bp(m) < p is fulfilled; here m = 1 , 2 , . . . ,me. Otherwise, the limiting state of the elements in f - c o l u m n propagates provided inequality ~bs(i) < s is fulfilled; here i < My - 1. Figure 7.22 presents the results of a computer simulation procedure illustrating the influence of parameter s on the dependence of the natural frequency of a specimen on cycle number. Introducing probability p into the procedure leads to the interruption, at some value of N, the damage accumulation process and this yields a statistical value of the ultimate cycle number, N~. We start the estimating of the values o f p and s and constants in eq. (7.7) to fit the results of physical experiments presented above, with an approximate evaluation of constant n (the value of an in eq. (7.7) may be fixed beforehand). We need such a value to use it further on as a first approximation. To do it, we simplify the model by assuming the damage being accumulated in the upper layer of the specimen (fig. 7.20) only. At N - N*, when the length of the damage zone reaches Xp, the macrocrack occurs that fractures the specimen. Neglecting a change in the shape of the centre line of the vibrating rod, we may write

1.00

0.99

8--0

0.98

0.97

0.96

0.95

0.0

0.2

0.4

N/N*

0.6

0.8

1.0

Fig. 7.22. The calculated natural frequency versus the cycle number for various values of probability s. After Mileiko and Suleimanov [457].

Fatigue and baH&tic impact

360

N* - f0 ~

d~o

O~(O'(Xp)/O'n) n

=

Ch. VII, w

(7.18)

60* O~(0" (Xp)/O'n) n"

We have also

O"(Xp) --

E o H Y tt (Xp)/2

Y" (Xp) --

(Xp).

ak 2 Y"

(7.19)

Substituting eq. (7.19) into eq. (7.18) and accounting for the approximate expression for the maximum initial stress amplitude in the specimen, i.e. ~ro ~ aEokoH/2, yields N* =

.

o~(ytt(Xp)(70/(Tn) n

(7.20)

Hence, the log-log plot of the experimental data (r0(N*) supplies an approximate value of n. Figure 7.23 presents these values for the composites described in Table 7.3. An optimum set of constants n, fl in eq. (7.7) is evaluated by comparing the experimental data with computer simulation results to provide a minimum to the target function

5

i

I

i

l

i

i

i

I

i

l

1o,

5 I

"

[p

0 0.0

i

i

B/(AI(F),

ooooo

r--,nmuu a / A l ( P 1 ) , ~..*..,..,.A B / S t e e l / A l ( F i

I

0.2

i

i

vl

i

I

0.4

.

.

) i

i

i

0.6

Fig. 7.23. The exponent, n, in eq. (7.7) as a function of the fibre volume fraction for the composites described in Table 7.3. (After Mileiko and Suleimanov [457].


Ch. VII, w

F(Xl,X2)-- F ( n , l o g ( ~ ) ) =

\'og\gi(e)/

361

(7.21)

-log\gi(n)//

i--1

where gi (e) -Ni(e)(o'~) and gi (n) -gi(n)(o'~)

are the ultimate cycle numbers in the physical and computer experiments, respectively. A search for the optimal set of (n,/~) starts with the first approximation already found and is carried out taking fixed values of other parameters of the model (s = 0, p = 0, O'n -- const). Then the set obtained is corrected for a value of s appropriate for the fibre volume fraction under consideration. A scatter of the results occurred for s-r 0 and this sets a limitation on the accuracy of the search for optimal values of the parameters. So a condition to end the search is set up, i.e. F _
O'**, the damage is accumulating continuously, the specimen fails at N < 106 cycles and E/Eo ,~ 0.8. The fatigue behaviour of the cross-ply composites is more complicated that reflects a more complicated tensile behaviour of such materials. However, the general description of the behaviour is the same: if the maximum stress does not exceed the stress corresponding to the matrix microcracking start, the fatigue damage is not accumulated; if the matrix cracking takes place, the specimen exhibits a sharp drop after the first cycle followed by a continuing modulus decrease and failure prior to 106 cycles [728]. When testing aluminosilicate-glass-matrix/silicon-carbide-fibre composites at elevated temperature, the effect of atmosphere appears to be pronounced. The oxidizing environment alters the fibre/matrix interface and causes matrix cracks to propagate into the fibres. Obviously such a process is speeding up by matrix microcracking, the cracks occur to be channels for oxygen going to the interface. Temperature of 900 ~ appears to be sufficiently high for the effects to be observed [552]. The fatigue damage initiation and growth mechanism from a circular hole in a plate of this composite was also studied [382]. In the case of a unidirectionally reinforced material, it was found that the shear cracks were initiated at the surface points of the hole where the shear stress concentration reached a maximum, at an average angle of 68 ~ from the loading direction. This took place at early stages of the fatigue loading. There was little or no growth of these cracks during further cycling (up to 106 cycles) if the maximum cycling stress is smaller than a fatigue limit defined by the authors as 80% of the ultimate strength. If the maximum cycling stress is greater than the fatigue limit these cracks grew under a mode II condition. In the case of a cross-ply laminate, the transverse matrix cracks in the 90 ~ plies arise at the beginning of the process. The behaviour of the crack system depends again on whether the stress is lower or higher of the fatigue limit. In the former case the cracks

Ch. VII, w

Concluding remarks

371

reach a saturation level, in the latter one they coalesce together at the interfaces, causing local delamination and transferring the whole load to the 0 ~ plies. This leads eventually to the failure of the specimen by fibre breakage and pull-out of fibres. Comparison of cyclic and monotonic fracture surfaces of SiCw/AI203 composites shows [118] that a crack under monotonic loading propagates in the transgranular mode leaving behind it cleavage steps, whereas under cyclic loading intergranular fracture is predominant. A larger number of pulled-out fibres leads to a suggestion that the effect of cyclic loading may be a progressive weakening of the interface. The dependence of the crack growth rate on the stress intensity range is approximated by the Paris law with an exponent m of 15. The value of the fatigue threshold, measured at a minimum crack rate of 10 -1~ m/cycle is about 60% of K*. For there exists a similarity of the fracture surfaces resulting from cyclic and monotonic loading, fracture modes similar to those under monotonic loading are presumed to be operating. This means that a clear dependence of the crack rate on the maximum stress intensity factor is expected to be essential. So the authors write crack-rate/loading-factors dependence as d c / d N = A (Kmax)n(AK) p

which can be reduced to the familiar Paris law by rewriting it in the form A

d c / d N -- ------------~ (1 - R) ( ~ ) n + p

where R = O'min//O'max. The experiments with Kmax = const and Kmin increasing yield an explicit dependence of d c / d N oc (AK) p with p = 4.8. The small cracks (fig. 7.27) grow at progressively decreasing rates with increasing in size. The growth rate appears to vary significantly which is typical for small-crack growth which is highly sensitive to local stresses and microstructure inhomogeneities.

7.5. Concluding remarks We see that a simplest composite, that is the laminated metal with non-ideal interlamina bonding, under a cycling loading reveals a typical composite behaviour: the non-ideal interface arrests the crack, and on the other hand, the interface emanates new cracks. Some configurations occur to exhibit the fatigue resistance higher than that inherent to the monolithic metal. The same is true for the perforation of a laminated plate by a projectile. This is because the perforation resistance is also determined by interaction of the cracks that go ahead of the projectile, with the interfaces. Incorporating rigid fibres into the lamina provides new possibilities to shift a balance between the arrested and generated cracks to larger values of the applied

372


Ch. VII, w

stress due to both the occurrence of new interfaces and lowering the stress in the matrix. Both arguments are also valid when the composite does not contain interfaces within the matrix. This is true for ceramic matrix composites as well in which case the properties of the fibre/matrix interface are of a special importance. A direct influence of the fatigue loading conditions on the interface, or the influence of the atmosphere through the fatigue microcracks in the matrix can lead to changes in the interface structure followed by a change in the crack resistance. Both metal matrix composite with respect to metals and ceramic matrix composites with respect to ceramics, exhibit enhanced values of the ratio of the maximum stress in a cycle to the ultimate tensile strength. A value of the threshold can be always found, below which no fatigue failure occurs within a reasonable cycle number. At the same time, the exponent in the Paris law for the fatigue crack growth rate is varied between 10 and 50 for composites as compared to 4 for metals. So actually the fatigue events in composites are going over a rather narrow interval of the loading conditions. This means that calculations of the fatigue life of structural elements with non-homogeneous stress fields, such as a turbine blade or fan, can be performed by assuming simple constitutive equations of the material. Mechanical models of the fatigue behaviour of composites either describe the real situation in a phenomenological way and this is a base for classification and interpretation of experimental results, or explain qualitatively interaction of the cracks with the interfaces. The first kind of model can be of a predictive nature as in the case of using them in computer simulation. Perhaps a most important goal of the future research in this field is to combine both types of models in one. This can be done on the basis of a systematic experimental study.

Chapter VIII COMPRESSIVE STRENGTH

Compressive strength is a propriety of the specimen under testing. The ultimate load is determined by a design of either the specimen or structural element. On the other hand, it depends on microstructural features of the material. So we shall analyze the uniaxial compression of a rod and a tube with the longitudinal reinforcement, and the behaviour of a cylindrical shell under external hydrostatic pressure accounting for microstructure of the material. In this chapter, we shall deal with metal-matrix composites only. We shall not discuss a problem of the compression of an infinite medium. Still, it is interesting to note that a paper by Timoshenko published in 1907 [655], although concerned with a single beam in an elastic medium, may be actually considered as the first in a series of investigations of the stability of a regular array of infinite elastic plates embedded in an infinite elastic medium. For example, Rosen [571] obtained a well known solution of the problem by using the Timoshenko's method. All the aspects of the problem were analyzed by Guz (see for example [217, 219]).

8.1. Rods under compression To illustrate the behaviour of metal-matrix composites under the compression we consider first a simple problem of a composite rod of a rectangular cross-section clamped at the ends [436, 437]. 8.1.1. Failure model

The composite consists of elastic fibres and ideally plastic matrix with yield stress 0 Such a rod may have a number of stable configurations, two of them are shown o-m" 0 the in fig. 8.1. F o r configuration (a) at sufficiently large average stress, o- >> o-m, elastic energy per unit area of the cross-section can be written as U1 ~

o-2 2Efvf

L

(8.1)

where L is the rod length.

373

Compressive strength

374

Ch. VIII, w

~q

/

(a)

(b)

Fig. 8.1. Two possible configurations of a composite rod under compression. (After Mileiko and Khvostunkov [436].)

The energy absorbed by the specimen in configuration (b), with two regions of plastic shear 7 - ~o in the matrix, will be

U2 ~ %~OVmg.

(8.2)

If the transition from configuration (a) to (b) takes place without a change in the distance between the specimen ends, then tr

L

Efvf 4 sin2 (tp/2) Assuming UI - U2 and the angle, q~ be sufficiently small, we obtain the critical angle 0

q~* - 2 ~ Vm. o"

(8.3)

It follows that for each value of a, in addition to the main equilibrium state there exists a 'kink' state with the kink angle qg* which determines shear strain in the matrix within the shear band zone ('kink'). However, to jump to the 'kink' state, the rod has to overcome a potential barrier. This will be possible if the rod has an initial imperfection. Suppose the equation of the central line of the rod before loading is

a0( )

y0=-~ -

1-cos

where a0 is an initial deflection at x - l/2. Neglecting the effect of the shear deformation, we obtain the equation of the neutral line under the load

Ch. VIII, w

Rods under compression

a0 sin2 ~zx. Y - 1 - a/aE L

375 (8.4)

Here ~rE is the Euler stress. Obviously, there exists the stress, ~, < CrE, at which the maximum angle reaches value q~*. At this moment the jump from configuration (a) to (b) may occur without a change in the distance between the rod ends. Differentiating eq. (8.4) yields or,-

(1

1)1

-~ 2 h LO'mVm

(8.5)

for the rod of thickness h. Parameter ao/h is the initial imperfection, its value being mainly determined by irregularities in the fibre system. Hence, this is a technologically determined parameter. Assuming now the value of ao/h being constant for a composite family, eq. (8.5) yields a non-monotonous dependence of the ultimate stress on the value of L/h. This corresponds to a competition between an increase in the Euler stress with decreasing L/h and increase in the shear stress simultaneously. The value of L/h corresponding to a maximum value of ~r, is

(L/h), =

~m

.

(8.6)

Equation (8.6) is obtained neglecting the matrix contribution into the effective secant modulus of the composite, that is taking the Euler stress for an elastic-plastic rod under the compression in the form

C~E-- k

Ervr

(8.7)

where k is a constant depending on the conditions at the rod ends. The fibre volume fraction corresponding to an absolute maximum of the ultimate stress cr, occurs to be constant, v~ - 1/3, and the maximum stress is 4

k

Er

O',max = 9 (gk)2/3 (~ff~_mr)2/3

(8.8)

Note that at large values of L/h the difference between values of ~E and a, is small. Moreover, eq. (8.8) is valid for sufficiently small values of L/h only. Therefore, the model does not predict the value of L/h that corresponds to the change in the failure mode.

8.1.2. Experimental results We shall now present experimental data supporting the failure model, but we start with the description of technique which provides a possibility to measure the stress/ strain curve in compression.


376

Ch. VIII, w

Usually, it is more difficult to perform an experiment to obtain the stress/strain curve in compression than that in tension because of obvious problems of designing an appropriate specimen. The most serious problem is buckling of a specimen loaded in compression. Buckling can be prevented by using sufficiently thick specimens, but this creates a problem of realizing homogeneous stress state in the specimen. It makes researchers to support specimens in some way to prevent them from premature buckling. For instance, a mini-sandwich specimen composed of two composite skins made of a material to be tested and a sufficiently thick core made of a material with significantly lower stiffness was proposed to perform the compression test [643]. In measuring compressive properties of a material, it would be natural to utilize the buckling phenomenon instead of eliminating it. So a Rabotnov's idea, that is to measure the tangent modulus, Et = d~/de, of a material as a function of the stress, by testing specimens of various values of the slenderness and then to obtain ~(c) curve by integrating the experimental dependence, was realized by Mileiko and Khvostunkov [436]. Because for some applications, we need the tangent modulus rather than the stress/strain curve and, secondly, the integrating of experimental data is preferable as compared to differentiating them, the procedure developed in [436] seems to be advantageous. In the elastic-plastic region, a rod of length l loaded by the compressive stress, o, is to buckle when the stress reaches the value [560] o,-

:n:i) 2 -~ Et

(8.9)

where i is a smallest value of the radius of inertia of the rod cross-section, the value of v depends on the end conditions, and tangent modulus Et corresponds to a point of the stress/strain curve with ordinate 0,. To prove the idea, the elastic modulus of an aluminium alloy was first determined by compressing long rods with well prepared ends between parallel plates of a testing machine. Assuming the rigid clamping conditions, v = 0.5, the value of Young's modulus equal to 70 GPa was obtained. A specially designed griping fixture was used to test shorter specimen including those of composites with hard fibres. The grip allows precise alignment of a specimen of the rectangular crosssection. For a particular grip used in [436], it was found that the specimen ends might be assumed to be clamped if the rod length is multiplied by a constant equal to 1.04.

We illustrate the method by plotting buckling stress versus specimen slenderness (fig. 8.2) and tangent modulus versus buckling stress (fig. 8.3) for boron/aluminium composites together with corresponding stress/strain curves. The results of the testing of steel/aluminium composites are shown in fig. 8.4. Note, that the stress values obtained are considerably larger than those usually being obtained in tension. When values of L/h are sufficiently small, specimens buckle with shear band occurrence as shown in fig. 8.5. The shear band is obviously formed in a jump

Rods under compression

Ch. VIII, w 3.0

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g

0.04

Fig. 8.2. The behaviour of a boron/aluminium composite, vf = 0.347. The matrix is 2024-T6 alloy. (a) Critical compressive stress versus specimen slenderness for boron/aluminium rods. (b) The stress/strain curve.

fashion. A specimen is unloaded during the shear band formation and the unloading time is about 30 ~ts [437]. The test data and the results of calculation according to eq. (8.5) are compared in fig. 8.4. The experimental data can be fitted into the theoretical prediction by choosing a proper combination of the matrix yield stress and initial imperfection value. Doing so, it is necessary to take into account that the values of shear strain in the shear band may be as large as about 1% (see eq. (8.3)). Also it should be noted that the value of ~0, increases with the fibre volume fraction decreasing. Therefore, 0 should be larger than that evaluated on a small plastic an effective value of o-m 0 _ 70 M P a is a good estimate for the effective yielding basis, say 0.2% . Thus, o-m yield stress of the matrix for the composites under consideration. The value of the

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Fig. 8.3. The behaviour of boron/aluminium composite under compression, V f - " 0.0476. The matrix is D16M alloy. (a) The tangent modulus versus critical compressive strength of a composite rod. (b) The stress/strain curve. (Experimental data after Mileiko et al. [451].)


378 I000

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Fig. 8.4. Critical compressive stress versus specimen slenderness for steel-wire/aluminium-matrix rods and stress/strain curve of the material. The matrix is pure aluminium. Open points stand for specimens buckled according to the Euler form, black points are for those buckled with the shear bands. The lines present dependencies calculated according to eq. (8.5), Er = 2 0 0 G P a . F o r vf = 0.440: Calculated dependencies correspond to a 0m = 7 0 M P a ; ao/h= 0.1,0.2, and 0.3 (from top to bottom). F o r 0 = 5 0 M P a , dashed lines correspond to a 0m = 70 MPa; ao/h vf = 0.305" solid lines correspond to a m = 0.1,0.2, and 0.3 (from top to bottom). F o r vf = 0.095: a 0m = 70 MPa; ao/h = 0.15, 0.3, and 0.45 (from top to bottom). (Experimental data are after Mileiko and K h v o s t u n k o v [436].)

initial imperfection being really of a probabilistic nature and so inducing a scatter of the data at low values of the slenderness, should be equal to about 0.2 to fit the experimental data. This is an average value which tends to increase with the fibre volume fraction decreasing. The original experimental data for boron/aluminium composites with various fibre volume fractions are given in fig. 8.6. A large scatter of these data implies variations of the initial imperfection in the specimens. Nevertheless, assuming am0 _ 180 MPa we obtain the average value ofao/h = 0.1. Also, these data show that the increasing of the fibre volume fraction from 27% to 41% does not yield a significant increase in values of the ultimate critical stress. This is in qualitative

Ch. VIII, w

Rods' under compression

379

Fig. 8.5. Three sequential photographs of a steel/aluminium rod under compression taken by a camera making 2500 shots per second. The exposure time of a shot is about 200 ~ts. The shear band is obviously being shaped during the second shot. (After Mileiko and Khvostunkov [437].)

a c c o r d a n c e with the b e h a v i o u r o f the m o d e l considered, which is characterized by a m a x i m u m critical stress at v f - 1/3. T h e m o d e l predicts an influence of the yield stress of the m a t r i x on the m a x i m u m stress as a, o~ (a0)2/3. This prediction is also confirmed by e x p e r i m e n t a l observations. We show just one e x a m p l e by presenting fig. 8.7. The ageing of the m a t r i x enhances the yield stress by a b o u t 30% . Hence, the m a x i m u m stress increases by a p p r o x i m a t e l y 10%.


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Fig. 8.6. Critical compressive stress versus specimen slenderness for a boron/aluminium composite. The specimens buckled according to the shear band mode mainly. The matrix is D 16M alloy. (After Mileiko et al. [451].)

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Fig. 8.7. Maximum critical stress of boron/aluminium specimens with 2024 alloy as the matrix versus fibre volume fraction. (After Mileiko et al. [451].)

Ch. VIII, w

Tubes

381

Fig. 8.8. Photograph of a boron/aluminium tubes failed under the compressive loads. 8.2. Tubes

There have been published theoretical evaluations of the ultimate loads for failure modes shown in fig. 8.8 [147, 335, 543]. We shall present the main results of these studies. 8.2.1. An idealized model

A simple model of the tube failure that includes the splitting of the tube wall into a number of strips and buckling of the strips was suggested in [543]. The authors


382

Ch. VIII, w

considered the energy balance for the processes involved in the failure of an idealized tube. The main assumption to idealize a real structure is that of the homogeneity of the tube. The energy accumulated in the tube loaded by compressive stress a is 0-2 Uo - ~ x 2rtRoLh

(S. 10)

where Ez is the longitudinal Young's modulus, R0, L, and h are the mean tube radius, tube length, and wall thickness, respectively. Denoting the Euler stress for a strip via aE, we write the maximum energy resulting from the strips compression:

4

U, - - ~ z

x nFL

(S.ll)

where n is the number of strips, F is the cross-section area of the strip (see fig. 8.9), F = 2o~Roh, o~ = rr/n, o~ A0

(8.21)

where

4x2I Ao-

Sl

"

Because we shall apply the relationship given by eq. (8.21) for A x/~ the strip length increases in a stable manner until the stress reaches the value o ' , - O'E(10)- 2 127ffh. N o w consider a case when the strip is formed by non-through cracks. Let us introduce dimensionless parameter of continuity, K, such that x = 0 corresponds to the case just considered, that is a case of the normal crack, and x = 1 corresponds to the absence of a crack. We shall call a defect as quasi-crack if 0 < K < 0. If the energy absorbed by new surfaces which stands in eq. (8.24), is replaced by value 4/htcT, then the energy balance yields

,

{7 E =

4rc2EI 18hs 7E SI----T- + .

(8.28)

Equation (8.28) replaces eq. (8.20). Substituting eq. (8.28) into eq. (8.26) we obtain the condition for a strip to be stable, that is


388

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I /32~cE 1 - tc V T ~ "

Ch. VIII, w

(8.29)

In this case, eq. (8.27) remains valid and the Euler stress, aE, is still defined by eq. (8.20). The behaviour of the strips in a tube is described by similar equations. They are not written down here because of the complicity, but they shall actually be used further on.

Computer simulation We assume that a tube with longitudinal reinforcement carries a system of quasicracks. They may be originated during both the fabrication process and service, the latter being normally accompanied with thermal cycling, corrosion of various types and so on. Delamination on the fibre/matrix interface, a chain of cracked fibres, irregularities in fibre packing, etc. are possible real shapes of the quasi-cracks. An example is shown in fig. 8.14, one can see chains of the longitudinal cracks in fibres, their surfaces coincide with radial planes of the tube. The length of such quasi-crack can be large.

Fig. 8.14. The cross-section of a boron-aluminium tube. The chains of longitudinal fibre cracks can be seen. (After Kovalenko et al. [335].)

Tubes

Ch. VIII, w

389

A batch of the tubes is characterized by a set of deterministic parameters, those being the tube length, L, wall thickness, h, tube radius, R, material longitudinal Young's modulus, E, and effective surface energy, 7- It is also convenient to introduce an ultimate value of the compressive strength of the material, #. The second set of parameters is of a probabilistic nature, those being the number of the quasi-cracks, N, the value of the parameter of continuity of the jth quasi-crack, tq, the length of the jth quasi-crack, lj, and the coordinates of its centre, zj and q)j. We may assume

(8.30)

(/91 < (/91 < "'" < q)N-1 < QgN"

The distribution function for N is assumed to be binomial, so the corresponding distribution density is (Nmax - Nmin)! ( . ( N ) - Nmin ) N-Nmin ( Nmax --: Nmin/ z ( N ) - ( N - Nmin)! (Nmax - N)! ~kNmax - Nmin

Nmax-N

(8.31) where Nmin, Nmax and (N) are the minimum, maximum, and average number of the strips in a tube. The standard deviation is

DN-- (N) Nmax-{~,N;.

(8.32)

Nmax - Nmin

The angle coordinate of the jth quasi-crack is taken in the form 2~

qgj - -~-(j + ~fl~o)

(8.33)

where N is a number realized in a particular computer experiment, parameter ~ takes a random value between -0.5 and 0.5 with equal probability, fl~0 characterizes the dispersion. To satisfy eq. (8.30) we should have 0 _< fl~0 < 1. The length lj is assumed to follow the Weibull distribution with the distribution density written as

f (x) where c~-

- (X~lXfll-1 e x p ( - ~ ~')

((

F 1+~

(x)

(8.34)

x>O,

/~l>o

and fll determines the standard deviation, that is


390

Ch. VIII, w

-1.

The distribution of the quasi-crack centres is assumed being of type given by eq. (8.34). If in a process of the computer simulation there occurs lj > L or/and zj E ( l j / 2 , L - / j / 2 ) , then the redetermination of lj or/and zj, lj is carried out. The parameter of continuity, •, being statistical by its nature, is assumed constant in the examples to be presented below. We are finishing with the designing of a statistical composite tube by the assumption that a pair of neighbouring quasi-cracks forms a strip that can buckle in the Euler fashion under the load and then grow in the Griffith fashion. The computer simulation process starts with the determination of a particular tube specimen to be tested in a computer experiment. As a result of the realization of all statistical parameters, the statistical tube transforms into a deterministic specimen. Now the external loads,/~, are calculated to correspond to the Euler loads for the strips. Then load Pk is applied to the specimen (Pk is a minimum value of Pj) and the Griffith condition is checked for the kth strip. If the strip may grow, then its new length, lk, is found. This changes the system configuration, so new lengths of the neighbour strips, /k-I and lk+l, are determined, new stress distribution in the strips is calculated, and a new set of values/~ is determined. If the stress in a strip reaches the ultimate value, ~, in this process, then the stress in this strip does not increase any more. Strips which are buckling in the Euler fashion may grow if the corresponding condition is fulfilled and the stresses in these strips may not increase. The loading of the tube continues until all the strips reach either the Euler load or the value of ~. The corresponding value of the external load is assumed to provide the tube strength.

Dependencies of the tube strength on tube parameters We have introduced a rather large number of the tube parameters in the model. Under some conditions, one combination of the parameters can prevail, under other conditions another one can dominate. In what follows we shall just point out some characteristic dependencies only. The computer experiments were conducted with the tubes of length L = 1000 mm, radius R = 30mm, wall thickness h = l mm, longitudinal Young's modulus E = 240 GPa, ultimate stress for the material ~ = 1.5 GPa. These material properties are characteristic for boron/aluminium composites. Each particular set of the statistical parameters was used to generate about 100 specimens to be tested. First, the dependence of the tube critical load, P*, on the number of the strips, N, was obtained. An example of the dependence is shown in fig. 8.15 for a particular statistical value of N. In fig. 8.16, the critical load is plotted versus the average value of N realized in the experiments with various statistical values of N.

Ch. VIII, w ~O0

391

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Fig. 8.15. The ultimate load of a tube versus the number of strips realized in a particular computer experiment. ~z/2, then it is said that the liquid does not wet the solid, and the liquid column does not rise in a capillary tube (see fig. 9.3b). If 0 < g/2, then the liquid wets solid and the liquid column does rise in a capillary tube (see fig. 9.3c). Equation (9.7) gives the height of the liquid column rise for r ~ 0: h -- 27Lv COS 0

(9.9)

pgr

where r is the radius of a capillary tube. To make a liquid overcome the surface tension in the situation illustrated by fig. 9.3b and enter into a capillary tube it is necessary to apply a pressure to the liquid. Bearing in mind an application of the result to estimating the necessary pressure, p, in liquid phase schemes of composite fabrication we rewrite eq. (9.9) in the form

VAPOUR~ - - ~ 0 ~'sv

LIQUID

SOLID

h

////////////

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(c)

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426

Interfaces and wetting

p = SLSTLVCOS0

Ch. IX, w (9.10)

where SLS is the interface area per unit volume of the liquid. Equation (9.10) gives a lower bound for the infiltration pressure because irreversible energy losses exist in a real process [475]. To measure a value of the angle of contact, a sessile drop method is often used, for which the experimental scheme is obvious (cf. fig. 9.3). In fact, the determination of the value of 0 is now being done assuming a particular shape of the drop and then connecting the linear dimensions measured on a photograph of the drop to the angle to be obtained [40]. Since the wettability depends on both the solid surface state and atmosphere, experimental data on the contact angle can occur to be unsuitable to predict whether a particular set of the infiltration conditions provides wetting of a preform filled by fibres or not. The next question is what value of the external pressure is to be applied to overcome negative capillary effect (fig. 9.3b). Again, because of the reason mentioned above, the answer formulated as a result of the sessile drop experiment can be misleading. Therefore, it seems reasonable to measure wettability in an experiment simulating the infiltration process. An example of such an experiment was given by Oh et al. [511] who infiltrated the packed bed of a ceramic powder held in a controllable atmosphere by a molten metal. They measured the infiltration distance versus the applied pressure (fig. 9.4) and found the threshold pressure, Pth, for the initiation of the infiltration may not be necessarily equal to the value of p given by eq. (9.10). In addition to the reason mentioned, the wettability can be improved because oxide layer on the molten aluminium can be washed away during the infiltration [67].

g

APPLIED

IJRi~'SS"URE

Fig. 9.4. Schematic of the plot obtained from the infiltration experiment according to Oh et al. [511].

Ch. IX, w

Surface energy and wetting

427

9.3.2. Some features of wetting The above consideration of the solid-liquid interface was of a thermodynamical nature. But the wetting kinetics can be quite complicated for particular pairs of the substances. It includes chemical interaction at the interface and formation of a new phase, mutual solution of elements both in the solid and in the liquid, solution of elements from the vapour in the liquid (if the process is not going on in vacuum), absorption on the solid surface of trace elements contained in the liquid, and so on. These factors make the use of a simple table for the values of surface energy quite difficult. Nevertheless we present such a table (see Table 9.1) which can be considered as preliminary information only. It could also be useful as a list of references for more complete information on experimental conditions as well as on features of the wetting kinetics. When planning to fabricate a composite by liquid infiltration method, the data in Table 9.1 and the following discussion provide a guidance in choosing a scheme of the infiltration. If 0 is sufficiently small, the pressure can be not necessary to promote the process. However, if 0 is close to 90 ~ or more than 90 ~ then one needs to use a pressure infiltration or squeeze casting method to ensure obtaining a poreless material. One of the many factors which influence wetting is surface roughness and chemical inhomogeneity. Wetting of a rough surface is determined by the local angles of contact which can differ markedly from the apparent angle. When surface roughness increases, wetting as a rule gets worse. To illustrate the situation we note that the value of dO/d(Ra/2a), where Ra and 2a are the values of amplitude and wavelength of the roughness, for the pair Cu-HfC, for example, is equal to 103 ~ at 1200~ The value of the true angle of contact is equal to 128 ~ under these conditions [252]. Ultrasonic agitation seems to improve wetting in this case. The contact angle in the A1/A1203 couple is shown to be sensitive to oxygen content at the liquid/solid interface. In the absence of the oxygen 0 - 90 ~ at 700~ the oxygen contamination of the interface zone rises this value to 105 ~ [681]. Kinetics of wetting a solid with molten aluminium is influenced by the behaviour of alumina layer enveloping a droplet of aluminium and preventing the formation of a true liquid-metal/solid-ceramic interface [168]. The thickness, stability and breaking-up of the layer depends on the temperature, partial pressure of oxygen in the atmosphere and other factors, so non-wetting/wetting transition depends on a particular system and experimental conditions. Under the experimental conditions of [168], this occurs between 900 and 1100~ Generally speaking, the kinetics of wetting is usually described by a curve such as that shown in fig. 9.5. Parameters of this curve depend on temperature and other conditions of the experiment. These can be conditions which provide wetting, i.e. the angle of contact reaches ~z/2 (a case shown in this figure), but there-can also be situations when the value of 0 does not reach 7t/2, and that is a case of non-wetting. The surface energy of both solids and liquids decreases with temperature increases monotonically and relatively slowly. Therefore, no abrupt wetting transition temperature is to be observed. But the measurements of angle of contact for

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TABLE 9.1 Wetting o f some pr~ssihlefibre and tihrc coating materials with some moltcn metals

9

S~

t~

9

L.._,

0

0

e" -t

--.

'-*

Notc.5: (i) somc data glvcn in [307] and 14801 h;~vu bccn tnkcn from thc third works; (illthc & l a taken from 13641arc avcragc ones from a numhcr ofsourccs; ( i i i ) the mi!jority of thc data IS ohta~ncdIn vacuum hut In somc cases ;in incr( arrnosphcrc was uscd


Ch. IX, w

429

several molted metals on various substrates have revealed such a transition temperature [475]. Because the phenomenon depends strongly on the oxygen partial pressure in the atmosphere as well as on alloying elements in the melt, it is also a kinetic feature.

9.3.3. Improvement of wetting Following considerations in Chapter 9.3.2 we see that wetting in a particular system can be improved by various methods, including treatment of the fibre surface, modification of the matrix, ultrasonic agitation, etc. Treatment of fibre surface

Fibre surface coating is a usual way to improve wetting. The data compiled in Table 9.1 provide some guidance to the choice of the coating materials. Coating procedures were described in Section 9.2. Also, the fibre surface energy 7sv can be raised just by changing the chemical nature of the atmosphere before getting into contact with liquid; for example, heat treatment of alumina, silicon carbide, and graphite particles is known to improve their wettability with aluminium melt [475]. It is explained by possible desorption of gaseous species from the reinforcement surface during the heat treatment. Matrix modification

A definite influence on the fibre surface during fabrication process can be achieved by matrix alloy modification. It can be important from the point of view of liquid phase techniques of composite fabrication. To promote wetting it is necessary to decrease the surface energy of a fibre. Therefore we are looking for dopants which would react with a fibre material producing reaction products which would form a thin surface layer (which is wetted by the molten matrix). Because it is difficult to

7V 2

~o

t'

t

Fig. 9.5. Schematic dependence of the angle of contact on time of liquid-solid contact. Wetting is thought to be achieved at time t'.

430


Ch. IX, w

predict the thickness of the layer and the bond at the layer-fibre interface, the procedure needs extensive trial and error. Hence, it is easier to illustrate the situation by a series of examples, a part of which can be obtained in patents. Let us start with a proposal made by Kalnin [291] to promote wetting of carbon fibres by a matrix containing large enough amounts of magnesium. The author has noticed that adding relatively refractory magnesium nitride Mg3N2 to a molten matrix markedly enhances wetting, provided that particle size is less than 2 ~tm. The improvement is such that the infiltration technique (see Section 14) becomes possible and high compressive and shear strength of the composites can be obtained. Certainly the result relies on formation of magnesium carbonitride (MgCxNy) or magnesium cyanamide (MgCN2) on the fibre surface. Doping the matrix by fine particles of magnesium nitride can be done in various ways. It is possible to add the particles into the molten matrix alloy in a concentration of about 0.2 to 25% by weight (preferably 1.0 to 10% ). But it is better to obtain the particles in situ by providing the reaction between the molten magnesium metal with the nitrogen ambient (preferably at a temperature between 800 and 850~ In such a process Mg3N2 particles tend to be very fine. Still another way of getting the necessary modification of the matrix is to add the matrix a metallic nitride capable of reacting with magnesium metal to form magnesium nitride. Such nitrides are silicon nitride, aluminium nitride, titanium nitride, etc. This procedure can be applied to metal alloys based on aluminium, zinc, titanium, chromium, and so on, containing at least 10% magnesium, according to the estimate Kalnin [291]. A further example is the modification of an aluminium matrix by alloying it with lithium to promote wetting of polycrystalline alumina fibres by an aluminium alloy matrix [71]. The resulting alloy can be infiltrated into a fibre bundle. X-ray diffraction studies of fibres extracted from a composite show the presence of small quantities of LiAIO2 on the fibre surface. A minimum concentration of lithium in the matrix and preferred values of technological parameters have been found by the authors after mechanical testing of composites obtained in this way. They found that, when the lithium concentration goes above approximately 3.5-4.0% by weight, the tensile strength of the composites goes down in a nearly linear fashion with the lithium concentration. Increasing the contact time with the molten alloy leads to increasing amounts of LiAIO2 on the fibre surface, but the tensile strength of the composite with fibre volume fraction equal to 60% appears to decrease by only 20% in the contact time interval from 1 to 10 minutes. (Lithium concentration in these tests was 3.3% , and melt temperature was 700~ The same is found in [523], where evidences of the existence of the LiA1508 crystals in the interaction zone were presented by using the electron diffraction technique. The kinetics of the interface formation in AlzO3/A1 + Li composites was studied by Hall and Barrailler [222]. They found that after the liquid phase fabrication of a composite with 35% fibres in the A1-25%Li matrix, the interface zone (~-LiA102 + LiALsO8) with a thickness of 50-200 nm arose. Certainly the zone was formed according to the following reactions

Ch. IX, w

431


6Li + A1203 -- 3Li20 + 2A1 Li20 + A1203 -- 2LiA102 LiA102 + 2A1203 - LiA1508 At least the first two reactions were going, because the corresponding changes of the free energy of the system are 21 kcal/mol, for the first one, and 12 kcal/mol, for the second. Lithium is also known to dope aluminium and magnesium to promote their wetting silicon carbide [475]. The influence of alloying aluminium with silicon on wetting behaviour in the SiC/A1 couple is reviewed and checked experimentally by Ferro and Derby [168]. It was found that at 1000~ value of 0 decreased from about 70 ~ to 35 ~ and at 1100~ from 55 ~ to 25 ~ when 12 to 16% silicon was added to aluminium. Generally, a strong correlation is found to exist between the free energy of oxide formation for the alloying element and its effect on wettability [512]. The larger the negative value of free energy formation, the smaller the contact angle. This is demonstrated in fig. 9.6 that presents dependencies of threshold infiltration pressure for a powder bed of SiC and B4C of aluminium-based alloys on the content of alloying elements. The corresponding values of free energy of Cu20, SiO2, and MgO are approximately -200, -700, - 9 5 0 kJ per mol O2 at 800~ respectively [512]. One can see that addition of magnesium decreases essentially the infiltration pressure. Another possibility for matrix modification is to dope the matrix by elements which are absorbed by the fibre surface, thus decreasing the fibre surface energy. To evaluate the kinetics of wetting in this case, it is also necessary to conduct a special experiment. The angle of contact depends on time as shown schematically in fig. 9.5. Time t' should decrease with increasing temperature. BOO

L

600

'

l

\

\

400

,

,

i

I

1000

,

B4C

\

.

0.

~

~

\ \ ..~

,

"

ooooo c'~,

800 a~

eeoee Cu, Ar ooooo M g , Ar ~ ..... Mu, A i r nAnLxn S~, A i r

~ --

600 SiC Cu, ooooo Cu, ooooo Mg, . . . . . Mg, nnnz~zx Si, aaaam Si,

ooooo

400

200

O0

i

I

1

i

I

2

M /

i

I

3 pcg

i

I

4

i

5

200

0

i

i

1

i

Air Ar Air Ar Air Ar t

2 M

i

/

I

3 pet

i

I

4

i

5

Fig. 9.6. Threshold pressure, Pth, for molten aluminium infiltration of SiC and B4C particulates in Ar and air atmosphere at 800~ versus the alloying elements mass content, M. Experimental data after Oh et al. [512].

432


Ch. IX, w

A review of experimental data on such matrix modification for the case of

graphite-fibre/aluminium-matrix composites has been presented in [307]. These data show that the value of 0 becomes less than ~z/2 at temperatures between 800 and 925~ (when aluminium starts to wet graphite fibres) if the matrix metal is doped by about 1% of such elements as Ga, Cr, Ni, Co [489]. The same occurs at temperatures between 900 and 1120~ if doping elements are V, Nb, Ti, Zr. Aluminium does not wet graphite at temperatures of at least 925~ if the doping elements are Si, Mg, Fe. Wetting of vitreous carbon and sapphire with molten copper is improved by doping the copper with titanium [622], and still better results can be obtained if titanium and tin are employed simultaneously (although tin itself does not improve wetting). Copper with 10% Ti starts to wet vitreous carbon at 1150~ and copper with 1.2% Ti and 2.8% Sn also wets vitreous carbon at the same temperature. To wet sapphire at 1150~ it is necessary to add 8% Ti or 3.5% Ti and 11.8% Sn. The role of interfacially active substances in wetting of sapphire by nickel has been investigated in a number of works. Kurkjian and Kingery [347] measured sapphire-nickel interface energy 7es when nickel is doped by such elements as Sn, Zn, Cr and Ta. They discovered that tin and zinc do not influence the value of Yes up to about 2 or 3 weight%. Doping by chromium leads to a decrease in the value of ~'es, starting with a chromium content of about 0.1%. At 10% chromium content ?LS decreases by about 1/3. But the most effective way to decrease ?LS is to dope copper with titanium. A titanium content of about 0.1% decreases 7LS by one half of the original value. This result was confirmed by Allen and Kingery [12], who also demonstrated the possibility of wetting sapphire by nickel alloys containing more than 1% of titanium. Sutton and Feingold [637] studied in detail nickel matrix modification with relation to wetting of sapphire filaments. They showed that doping nickel with chromium and zirconium increases the angle of contact at 1500~ while titanium doping decreases the value of 0 down to 95 ~ at 1500~ These experiments were carried out in vacuum. Sometimes it seems to be useful to dope a matrix material with two elements. The wetting of sapphire by copper doped by titanium is greatly improved if indium is added, slightly improved by adding aluminium, and is not improved by additional doping with nickel and gallium [491]. Good wetting of AIzO3/CaO fibres of the eutectic composition with A1 + 12wt%Si was observed at 700~ as a contrast to the non-wetting behaviour of that of fibre and pure aluminium [633]. The wetting is possibly promoted by large (about 30 wt%) accumulation of silicon in the interface regions.

9.4. Interface properties We know (Chapters 4 mechanical behaviour of composite philosophy and coming back to the subject

and 5) that the interface characteristics determine composites. This is a very important point of the practice. That is why we discuss it now, and will be further on.

Ch. IX, w

Interface properties

433

9.4.1. Bond strength As was mentioned above, measurements of the values of the angle of contact and/ or the interface energy are to be conducted, not only to estimate the feasibility of liquid phase fabrication methods, but also to attempt to evaluate the interfacial bond strength, which is sometimes called an adhesion strength. In the latter case, the interfacial energy is assumed to have no strong temperature dependence. However, such an approach can lead to an upper limit of the bond strength similar to the results of calculation of the ideal strength of a defect-free solid. A real stress state in the vicinity of an interface is perturbed by defects of the interface and it determines a real limiting load of an element containing the interface. The evaluation of the bond energy is very simple. If the interface energy is 7LS and the values of surface energy of the two phases are 7L and 7s, then the work to fracture the interface is WLS = (~'L-Jr- ~S) -- ~LS"

(9.11)

Introducing the angle of contact according to eq. (9.8) we obtain WLS = ?L(1 + COS0).

(9.12)

Now experimental data for the values of 7L and 0 (see, for example, Table 9.1) enable us to estimate the work to fracture of some particular interfaces. Note that besides the already-mentioned reasons for the difference between real values of the interface strength and those given by the values of WLS, there are some other reasons which will be discussed in Section 11.3. However, WLS is, in fact, the work of adhesion and the critical energy release rate for the interface is determined by a mode of debonding. Evans and Dalgleish [160] presented an analysis of those effects for the case of metal/ceramic interfaces. They have shown a reality of a debonding mode for which plastic dissipation in the metal layer is essential. In addition to the dissipation within a close vicinity of the interface crack tip, the periodic microcracking of a brittle layer can enhance the output similar to that occurred in the fracture process zone of a brittle-fibre/ductile-matrix composite (Section 5.4). On the other hand, there can be observed fracture surfaces of the former interfaces of a brittle type: they reveal neither metal attached to ceramics nor ceramics attached to metal. In such a case, the critical energy release rate for the interface can also be different from Wes because the former is again determined by features of the stress field ahead of the crack and the energy dissipation ways. Hence, mechanical tests to obtain real values of the interface strength and effective surface energy is necessary.

9.4.2. Engineering interface strength The interface strength characteristics can be measured by testing (i) plane model interfaces, (ii) composite model specimens, (iii) real specimens. The latter is done by


434

Ch. IX, w

using an appropriate mechanical model of a composite behaviour. Examples of such an approach to the assessment of the shear strength of the interface in metal-matrix composites were presented in Section 5.3.2. In the case of a brittle-matrix composite, macroscopic values such as the matrix cracking offset, the average distance between matrix cracks in a saturated state, etc. depend on the interface shear stress. These dependencies, e.g. that given by eq. (4.114), can be used to evaluate the interface shear strength. In this section, we shall give a description of special experimental schemes developed to assess the interface properties. Plane models

Two main problems arise in measuring interface properties on plane models: one has to be sure that strength and fracture toughness characteristics of the plane model are the same as in a composite and, secondly, interpreting the experimental data obtained needs to be based on known stress state at the fracture site and an appropriate fracture criterion should be used. At least four experimental schemes to measure the strength properties of the plane model have been described in literature. A modified Hertzian indentation test (fig. 9.7a) and a mixed mode flexure test (fig. 9.7b) were developed by Evans and co-authors (see [162]), a double-cantilever beam (fig. 9.7c) was used by Gupta et al. [213], a laser spallation scheme (fig. 9.7d) was introduced by Cornie et al. [ 101]. ~

~

Spherical indentor Cone-shaped crack Interface crack

////////////////////)

~//////////////~ Notch

In terf a ce ( a) Interface

Substrate Coating...~ Glue ~ CoatingJ#r S u bs tra tef l

~Q ] _~

Gold f i l m f~

L

4q

(c) --

Coating

Laser b e a m ~ _ ~

(d)

Quartz f u s e d plate/: a [Substrate

Fig. 9.7. Schematics of the plane models to evaluate debonding characteristics: (a) modified Hertzian indentation test [162], (b) mixed mode flexure test [162], (c) double-cantilever beam [213], (d) laser spallation [ 101].

Ch. IX, w


435

The modified Hertzian indentation test is to be carefully calibrated. It is based [119] on the solution of the Hertzian problem formulated for a non-homogeneous fracturing solid. But first, it is noted that in the case of homogeneous solid, the crack under the spherical indentor, after a small path normal to the surface, deviates into a trajectory having constant angle,/~, such that

ETR3/Q 2 = ~ ( v ) c o s fl(v) where r is the cone crack radius, 7 the effective surface energy, v the Poisson's ratio, and ~ a coefficient. The appearance of an interface changes the crack configuration and the corresponding stress analysis is performed numerically. An iterative procedure seeks for such a cone angle which gives K I I = 0. Then a possibility for crack to propagate along the interface in a stable manner is admitted in the calculation that gives a dependence between the critical energy release rate for the interface, applied load, cone angle, and the crack radius, R.

The double-cantilever beam experiments [213] are interpreted by using simple beam theory and the Griffith criterion that yields critical energy release rate as

G = Q2L2/EIb where I is the moment of inertia, b the width, and E the elasticity modulus of the substrate.

The laser spallation scheme [101] gives the tensile strength of the interface. The collimated laser pulse impinges on thin gold film which is located between the substrate and a confining quartz plate transparent for the laser beam. On absorption of the laser energy, the gold film expands generating a compressive wave. The wave, after reflection from the free surface of the coating, goes back as a tensile pulse. If the amplitude of the pulse is sufficiently high, the coating debonds from the substrate. Calibration of the experimental setup by comparing results of numerical calculations of both the process of conversion of the light pulse into a pressure pulse and wave propagation with measurements of transient pressures with a circuit of high time resolution allow to obtain a local strength of the substrate/coating interface. Pull-out and push-out tests Pull-out and push-out tests are really the same experiments when a single fibre embedded in the matrix is loaded along its axis (fig. 9.8). Pull-out test is widely used for an analysis of the behaviour of the interface in polymer matrix composites [539, 663]. The push-out test is conducted usually on a thin slice of the composite cut normally to the fibre direction. Such a test is more suitable for sufficiently rigid matrices which is a case of ceramic and metal matrices. In the case of ductile behaviour of the interface (a metal-matrix composite without brittle reaction zone), the dependence of the fibre stress to pull the fibre out

436

Interfaces a n d w e t t i n g

I

tttttttttttt

_ -

Ch. IX, w

tttttttttttt

PUZ% - 0 LIT P U S H - 0 LIT % _-] Debonded

,. '-"

V/////////A

V [Debondingstarts %/y

Displacement

Displacement

Fig. 9.8. Pull-out and push-out tests.

upon the length of the embedded part of the fibre is linear up to the ultimate stress of the fibre, a~ (see Section 3.5.3). When the interface can debond, the behaviour of the system is determined by friction, the friction stress, :i, being determined by interfacial radial stress ar, that is 75i = --]~ar =

--fl(aT-~- 6 v )

(9.13)

where p is the friction coefficient, aT and av are the residual radial stress due to a difference in thermal expansion coefficients of the components and the stress due to Poisson's effect, respectively. The latter contribution is negative for pull-out and positive for push-out.

The fibre push-out test suggested by Marshall about two decades ago [384] is now a most valuable method to evaluate fibre/matrix interface shear strength characteristics. A simple stress analysis of the push-out scheme (as well as pull-out) can be based on the shear lag approach as was discussed with regard to the analysis of the stress/strain state at the vicinity of the fibre end (Section 3.6). In particular, the push-out scheme when just interface friction resists to pushing fibre out yields [597] the axial fibre stress as az-~

( a T + k p ) exp - - ~

rf /

--aT

(9.14)


Ch. IX, {}9.4

437

where k

~_

Em l~f Ef 1 + Vm

and Ef >> Em. For a given applied stress p, there is a finite length z = l at which ~rz = 0, so eq. (9.14) is valid only if the embedded length L > l. When the length l reaches L, the applied stress reaches a maximum:

~zr~aT(

Pmax - - ~

exp

(21*kL'] ) - 1 . \rf/

(9.15)

Equation (9.15) can be used in the interpretation of the push-out experiments. The shear-lag analysis yields also dependence of the displacement of the fibre on the applied load as ) u - -rf~ ( ~p -- fl ~ T ln ( l P+ )k --~T

(9.16)

where ~ and fl are known constants. Push-out and pull-out test schemes have been analyzed in a more precise way many times, recent results are given by Desarmot and Favre [123] and Hsueh [259, 260, 261,262]. The real push-out experiments are based normally on using a conventional microhardness indentor, but testing can also be instrumented with a variety of physical tools providing identification of fibre debonding and sliding events. Acoustic emission and in situ video microscopy [153] is an example of such tools. The instrumentation used by Ma et al. [375] involves optical setup to measure the characteristic fluorescence line shift, A, for Cr +3 ions in sapphire. So it is used to analyze the stress state in a single crystalline sapphire fibre embedded in a TiA1 matrix during push-out experiment. It is known that for the axisymmetrical situation when the fibre axis coincides with the c-axis of sapphire: A = 2I-Iao-r + 1-Ico'z where Ha and Ilz are the measured piezo-spectroscopic coefficients for a and c directions, respectively, and ~rr and az the stress components. The optical device used allows to measure average A over sufficiently small volume so to observe an evolution of the stress state in the fibre during loading. Actually, the authors assume such an initial configuration of the sample under testing as to make stress distributions derived by numerical calculation to be identical to that obtained using the optical method. The former is done by presenting the stress components as superpositions of several independent terms, that is O'z -- pf(1)(~, ~) + aof(2)(~, x) + zof(3)(~, ~),

(9.17)

438

Interfaces and wetting 0 (2) O'r -- --firo -+-pg(1) (~, ~) -+- O'zg (~, ~) + "Cog(3) (~, ~)

Ch. IX, w (9.18)

where p is the applied load, a zo and o-or are the axial and radial residual stresses in the fibre, z0 the frictional shear stress in the d e b o n d interface region, ( = z/rf, ~ = l/rf, rf is the fibre radius, l the length of the d e b o n d crack (fig. 9.9), and f~(-), gg(-) function to be determined. The authors claim that they have used a modification of eqs. (9.17) and (9.18) to account for the coating thickness, which has been in the physical experiment either carbon or m o l y b d e n u m with thickness h ~ 10 gm. Such a kind of the c o m b i n a t i o n of numerical calculations and physical measurements brings an excellent assess to the stress state in the fibre undergone to pushing out. The effective energy of debonding surface can also be obtained [375] after a dependence of the d e b o n d length on the applied load is measured. Assuming the stress balance as 2l Peff -- P - azo - ~ zo rf

(9 19)

where Pelf is the effective load applied on the d e b o n d crack and taking into account the dimensional considerations yield the expression for the elastic energy change via the effective load and the d e b o n d length as 2

Peff nr~l

(9.20)

where fl is a constant. Therefore, the energy release rate is G - 8U rf _2 - ~ - - - fl ~fPeff"

(9.21)

It was taken into account that A = 2nrfl. N o w the effective surface energy of debonding surface is

Debond

crack

,~ I n t e r f a c e

'f

layer

z

Fig. 9.9. A schematic of the push-out experiment.

Ch. IX,{}9.4 flrf(

7d . ~ .


.P

o21)

. az

rf z0

2

9

439 (9.22)

Debonding can take place either on the matrix/fibre-coating or on coating/fibre interface. Fracturing of the coating itself can also be expected. In the experiment, the values of 7d for carbon and molybdenum interlayers in AlzO3/TiA1 composites were determined as 7c - 0 . 0 1 5 J/m 2 7~I~ - 0 . 2 8 J/m 2 In general, comparison between numerical simulation and physical experiment provides an estimation of the parameters involved. For example, Evans and Zok [161] presented the results of computer simulation of the effect of roughness, residual stresses, and friction coefficient on the load-displacement curve in push-out tests. This yields, in particular, the values of bt equal to about 0.1 for either ceramic/ carbon-coating or ceramic/boron-nitride-coating. In the case of oxide coating, ~t ~ 0.5. Chandra and Ananth [75] went along a similar road to evaluate residual stresses in silicon carbide fibre reinforced titanium based intermetallic matrix at various temperatures.


Chapter X DIFFUSION THROUGH FIBRE/MATRIX INTERFACE

If there is no thermodynamic equilibrium between the fibre and the matrix, then physical-chemical interaction between them is inevitable. The consequence will be a structure of the interface region which can contain substances with properties different from those of the component materials. In principle, the situation is quite clear but there exist some technical difficulties in describing all the details. The first difficulty arises because the phase diagrams of complex systems for elements presented in matrix and fibre materials are usually unknown. Secondly, the kinetics of the interaction in complex systems cannot be described exactly without special experiments. This means that it is difficult to evaluate interaction processes in a composite without making specimens and studying them. In fact, almost any matrix modification demands a special study. This explains a large number of publications on this subject. Thirdly, the influence of the interface region on the mechanical properties of a composite can be varied, depending on the type of loading. This can hardly be described systematically at present. With regard to the first difficulty it should be noted that a lot of data has been collected and published (see, for example, [621]). We have to refer to these publications because it seems impossible to present this experimental information in a compact form.

10.1. K i n e t i c s - a simple case

A simple case occurs when in the interfacial zone no new chemical compounds arise. The diffusion kinetics is governed by Fick's second law for the concentration, c, that for planar case and a constant diffusion coefficient D is written as ~C ~2C St = D Ox---5.

(10.1)

The integration of eq. (10.1) for a concentration profile change with time yields [592] c-

C2 q- CI ~

C2 -- CI ~rfI( ~ O l l 2

2

(10.2)

441

442

Diffusion throughfibre~matrix interface

Ch. X, w

where Cl and C2 are the initial concentrations at x > 0 and x < 0, respectively, and is

V(z)

2

/o z exp (_~.2) d~."

It follows from eq. (10.2) that the plane with a constant concentration, c, moves in such a. manner that X

2x/-~

- const.

(10.3)

This is widely used in studying the interface zone growth in composites although Fick's second law for composites with circular fibre should be written in cylindrical coordinates. Hence, if the fibre volume fraction in a composite is small enough for diffusion in the vicinity of one fibre to be unaffected by concentration gradients around other fibres and assuming radial and axial symmetry we have

~c 1~ (rD(c)~rr ~C)

a t = r ar

(10.4)

where the diffusion coefficient, D, depends on the concentration. Equation (10.4) is solved analytically assuming D = const, otherwise a numerical analysis is applied. With respect to the situation in a fibrous composite, such an analysis was done in [244] for Cu-Ni-system. Non-axisymmetry of the problem was neglected and the concentration changes only along definite radii in a periodic fibre array aligned parallel to the fibre axis were considered. The next difficulty was to choose a dependence of the diffusion coefficient on concentration of nickel in copper (because the data published previously had the scatter of one order of magnitude). Nevertheless, the solution appeared to correlate with the experimental results for the Cu-Ni-system. The solution procedure was a finite-difference one with the obvious boundary conditions. Also the auxiliary condition

~ RAc(r)dr

coRf

was to be satisfied. Here RA is the radius of a cylindrical zone of influence associated with a single fibre, co the initial concentration of the diffusing element in the fibre. Note that in the experiments with the Cu-Ni-system, the porosity in a reacted zone of the matrix has been clearly observed which corresponds to the KirkendallFrenkel effect. Of course it has not been taken into account in the calculations. 10.2. K i n e t i c s in the c a s e o f c h e m i c a l r e a c t i o n s

High temperature treatments involved in the fabrication technology of metal- and ceramic-matrix composites as well as high temperature exposure at the service yield chemical reactions between matrix and reinforcement.

Ch. X, w

Kinetics in the case o f chemical reactions

443

10.2.1. A n a n a l y s i s

The kinetics of diffusion and chemical reactions are obviously mutually dependent. An analytical or numerical solution of the corresponding problem, although simple in principle, seems to be too complex because of the absence of much of the necessary data including the diffusion coefficients for reaction products. So the investigations remain to be time-consuming, not too effective but still unavoidable. The data are presented usually [621] in the form of the constants in the equations of a phenomenological nature which are written, accounting for eq. (10.3), as h 2 - Kt

(10.5)

K -- Ko e x p ( - Q / R T )

(10.6)

Here Do and Q are constants. Nevertheless, Caulfield and Tien [70] analyzed the reaction zone (RZ) growth in tungsten-fibre/nickel-alloy-matrix composites by applying the moving boundary equations to the problem and formulating some additional conditions derived from the experimental observations of the behaviour of the same system. The model is depicted in fig. 10.1 with r0 as the initial radius of the fibre. The notations to be used are: ~f-

r 0 -- rf,

~m - - r m - - tO,

(10.7)

h - - ~m -1- ~f.

The system of equations described diffusion in the model is obvious. First, two flux balance conditions at the fibre/RZ and RZ/matrix interfaces:

9. , " .,, :, '

Matrix

,.... . 9. . .

..,. . .... Pf-fibre tion

i

Zone

i

.

.

.

.

.

.

.

.

.

Fig. 10.1. A m o d e l o f the interface z o n e in the a n a l y s i s d u e to C a u l f i e l d a n d T i e n [70].


444

d~f IRZ+ dt c~ Jx~ - ~'w

Ch. X, w

(10.8)

and d~m

dt cx

jRZ_

m+

- Jw 9

(10.9)

Here J is the flux, subscript W relates a flux and, later, concentration to tungsten atoms, superscripts f_ and m+ relate a flux to that occurred in the fibre in the vicinity of the fibre/RZ interface and in the matrix at the RZ/matrix interface, respectively; superscripts RZ+ and RZ_ relate fluxes to those occurred in the reaction zone at the vicinity of fibre/RZ and RZ/matrix interfaces, respectively. So the interfaces move due to the difference in the interdiffusional fluxes across them. By use of Fick's first law and conservation of atoms across the interface, eqs. (10.8) and (10.9) become (CfRZ -- CRZf) d~f _ o f (T) ~ r

- O RZ (r) ~cRZ

,

d Cm - O RZ ~cRZ - - O m ( T ) ~c~ (CRZm -- CmRZ)---~ (T) ~r ~r

(10.10) (10.11)

where CfRZ, CRZf, CRZm, and CmRZ are the equilibrium compositions, and D~(T) the interdiffusional coefficients for each phase involved (~ = f , m or RZ). Two equations obtained are complemented with three equations describing Fick's second law, eq. (10.1). For a particular composite system characterized by a very low solubility of matrix element in the tungsten fibre and a matrix containing tungsten in excess of the solubility of W in the matrix, the authors assumed

~r

Or

=0.

Hence, combining eqs. (10.7), (10.10) and (10.11) yields dh = DRZ(T) ~c Rz CRZm -- r dt

Or (CfRZ -- r

q- r

-- CRZf -- CmRZ)

(10.12)

which means that in this particular case, the reaction zone growth is independent of interdiffusion in the fibre and matrix, but solely dependent on the interdiffusion across the RZ. At this point, the authors introduce two additional conditions taken from the experimental data. They assumed, first, a linear approximation of the compositional profile across the RZ and, second, a parabolic law given by eq. (10.15). This yields

Ch. X, w

DRZ(T) --


1 -~F(ca~)K(T)

445

(10.13)

where F(c~) is a known function of all the equilibrium compositions. Actually, the main result of the study is the determination of the temperature dependence of the coefficient of diffusion of tungsten in a material of the reaction zone, and hence the diffusion constant Do and the activation energy. Further experiments with various metal matrix alloys [653] allowed their interpretation based on the results just described and led to the conclusion that a definite correlation between matrix composition and DRZ(T) exists through the effect of the matrix chemistry on the reaction zone chemistry. It was shown that increases in the Ni contents in the matrix and decreases in the Fe, Co, and Cr contents in the matrix resulted in decreases in the DRZ(T) values and, hence, made the reaction zone grow more slowly.

10.2.2. Experimental data We shall present here just limited information on the subject trying to choose typical examples illustrating the problem. Results of the interaction of the components when one of them is liquid, will be also considered. Effects of the interactions on mechanical properties of components and composites shall be discussed in the next section.

Aluminium-matrix composites

Boron/aluminium composites A

sequence of the events on the interface between boron fibre and pure aluminium matrix is observed in [223] by using transmission electron microscopy, electron diffraction and X-ray microanalysis. An initial structure of the interface obtained as a result of hot pressing in vacuum of a stack of aluminium foils and boron fibres was changing by the heat treatment at temperatures 450 and 500~ for up to 72 h. The study performed gives the following picture of the process. At an early stage of the interaction, between fine oxide particles on the interface, which remain further on to be markers of the initial interface, aluminium intrusions into the fibre appear (fig. 10.2, the top picture) which have the same crystallographic orientation as the mother volume of the matrix. Then (fig. 10.2) the intrusions coalesce pushing, in fact, the interface to the direction of a component which diffuses faster in a given pair. The characteristic size of this move is about 0.1 ~tm. No borides at this stage are observed. At the next stage, the A1B10 borides are revealed near oxide inclusions on the interface (fig. 10.2), their size is about 0.1 l~m. With the heat treatment continuing, the boride inclusions tend to form a continuous interface layer as shown in fig. 10.2. The interface seems to be composed of all known borides, A1B2 (mainly), A1B10, and A1B12. A difference in the interface reaction in the boron-aluminium composites with pure aluminium and an aluminium alloy matrix has been emphasized by Kim

446

Diffusion through fibre~matrix &terface

@

(~

~

Ch. X, w

~-w

|

./@

A1Blo

./

@ (~

Borides

......

Fig. 10.2. Schematic sequence of the events on the boron/aluminium interface according to Hall et al.

et al. [315]. The first composite is that with commercially pure aluminium (1100 alloy) as a matrix. The matrix of the second one is the aluminium-magnesium alloy containing small quantities of silicon, copper and chromium (6061 alloy). Specimens have been investigated after isothermal exposure at temperatures of 350 and 500~ The investigation of a layer remaining on the fibre surface after extraction from the l l00-alloy matrix by SEM-observation and X-ray diffraction shows that separate particles of AIB2 start to appear after exposure at 500~ for 2.5 h. The number of such particles increases with time and after exposure for about 30 h a continuous layer of AIB2 covers the fibre surface. In the case of the aluminium-magnesium alloy, X-ray analysis of the reaction products reveals AIBI2. The particle size reaches about 5 lam after exposure at 500~ for 2.5 h and the surface layer becomes continuous after exposure for about 7 h. The exposure at 350~ for about 14 h corresponds (in terms of appearance of the particles) to that at 500~ for 2.5 h. In both cases only aluminium and boron are present in the final reaction products, but the compositions of the products and the rates of reactions are quite different. The authors note that in the binary AI-B phase diagram both carbides are present, A1B2 being stable up to a relatively low temperature 975~ and A1BI2 being stable up to 2070~ Then the ternary AI-Mg-B system shows, besides the two borides mentioned above, three aluminium-magnesium intermetallics and four magnesium borides. Because of the presence of alloying elements in the matrix the possibility of formation of aluminium and boron oxides as well as complex oxides of A1-Si-O and A1-Mg-O types, and other compounds also exists. A thermodynamic analysis of the stability of the possible compounds gives the following chain of compounds, in order of growing stability at temperature 575~ A1B2- M g B 2 - A1BI2- B N AlN

- MoB 4 - B203

- Al203

- Al2Mg04

- AIzSi05.

To start the analysis of the kinetics, the authors assume that the diffusion of various elements in less stable A1B2 proceeds faster than in A1BI2 at comparable

Ch. X, w


447

temperatures (JAIB2 > JA1BI2), although a real situation can be complicated by defects of the structure, diffusion mechanism and ternary additions. Then, on the basis of the comparison of ionic and atomic radii they place the elements of interest in the following sequence of increasing diffusion coefficient: Mg-A1-Si-B. If one considers these assumptions and the experimental data together, the following conclusions can be produced. In the case of the 1100 aluminium alloy, the growth of the interface layer starts with the formation of A1B2 on the matrix side and A1BI2 on the fibre side. Then, because the diffusional flux of aluminium through A1B2 is larger than that of boron through A1BI2, the A1Bz/A1BI2 interface moves in the direction of the fibre surface. It leads to disappearance of the A1BI2 phase. The whole process can be influenced by the formation and fracture of oxides of aluminium and boron. This decreases the growth rate of the boride layers. To explain the opposite result in the case of the 6061-matrix, the authors have to a s s u m e JA1BI2 ~> JAIB2 because of the influence of ternary additions. In fact, preliminary Auger electron analysis indicates the presence of magnesium and silicon at the extracted fibre surface.

Carbon/aluminium composites Kinetics of the formation of an interface structure in C/A1 composites and its influence on the fibre strength is studied by many authors. Kohara and Muto [320] supplemented the strength versus temperature dependence of both PAN and pitch fibres kept in either pure aluminium or in A1-10% Si alloy at temperatures 680-800~ for up to 10 min with the electron diffraction study of the products appeared on the interface. They found that both types of fibres being originally very different in strength values (about 2500 MPa and 700 MPa for PAN and pitch based fibres, respectively) became about the same (~ 500 MPa) after being extracted from the composites. The electron diffraction patterns of the interface products at the initial stage of their formation were characteristic for polycrystals, at the later stages the products are transformed into single crystals. The A14C3 crystals started to grow into the matrix and then entered the fibre volume turning to be stress concentrators. It is important to note that the results of the experiments in pure aluminium melt and aluminium-silicon alloy melt occurred to be identical. An interesting way of the formation of silicon carbide interface in a C/A1 composite was discussed by Okura [515]. The samples obtained by hot pressing of prepregs, which had been prepared by either plasma spraying or ion-plating of the matrix, were heat-treated at temperatures between 300 and 650~ for time up to 400 h. The heat treatment was performed in a quartz tube sealed under vacuum. A part of the quartz tubes contained a stainless steel shield between the wall of the tube and sample. The Auger analysis of specimens revealed relatively large quantities of silicon on the fibre/matrix interface in specimens heat-treated without shielding from the quartz wall. No silicon was discovered if a specimen was heat-treated in a container with the shield. The author assumed the presence of silicon dioxide in the atmosphere surrounding a sample. The dioxide, SiO2, is reduced by aluminium to oxide, SiO, and then to silicon.

448

Diffusion throughfibre/matrix interface

Ch. X, w

SiC/aluminium composites Liu et al. [372] measured the strength of silicon carbide fibre of Nicalon type heat treated in pure aluminium matrix as well as in A1-1 at % Si and A1-5 at % Si matrices. They found that in a definite time/temperature interval the fibre strength degradation follows eq. (10.27) with Q ~ 180 kJ/mol. The presence of silicon makes the degradation rate decrease. It is suggested that the degradation is due to formation of aluminium carbide according to the reaction: 3SiC + 4A1 ~ A14C3 -k- 3Si. Because the value of Q obtained is fairly higher than the activation energies for silicon diffusion through both aluminium and aluminium carbide, the reaction takes place at the fibre/reaction-zone interface.

Magnesium-matrix composites It is interesting to note that an attempt to use magnesium-lithium alloys as a matrix [388] revealed an intensive chemical reaction of the matrix with Saffil A1203, Nicalon SiC, whisker SiC, and carbon fibres at the stage of the pressure infiltration when making composites. Only CVD S I G M A SiC filaments appear to be stable enough in the infiltration environments. The formation of aluminium magnesium carbide AIzMgC2 on the fibre/matrix interface in C/Mg composites with high modulus fibres M40 (Torayca) and FT700 (Tonen) and AZ61 alloy as a matrix was revealed after annealing at 650~ for 20 h, but this led to a decrease in the mean fibre strength by no more than about 20% [587].

Titanium-matrix composites Boron fibres are completely unstable in titanium matrix [33, 355]. Carbon fibres would be of great interest in this aspect but the problem to overcome the reactivity in this system remains to be solved despite attempts in this direction are known. Therefore, carbide fibres are now considered to be most promising reinforcement for titanium matrix.

Silicon carbide~titanium Interactions at the interface in SiC/Ti composites are severe. They were studied by Dudek et al. [139] by observation of Auger electron spectra in the interaction zone mechanically magnified by cutting a specimen under small angle to the reinforcement direction. In a specimen that had undergone heat treatment at a temperature of 950~ for 1 h, which corresponds to a maximum shear strength of the composite, silicon, carbon, titanium, and some oxygen are revealed. The interaction zone can be reconstructed as shown schematically in fig. 10.3. The process seems to start with the reaction Ti + SiC ---, TixSiy + TiC.

Ch. X, w


449

TiC

Fig. 10.3. Schematic of the interaction zone in a SiC/Ti composite. Heat treatment is 950~ - 1 h, the thickness of the TisSi3 + TiC-zone is about 1 ~tm. Experimental data after Dudek et al. [139]. With the thickness of the TiC-layer, which adheres to the fibre, increasing, the rate of the reaction decreases, free silicon diffuses into the matrix forming a layer of TisSi3. Another silicide (TiSi2) does not occur due to the shortage of silicon; the process is limited by diffusion of silicon through the titanium carbide layer. An intensive chemical interaction of titanium matrix with silicon carbide fibre was a main reason for developing SiC fibres with carbon-rich external layer (SCS-2 and SCS-6, Section 2.2.3). The structure of the interface zone changes as a result of the presence of free carbon on the fibre surface. Jones et al. [288] analyzed carefully the interface structure in 'as-received' specimens of SCS-6/Ti-6A1-4V alloy of (~ +/~) structure, their result is shown schematically in fig. 10.4. The picture is obviously different from that in SiC/Ti, composite presented in fig. 10.3, by a more complicated composition. More and more complicated structure of the reaction zone appears with more sophisticated instruments being used to study it. F o r example, Shyue et al. [603] distinguish six layers in the reaction zone of a SCS-6 SiC-fibre/Ti-15V-3A1-3Cr-3Snmatrix composite obtained by hipping at 982~ for 2 h and then heat-treated. A Tix Siy(C)

a

fl

TiC

SiC

C

SiC

Fig. 10.4. Schematic of the interaction zone in a SCS-6/Ti-6A1-4V-alloycomposite. After experimental data by Jones et al. [288].

450

-Diffusion through fibre~matrix interface

Ch. X, w

continuous titanium carbide layer at the C / T i interface as well as precipitation of titanium carbide and titanium silicides are found in the reaction zone. In the multi-layered reaction zone of a SCS-6/Ti-6AI-3Sn-2Zr-0.2Mo-0.75Nb composite, both TiC and TiSi2 crystals are found. The zone appears to grow as a result of consumption of the fibre and matrix [117]. A total thickness of the reaction zone in the as-received composite (hipping at 870~ for 4 h under 280 MPa) is about 260 nm. The thermal exposure at 975~ causes a growth of the zone to about 2.5 lam after 4 h and about 4.2 l,tm after 16 h. The interaction kinetics in SCS-2- and SCS-6 fibres/titanium-matrix composites has been also studied and the first findings of these studies was that the interaction zone in the case of both SCS-2 and SCS-6 fibres grows much more slowly than in the case of unprotected silicon carbide fibre [355]. Most studies till date have been carried out on the (~ +/3) titanium alloy, however some experimental data obtained for alloys of various phase and chemical composition reveal a broad interval of possible variations in the activation energy as well as in the pre-exponential factor for a variety of alloys studied. The result obtained by Gundel and Wawner [212] are presented in Table 10.1. We see that the composition of an alloy influences the interface growth kinetics. It was postulated that this was both due to a higher diffusivity and lower solubility of carbon in //-phase than in 0c-phase and due to a purely compositional effect. The measurement of the reaction zone thickness at an isothermal, at 1000~ heat treatment was conducted by Hall et al. [224] also on a number of titanium alloys and both SCS-6 and SCS-2 fibres. In the as-fabricated specimens, the thickness of the interfacial layers was typically 0.3-0.4 lam for all [~ alloys studied and about 1 lam for (~ +/3) Ti-6AI-4V alloy. A reduction in the growth rate was found at some stage of the reaction which coincides with the complete consumption of the protective carbon coating. At this point, the reaction layer consists of both TiC and TisSi3 in SCS-6/Ti-6AI-4V, as was shown earlier, and of mainly TiC in SCS-2//%alloy composites. It was found that the interface growth in heavily alloyed matrices went T A B L E 10.1 Characteristics of the kinetics of the reaction zone in SCS-6SiC-fibre/titanium-alloy-matrix composites according to Gundel and Wawner [212]. Chemical composition

Phase composition

Activation energy, Q kJ/mol

K0-104 m/s I/2

Ti Ti Ti-6AI-4V Ti- 15V-3Cr-3AI-3Sn Ti- 15Mo-2.7Nb-3AI-0.2Si Ti-6AI-2.8Sn-4Zr-0.4M o-0.45Si Ti- 14AI-21Nb

~ /~ ~ + [~ * /~ ~

197 148 252 16 I 232 288 269

4.24 0.36 48.8 0.35 12.2 121 29.0

*~ ~ [/ transition at 760~ **0r 2 is an intermetallic Ti3AI phase.

~* + [t

Ch. X, w

Kinetics & the case o f chemical reactions

451

on with a lower rate than in the case of alloys normally used for studying SiC/Ti composites. These observation lead to the following suggestions: 9 It is necessary to pay very serious attention to using/~- or metastable//-alloys such as Ti-10V-2Fe-3A1, as matrices. Such alloys give minimal reaction layer thickness and can be heat treated to high strength level. The use of alloys such as Ti-15V3Cr-3Sn-3A1 is an important step in this direction. 9 It is useful to protect a protective carbon layer by additional protection by using, for example, deposition of a layer of titanium carbide on the C-rich layer. Actually, chemical reactions and diffusion processes at the interface in an ~ +/~ titanium alloy reinforced with SiC fibre coated with carbon/titanium-diboride layer have been studied by Badini et al. [32]. Each sublayer (carbon is the inner one) has the initial thickness of about 1 ~tm. By using results of scanning electron microscopy, energy dispersive X-ray analysis, and Auger spectroscopy of specimens after hot pressing and those after heat treatment in vacuum at temperatures between 600 and 1000~ for the times from 100 to 1000 h, Badini et al. have made observations that lead them to the following representation of the events and the microstructures obtained. At the TiBz/matrix interface, the reaction produces elongated crystals of about 0.1 ~tm in diameter and with a length of some micrometers for as received specimens to tens micrometers for those annealed for 100 to 200 h at 600~ With the annealing temperature and time enhancing, the aspect ratio of the crystals decreases and their density increases, an around-fibre zone, occupied densely by the crystals, expands. It is important to note that the boride crystals have been found far from the TiB2/ matrix interface. Hence, similar to the process of boron diffusion in A1-Mg matrix and subsequent formation of the magnesium oxide precipitates, boron which has a low solubility in both titanium phases, forms boride particles in the matrix. At the TiB2/C interface, titanium coming from the matrix by diffusion through the TiB2 layer, reacts with carbon at the beginning. The output is TiCx. Then the diffusion of titanium and carbon in opposite directions yields the formation of a multicomponent zone which probably contains TiByCl_y, TiB and TiCx.

IntermetaUic matrix composites Fibre/matrix reactions in the case of an intermetallic matrix involve elements of the intermetallics so the corresponding kinetics and the composition of the reaction zone is often similar to one of those considered above. Titanium aluminides Thermodynamic considerations of compatibility of various fibres with titanium aluminide (Ti-40A1) containing ternary additions based on binary phase diagrams predict formation of FeSi and A13C3 at the interface in the case of SiC-fibre, and FeB and free carbon in the case of boron fibre coated with boron carbide, and no reaction in the case of sapphire fibre [465]. The experiments carried out by Draper et al. [135] supported these predictions and revealed some

Diffusion throughfibre/matrix interface

452

Ch. X, w

features of the interactions. In particular, extensive reaction occurred between SiCfibre and the matrix after 5 h at 1023~ Iron and aluminium diffuse into the fibre from the matrix and form a ternary compound of Fe, A1, and Si. Carbon and silicon are also found in the reaction zone. As in the case of titanium matrix composites, using silicon carbide fibres protected by C-rich layers yields a favourable change in the reactions going on at the fibre/ matrix interface. Goo et al. [199] studied the structure of the SCS-6 SiC/TiA1 interface after keeping the fibre/matrix mixture at 1100~ for 1 h under a pressure of 50 MPa. Again, the carbon excess at the interface yields formation of the titanium carbide protective layer which prevents an accelerating interaction. The structure of the layer is presented schematically in fig. 10.5. The reaction zone microstructure in a SiC/Ti3AI + Nb composite after hotpressing performed in such a way as to achieve full consolidation with a minimum of the interactions at the interface, was studied by Baumann et al. [42]. A schematic reconstruction of the microstructure obtained is presented in fig. 10.6. Note that the matrix in the vicinity of the reaction zone is depleted of//-phase, possibly due to ~2 stabilization by the elements occurred there. The decrease in the/~-phase content yields brittleness of this zone of the matrix [465]. In the case of B/B4C-fibre [135], the reaction, being also extensive, proceeds into the matrix. Two different reaction products are detected after heat treatment. Sapphire fibre does not react with the matrix even after 25 h at 1223~ On the other hand, processing an alumina-based-fibre/(~2 + 7)-titanium-aluminide matrix composite by using pressure infiltration method [495] is accompanied with an extensive fibre/matrix interaction (see Section 10.3.4). Nickel aluminides Chou and Nieh [84] when working with a plane SiC/Ni3A1 interface configuration, obtained by hot pressing of SiC/Ni3AI couples, observed, after annealing specimens at 1000~ for various times, the reaction zone composed

F--t~-.-i | | |

~

[..,

TiA1

|

@

-% ~z

< E--,

@ Ill

II

I

Fig. 10.5. Schematic of the interaction zone in a SCS-6/TiA1 composite. After experimental data by Goo et al. [199].

Ch. X, w


453

MICROPOR O S I T Y C - R I C H LAYER

Ti3A1

SiC - F I B R E

MATRIX 0( 2 + t~

~ r (TiNb)C 1 x+(TiNbal)5 Si -

3

(TiNb) 3A1C+ (NiNbA1)5Si3 Fig. 10.6. Schematic representation of the reaction zone in a SiC/Ti3A1 + Nb composite. After Baumann et al. [42].

of three layers. The layer neighbouring Ni3A1 is NiA1, then a layer of Nis.4_xA1Si2 follows, and closer to SiC, a layer composed of Nis.4_xA1Si2 and modulated carbon bands exists. An interesting observation is that the kinetics of the NiA1 layer formation obeys the parabolic law which means a diffusion controlled mechanism as contrary to the formation of the two layers adherent to the SiC side. The total thickness of these two layers is always less than that of the first one and no parabolic growth rate is obeyed. This suggests that the decomposition of silicon carbide may be a rate limiting stage for the whole kinetics of the reaction zone in this case. The carbon-rich layer in SCS-6 SiC fibre can slow down diffusion of Ni and other elements from the Ni3A1 matrix alloyed with Cr, Zr, and B, into the fibre [719]. This is a result of formation of a rather complicated layered structure of the reaction zone arising during heat treatment of the composite at 780-980~ for 1 to 100 h and containing a number of subzones which consists of nickel silicides and discrete graphite particles. Iron aluminides Zirconia-toughened alumina fibre, PRD-166, reacts with interme-

tallic alloy Fe-28A1-2Cr-lTi (at%) yielding formation of the Fe2A1Zr phase which occurs at the fibre/matrix interface after infiltration of a fibre bundle with the molten matrix [494].

Ceramic-matrix composites For most ceramic/ceramic systems, at least for continuous fibre composites, introducing a special interface layer to ensure sufficiently weak bond, which is necessary to obtain reasonable fracture toughness value of a composite, is unavoidable [161]. The existence of a large variety of couples, although the choice of possible fibre coatings seems to be rather limited at present to refractory metals (mainly molybdenum), boron nitride, and porous oxides [69, 120, 161], makes a

454

Diffusion throughfibre~matrix &terface

Ch. X, w

routine thermodynamical analysis be necessary at an early stage of the development. Corresponding examples are known (for example, [204, 466]). The chemical stability of prospective carbon and boron nitride fibre-coating in silicon-based matrices was considered in [358]. It was shown that at the C/Si3N4 interface, the following reactions may occur: Si3N4 + 3C = 3SiC + 2N2(g), Si3N4 - 3Si + 2N2(g)

(10.14) (10.15)

and the equilibrium gaseous pressure of 1 atm can be reached for the reaction given by eq. (10.14) at about 1730 K. This would lead to a danger of damaging the composite structure by high gaseous pressure. However, the kinetics of a real interaction appears to be strongly influenced by formation of solid SiC at the interface which prevents further reaction and thus reduces nitrogen pressure by three orders of magnitude. In the case of BN/SiC(with excess of C)-interface, the potential reactions include SiC + 4BN = B4C + Si(g) + 2N2(g), 3SiC + 12BN = Si3N4 + 3B4C + 4Nz(g), C + 4BN = B4C -+- 2N2(g).

(10.16) (10.17) (10.18)

Despite the real reactions not generating any significant amounts of gaseous species, the experimental indications of a reaction zone at the interface exist. This is certainly due to the interaction according to eq. (10.18). Multi-component glass-ceramic matrices provide more ways for the fibre/matrix interactions. For example, Nicalon type SiC fibres being embedded in a ( B a O - SiO2 - A1203 + Si3N4) matrices degraded during the fabrication process (powder metallurgy) as a result of diffusion of barium into the fibre [245]. Interfaces in ceramic-matrix composite are subject to changes when heat treated in oxidation environments. For, example, heating composites with reaction-bondedsilicon-nitride matrix and SCS-6 SiC fibre with a coating of SiC on top of carbon coating causes a drop in the interfacial shear strength due to changes in the fibre outer coating [210]. The oxidation results in transformation of the initial coating into highly porous one.

10.3. Effects of component interaction on composite properties Physical and chemical interaction at the interface occurred at the fabrication stage and also during service under some conditions (high temperatures and corrosive environments) can influence the mechanical properties of composites in various ways. In particular, the interface strength can be changed, a new phase can contribute to the composite strength, mechanical properties of the fibre can change, and the mechanical behaviour of the matrix can also be altered. Most of the

Ch. X, w

Effects of component interaction on composite properties

455

consequences of these changes can be predicted on the basis of results presented in Chapters 5 to 8.

10.3.1. Interface strength changes Note, first, that the observation of a structure of the interface (Section 10. l) does not bring any reliable quantitative information about interface strength, and neither do thermodynamic considerations (Section 10.4.1.). We should also remember that the measurement of the interface strength (Section 10.4.2) has to be always connected to a particular problem or type of loading.

Metal matrix composites When a metal-fibre/metal-matrix composite is loaded in the fibre direction, the problem has got a very simple solution (see Sections 5.6 and 5.7); namely, increasing the interface strength always leads to a better result, although quite good results (high strength and fracture toughness) can be obtained with a relatively weak interface. The choice of technological parameters does not appear to be very critical. Formation of a brittle intermetallic layer on the interface does not lead to a decrease of the composite strength and some increase of the effective fibre strength can be observed due to a direct contribution of the reaction layer to the composite strength (see below).

A brittle-fibre composite (Sections 5.2 and 5.4) is affected by interface strength changes in a more complicated fashion. If its failure is accompanied by fibre breakage at weak points (line (a~)up to point A in fig. 5.2), then increasing the interface strength leads to decreasing the critical fibre length and so that to increasing fibre stress contribution to the ultimate composite stress (as a result of the length dependence of the fibre strength), fig. 10.7. This leads also to an increase in the plastic dissipation contribution to the effective surface energy of a composite and to a decrease in the energy dissipation at the interface. The latter should be favourable in the case of a tough matrix, and less favourable in the case of a less ductile m a t r i x - cast alloys and plasma sprayed matrices being examples of the latter kind. Certainly in this case an optimal interface strength and thus an optimal set of fabrication parameters are expected to exist. An example is supplied by bending tests of a graphite-fibre/aluminium composite with various contents of the aluminium carbide phase at the interface (fig. 10.8). The strength of such composites goes up with the carbide content at small volume fractions of the carbide when obviously the interface strength may increase. But then the composite strength goes down, certainly because of the formation of a brittle layer decreasing the effective fibre strength (see Section 10.3.3). A similar result was obtained in testing composites with the tungsten fibre and copper matrix doped by manganese, which is soluble in tungsten (fig. 10.9). As mentioned above (Section 10.3.3), a modification of the matrix can yield the interface strength increasing because of improving the wetting conditions during the


456

Ch. X, w

O-*

J O

vpJ

v/'J

v/~

vI

Fig. 10.7. Brittle-fibre/ductile-matrix composite: with the fibre/matrix interface increasing, the composite strength goes up (OA0 ~ OA: ~ OA2) provided it fails by the fibre break accumulation mechanism. However, increasing the interface strength makes more narrow the interval of fibre volume fractions in which such a mechanism operates "~vf(0) ~ vf(l) ~ vf(2),).

infiltration. It corresponds usually to composite strength increasing. For example, alloying the 6061 aluminium matrix with 1% Li increases the strength of a composite with 30% SiC whiskers by 10% at a temperature of 200~ and by 50% at 300~ [5781. On the other hand, if the fibre volume fraction is sufficiently high, the fibre/matrix debonding becomes an important mechanism of crack arrest, that is a situation characteristic to vr > v~} in fig. 5.2, then there should be an optimum strength .2

9

,

~ 1.0 %

9

,

9

x / /

0

"

t

0.8

t t

O

~

O

0.6

0.4 I

0.00

'

0.;2

'

0.04 v

|

AI4C3

0.06

Fig. 10.8. Dependence of the bending strength of the graphite-aluminium composite on the volume content of the carbide phase at the interface. The matrix is commercially pure aluminium, vf - 0.46, melt temperature at fabrication process is between 670 and 760~ pressure on the melt is about 2 to 4 MPa, process time is less than 60 s, a~ -- 570 MPa. The experimental data after Portnoi et al. (1981).

Effects of component

Ch. X, w

interaction on composite properties

,

457

,,

'

I

*

I

"

,.0,.

..500 I

%

l l l

l

I

S

S

i l

i

I

S

i

I

i

400 I 0...,

300

0

I

/

s

i

i

i

i i

i

,.I

'

' 5

'

' 10

'

81~rn

15

Fig. 10.9. Dependence of strength of the W - (Cu + Mn) composite on the thickness of the interface zone arising during annealing, at 850~ vf - 0.148. The experimental data after Umakoshi et al. See [419].

characteristics of the interface. In experiments by Naik et al. [481], carried out on SCS-6-SiC-fibre/Ti- 15V-3Cr-3A1-3Sn-matrix composite, it was revealed that annealing the composite at 1000~ for 1 h led to a 25% decrease in the composite strength and to a drop in the fatigue strength as compare to as-received specimens. The observation of fracture surface made an assumption on strengthening the interface as a result of the annealing be reasonable. Obviously the interface strength strongly affects the transverse strength of a composite. This property seems to be very sensitive to fabrication parameters. To give an example, we refer to the results of testing boron-aluminium composites [256]. The transverse strength of the composite, with commercially pure aluminium as a matrix and fibres coated with silicon carbide, changes from about 22 to 92 MPa depending on fabrication parameters. Hot pressing in an argon atmosphere gives better results than in air. Ceramic-matrix composites As shown above (Sections 4.5 and 5.8) changes in mechanical properties of the interface in ceramic-matrix composites can lead to most serious changes in composite strength and fracture toughness. The theoretical conclusions are supported by numerous experimental observation. The interactions of the interface with matrix cracks are crucial events determining the fracture behaviour of brittle-matrix composites. To validate further this conclusion, we give some experimental results. Evans [159] presented and analyzed modes of the matrix crack behaviour in a Nicalon SiC-fibre/C-fibre-coating/LAS-matrix composite after various heat treatments in air. In the as-received state, the carbon interface layer exists and prevents the matrix crack to penetrate the fibre. The composite exhibits a non-brittle failure

458

Diffusion through fibre~matrix interface

Ch. X, w

mode. After heat treatment at 800~ for 4 h, a partial SiO2 layer occurs at the interface yielding just a limited fibre pull-out. With the heat treatment time increasing up to 16 h, a continuous silica layer occupies the interface and the crack propagates through the fibre without debonding the interface. Bender et al. [46] produced Nicalon-SiC-fibre/ZrTiO4-matrix composites with vf = 0.5 and reported effects of the interface on strength and fracture toughness of the composites. Sintering of the composites was performed in CO atmosphere (to retain the fibre strength after heating above 1300~ at temperatures shown in Table 10.2. In batches A to C, fibres were coated with a layer of amorphous BN (about 135 nm thick), samples D were produced without any fibre coating, and samples E contained fibres with a triple SiC-BN-SiC coating. A schematic drawing of the results of TEM observations is presented in fig. 10.10. We see that with sintering temperature increasing above 1270~ the thickness of BN layer rapidly decreases and the layer disappears after sintering at 1330~ This leads to intensification of the fibre/matrix reaction that certainly corresponds to an increase in the interface strength. As a result of this, the strength and fracture toughness decrease to a level of those for composites containing uncoated fibres or fibres with triple coating which occurs to be unstable under conditions of the experiment. The observation of the failure mode is in a qualitative agreement with the properties measured and the microstructures observed. An important information is supplied by measuring shear strength of the interface (Section 10.4.2) that has been effected by various heat treatments or environmental conditions (oxidation and similar effects). Such experiments can reveal a degree of the fibre-matrix interaction and the role of the interface layers. For example, boron nitride coating (~1 lam) prevents severe chemical interaction in the SiC/mullite system and keeps the interfacial shear strength on a level of tens MPa that is an order of magnitude lower than the shear strength in the SiC/mullite without interface coating [613].

Carbon/carbon composites Carbon/carbon composites are typical examples of a brittle-matrix composite. So its mechanical behaviour is highly dependent on the interface properties. On the other hand, as shown above (Section 10.1), variations in fibre type and matrix TABLE 10.2 Sintering temperatures, bending strength, a*, fracture toughness, K*, and fracture modes of SiC/ZrTiO4 composites. After Bender et al. [46]. Batch

T~

a* MPa

K* MPa 9m 1/2

Featuresof the fracture mode

A B C D E

1270 1330 1400 1270 1270

960 504 315 387 398

22.4 11.5 7.2 6.7 6.8

Fibre microcracking, debonding No microcracks, some debonding and pull-out Short pull-outs Flat surface, little pull-out Flat, some debonding

Ch. X, w


459

precursor allow to vary the microstructure and thus properties of the carbon/carbon interface in broad intervals. It has been also shown [338] that if to exclude chemical bonding in composites obtained by chemical vapour infiltration of a 2-D skeleton of fibre with no surface treatment, then a composite with purely mechanical bonding will be obtained. The strong mechanical bonding is a result of the compressive stress exerted on the fibre by well oriented CVI-matrix. A degree of the crystalline orientation of the matrix depends on the processing conditions. In contrast to that case, 2-D composites obtained by resin infiltration route, have amorphous carbon interfacial layer which does not provide strong bonding. As a consequence, the former composites have high bending strength and poor fracture toughness, the latter ones have lower strength and higher fracture toughness. 10.3.2.

The additive contribution o f a new p h a s e

An attempt to take into account a contribution of the brittle phase at the interface directly was made in Ref [177]. The authors assume the strength of the interface layer of small enough thickness h to be higher than that of a part of the fibre

A BN

g,

~

B

~ c aRe a c tio n layer

D

Fig. 10.10. Schematicpresentation of the results of TEM observation by Bender et al. [46]of the interface regions in SiC-fibre/ZrTiO4-matrix. The batch notation are the same as in Table 10.2. The scale is approximate: the thickness of the BN layer in A is about 100 nm.


460

Ch. X, w

replaced by the layer, so that Oa*/Oh > 0 as h ~ 0. However, when thickness h increases, the strength of the layer decreases proportionally to h -1/~, where/3 is the Weibull's parameter for a material of the brittle layer. Hence the layer's contribution to the composite strength varies as Aa* - af

- 1 .

Here # is the strength of the brittle layer of thickness h. So the strength of a composite goes down, starting with a particular value of h. Therefore, the dependence of a*/a~ on h (a~ is the strength of the composite at h - 0) should have a maximum.

10.3.3. Changes in effective fibre characteristics The influence of the physical-chemical interaction on the fibre strength can be observed in various ways. Changes in the fibre strength effect the composite strength directly. Mechanisms

Two mechanisms of the effect of the fibre/matrix interaction are known and at least two ones can be expected to exist.

Recrystallization and other structural changes This was revealed soon after carbon fibres were started to be studied. Jackson and Marjoram [276] discovered that the recrystallization of carbon and graphite fibres at a temperature of 1000~ was stimulated by the presence of nickel. This led to a drastic drop of the fibre strength [273]. However, Barclay and Bonfield [38] believed that this effect had been caused not by nickel but elements soluted in nickel. They deposited pure nickel from the vapour and conducted the experiments in a good enough vacuum and did not observe recrystallization of nickel. But from the technological point of view the first result remains important. A similar situation was then observed in other fibre-matrix systems. The tungsten fibres have been investigated most widely. Diffusion of elements of a matrix to the tungsten fibre, its recrystallization and degradation have been studied in detail. It seems that unalloyed tungsten exhibits slower recrystallization rate than solid solution strengthened and dispersion hardened tungsten wires [293]. Grain growth in zirconia toughened alumina fibre (PRD-166) in contact with molten intermetallic matrices and ZrO2 depletion in an outer layer of the fibre has been observed by Nourbakhsh et al. [495, 496, 497]. Brittle interfacial zone. A brittle layer containing chemical compounds of the fibre and matrix elements can influence the fibre strength strongly. A circumferential crack at a ductile fibre can cause a local increase in the fibre yield stress if the


Ch. X, w

(o)

461

(z)

RZ

B

A

A_

BH

BI

Fig. 10.11. A schematic drawing of stages of the reaction zone growth. F denotes the fibre and RZ, the reaction zone. The microcracks in the fibre core and the reaction zone are shown.

interface between the brittle layer and the fibre is strong enough [501]. The same situation in the case of a brittle fibre can obviously be unfavourable [669]. The effect of a brittle layer, which can arise due to an interface reaction or be applied as a fibre coating, on the fibre strength has been a subject of a rather thorough study [372, 404, 503, 602, 669]. Let us consider changes in the strength of a single fibre with growing the reaction zone (RZ in fig. 10.11) following the ideas of the above mentioned references. In the initial state, (0), in fig. 10.11, the fibre diameter is d~~ the fibre strength, a~~ is determined by a population of fibre defects, or by a defect of length cf. Occurring in the reaction zone, states (1) and (2), the defects do not effect the fibre strength until h < cf. At the same time, they are certainly transforming into cracks and the fracturing of the zone occurs. These events can be described by using the Weibull statistics, just the size parameter, l, in eq. (2.7) should be replaced with the volume of the brittle layer, gldfh, so we have

(~RZ)- ~~

h0

~-l/~v(1 + 1//~)--A

2)~

V(1 + 1/~)

(10.19)

where ( - h/dr and (Cr~ h0, fl and A are constants. Hence, the zone has been cracked when, on average, the fibre stress is

,Rz,

f

- - O'f

1 (1 + ~)2

+ (O'RZ)

1 --

(1 +

.

(10.20)

462

Ch. X, w


Because

O'f Ef

(O'RZ) ERZ

where Ef and ERZ are the Young's moduli of the fibre and reaction zone, respectively, we can write

(RZ)--A (d~2)~) - 1/13F(1 +

O'f

[ 1 lift) 1 + (1 + r

/Ef

~zrz-1

/1

.

(10.21)

Neglecting a dependence of d~2) on ~ we have now

a(RZ) f = a~~

21~ -1/l~

(10.22)

When the reaction zone thickness reaches the length of a dangerous fibre defect (state (2)), the microcracks in the zone, if they have occurred, can either enter the fibre core (situation Bi in fig. 10.11) or delaminate the reaction-zone/fibre-core interface (situation Bii). Crack entering the fibre core occurs when

O'f(2) -- A~ 1/2.

(10.23)

This follows from Irwin's criterion and constant A includes the stress intensity factor for the axisymmetrical problem. If the initial fibre strength, a~~ is known then A cx a~~ and ,(2)

O'f

= ,~2~-1/2.

(10.24)

In the case of interface delamination (situation BII), the brittle layer does not effect the core strength. To find the conditions for each of the two cases to be realized, an analysis similar to that described in Section 4.5 is needed; however nobody seems to have performed such an analysis yet. We should note that the diameter of the fibre core to remain unreacted, dr, depends on the nature of the reaction yielding formation of the zone; generally, df > d~~ - 2h. The dependencies of 0"~(2) and a~Z on ( are drawn schematically in fig. 10.12a. Consider a possible dependence of the fibre strength on the relative zone thickness ~. If df - d{~ - 2h, then in interval OA, the fibre strength, a~, will be determined by the core strength. For Ef - ERZ (this is a case in fig. 10.12b), a~ does not depend on ~"in this interval. In interval AB (state (1) in fig. 10.11), the reaction zone does not carry a load, but the microcracks in the brittle layer do not compete with the core microcracks, which determine the fibre strength, so that

Ch. X, w

463


(a)

a/

O ~A,~B ~ (')

r o7

(b) o

Fig. 10.12. Schematic illustration of the fibre-strength/relative-reaction-zone-thickness dependence.

O'f

id~O) j

.

(10.25)

At ~ > ~B (interval BC) the microcracks in the brittle layer act as the dangerous defects for the fibre core, and the fibre stress follows eq. (10.23). A t ( > ~c, the layer microcracks when occurring appear to be unstable (afRZ > af ~ J) and the fibre fracture takes place along the line CD. The general dependence of the fibre strength on the reaction zone thickness is depicted schematically in fig. 10.12b as the line OAIBCD. We see now that a severe drop in the fibre strength follows either the dependence given by eq. (10.22) or that by eq. (10.23). Thus, it is interesting to express the time dependence of the fibre strength in a corresponding heat treatment time interval. Substituting eq. (10.6) into eqs. (10.22) and (10.23) yields

0-~(2)

= A1 t -1/2~ exp(-Q/2~RT)

(10.26)

and

0-~(2)

= Azt -1/4 exp(-Q/4RT) a~~ .

(10.27)


464

Ch. X, w

where A1 and A2 are the constants. The equations written allow to derive the activation energy Q of the process controlling the reaction zone formation from an experimentally obtained temperature dependence of the fibre strength. Two remarks are necessary with regard to the scheme just described. First, there can be imagined a variety of the relative locations of the characteristic curves in the ~ / ( - p l a n e , so that a variety of the corresponding dependencies can be expected. Second, there are known various improvement of the scheme. In particular, Ochiai and Osamura [508] have used a modified shear-lag analysis (see Section 3.6) to calculate the stress intensity factors for multiply cracking in the brittle layer. They have found that such cracking decreases values of the stress intensity factor and, therefore, increases the fibre strength as compared to the solution for a single microcrack. The effect depends on the crack spacing, the ratio of the Young's moduli of the fibre and the layer and the layer thickness.

Fibre healing This can occur as a result of filling a cavity on the fibre surface with matrix material. This reduces the stress intensity factor at the tip of the cavity that should yield an increase in the fibre strength. A similar situation was observed in the behaviour of boron fibres, uncoated and coated with a layer of Y203 of a thickness from 1.5 to 5 p [341]. The coating of the fibres appears to discharge surface defects of a size from 0.15 p and larger from their role to reduce the fibre strength. Figure 10.13 illustrates dependence of strength characteristics of Nextel mullite fibre on the BN coating thickness. One can see that thin coating acts as a healing agent, but with the thickness increasing the fibre strength degrades as described in the

2.5

A'~ 2.0 vb

1.5

1.0

0.0

~

0.2

i

t

0.4

,

I

0.6

h/~m

,

I

0.8

j

1.0

Fig. 10.13. Mean fibre strength versus coating thickness. The experimental data were obtained by Chawla et al. [78] in testing mullite-based Nextel fibres coated with boron nitride.

Ch. X, w

Effects of component interaction on compositeproperties

465

above analysis. However, such a mechanism remains to be a hypothetical one as the present author has not still found any strict experimental justification of it. Changes in fibre composition There have been observed drastic changes in fibre phase composition as a result of fibre/matrix interactions at high temperatures. An example is supplied by Nourbakhsh et al. [495] who have observed dissolution of ZrO2 from an outer layer of zirconia toughened alumina fibre (PRD-166) that takes place in contact with a titanium aluminide matrix. This definitely yields a change in the fibre strength characteristics, but nobody seems to have measured that change yet.

Experimental data A decrease in the characteristic fibre strength is usually measured by testing fibre specimens of a particular length. The schematic dependence of the fibre strength on time of exposure in contact with a matrix material are shown in fig. 10.14. Different mechanisms of the fibre strength degradation leads to the different dependencies. Table 10.3 compiles some experimental data relevant to the subject. 10.3.4. Change & effective matrix character&tics A detailed analysis of a possible effect of a change in the matrix microstructure around the fibre as a result of the diffusion of fibre element into the matrix, of the composite strength was carried out in Section 5.2.5, further discussion of the subject will be given below when describing mechanical properties of boron/ aluminium composite in Section 11.6. So here we are just to remind that precipitation of intermetallic compounds formed by a fibre element diffused to the matrix and those dissolved in the matrix, changes properties of a zone around the fibre. The zone is strengthened and this yields to an increase in the composite strength, but at the same time, the zone becomes brittle and this lowers a value of the Different mechanisms of fibre degradation

7e 7' Fig. 10.14. Schematic dependence of the fibre strength on time of exposure with a metal matrix.


466

Ch. X, w

T A B L E 10.3 Stability of some fibres (coatings) in metal matrices. Fibre/ coating C C C C C C C C C C-I C-II C SiC SiC SiC SiC SiC SiC SiC SiC SiC SiC TiC A1203 A1203 A1203 A1203 A1203 A1203 B B B B B B B B B B B B/BnC B/SiC B/SiC Mo W

Fibre structure

High strength High modulus

Nicalon

SIGMA SCS-2 SCS-6 SCS-2 SCS-6 MC SCS-6 coating SC SC SC SC SC SC MC MC MC MC MC MC MC MC MC MC MC MC MC MC PC PC

Matrix

tc h

Tc~

Source

A1 AI A1 Ni Ni Cu Ni Ni Ni Ni Ni Ni-Cr AI AI AI-3Mg A1-10Si AI- 10Si AI- 10Si Ti Ti Ti Ti- 15V-3AI-3Cr-3Sn Ti Ni 80Ni-20Cr Ni-Cr-Fe NiAI W W AI-3Mg AI-3Mg AI-3Mg AI-3Mg AI-3Mg Ti Ti Ti Ti-6AI-4V Ti-6AI-4V Ni Ti Ti Ti-6AI-4V Ni Ni

24 100 100 1 5 24 < 1 24 1 1 1 24 0.5 24 10 "0.002 "0.002 ,-0.002 0.25 > 1 < 0.5 1000 > 0.5 < 1 < 1 < 16 0 < 16 100 > 10 1 0.1 1000 1200 0.15 < 0.5 1500 4.3 24 < 0.5 0.5 0.5 < 100 < 1

580 550 475 600 600-800 800 900 1000 > 1270 1230 900 500 580 700 580 < 700 > 800 > 800 850 950 750 600 750 1000 1000 1000 > 1700 1320 1000 400 500 540 580 230 630 870 750 640 760 400 1000 870 850 1100 1100

[273] [34] [34] [689] [137] [273] [137] [273] [683] [34] [34] [273] [372] [273] [394] [723] [723] [723] [355] [355] [355] [233] [86] [660] [660] [660] [650] [660] [660] [394] [394] [394] [394] [394] [404] [404] [355] [404] [404] [273] [355] [404] [404] [545] [545]

SC-single crystal, PC-polycrystal, MC-microcrystal.

Ch. X, w

Effects of component &teraction on composite properties

467

fibre volume fraction at which a transition from the fibre breaks accumulation mechanism of composite failure to that determined by some fibre breaks at weak points. Also we present here some experimental evidence of the formation of the "influence zones" in the matrix in addition to those given in Sections 5.2 and 11.6. An above mentioned evidence was supplied by a study of the near-reaction zone composition of the matrix in a SiC fibre-reinforced titanium alloy containing, in particular, Zr and Nb, which was carried out by Das [117]. He found (i) a rapid increase in silicon, zirconium, and niobium concentration in the matrix adjacent to the reaction zone; (ii) enhanced hardness of the matrix in the same region. The latter was attributed to a combination of solid solution hardening of the matrix by Zr, Nb, and Si and precipitation hardening through formation of zirconium and niobium silicides. It should be noted that Zr and Nb are stronger silicide formers than Ti. Das' observation of a crack around the Knoop indentation reveals a drastic loss in the fracture toughness of the matrix around the reaction zone. Precipitation of the titanium boride crystals in the matrix of TiC-fibre/TiBzcoating/Ti-matrix was observed by Badini et al. [32] (see Section 10.2.2). A global change in the matrix microstructure occurs when alumina-based-fibre/ Ti-50% Al-matrix composite is processed by infiltration of the molten matrix into the fibre bundle [495]. The unreinforced intermetallic matrix alloy has a structure consisting of alternating lamella of 7-TiA1 and ~2-Ti3A1 which is known to exhibit a high fracture toughness that of pure TiA1. An investigation of the as-cast composites reveals the lamellar structure of the matrix and zirconia free outer layer in the fibre. ZrO2 particles appear in both matrix phases. Vacuum annealing of the composite at 950~ for 175 h leads to the occurrence of a zone around the fibre occupied entirely by ~,-phase. In that zone, particles of both ZrO2 and AlzZr are observed, the former being much larger in size than the particles in the as-cast microstructure. No doubt, such a change in the matrix structure produces the "influence" zone being much more brittle than the initial matrix alloy. The microstructure of composites annealed at the same temperature in air is strongly affected by oxygen diffusion along the fibre. Although a finding by Hong and Grag [255] is related to SiC-particulate reinforced an A1-Zn-Mg-Cu alloy, the microstructure obtained can be certainly observed in a corresponding fibrous composite as well. They discovered the aging deceleration in the matrix as compared to the unreinforced alloy. On the other hand, there are clear evidences of acceleration of aging processes in aluminium alloys when they are matrices in composite. For instance, Christman and Suresh [89] have shown that a SiC-whisker/A1-Cu-Mg-alloy-matrix composite reaches peak hardness in 4 h at the aging temperature of 177~ while the equivalent hardness of the unreinforced alloy was attained in 12 h. The effect is attributed to a higher dislocation density in the matrix as compared to the pure alloy. The difference in the dislocation density is found to be an order of magnitude. A change in the matrix composition which can be a result of diffusion of an alloying element into the matrix, can change microstructure and properties of the matrix completely. Smith et al. [616] found that a precipitation phase containing Mg in an unreinforced aluminium alloy had been replaced with another phase in the matrix reinforced with alumina-based-fibre since magnesium seemed to diffuse to

468


Ch. X, w

the fibre during composite processing. Exposure of a A1203/A1 composite at temperatures above 570~ for 10 min results in magnesium diffusion from the AA6061 matrix to the fibre/matrix interface to form spinel MgA1204 [628]. This is accompanied with a decrease in volume fraction of the Mg2Si precipitates and corresponding decrease in the strength of both matrix and composite. In the case of short alumina fibre composite [628], this sets limits for secondary processing such as extrusion or brazing. The aging of SCS-6 SiC-fibre-reinforced /~-titanium alloy composite can be accompanied with oxygen diffusion from the fibre/matrix interface along the grain boundaries of the matrix. This yields formation of a-titanium on the grain boundaries and deterioration in the mechanical properties of the matrix. This process can contribute to a drop in the composite strength after exposure at 600~ for 1000 h [233]. Certainly, an increase in the dislocation density in the matrix due to a difference in the coefficients of thermal expansion of the components is also a possible way of the direct influence of fibre/matrix interaction on mechanical properties of the matrix through a work hardening of the matrix material.

10.4. Diffusion barriers

A fibre-matrix combination desirable from the point of view of mechanical properties of the components is often characterized by a tendency to unwanted physical or chemical interaction. An obvious way to restrict this interaction is to introduce an interface layer to serve as a diffusion barrier which also should provide a necessary value of the interface strength. A number of the requirements to the diffusion barrier (DB) should be considered when choosing a particular material and a way of applying it. Some of them are as follows: 9 DB has to prevent transport of reactants through the interface; 9 DB has to be thermodynamically stable; 9 the fibre strength should not decrease due to the presence of the DB adherent to the fibre surface; 9 DB should be compatible with the fibre and matrix on its thermal expansion coefficient; 9 the application of the DB should not rise drastically the fibre cost. The first three requirements are obvious. With regard to the fourth one, it should be pointed out that in addition to an effect of the residual stress caused by the thermal expansion coefficients mismatch, the latter can also cause the breakdown of the barrier and, consequently, get the diffusion to occur. Obviously, it is difficult to satisfy all these requirements. Constructing multilayered diffusion barrier can help to achieve a necessary compromise. Also it should be noted that the above mentioned construction cannot be performed without special experiments. For example, as was mentioned above, in connection to the diffusion in the presence of chemical reactions (Section 10.2.1), the corresponding analysis is hindered by the lack of necessary data on the diffusivity.

Ch. X, w

Diffusion barriers

469

In principle, the problem of choosing and forming diffusion barriers can be solved on the basis of the research results described above in Sections 9.2, 10.1, 10.2, and 10.3). Hence, we shall give just some examples to illustrate possible solutions to the problem.

Titanium-matrix composites Formation of carbon-rich layer on CVD SiC carbide layer (see above) can be considered as a protective barrier for titanium alloy matrices. Still, different attempts to improve resistivity to the chemical interaction are known. A promising stability of carbon coating preserved from a direct contact with titanium matrix by a thin layer of titanium boride was reported (see Ref [688]). Also it was shown above that either C/TiB2 or C/TiC layer of a thickness of about 2 gm on CVD silicon carbide fibre is an effective protection against the reactions in a titanium matrix. Listovnichaya et al. [371] and Kieschke et al. [311] analyzed thermal stability of various oxides as a diffusion barrier for titanium matrices. They found yttria as the most stable of the candidate oxides. From the point of view of an ability to impair the transport of reactants through the interface, zirconia and probably haffnia appear more attractive than yttria [311], but choosing the barrier material, one has to remain the stability consideration as the first priority. With some simplifying assumptions, numerical calculation of the total flux of reactants transported through a yttria barrier yields, after conversion of the flux into a reactant product, dependencies of the reaction zone thickness on the barrier layer thickness for SiC/Ti composites [311]. The results allow to estimate a minimum barrier thickness which prevents formation of a dangerous thickness of the reaction zone. For SiC/Ti composites, it should be at least about 500 nm. Still, to decrease a danger of the penetration of crack from the oxide layer, a duplex barrier with an inner layer of metallic yttrium is recommended. Such a barrier has also a capacity for "selfhealing" of damage to the yttria layer due to gettering of oxygen dissolved in the titanium matrix. The authors developed a method of sputter deposition to apply to the coating (see Section 10.2.5). An experimental examination of stability of silicon carbide fibres with the duplex coating was conducted by Kieschke et al. [312] on SiC/Ti composites made with using plasma spraying the matrix under conditions when the matrix is doped with hydrogen. It was found that yttrium diffuses into the fibre before the structure of the protective layer to be formed in situ, occurs. To improve the situation, the authors had to make a "triplex" coating by fibre preoxidation prior to Y coating. This inhibits yttrium penetration into the fibre. Hence, the triplex diffusion barrier is necessary if one relies on yttria as a main layer to impair the diffusion. It should be noted that the approach just described is a good example of the step-by-step procedure of construction of the diffusion barrier which includes thermodynamic analysis of the barrier stability, numerical calculations of the transport phenomena, and heavy experimentation. As shown in Section 10.2.2, a good diffusion barrier for SiC/Ti system is TiB2. It should be noted that a graded titanium boride coating on SCS-6 SiC fibre with high

470


Ch. X, w

concentration of boron near the carbon layer and a high concentration of titanium near the titanium matrix occurs to be more attainable than a stoicheometric coating [138]

Nickel-matrix composites The difficulties that arose in the case of sapphire-nickel composites are well known (see Table 10.3). A great number of the possible barriers has been studied including refractory metals, carbides and so on. Quite a good result has been obtained by using a combined fibre coating containing YzO3-W-Ni [660]. Many publications can be found on diffusion barriers in tungsten-nickel composites. Cornie et al. [100] have conducted a most detailed study of the thermal stability of the system tungsten-a barrier-nickel. Among the following substancesHfO2, Y203, A1203, TiC, ZiN, HfC - they have chosen hafnium carbide as the appropriate candidate. The HfC-layer provides the stability of a composite at temperature 1175~ for at least 2000 h. However, the authors conclude that their study has raised as many questions as answers. In particular, it is not clear how matrix impurities affect the barrier stability, how barrier stoichiometry affects the barrier stability, and what is the efficiency of the barriers during thermocycling. Other examples of diffusion barriers for nickel-matrix composites are given by Portnoi et al. [545] who also give recommendations on matrix alloying to decrease its interaction with fibres.

Intermetailic-matrix composites A need for diffusion barriers for this class of composites has been shown [467, 495]. However, a systematical study of the subject remains to be performed. Just separate results are being published, an example being that by Majumdar and Miracle [381] who tested a coating of a sapphire fibre to be used in TiAI matrix, with double metal layers. A real diffusion barrier was provided by an yttrium layer, whereas a niobium layer between the fibre and diffusion barrier was a ductile interface. The measured shear strength of the alumina/niobium interface occurred to be about 145 MPa. In general, the problem of a choice of the diffusion barrier for such a composite is exactly the same as that for a metal-matrix composite containing elements of the intermetallic matrix.

Ceramic-matrix composites The main function of the diffusion barrier in a ceramic-matrix composite is to keep the interface bond relatively weak. This was discussed above, in Section 10.2.2.

10.5. Sintering Sintering is a controlling mechanism in all powder metallurgy processes. Sintering is a process that starts with a mechanical mixture of powder material (or powder/

Ch. X, {}10.5

471

Sintering

fibre) and results in a compact solid with sufficiently low pore content. Obviously, that process is really a combination of various mechanisms, some of them being dominated under some conditions, others become dominant under other conditions. It seems to be useful, from the point of view of understanding of how to choose and optimize fabrication parameters, to present some very simple models of sintering, although they have been developed long time ago. The problem of the behaviour of a single void in an infinite linear-viscous body has been formulated by Frenkel [173]. He considers a spherical void of radius R(t) decreasing with time. A driving force of the process is the surface tension, while the viscosity of the material resists the decrease in radius. Because of the spherical symmetry, the displacement rate v has only a radial component Vr-

fl/r 2

where 13 is determined by the rate of void diminishing, namely, fl/R 2 -

dR/dt.

Then we have err --

dvr/dr- -2fl/r 2,

and the density of the rate of energy dissipation

W - 2qeZr where r/is the viscosity coefficient, and the total rate of energy dissipation is

Wl --8'lfR ~ a:er2rr2dr =

32

f12

-~- a:q ~--g.

The rate of free energy decrease due to the decrease in the surface area is W2 - ~ d (47zTR2)_

_8~z~/RdRdt

where 7 is the surface energy. The condition I~1 - I~2 leads to dR dt

37 4q

and finally the time necessary to remove the void is 4q

t, - TRo where Ro - R]t=o.

(10.28)


472

Ch. X, w

To consider the thermodynamic equilibrium between a void and a gas of vacancies in a solid, the treatment will not differ from an analysis of vapour equilibrium around the curved surface of a solid. In both cases the equilibrium gas pressure q increases when the radius of curvature decreases, namely, 27 f~) q -- qo 1 + -~- ~-~

(10.29)

where q0 is the equilibrium pressure around a plane surface and f~ the atomic volume. Therefore, a void can evaporate into a solid and this tendency becomes more pronounced as the radius of the curvature decreases. The process will be limited by the rate of vacancy diffusion away from the void. This problem was considered by Pines [185]. The final result is

kT

t, -- 6D7-----~R~

(10.30)

where D is the self-diffusion coefficient. An important contribution to the densification process can be done by the volume vacancy diffusion from the regions of high pressure in the vicinity of the concave surface of the neck between two particles to those of a low pressure in a vicinity of the convex surface of the particles [185]. This is actually a creep mechanism called the Nabarro-Herring creep [184] with the linear dependence of the strain rate on the stress: f~

~ oc D - ~ a .

(10.31)

Applying an external pressure accelerates the process of void radius decreasing. Let us find, following Wilkinson and Ashby [703], the rate of decreasing the radius, R, of a spherical pore surrounded by a spherical layer of the creeping material such that the external radius of the model is Re, under the applied pressure qe. Let qi be pressure inside the pore. In fact, this can be a model of a porous body (porosity p - R/Re and the initial porosity is p0 = Ro/Reo) under the hydrostatic pressure and internal pressure qi. The problem is characterized by the spherical symmetry, so that only stress and strain components a0, O'r and ~0, ~r are different. The equilibrium and compatibility conditions are do-r 2 d---;- + - ( a r r - ao) - 0

(10.32)

and d

d--7(r~:o) -

~r.

(10.33)

Ch. X, w

473

Sintering

The boundary conditions are

O'r(R)---qi,

(10.34)

O'r(Re) -- - q .

The material itself creeps without changing its volume, so

~r = -2~o.

(10.35)

The unidirectional creep law, eq. (6.1), that is

= ~(O'/O'm) m,

(10.36)

is generalized as

( )m ( )m ~r -- /I [O'r -- aO[ sign(ar -- 0"0) -- er -- /1 [O'r -- 0"01. sign(q -- qi) O'm O'm

(10.37)

where r/is a constant, not the viscosity coefficient used above. If q >> qi, then the solution of the problem formulated by eqs. (10.32) to (10.37) written in terms of the current porosity is 3 /~ -- -sign(q - qi) ~ r/

p(1-p)

(~ [q - qil) m

(1 - pl/m) m

mO'm

(10.38)

"

When the internal pressure cannot be neglected, 3

p - - -sign(q - qi) ~ r/

p(1-p) (1 - pl /m ) m

(

3 2m6m q -

( P0

l-pol-p

t7 qi0 ) 1 ) m 9

(10.39)

Wilkinson and Ashby [703] considered the model just described as that for a final stage of sintering. An intermediate stage is better modelled by a cylindrical pore yielding a larger rate of void removing due to a larger deviatoric components of the stress tensor in that case. Actually, because of a non-symmetrical configuration, the deviatoric stresses at the vicinity of a spherical pore can be essentially higher than in the situation considered by Wilkinson and Ashby. This yields to substantial acceleration of the densification kinetics [143]. At the initial stage of sintering, the process is determined by the mechanical interactions between separate particles yielding plastic deformation of both bodies. Bonding of dissimilar particles includes more complicated processes, some of them have been considered above. Here we just note that if materials of the particles are mutually insoluble, combining two particles A and B becomes themodynamically possible if only

TAB < 7A -+-7B where ])AB is the surface energy of the A/B-interface [540]. If we denote


474

Ch. X, w

A~/AB = ~/A -- "YB

then the kinetics of particle bonding will depend on which inequality is fulfilled: TAB > ATAB

or

TAB < ATAB-

In the first case the equilibrium shape of the final particle should be that of the concentric sphere with particle B in the center. Otherwise, the particles will be bound together by forming a neck between them which grows in the diameter. If the material of the particles is mutually soluble, then the bonding can be accompanied by new events [185]. Those of most interest are determined by a difference of the diffusion coefficients. If for example DA > DB then fluxes JA and JB of atoms A and B through the surface of contact will be different. Hence, extra vacancies will arise in body A near the interface and the rate of vacancy generation will be proportional to JA--ORB. These vacancies will be absorbed either by dislocations (then creep will occur, moving the interface in the direction of body A), or by immobile defects and inhomogeneities, leading to the formation of macroscopical voids in body A in the vicinity of the interface. The first event is usually called the Kirkendall effect, the second is called the Frenkel effect in Russian literature and also the Kirkendall effect in English literature.

Chapter XI HOT PRESSING

There is known a variety of hot-pressing methods for production of structural components. Normally, semi-fabricated materials are processed by using such methods. Therefore we start with a description of fabrication of composite precursors. At the same time, such methods as rolling and explosive welding are discussed in this chapter, although they are not hot-pressing in a strict sense. Some composite materials normally obtained by hot pressing are also described in this chapter.

11.1. Fabrication of composite precursors Hot pressing like many other technological schemes is performed by using premanufactured semi-fabricated products. It is similar to well known prepregs in the fibre reinforced plastics technology. Preliminary bonding of fibres to a matrix appears to be convenient for further fabrication of structural components. A number of methods can be used to prepare composite precursors. Some of them, liquid infiltration being an example, is presented in other chapters.

11.1.1. Plasma sprayed tapes A jet of low temperature plasma had been used for deposition of metals and various coating on a solid surface long before the necessity of obtaining composite semi-fabricated products arose. Hence the suggestion to use such a process to deposit a layer of a matrix onto a set of fibres by Kreider [340] was a natural step. The process is usually conducted in the following way. Fibres are wound onto a cylindrical mandrel with a fixed pitch. Then drops of the molten matrix material are carried by a low temperature plasma jet to the mandrel surface, coating the fibres with a matrix layer. Then a composite layer with a weak matrix is cut along a generatrix of the cylinder and a precursor sheet or tape removed from the mandrel. Normally, the tape has one surface, the bottom one, being smooth, and the other one rough. A stack of monotapes is to be densified and sintered in a process of producing a particular structural component. Besides the factors important in assessing any fabrication method (output, energy consumption and so on), in this particular method we have obviously to be 475

476

Hot pressing

Ch. XI, w

interested in the following factors: properties of the matrix obtained, the influence of the process on the mechanical properties of the fibres, the quality of fibre packing, and the possibility of effective processing of the semi-fabricated material. We shall consider some of these factors, looking mainly at the process of making aluminiummatrix composites. Porosity of the plasma sprayed tape is inevitable; in some cases it can be useful the oxide layer at the fibre/matrix interface is being broken and an interface bond can then be formed as a result of the large displacement at the interface during densification of the matrix [548]. The porosity depends on spraying parameters, such as the electric power, the powder size and so on. The dependence of the porosity on the powder size has a minimum [297]. It should be noted that the porosity of a tape is larger than the initial porosity of a single tape due to roughness of one surface of the monotape mentioned above. During spraying, the chemical composition of a matrix can change due to oxidation if the process is conducted in air or in atmosphere of an impure inert gas. Measurements of oxygen content in aluminium-magnesium alloys after spraying conducted by Rycalin et al. [577] have shown that it changes from the initial value equal to 0.06-0.09% to 0.7-0.8% after spraying in air, to 0.27-0.31% after spraying with a local protection by argon gas, to about 0.25% after spraying in a box with an argon atmosphere, and to about 0.22% after spraying in the same box with a zirconium getter. The same authors studied also the formation of a structure of the aluminiummagnesium alloys after spraying and found temperatures between 560 and 580~ to be optimal for hot pressing. The alloys hot pressed in this temperature interval have the highest tensile strength, although such temperatures are too high for the boron fibres to preserve their strength (see Table 11.3), so the optimal temperature interval would appear to be lower. Decreasing the oxygen content in a matrix permits a reduction in hot pressing temperature (corresponding to a maximum value of the matrix tensile strength) by about 30-40~ Titanium matrix can be sprayed under carefully controlled atmosphere to prevent gettering oxygen or hydrogen from the environments. Kieschke et al. [312] conducted plasma spray deposition of titanium onto silicon carbide fibres using the following spraying conditions: Chamber pressure (Ar)." 150 mbar. Gun-substrate distance." 330 mm. Gun current." 750 A. Plasma gas flow rates." Ar: 11 1/min, He: 25 l/min, H2:81 1/min. Powder size range." 45-63 ~tm.

Processing under these conditions lead to absorption of hydrogen by titanium droplets during their flight life. This causes a specific behaviour of the diffusion barrier mainly composed of yttrium oxide.

Ch. XI, w

477

Fabrication of composite precursors

11.1.2. Powder metallurgy prepregs Although powder metallurgy techniques shall be discussed in the next chapter, we describe here some processes based on using either powder or some other kind of the precursor as a source for the matrix material for making prepregs to be used in hotpressing fabrication routes. These routes are normally used for making ceramic matrix composites.

Slurry impregnation This is perhaps an oldest process for producing a prepreg by the powder metallurgy route. It was described in detail by Phillips [533]. A general scheme of making a tape prepreg from a fibre tow is shown in fig. 11.1. Either a row of single filament or single filament or a tow of the fibres is passed through the tank containing slurry of powdered matrix material suspended in an organic solvent with an organic binder. Slurry impregnates the tow or row of fibres, powder adheres to the fibre and the prepreg is then wound onto a take-up drum and dried. In the case of fibres supplied as a slightly twisted tow, the tow is to be converted in a tape by moving it, for example, through a device which consists of a series of rollers and nozzles producing air jets to fan the fibre out. To stimulate impregnation, the slurry may be agitated by air flow from the bottom of the tank or by some other method. The fibre volume fraction, vf, in the prepreg is controlled by the content of the powder and binder in the slurry, the values of vf between 20 and 60% can be achieved.

Slip casting A powder matrix precursor can be obtained as a layer containing fibres by a method of tape casting. In a particular disclosure [565], to decrease porosity of a silicon nitride matrix obtained by reaction bonding, the matrix precursor contains silicon carbide powder in addition to silicon powder. A tape of the casting slip is

FIBRE

FIBRE

SPOOL

!

TO W

TO W ~

TAPE C O N V E R S I O N

\

i

TANK

T A K E - UP D R A M

I

++':+:+:+:+;+:+:+:+:+;+:':':+;+; :+

Fig. 11.1. A scheme of producing a tape pre-impregnated with slurry.

Hot pressing

478

Ch. XI, w

prepared by mixing two materials mentioned and corresponding sintering aids with a solvent, a surfactant to help to disperse the powder, a polymer binder, and associated plasticizer. The slip is cast over fibres (SCS-6 SiC) wound onto a drum. An adjustable gate is used to control the tape thickness. Before nitriding to convert silicon into silicon nitride, which is done in alumina tube furnace using a flowing mixture of 90% N2/10% H2 at temperatures up to 1370~ the prepregs are taken out from the drum, stacked in a graphite die and hot pressed to remove polymer binder and partially sinter silicon with the matrix. 11.1.3. Other processes

Ion-plating The ion-plating process can be conducted at low temperatures and that is its advantage when it is used to make a precursor from two or more reactive components. On the other hand, it is a slow and energy consuming method to be considered as a real candidate for a large scale production. A schematic drawing of the apparatus suitable to execute ion plating (due to Ohsaki et al. [513]) is presented in fig. 11.2 without showing a mechanism for making the continuous precursor tapes.

Vapour condensation Condensation of a vapour of the matrix material on the relatively cold fibre can have an advantage of a low temperature process. One can expect to exclude chemical reaction during the deposition. Ar~

VImuum out

l fibre

--

High (') AI in

voltage

Fig. 11.2. Schematic draw of the ion-plating apparatus according to Ohsaki et al. [513]

Ch. XI, w

Fabrication of composite precursors

479

The process was carried out for coating thick SiC-fibre with a titanium alloy, a titanium alloy strengthened by disperse yttria particles, intermetallic compounds Ti3A1 and TiA1, and an aluminium alloy (A1-4.3Cr-0.3Fe) [682]. Metal matrices were evaporated by an electron beam by using electron beam accelerated by 10 kW to heat a double evaporation source. The deposition rate occurred to be approximately 5 to 10 ~tm min -1. Coating with titanium and aluminium alloys were performed from a single evaporation source. Dispersion of Y203 in a titanium alloy was produced by evaporating yttrium from the second evaporation source that combines with the oxygen in the solid solution. Coating with titanium aluminides was organized by evaporating Ti-6A1-4V-alloy from one source and pure aluminium from another one. Quench rates for vapour deposition were estimated to be 1013 K/s, compared with 104 to 108 K/s for liquid-quenching methods. Therefore, the potential is expected with vapour quenching for producing matrix alloy with extended solid solubilities, or very fine dispersions of reinforcing phases that can improve matrix properties.

Magnetron sputtering Magnetron sputtering described above (Section 10.2.5) as a method of fibre coating is also used by Dudek et al. [138] as a method of producing composite precursor. The advantages of this method are homogeneity of coating thickness which results in homogeneous fibre distribution in the composite structure, a possibility of a strict control of chemical composition of the coating, and small grain size of the matrix material with corresponding benefits for the composite properties (Section 5.2.5). In particular, this method was used to produce titanium aluminide matrix composites.

Foil precursor Using a foil as matrix precursor has an advantage of retaining during a composite fabrication process of a microstructure of the matrix material normally yielding such characteristics of plasticity and fracture toughness that are wanted to attain sufficiently high values of admissible fibre volume fraction and, correspondingly, high strength and stiffness values of the composite (see Section 5.2.5). To join boron fibres and aluminium alloy matrix, Mileiko and Gryaznov [429] used simple techniques. One of them consists of winding the fibre on a drum similar to that used for the matrix plasma-spraying, then covering the fibre with the foil, and finally pressing periodically the foil into the fibre array (fig. 11.3). The length of the stamp and the pitch of stamping are chosen so as to ensure a minimum fibre damage and not to affect the composite strength. To arrange fibres unidirectionally and uniformly and retain the arrangement during consolidation is possible to punch a ductile wire into a fibre array wound onto a drum or to crossweave a fibre layer with a wire. A problem arises when an alloy which is difficult to obtain in a foil form, is chosen as a matrix. In such a case, a technique called "powder cloth" can be used [59, 345]. The powder of a necessary composition is mixed with an organic binder and a

480

Hot pressing

Ch. XI, w

Drum Fig. 11.3. Joining fibre and matrix foil.

wetting agent and then the dough-like mixture is rolled into a foil-like material. During the processing the wetting agent and binder are evaporated, although it is difficult to ensure that a complete removal of the excess material has taken place. Dobbs et al. [131] reported hot isostatic consolidation of powder to produce titanium aluminide, Ti2AINb, foil which needs just a small degree of cold rolling to achieve a final thickness of about 0.125 mm.

11.2. Processing parameters Choosing hot pressing parameters, one should consider the following processes: (1) removing long wave roughness of either the matrix surfaces or both the matrix and fibre surfaces; (2) removing short wave roughness on surfaces of interest; (3) removing voids at the interfaces; (4) development of matrix material properties; (5) formation of either physical or chemical bonding at the interfaces; (6) growth of the reaction zone at the fibre/matrix interface. The first two processes yield the mechanical consolidation or densification of a fibre/ matrix mixture. The third process is actually sintering that was considered in the previous chapter (Section 11.5). Sintering is also involved in the formation of the microstructure of a plasma-sprayed matrix and the development of matrix properties. The fifth and sixth processes are normally accompanied by the former ones and provide, at first, occurrence of the interface bonding and its increase, then either degradation of the bond or a negative contribution to the composite properties (see Section 11.3.1). Thus, a choice of the process parameters is to be performed taking into account the kinetics of the processes mentioned. At the same time, there can be some limitations to the parameters to be chosen. First, hot pressing can cause an excessive fibre breaking which yields a decrease in

Ch. XI, w

Processing parameters

481

composite strength. Some purely technical or facility limitations can also be imposed.

11.2.1. Densification An instructive experiment was carried out by Shioiri who studied bond formation between two titanium surfaces (see [419]) by measuring the intensity of an ultrasonic pulse reflected from the interface. There were revealed two characteristic rates of decreasing the intensity (fig. 11.4). The first characteristic rate relates to disappearance of long wave roughness of the surfaces and the other one relates to short wave roughness. If the load is removed at t < t t and the specimen kept at high enough temperature, we will never observe the strength of the interface equal to the strength of the bulk material. But if the same is done at t > t I, then the strength of the interface inevitably reaches the strength of the bulk material. Plastic smoothing of the surface is usually modelled [294] by deformation of a series of wedges by rigid smooth surface (as shown schematically in fig. 11.5a). A first study of this problem was undertaken by Ushizki [668] (see also [419]) who obtained asymptotic expressions for the pressure and contact area in the framework of rigid/ideally-plastic formulation of the physical problem. When considering a case of the densification of a plasma-sprayed monotape, Elzey and Wadley [154] performed a study of the problem in a way more suitable for numerical procedures. They modelled an interface between two monotapes as a layer of height h between a smooth plane and a surface composed of a stochastic set of the asperities (fig. 11.5b). The probability density functions describing distributions of the asperity heights and radii are q~h(h) and q~r(r), respectively. Assuming the mutual independence of these functions, the probability density of an asperity of height h and radius r is ~c(h,F) = q)h(h)q~r(F).

(11.1)

Fig. 11.4. Dependence of intensity of a reflected pulse on pressing time for diffusion bonding of two titanium surfaces in Shioiri's experiment (a scheme).

Hot press&g

482 ////////////.

Ch. XI, w

"////////////

~///////////////////~

~////////////////////A

V Fig. 11.5. Schematic of the plastic smoothing during hot pressing. (a) A rigid surface acting on a plastic wedge (~). A real picture of large deformation (fl) is replaced by a simpler scheme (7) to be analyzed. (b) Densification during plastic deformation of a stochastic set of the asperities. (c) A matrix material yielding through a lattice of rigid fibres.

In particular calculations by Elzey and Wadley [154], the normal distribution for the heights and the exponential one for the radii were adopted with justifications found in experimental observations. Hence, q>h(h) - x / ~

exp - ~

~

(11.2)

and q~r(r) -- 2exp ( - 2 r )

(11.3)

where h and h are the mean height and the standard deviation of the heights, respectively, and 2 is a parameter of the radii distribution. The probability density distribution of forces required to cause plastic deformation of asperities will be ~f(h, r) - ~c(h,

r)Fc(h, r,z,~)

where Fc is the contact force required to deform a single asperity.

(11.4)


Ch. XI, w

483

Integrating eq. (11.4) over all asperity heights and radii encountered in compacting the layer from z0 to z, the total force required to continue the smoothing will be F ( z , ~ ) ---

f Z0jr0~176 ~f(h,r)

/zZO/o

(11.5)

drdh

~c(h,r)Fc(h,r,z,~.)drdh

(11.6)

and the corresponding pressure is (11.7)

q(z,k) = n . F(z,k)

where n is the number of asperities per unit area. As in the Ushizki's solution, the theory of perfect plasticity is applied to find the force acting on a single asperity to cause plastic yielding, that is

(ll.8)

Fc = acac = ]7oyac ~ 2?+r(h - z ) f l o ' y

where ~rc is the contact stress, cry is the yield stress,/3 is a constant derived from a plasticity theory solution,/3 ~ 3, ac is the contact area, ac ~ 2 g r ( h - r). Now eq. (11.7) gives the integral equation for the pressure to cause plastic smoothing of the surface as a function of displacement z. However, solid state bonding is normally conducted at elevated temperature when creep of metals is essential. If one uses a power creep law, eq. (10.36), then it should be noted that at relatively low temperatures the value of O"m is high and the value of the exponent m is large. (Examples of temperature dependencies of O'm and m for some alloys which can be used as matrix materials, are shown in fig. 11.6.) So at low temperatures the rigid-plastic analysis can be considered as a good approximation, if the characteristic stress O'm for a characteristic time t oc qm1 is taken as the yield stress ~rv. But at high temperatures, which is the usual case, we have 1 < m < 3 and a creep problem for large deformations has to be solved. Actually, to analyze a temperature dependence of the consolidation process determined by the matrix creep, well known creep-rate/temperature/stress dependencies can be used. Dorn equation [184] is appropriate: -- B

exp(-Qc/kT)

- A

exp(-Qc/kT)

(11.9)

where E and p are the Young's and shear moduli of the creeping material, respectively, A and B are corresponding constants, and Qc is the activation energy for the creep process. In addition to fig. 11.6, we present values of parameters involved in eq. (11.9) for some titanium based alloys (Table 11.1).

Hot pressing

484

Ch. XI, w

10.0

4

E 3

o ~ 2024-T6 - - o ~ AI-6%Mg

a m

o

7.5 5.0

2

E

2.5

1 450 5

b

1

500 ',,

.

,

.

,

4

0.0 550 9 1000 100

E 3 ....

-.......

2 -

..

rJ-3A~-~. Ti-4AI-3Mo- l V I

600

,

I

,

I

800 T / o c 1000

Fig. 11.6. Temperature dependencies of the creep parameters of two aluminium and two titanium alloys. The Creep law is ~ = rh,(a/t~,,)", r/,, = 10-4s -1. After Rabotnov and Mileiko [559], characteristics of A16Mg alloy at high temperatures were obtained by S.V. Trifonov.

Smoothing the surface carrying a stochastic set of the asperitites (Elzey and Wadley's model, eq. (11.1)-(11.7)) was used to evaluate a kinetics of densification due to removing rough waviness of the surface as a result of creep [154]. The stress, ~, in a creeping asperity under a contact stress, ~ c - Fc/ac where ac - ~zx2, is assumed to be

Ch. XI, w 1.2

Processing

485

parameters

TABLE 11.1

The creep parameters of titanium based alloys. After compilation by Elzey and Wadley [154]. Alloy composition

Young's Modulus, E(T) GPa

Ti-6A1-4V

T < 500~ 9115-0.056. T T > 500~ 9172.4-0.16- T T _< 500~ 9100-0.04. T T > 500~ 9140-0.12. T 172-0.03 9T

Ti3AI + Nb (Ti-24Al-11Nb) TiA1

A h -1

m

Qc kJ/mol

8.4.1024

4.0

280

6.0.1017

2.5

285

7.6 91022

4.0

300

(11.10)

O " - C10"c

where Cl is a constant. The displacement rate, ~, must scale with strain rate ~ and with the radius of contact, x, which yields ~ - c2~x. Therefore, "Z - - C 2 YlX

( C l tT-------~c

(11.11)

.

\ O'm //

The constants e l and r are determined by satisfying the condition of reducing eq. (11.11) to the perfectly plastic solution at m ~ c~ and to the elastic Hertz solution when m - 1, ~ - e and (r/q0)-ltrm - E, where E is the Young's modulus and r/0 is the time unit. This yields Cl = fl and c2 = 1.36rcfl, where fl was already determined. Hence, ~_ --

O-c

1.36~zfll-mxrl

m

(11.12)

.

Taking into account that ac - ~ x 2 on an asperity into eq. (11.7) yields

~

27tr(h -z)

and substituting the force acting

Z' ' n mexp[ m (ll.13)

}-m •

J/O"

r 1-1/m exp(--2r) dr

.

Here eqs. (11.2) and (11.3) have been used and

--

1.36.21/2-mrl(Tzfl)l-1/m.

Equation (11.13) is to be solved numerically. When considering the hot-pressing of a fibrous composite made of a stack of fibres and foil, we see that the matrix, at least at the initial stages of the process, flows through a lattice of the rigid fibres (cf. fig. l l.5c). At the matrix/fibre

Hot pressing

486

Ch. XI, w

b o u n d a r y there exist both normal and tangential components of stresses. Such a process can be modelled as flowing of a rigid-plastic matrix in a convergent channel and a result of one such solution can be found in [658]; namely, the dependence of pressure q on the ratio of df to distance L between fibres is q c~ am ln(1 - d f / L ) -1.

(11.14)

The value of the logarithmic term changes with a factor 2 when the fibre volume fraction changes from 0.2 to 0.5. Goetz et al. [197] derived a closed form expression to estimate the consolidation time of an assemblage depicted schematically in fig. 11.7 when the matrix creeps according to eq. (11.9). Their result can be written as

l)f (df -k- r

{ cx Lm_l

d~

1 exp(Qc/kT)

A---N q

"

(11 15) "

Same authors performed a numerical simulation of the process accounting for the effect of shear friction coefficient, M, at the fibre/matrix interface. They found that time t" to complete 97% of the consolidation is approximately equal to the time to complete the final 3% of the consolidation which is in qualitative agreement with Shiori's observations of the behaviour of plane models (fig. 11.4). It occurred to be convenient to introduce an average effective flow stress # dependent on the effective strain rate ~, which was taken as the quotient of the average effective strain, ~, for a particular geometry, and time t". The simulation results show that for a wide range of the hipping parameters ( Ti3A1 + Nb-matrix, SCS-6 SiC-fibre, q = 10 - 500 MPa, T = 760 - 980~ M -- 0.1 - 1), the ratio, p, of isostatic pressure, q, to the average effective flow stress was a function only of M, so

d

/

Q

Q

,,,

Fig. 11.7. Schematic of the consolidation process of a fibre/foil assemblage.


Ch. XI, w

487

p = p(M) = q/~. It occurred that p = 1.27 for M = 0.1 and p = 1.61 for M = 1. In order to evaluate the soak time tt (cf. fig. 11.4) necessary to form the complete physical contact, we need to obtain a combination of temperature T and pressure q which provides the result without unwanted effects like fibre degradation (dissolving, breaking) or formation of too thick an interface layer containing products of chemical interaction between the fibre and the matrix. If an optimum combination of fabrication parameters (To,qo, to) is known for a composite with one matrix material, then a first approximation to an optimal set of parameters for a composite with other matrix material can be obtained assuming creep of the matrix being a determining process. Note that for the power creep law, the solution of a creep problem is characterized by an interesting property [558]. Namely, if all external loads increase proportionally to one factor, say 2 and for one value of 2, say 20, a solution of a problem has been obtained, and if we know the stress field r 1j and the dist~lacement field u!1j~ (X), then for an arbitrary value of 2 the stress field will be (2/20)~r,, and the displacement field ()0 m (.), 0 9 ij . 9 be ((2/20) u,, ). Therefore, if creep parameters for two matrix materials O'm ( T ) , a~)(T), m0({), ml(T), and rt0--rtl = rt are known, and it is possible to assume m0 = ml = m in a temperature interval of interest, then temperature T1 is to be chosen such that Cr(m ~ - a(m1). If the possibility of changing the temperature is restricted by the chemical interaction, then the following equation has to be satisfied:

qoa~ ) -

(,0) "m ~

.

(11.16)

If m0 ~: m l then a consequence of the well known Calladine-Drucker's theorem in the mathematical theory of creep [558] should be used. The only generalized force here is q, so choosing the parameters, inequality ql

< 1 \q0j

(11.17)

-

has to be taken into account if ml > m0. This inequality can be especially useful if optimum sets of hot pressing parameters for a number of matrix materials with various values of the exponent m are known.

11.2.2. Sintering stage A final stage of the densification can be influenced by the process of removing small voids at the interfaces (matrix/matrix and fibre/matrix) according to stress induced diffusion as discussed in Section 11.5. An evaluation of the corresponding contribution can be done again in the framework of the geometry depicted in fig. 11.5b and taking the Nabarro-Herring

Hot pressing

488

Ch. XI, w

equation, eq. (10.31), as a base for a description of the diffusion flow [ 154]. Assuming the total flux of matter to be the sum of grain boundary and volume fluxes, that is ~2

~r O( ( 6 D b nt-

2pDv)~--~c

(11.18)

and the void has a special geometry, one obtains r3

(z - -~ 9(z)~.

(11.19)

Here Db and Dv are the grain boundary (along the interface) and volume diffusion coefficients, respectively, 6 and p are the boundary thickness and the radius of curvature of the neck around a sintering front, r is a contacting sphere radius in the geometry assumed, 9(z) is a function determined by the geometry. Combining eq. (11.18) and (11.19) gives the dependence between the force applied to an asperity and the displacement rate:

r2 kT Fc -- ~ 9(z)ac --~ 6Db -~ 2pDv

(11.20)

which should be also substituted into eq. (11.7) to obtain an integral equation for z. In the case of a plasma-sprayed matrix, densification and sintering at the monotape/monotape interface is accompanied by densification and sintering of the matrix. A study of sintering mechanisms considered in Section 10.5 was also conducted by Elzey and Wadley [154].

11.2.3. Development of matrix properties Formation of the matrix microstructure during hot pressing of a plasma-sprayed matrix precursor is determined by densification and sintering processes discussed above. At the same time, melting and subsequent rapid cooling of a matrix alloy during plasma spraying can essentially change a structure of the future matrix, when compared with the structure of a nominal alloy (which is usually taken from a series of wrought alloys). A possible deterioration of mechanical properties of the alloy can lead to a corresponding deviation of composite properties from those predicted assuming nominal alloy characteristics. That is why the investigation of alloy properties after its spraying and various treatments is of importance. Alipova et al. [11] carried out such a study of the aluminium-zinc-magnesium alloy which could be strengthened by heat-treatment. They directed the study towards fabrication of boron-aluminium composites by hot isostatic pressing. The specimens were obtained by depositing the matrix layer-by-layer in air, the density of the sprayed material being from about 10 to 15% less than that of the nominal alloy. The oxygen content in the original alloy was 0.01%, in sprayed specimens about 0.11% by weight. The strength of these specimens was about 50 MPa. Then the specimens were placed into a vacuum container and isostatically hot pressed in a high


Ch. XI, w

350

.....

,-

, , , , ....

489

, ~ , , , _

o

5

300

5

b~ 250 200

150

I000

tO0

200

300

~- /

400

m4,n

Fig. 11.8. Dependence of the strength of a plasma sprayed A1-Zn-Mg-matrix alloy on equivalent time of hot pressing. Experimental data by Alipova et al. [11].

t e m p e r a t u r e autoclave. The a u t h o r s gave a table of the pressing p a r a m e t e r s , n a m e l y t e m p e r a t u r e T, pressure q a n d time t, as well as strength a m of final materials. A n analysis of the e x p e r i m e n t a l d a t a was u n d e r t a k e n in [419] a s s u m i n g the existence of the equivalent time expressed as z-

t

q

exp

-

(11.21)

where n, a . , a n d T. are c o n s t a n t , the latter being d e t e r m i n e d by the activation energy of a process to p r e d e t e r m i n e the result. A set of the c o n s t a n t s can be o b t a i n e d which supplies a best c o r r e l a t i o n between the m a t r i x strength, a m, a n d the equivalent time, z. The c o n s t a n t s are f o u n d to be n = 2, a. = 40 M P a , a n d T. = 1000K. The e x p e r i m e n t a l points on the am-Z plane are s h o w n in fig. 11.81. The e x p e r i m e n t a l d e p e n d e n c e calls for either p o w e r or e x p o n e n t i a l a p p r o x i m a tion. T h e p o w e r a p p r o x i m a t i o n s h o w n in the figure, t h a t is , a m -- a 0

(~00)r

,

fits the e x p e r i m e n t a l d a t a if a0 = 153.9 M P a , z0 = 1 min, a n d r = 0.1254. 1 A quantitative error in calculating the equivalent time in fig. 37 of [419] exists.

(11.22)

490

Ch. XI, w

Hot pressing

The ultimate elongation can be made higher by annealing in vacuum, and a standard heat treatment after annealing leads to a strength increase and an ultimate strain decrease. Similar results are presented in fig. 11.9. One can see that there are optimal sets of the processing parameters for the hot-pressing of a plasma-sprayed matrix depending on particular requirements to the matrix. F r o m a very general consideration supported by the results presented in the previous sections, we could write down eq. (11.21) in the form: -- to (~m'm) q

(11.23)

mexp(-Qc/kT).

However, as we have just seen, a number of the processes are really acting and we do not know a priori which one dominates. Therefore, we could change to a consideration of the corresponding rates determined by different activation energies and pre-exponential terms, then to sum the rates (under some assumptions) and finally to derive the total time. Without doing so and going to point out an empirical nature of eq. (11.21), we write it in terms of purely empirical constants, although for a more general discussion, it is perhaps more convenient to use eq. (11.23). Note that if temperature is changing during the hot pressing procedure and the pressure is constant, we can write a cumulative equivalent time as

0~0t e x p ( - Q / k T ( t ) )

7 cx

(11.24)

dt

where Q is an appropriate activation energy.

l

350, 300 \ *b~ 250

00000

Pre.sed i.n air

~

I

200 150

-

i

-

i

~

-~

~

-

O"

\

Yield s~ress Ultimate s~rength

100

527

5

J

1

500

I T /

I

~

525

550

Fig. 11.9. The yield and ultimate strength of the AI-6Mg alloy after plasma spraying and hot pressing versus temperature of pressing. Experimental data after Gukassyan et al. [211].


Ch. XI, w

491

TABLE 11.2 Interfacial shear strength in SiC/Ti monofilament composites Matrix

Filament/ coating

Temperature/time/pressure ~

Shear strength MPa

Source

Ti-6A1-4V Ti-6A1-4V Ti-6A1-4V Ti-6A1-4V Ti-6A1-4V Ti-6A1-4V Ti-25Al-10Nb-3V Ti-25Al-10Nb-3V Ti-25Al-10Nb-3V Ti-15V-3A1-3Cr-3Sn Ti-15V-3A1-3(~r-3Sn Ti-15V-3A1-3(3r-3Sn Ti-15V-3A1-3(3r-3Sn Ti-15V-3A1-3('x-3Sn

SiC/C/TiB2 SiC/C SiC/C/TiB2 SiC/C SiC/C SiC/C SiC/C SiC/C SiC/C SiC/C SiC/C SiC/C SiC/C SiC/C

925-0.5-30 925-0.5-30

350 -+-35 192 + 20 90-120 156 165 167 111 93 99 124 167 154 119 131

[469] [469] [693] [718] [718] [718] [718] [718] [718] [718] [718] [718] [718] [718]

As received Annealing 800-50-0 Annealing 800-100-0 As received Annealing 800-50-0 Annealing 800-100-0 As received Annealing 500-50-0 Annealing 500-100-0 Annealing 800-12-0 Annealing 800-50-0

Test methods: fibre fragmentation for the first two lines, push-out for the remaining results.

11.2.4. Bonding The interface strength in a composite is determined by all the processes listed above. The importance of reactions taking place at the interface was already pointed out above, in Section 10.3.1. We refer also to diffusion bonding experiments in SiC/Ti system (Table 11.2).

11.2.5. Fibre damage During the densification of a plasma-sprayed tape, fibre can be bent if it is located between asperities. This can lead to a random breaking of the fibre. Obviously, with the processing temperature increasing and pressure decreasing the risk to break fibre decreases. Actually, such a dependence was observed in hot pressing experiments with SiC/Ti3A1 + Nb [209] and SiCw/A1 composites [254]. In the former case, the direct measurements of the density of the fibre breaks were conducted; in the latter case, the qualitative observation were supported by measuring tensile properties which increased with increasing hot pressing temperature, since aspect ratio of whiskers and the density of composites were improved with increasing fraction of liquid phase. A model to incorporate statistics of the fibre strength and that of a geometry of the structure of a plasma-sprayed matrix [155] as well as an attempt to evaluate a dependence of the composite strength on the fibre damage during the consolidation are also known [144]. Clearly, a severe fibre damage can occur in the case of a nonunidirectional fibre distribution. Also, the problem becomes very important when non-plane elements are densified (see below, Section 11.3.3).

Hot pressing

492

Ch. XI, w

11.2.6. Concluding remarks A knowledge of the parameters determining kinetics of the processes in hotpressing fabrication routes allows to design the fabrication process sufficiently quickly and choose a first approximation to necessary fabrication parameters a priori. Experiments to correct the parameters are necessary, especially if we remember that wanted microstructures of a composite can be different for different loading patterns (see Part II). Still, research efforts are directed at lowing a necessary experimental work. An example is given by Nicolaou et al. [492] suggested to plot time/temperature contours corresponding to full consolidation at various pressures and those corresponding to a given thicknesses of the reaction zone (fig. 11.10). The former family of the contours is given by eq. (11.15), the latter one is a graphical presentation of eq. (10.5). Lines for largest admissible values of the processing time (tmax) and temperature (Tma• are also plotted. There can be plotted regime contours corresponding to a fibre damage that should be avoided. Suppose we are looking for a composite with the reaction thickness zone h2. Then sets of the parameters such as q2, t2, T2 and q3,tl, T1 are possible physically and permissible technically, set q2, t2, T2 is possible physically but prohibited technically. It should be noted that a realization of the process with parameters yielding a processing time longer than it is necessary for full consolidation is obviously

hS

h~

h~

7"

>h2 >h~

Tmax

qt ~--

t2

tl

t, na=

q2

t

Fig. 11.10. A schematic of the processing parameters map suggested by Nicolaou et al. [492]. The lines corresponding to various pressures q present combinations of time, t, and temperature, T, which provide the full consolidation of foil/fibre/foil assemblage. The lines corresponding various values of h give combination of processing parameters which provide given values of the reaction zone thickness, h.

Ch. XI, w


493

permissible provided it does not yield the reaction zone thickness larger than has been prescribed. With regard to consolidation of a plasma-sprayed semi-fabricate, an analysis by Elzey and Wadley (see above) yields [154] density/pressure maps constructed for various processing temperatures (fig. 11.1 l a) as well as those plotted in density/ temperature coordinates for various pressures (fig. 11.1 l b). These maps are to be calculated accounting for densification mechanisms considered above. It can be also advantageous to choose technological parameters to produce a new material in a composite system based on both an approximate evaluation of the creep process in a composite during the consolidation time (eqs. (11.16) and (11.17)) and a known set of the parameters for a composite of the same system. An example is presented in [419]. The temperature dependencies of creep parameters for aluminium alloys D 16 (analogous to 2024) and A1-6Mg presented in fig. 11.6. F o r a boron-aluminium composite with the D16-matrix the set of the parameters of hot pressing close to an optimal one is

I

~,o~,...------~,~ - _

,~r ~,~

" I

~

I# / /

~'~ .~,

T = const

Ini~at density I

log

"-

I

I

l

~ ~ . ~

.~9"//

(ql(~y)

/g

/;

q = const Initial density

T/T m

Fig. 11.11. A schematics of the calculated densification maps due to Elzey and Wadley [154] showinghow the density (l-p) grows during the process (p is the porosity). Plastic yielding is supposed to occur instantaneously upon pressure application; contours corresponding to non-linear creep are calculated for various times t; diffusion flow (linear creep) contributes to the densification when the total porosity is sufficiently small.

Hot pressing

494

485~

25MPa-

Ch. XI, w

1.5h

In an experiment [419], a set of optimum parameters for the A1-6Mg-matrix (to maximize the tensile strength) was found as 530 ~ - 30 MPa - 1.5 h These two combinations of the parameters nearly satisfy eq. (11.16). It should be noted that a deviation from these parameters can give composites with lower tensile strength values. An example is the dependence of the tensile strength of boronaluminium composites with the A1-6Mg-matrix (vf - 0.28 + 0.02) on temperature of hot pressing at constant pressure q = 30 MPa and time t = 1.5 h. We have the following: Temperature of hot pressing Average tensile strength

~ MPa

520 396

530 496

550 402

It occurred that the same hot-pressing parameters are nearly optimum for compressive strength. Figure 11.12 illustrates the dependence of the critical compressive stress, cr,, of the same composite on temperature of hot pressing. It can be seen that the difference in critical compressive stresses for the composites fabricated at temperatures 530 and 550~ is most essential at small ratios of thickness h to length l of specimens. It is supposed to be a consequence of decreasing either the yield stress a m of the matrix or the shear strength of the interface (see Section 8.4). Finally, we should remember that using the equivalent time given by eq. (11.21) or eq. (11.23) is a helpful method to adjust fabrication parameters to a particular

!

'

'

!

I

'

'

!

"

0

~ o~ oo

%2

oO~

,,~176 %o

9

T = 520~

9

T = 550~

o

T = 530~

vO

i o

0o

olO 0

o

~

'

'~ 60

I/h

8'o

'

100

Fig. 11.12. Critical stresses at compression of boron/aluminium composite with AI-6Mg-matrix (vf =0.28 +0.02 obtained at various temperatures of hot pressing; q - - 3 0 M P a , t - 1.5 h). After Mileiko [419].

Ch. XI, w

495

Techniques

technological conditions. This is especially useful if one has to follow a fabrication regime with changing temperature or pressure as can be observed in a case of making sufficiently large components in a large die (see below, Sections 11.3.2 and 11.3.3).

11.3. Techniques The techniques of hot pressing used for producing metal and ceramic based composites are simple and sufficiently versatile. They provide the possibilities to fabricate plates, as well as structural elements such as tubes, open profiles, shells. 11.3.1. Plates

To obtain plates, it is possible to use a rigid die (see fig. 11.13). The temperature should be about 500~ for aluminium, about 800~ for titanium and titanium based materials, and about 1200~ for nickel based matrices including nickel aluminide Ni3A1. A pressure required is normally between 20 and 100 MPa. So corresponding materials have to be chosen for the die with either a built-in or external heater. A chamber should provide the necessary displacement of a die, and a vacuum or protective atmosphere. Using plasma sprayed precursors, it is possible to conduct hot pressing of aluminium matrix composites in air. Large relative displacements of the surfaces coated with oxide layers lead to fracturing of such layers and provide conditions for FLeXiBLe eLeMeNT

CHAMBER

(VACUUM OR PROTeCTIVe

CHAMBER 301NT .,

THERMOCOUPLe

ATMOSPHERE PLATe WITH HeATeR

BLANK PLATe WITH HeATeR

Fig. 11.13. A sketch of a chamber for hot pressing a composite.

496

Ch. XI, w

Hot pressing

bonding. Prewo [548] has shown that the process parameters in this case can be chosen such as to decrease pressing time down to 10 minutes. Using dynamic hot pressing studied in detail by Karpinos et al. [297] also leads to a decrease in the total process time. 11.3.2. Tubes

For consolidation of structural elements like shells and tubes, 'soft' die should be used [419] to follow changes in the curvature of the blank surface during consolidation process. But simple liquid autoclaves are of no use because pressing temperatures are too high for metal- and ceramic-matrix composites. Hence, they should be produced in either more expensive high pressure gas isostat or a kind of quasi-isostat . Weisinger [696] was certainly the first to suggest applying gaseous pressure to densify and sinter boron-aluminium tubes according to the scheme shown in fig. 11.14a. Precursor boron-aluminium tape is rolled on a thin-walled steel mandrel. Then it is inserted into a thick-walled outer steel tube. The steel tubes are welded together at the ends to give a vacuum-tight assembly. Evacuation can be done via a special tube welded to the thick-walled steel tube. After diffusion bonding in a gaseous isostat, the outer tube is machined to almost the same thickness as the inner tube and then both steel tubes are etched in nitric acid.

Z

a

\ \ \ - 1//

\ 1 ..~.~ q

.~,f~ q

b

/

\

Fig. 11.14. A scheme [419] of making a composite shell in a gas isostat. (a) Densification of a blank with the outer rigid wall; (b) The same with the inner rigid wall.

Techniques

Ch. XI, w

Mandrel

Diaphragm .\

d

Segment

497

die

.

.

/B lank

~ ~ ~ \ ' { ~ \ ~ \ \ \ \ \ \ S x N ~1 Ib,~\\\\\\\'Lxt' v2. See text for details.

specimens tested by computer to model the failure according to the weakest link scheme. In such a case, one observes a maximum on the strength/fibre-volumefraction curve at some blending time (see a plot for n = 4). Otherwise, a reason for the maximum on experimental curves can be a change in the blending time (or, in structural terms, inthe fibre length) with changing fibre volume fraction. The situation is illustrated schematically in fig. 12.6. Suppose in making a particular series of the composite specimens and aimingat a "homogeneous" fibre/powder mixture, one uses a blending time equivalent to n - 16 in fig. 12.5 for specimens with vf < @) and that corresponding to n - 32 for specimens with vf > v~2). Then curve OA will represent the a(vf) dependence for low vf and curve OB will do it for high yr. Without considering what happens in the (v~'),v~2)) interval, we can imagine the overall dependence as that represented by the curve OCDB. 12.4.2. Metal matrix composites We shall discuss some details of microstructure and mechanical properties of two types of the composites making an accent at titanium matrix composites.

Aluminium matrix composites Two characteristics of aluminium alloys can be improved by using the alloys as matrices for discontinuous reinforcement. The first is rigidity, and the second is a maximum service temperature. Actually, the Young's modulus of shortfibre/aluminium-matrix composites can be enhanced, under some conditions, up to two times [190]. High-temperature properties are determined by shear characteristics of the matrix (see Chapter 6); hence, the service temperature of the composites is certainly limited by a temperature of about 300~ [589].

Powder metallurgy methods

532 500

,

.

0 i

,

,

'.

.

.

Ch. XII, w

.

{-x..

13

400

300

200

0.0

,

.1

l

0.2

V,

'

0.3

Fig. 12.7. Tensile strength of various short-fibre/aluminium-matrix composites versus fibre volume fraction. The curve averages experimental data compiled by Girot et al. [190].

As we have seen, the dependence of the room temperature strength on fibre volume fraction is influenced by a number of fabrication and microstructural parameters. So if to plot experimental data on one graph, as done by Girot et al. [190], the scatter will be enormous. Still, such plotting is instructive. It is repeated in fig. 12.7 although in a form different from the original. Instead of experimental original points, an averaged dependence is plotted. The dependence looks similar to a schematic curve shown in fig. 12.6.

Titanium matrix composites Reinforcing a titanium based matrix with continuous fibres like SiC (see Section 10.2.2) aims at obtaining high performance composites for elevated and high temperatures. At the same time, it is wanted to get relatively low cost titanium matrix composites reinforced with short fibres to enhance elastic properties and creep resistance of the materials at elevated temperatures. High reactivity of titanium makes powder metallurgy processes to be perhaps the only possibility to produce such composites. We describe here a composite with short carbon fibres as a reinforcement. Using powder metallurgy methods to fabricate C/Ti composites is expected to promise positive results if one does not aim at a pure carbon/titanium composite. The aim should certainly be to find possibilities to make use of materials containing rather large quantities of titanium carbide. A routine powder metallurgy scheme

Short-fibre composites

Ch. XII, w

1200

"

I

-

~.,

I

9

9

.9O

'

. o

1000 -o

o ~

~

o

.. ~ o

0

I I

~

533

"

_

I

vf = 0.05

-

v,= 0.10 .. -.

~

o

800 0 0 600

0

,

I

20

,

I

40

,

t/rnin

I

60

Fig. 12.8. Bending strength of the C/TiC/Ti composites at room temperature versus blending time. Sintering temperature is 900~ sintering time is 30 min. The container was not pumped out. After Mileiko et al. [425].

consisting of blending, cold pressing, and hipping or hot pressing was used by Mileiko et al. [425] to produce specimens of three various shapes, those being discs, cylinders and rectangular plates. Blending is done in a ball mill. The dependence of the composite strength on blending time is shown in fig. 12.8. We see that the blending of the raw mixture, which homogenizes the mixture and, at the same time, changes the histogram of the fibre lengths, results in a non-monotonic mixing-time/strength dependence (compare with fig. 12.5). This yields a necessity to choose an optimum blending time for any particular case of the mixture to provide a compromise between the mixture homogeneity and mean fibre length. Cold pressing performed in a closed die leads to a decrease in the corresponding dimension by about a half of the original value. This can cause a definite texture of the future composite. Sintering disc specimens is done by hot pressing in a closed die under uniaxial external load in a vacuum furnace. The cylinders and plates being enveloped in steel cans are sintered in a hot isostatic process. After sintering at 850~ for 60 min, the X-ray phase analysis does not reveal an occurrence of titanium carbide in the composites. However, after relatively short heat treatment at temperatures of 880-900~ the carbide does clearly exist (fig. 12.9). A change in the measured density of the composites as a result of the heat treatment shown in fig. 12.10 allows [425] to calculate the carbide layer thickness also shown in fig. 12.10. The occurrence of high modulus carbide layer yields an


534 1200

C~T~,,

Vt=O.20,

880~

-

30

Ch. XII, w

"m.~.'R

1000

800

aoo i i

400

I I

200 0 1500

........

i .........

a,~..,l

,

I,

C/T~. vt=o.o5, 9o~ "C

...| ......... -

30

i

m,i~ T~tan

Kulon f~bre

~

T~,C

1000

500

25

30

35

40

45 2

50 theta

55

60

65

70

Fig. 12.9. Typical X-ray patterns of the C / T i C / T i composites. After Mileiko et al. [425].

increase in the Young' modulus of the composites illustrated in fig. 12.11. The increase can be essential. It should be noted that the initial elastic modulus of randomly three-dimensionally reinforced C/Ti composites is lower than that of the titanium matrix since the spatially averaged Young's modulus of graphite is known to be about 70 GPa which is less than that of titanium (110 GPa). A particular feature of the failure behaviour of the C / T i C / T i composites is an occurrence of a maximum on the strength/fibre-volume-fraction curve at very low fibre volume fractions, about 5% , see figs. 5.17 and 12.12. It is important that a thick carbide layer around the fibre does not lead to a decrease in fibre contribution to the composite strength, the strength is increasing with the thickness increase within the interval studied (fig. 5.17). The effect of the carbide layer on the creep resistance of the composites can be very pronounced. This is illustrated by a creep curve presented in fig. 12.13. We see that increasing ageing time which leads to an increase in the layer thickness yields a decrease in creep deformation.


Ch. XII, w12.4 4.4

535

0.5

I O

0.4 ~E

4.3

E :::L 4.2

0

%7

V

)E

0.3~ ~O

- 0.2

4.1

Density 9

o 4.0

,

0

Layer thickness-..

vf = 0.1

w(

vf = 0.1

vf = 0.2

v

vf = 0.2

I

,

100

I

200

,

t/

~

300 mm

0.1

0.0

,

400

Fig. 12.10. Measured density of the C/TiC/Ti composites with initial fibre volume fraction 10% and calculated carbide layer thickness versus heat treatment time. The sintering temperature-time pattern is 850~ - 60min. The heat treatment temperature is 1075~ After Mileiko et al. [425].

1,3

1,15

1,10 o1,2

~|| 1,o5 1,00

1,1

0,95

0,90 0,00

'

' 0,04

'

~

~ 0,08

'

0

12

1,0

0

,

/

50

,

I

100

,

I

,

I

150 tlmiRO0

,

250

Fig. 12.11. (a) Dynamic modulus of heat treated C/TiC/Ti composites normalized by that of the same composites before heat treatment versus initial fibre volume fraction. The heat treatment temperature and time are 1075~ and 60 min, respectively. (b) Normalized dynamic modulus of C/TiC/Ti composites with initial fibre volume fraction 10% heat treated at 1075~ versus heat treatment time. Both sets of the specimens were obtained under sintering conditions as shown in fig. 12.10. After Mileiko et al. [425].


536 300

.

,

.

,

.

,

Ch. XII, w

9

-x..

b

250

200

o

VT-6 9 VT-23

1

00.

,

O0

I

0.05

,

I

,

0.10

v,

I

,

0.15

0.20

Fig. 12.12. Tensile strength at 593~ versus fibre v o l u m e fraction for c o m p o s i t e s with titanium alloy matrices sintered in v a c u u m . After Mileiko et al. [425].

5

/

_

Stresses: 01: 103 12:103 2 3 : 103 34:190 45:258 5: 292

c~5

MPa MPa MPa MPa MPa MPa

Heat treatments: l" 1 0 2 5 ~ 2 h A 2: 1 1 5 0 ~ - 1 h i

3

1

0

~

0

l

50

J

I

100

l

~

J

J

1

150

I

I

l

I

t

200

Fig. 12.13. S h o r t - t i m e creep curves o f C / T i C / T i composites. T h e creep tests were interrupted by heat t r e a t m e n t s s h o w n in the fields. T h e values of the stress are also shown. After M i l e i k o et al. [425].

Ch. XII, w


537

12.4.3. Ceramic matrix composites

Graphite fibre/carbide matrix composites Results of two experimental series of fabricating and testing C/B4C composites show clearly, first, an importance of an appropriate choice of fabrication parameters and, second, a possibility to qualitative interpretation of the results by using a model derived above, Section 12.4.1. In the first series, the fabrication route was as follows [186, 187] 9 Adding the fibres chopped to a length between 1 and 1.5 m m to the carbide powder. 9 Blending the mixture, to which ethyl alcohol is added, the procedure being performed by hands until the mixture reaches a uniform state controlled by both its colour and visual texture. 9 Sintering the mixture in a vacuum chamber (10 -4 Torr) at an applied pressure of about 30 M P a and a sintering temperature was between 1700 and 1950~ 2 The second fabrication route included 9 Blending of a mixture of the powder and fibres chopped to the length of about 20 m m in a ball-mill with titanium balls. This is done in a distilled water environment. 9 Drying of the product at a temperature of about 60~ for 5 h. 9 Cold pressing of a stack of dried mixture layer in a closed die. 9 Sintering the stack by hot pressing in a graphite die installed in a vacuum chamber (10 -4 Torr) at an applied pressure of 40 MPa. Hence, the main difference between the two series is in a way of controlling the blending process. In the first one, it is controlled by an external appearance of the mixture; in the second one, the blending time is a controlled parameter. In the first case, the mixtures before sintering are certainly those of about the same degree of homogeneity but of different fibre lengths. In the second case, both structural parameters are not fixed. On the other hand, the external appearance of the mixture is a rather subjective factor. The behaviour of the matrix and composites with low fibre volume fractions prepared by the first fabrication route is described in fig. 12.14. Note that the pure matrix material becomes extremely brittle after sintering at sufficiently high temperatures. An addition of even small quantities of fibres (only 5% ) changes the fracture behaviour radically. The composite fracture toughness reaches a maximum at some sintering temperatures, for 30 min sintering the optimum temperature seems to be about 1850~ Strength/sintering-temperature curve follows the same pattern. About the same behaviour of the strength/sinteringtemperature curve is observed for the second fabrication route (fig. 12.15), with the 2 In Refs [186, 187] there is an error in temperature measurements. All the temperatures related to composites with boron carbide matrix are, actually, higher than it is shown in the papers cited by 150~


538

Ch. XII, w

10.0 9 E

"bc

o

Matrix 9 Vf = 0.05

7.5

\

5.0 9

8

2.5

.0

| 1800

=

1700

m

I

2000

T / .19 oC O0

Fig. 12.14. Fracture toughness versus sintering temperature of boron carbide and C/B4C composites versus sintering temperature. The mixtures were blended until approximately the same degree of homogeneity. Sintering time is 30 min. After Mileiko et al. [439].

14 400 12

( 10

300

200

100

7'oo

.

8'oo

.

. 9'oo TI~ ooo.

.

,;00

.

8'oo

9'oo TI~ ooo

i

Fig. 12.15. Strength and fracture toughness of C/B4C composites with 1)f = 0 . 2 versus sintering temperature. Mixing time is 180 min, sintering time is 60 min. After Mileiko et al. [439].

maximum at about the same temperature. Note that specimens with an enhanced fracture toughness reveal distinct pull out of the fibres as shown in fig. 12.16. The dependencies of both strength and fracture toughness of the matrix and composites on mixing time at constant sintering conditions are presented in fig. 12.17. The following observations are of importance. First, each dependence has a maximum. Second, points of the maximum for the strength and fracture toughness are observed at about the same mixing times; this is clearly seen for low fibre volume fraction, vr = 0.1,

Ch. Xll, w

Short-Bbre composites

539

Fig. 12.16. Fracture surface of a C/B4 composite. Sintering temperature and time are 1700~ and 30 min, respectively. Courtesy to M.V. Gelachov.

with rather sharp maximum, and less clear for larger yr. Third, with the fibre volume fraction increasing, a point of the maximum shifts to larger mixing times. The observations regarding strength characteristics are in qualitative agreement with the model presented in Section 12.4.1, although we can see that a real failure behaviour is certainly effected by a number of factors outside the model. Hence, the model can be considered as that of a heuristic nature. The observations of the fracture behaviour are in accordance with general views of an effect of the fibre/matrix interface on the fracture toughness of brittle matrix composites discussed in Section 4.2. Let us compare dependencies of the critical stress intensity coefficient for the composites and matrix on sintering temperature presented in figs. 12.14 and 12.15. We see, first, that at sintering temperatures higher than 1900~ the reinforcement contribution (at v~ = 0.05) provides nearly the whole value of the effective surface energy of the material, unlike the situation at lower sintering temperatures (fig. 12.14). Second, a tendency of fracture toughness of the

540

Ch. X l I , w 2.4


400

10

300

.

,

9

,

9

,

9

8

oo O, 9

loo F

0

~

,

,

o

Matrix

I o v,-o., I

9

loo

,

,

~

200 1

,

,

,

soo

v

t

,

I

o

J

2 t

400

9

, ,

o

,oo

,

9 VM,,nx f= 0.1

.

,

s6o

,

200

,

I

,

400

!

lo

400

300

.~

9

ai

9

6 200

/|

V, = 0.3

V,

=

03

4 100

.

.

.

.

400 t / m/ShOO

600

0

l

,

I

,

,

I

,

,

t / rain

Fig. 12.17. Strength and fracture toughness of C/B4C composites versus mixing time. Sintering temperature is 1800~ sintering time is 60 min. After Mileiko et al. [439].

composites to go down after some sintering temperature revealed in fig. 12.14 occurs actually to be a tendency to reach a minimum at a temperature of about 1900~ and to go up with further increase in sintering temperature (fig. 12.15). Such a behaviour can be interpreted in terms of non-monotonic change in the interface strength with changing sintering parameters. The interface strength increases at first with increasing a temperature and time of a sintering process, then it reaches a maximum and starts to decrease certainly because of occurrence of the interface porosity which may be related to the diffusion of carbon from the fibre surface into the matrix volume. Measuring electrical conductivity of the composites with various fibre volume fractions and then calculating the specific conductivity of the composite components including that of the interface, Sfm, reveals also a non-monotonic dependence of sn,~ on the time/temperature conditions of sintering (Table 12.2). To illuminate a possibility to reach high values of the strength, a*, and fracture toughness, K*, of short-fibre/ceramic-matrix composites, we plot, in fig. 12.18, maximum values of c: and K* reached at sintering temperature-sintering t i m e mixing time being 1800~ min-180 min (fig. 12.17), versus fibre volume fraction. Such a plot does not have more physical meaning than the usual representation of a type shown in figs. 12.14 and 12.15. All these plots are rather illustrations. Presenting fig. 12.18 we emphasize both a possibility to optimize the microstructure of a short-fibre composite by performing a systematical experiment and the necessity

Ch. XII, w12.4

541

Short-[ihre composites

TABLE 12.2 Electrical conductivity of the matrix, Sm, and fibre/matrix interface, Srm, for C/B4C composites sintered at various temperature/time regimes [187]. (Sintering parameters) T (~

Smo

Sfm

t (min)

(ohm-m) -I

(ohm-m) -I

1600 1650 1700 1700 1700 1800

30 30 10 30 60 30

1910 3670 2180 1960 2880 2360

425 721 1420 1960 824 847

- 10

400 -

Strength

9

o

300

200

o

100

0

0,0

~

I

0,1

=

Fracture toughness

I

0,2

=

I

0,3

5

Fig. 12.18. Maximum values of the strength and fracture toughness of C/Bc composites attainable at the fabrication parameters shown in fig. 12.15. After Mileiko et al. [439].

to build up a structural m o d e l of a predictive n a t u r e to reduce a n u m b e r of experiments w h e n d o i n g the o p t i m i z a t i o n . Whisker/ceramic

matrix

composites

N o r m a l l y , for processing w h i s k e r - r e i n f o r c e d / c e r a m i c - m a t r i x c o m p o s i t e s a general scheme described in Section 12.1 is used. Sintering p a r a m e t e r s can vary d e p e n d i n g on whisker content. W r o n a et al. r e p o r t e d [708] that for m a k i n g SiC/A1203 composites with vf < 0.15, pressureless sintering was p e r f o r m e d at 1400-1800~ in an inert a t m o s p h e r e . A particular heating regime s h o u l d be c h o s e n d e p e n d i n g on the size a n d shape o f the article. W i t h whisker c o n t e n t increasing, the necessary regime

542


Ch. XII, w

becomes more sophisticated. It could be advantageous to introduce a soak time at an interim temperature, to use a controlled atmosphere, hipping under the pressure of 35-200 MPa, etc. Powder metallurgy processes involve dealing with whiskers, which are materials being toxic (see Section 2.2.6). Therefore, a modification of the routine scheme aimed at the formation of whiskers in situ [680, 716] can be of practical interest. The modification is called "chemical mixing process" and involves a stage of whiskerization reaction in a mixture of the powder of a matrix material and necessary reagents to form the whiskers. In particular, Yamada et al. [716] did blending of silicon nitride powder (the matrix material), carbon black (the carbon source for SIC), silica (the silicon source for SIC), and COC12 as a catalyzer for the reaction wanted. Also NaCI was added to the mixture as a space-forming agent for the growth of the whiskers as its boiling point is equal to the reaction temperature, so that porosity formed provides necessary space for whisker growth. The reaction was carried out at 1600~ for 1 h. After drying at 110~ and burning out of unreacted carbon, the reacted material underwent ball milling to deagglomerate the woolly balls with the strongly entangled texture. The mixture obtained was first pressed under 15 MPa at room temperature and sintered at 1800~ for 1 h in argon gas atmosphere. It is also possible to obtain silicon carbide whiskers in silicon nitride matrix by reaction of carbon added to the matrix with the matrix material [680]. The composite obtained in Ref. [716] had a fibre volume fraction from ~ 6 to ~ 3 2 % and its bending strength was ~440 MPa as compared to ~ 5 0 0 M P a for composites produced in a usual way and ~ 740 MPa for pure silicon nitride obtained under about the same conditions. The authors attributed the low values of the strength to an excess of carbon, another reason for this can be non-monotonic strength/fibre-volume-fraction dependence discussed above. We have seen that mechanical properties of composites with short fibres randomly oriented in space are very sensitive to fibre volume fraction, especially at low values of yr. Still, a majority of the published data presenting strength and fracture toughness values of such composites have been obtained in rather narrow intervals of the fibre volume fractions. This makes it difficult to draw reliable conclusions by analyzing the experimental data. An attempt to estimate an effect of whisker coating on strength and fracture toughness of SiC(w)/~i3N4 composites was reported by Matsui et al. [391]. Using the only sintering temperature (about 1700~ and the only composition of the material they followed the routine powder metallurgy way to produce the composites without whisker coating, and with various coatings. Mechanical testing of the material gives the results presented in Table 12.3. It should be noted, first, that the alumina coating enhances both room temperature strength and fracture toughness of the composites. The effect of the carbon coating is smaller. Zirconia coating decreases mechanical properties. The microstructure observed by using TEM reveals certainly no new phases at the interface in composites containing whisker without coating and with carbon coating. On the contrary, alumina coating yields formation of a new phase on the interface. Perhaps the only definite conclusion from these data is that for such composites strength and fracture


Ch. XII, w

543

TABLE 12.3 Average values of bending strength and fracture toughness of SiC(w)/Si3N4 composites: Experimental data after Ref. [391].

Non-coated fibre A1203 coating C coating ZrO2 coating

RT strength GPa

Strength at 1 2 5 0 ~ GPa

Fracture toughness MPa. m 1/2

0.99 1.11 1.05 0.86

0.79 0.67 0.92 0.60

8.7 10.2 7.8 7.4

toughness values change in the same direction when changes in composite microstructure occur. Actually it is now clear that the most important point in short-fibre/ceramicmatrix fabrication strategy is controlling the interface by either introducing a special coating or adjusting properly fabrication parameters. A strong interface can yield a high strength composite bat never sufficiently tough material. This makes the authors dealing with such materials to treat them as brittle ones as, for example, in the case of SiCw/MoSi2 composites [656].

12.5. Continuous fibre composites Composites of such a type were discussed in Section 11.8 (those with glass- and glass-ceramic matrix) and will still be discussed in Section 14.2.7 (oxide/oxide composites). In the present section, we shall consider oxide/oxide and SiC/SiC composites produced by using methods described in the present chapter.

12.5.1. Oxide~oxide composites There are a known number of oxide/oxide composites. Certainly no composites with properly designed interface have been obtained up to now. However, such a type of the composites is a very promising high-temperature material due to both potential high temperature strength of the components and their high oxidation resistance. So we shall definitely see oxide-fibre/oxide-matrix composites of an optimized microstructure in the near future. An example of producing oxide/oxide composites by using powder metallurgy procedures has been reported by Mah et al. [379]. Single crystalline sapphire fibres coated with pyrolytic carbon in a CVD process (coating thickness was about 2-3 ~tm) are aligned in the graphite die between the yttrium-aluminium-garnet (YAG) amorphous powder. Then it is heated up under a pressure of about 0.7 MPa in an argon atmosphere. As heating proceeds, the amorphous garnet begins to crystallize. At a temperature of about 1450~ a final pressure, that is 7 MPa, is applied and sintering takes place at 1650~ during 0.5 h. At that temperature, hard

544


Ch. XII, w

garnet grains penetrate the softer sapphire fibre. Hence, to avoid the indentation of the matrix grains on the fibres, a second fabrication regime is also used, that is sintering at a temperature of 1450~ However, in the latter case only 80% of the calculated density of the composite is reached as compared with 90% in the former case. The mechanical parameter obtained in these experiments was the interface shear strength, r~m, measured by push-out tests. It happens that the value of r~m varies from 2 to 85 MPa for as-hot-pressed composites with carbon interface coating that is much lower than that for composites without fibre coating. After oxidation at 1000~ for 2 h in air, r~m- 10-13 MPa. Heat treatment of the composite at 1500~ for 60 h yields r~m- 2-28 MPa. This is a promising result in the hope to preserve the carbon interface of oxide/oxide composites in an oxidizing atmosphere, to ensure the long-term existence of the microstructure that provides the crack resistance. In the same fibre/matrix system, both molybdenum and palladium interface were also studied [238]. Single crystalline sapphire fibres were coated with Mo and Pd by either magnetron sputtering or extracting the metals from metal containing compounds delivered to the fibre surface with appropriate solvents. Microstructural studies shown that it occurred to be possible to obtain palladium interface as the palladium coating could be preserved during hot pressing. This is not a case if molybdenum coating is used. Being observed, a metal interface seems to act as a crack deflector. Dealing with oxide/oxide composites it is always required to use non-oxidizing substances as interface materials. Hence, highly anisotropic oxides with mica-like cleavage characteristics look attractive. A family of oxides, so called magnetoplumbites with such properties is known and one of them, hibonite, CAA112019, has been used as the interface material [90]. The best microstructure, that is a layer of the oxide of a thickness of 1 to 2 lam, with a cleavage plane parallel to the fibre surface, occurs to be formed in A1203/AIsY3O12 composites produced by hot-pressing of the fibres with a sol-gel-derived hibonite and YAG powder doped with CaO. Shear strength of tubular specimen of polycrystalline-oxide-fibre/oxide-matrix composites with a tin oxide interface, which is partly determined by the interface strength is reported to be about 25 MPa [722]. A brittle nature of the failure behaviour of the composites yields a suggestion that the value cited is too high to trigger toughening mechanisms in the composite.

12.5.2. SiC/SiC composites SiC/SiC composites are mainly obtained by CVl-processes, although some examples of the fabrication of such composites by using pyrolysis of a liquid matrix precursor are also known. Such composites with 2D-structure produced by CVI were the first ceramic matrix composites of high fracture toughness, with effective surface energy at room temperature approaching values two orders of magnitude higher than that for unreinforced ceramics [484].


Ch. XII, w12.5

"

"

!ij, rlIlill

I 1 ' ' ' ~ 1

i i i i i i i

I

545

B

i

Fig. 12.19. Schematic interpretation of the microstructure of the interface zone in SiC/SiC composites produced by CVI. Fibre of Nicalon type containing oxygen are coated with carbon to form the interface. Total width of the interfacial zone is of order of tens nm. Experimental observation due to Naslain [484].

The stress/strain behaviour of the composite is clearly non-linear which is caused by non-reversible processes discussed in Chapter 4, those being fibre pull-out, crack bridging, interface debonding etc. Naslain [484, 485] related microstructure of the interface zone to mechanical behavior of the composites. His interpretation of TEM study of the interface is presented in fig. 12.19. Comparing the interface structure and mechanical behaviour of the composites yields Naslain to the following conclusions. First, it should be noted that the layer of carbon (or boron nitride) introduced into the structure predeterminely is not homogeneous. A sublayer near the fibre surface occurs to be very anisotropic. Under some fabrication conditions, either continuous or discontinuous layer of silica arises between silicon carbide core of the fibre and anisotropic carbon layer. Second, the occurrence of a continuous glassy silica layer in the interface (case A in fig 12.19) makes the weakest interface. The disappearance of a continuous anisotropic carbon layer yields the strongest interface (case C). Case B is an intermediate one. Third, the stress/strain behaviour changes with changing the microstructure and properties of the interface as shown schematically in fig. 12.20. The strong interface corresponds to a quasi-brittle behaviour, the weakest one corresponds to a rather low ultimate stress, and intermediate one corresponds to non-linear behaviour with relatively high ultimate stress. Room temperature fatigue limit on the base of 106 cycles under R -- 0 loading of a two-dimensional structure in a fibre direction is estimated [485] as about 85% of the tensile strength. A type of the damage under fatigue loading is nearly the same as that at monotonic loading. High temperature behaviour is mainly controlled by both the fibre and interface stability. The former is determined by oxygen content to a large degree (see Section 2.2.2), so the fibres with high oxygen content yield the composites that retain their strength as well as all the features of mechanical behaviour up to about 1000~ only [484, 485]. Degradation of the interface is determined by oxidation of the carbon (or

546


Ch. XII, w

B

Fig. 12.20. Schematic of the stress/strain behaviour of SiC/SiC composites with different interface properties. A, B, and C cases are illustrated in fig. 12.19. Representation due to Naslain [484].

boron nitride, at higher temperatures) interlayer. Certainly, the oxidation resistance of carbon layer can be enhanced if the total thickness of it is divided into thinner sublayers surrounded by SiC layers [485]. SiC-Nicalon/SiC-CVI-matrix composites exhibit good thermal shock resistance: they retain the room temperature strength following up to 50 thermal shock cycles of 1 s time at the temperature difference, AT equal to 1700~ At AT = 1900~ the retained strength depends on number of cycles [150]. This is in contrast to the behaviour of monolithic ceramics under the same conditions which fail catastrophically during either the first or second cycle. Unlike carbon/carbon composites, SiC/SiC composites have good oxidation resistance obviously due to the corresponding properties of the components.

Chapter XIII LIQUID INFILTRATION

Liquid infiltration can be done by a number of ways. It can be pressureless, vacuum, pressure infiltration, squeeze casting, compocasting, etc. Such fabrication methods can be expedient for at least two reasons. Firstly, filaments of a small diameter (carbon, silicon carbide, polycrystalline sapphire, various whiskers) can hardly be introduced into a solid matrix unless powder metallurgy methods are used with limitations inherent to such methods. Secondly, liquid infiltration methods are supposed to be simple. The simplicity of the method leads also to the development of the fabrication route for producing prepregs, based on the infiltration of the molten matrix into the fibre tow or sheet. Wetting conditions and fibre stability in a molten matrix, possibilities of fibre coating to promote both wetting and fibre protection as well as possibilities of matrix modification to make easier liquid infiltration processes were analyzed above, in Chapters 9 and 10. Such questions as kinetics of the infiltration and peculiarities of the matrix solidification in the presence of the fibre are to be considered in this chapter. Particular liquid phase technologies mentioned above as well as typical materials produced by such technologies shall also be described.

13.1. Infiltration mechanics

A general uni-dimensional picture of the infiltration of a fibre preform with a molten matrix is presented in fig. 13.1. Infiltration processes involve (i) capillary effects; (ii) flowing of a viscous liquid in a fibre preform (the infiltration itself); (iii) thermal effects; (iv) solidification of the matrix.

13.1.1. Capillary effects As shown above (see Section 9.3.1), to start the infiltration of a fibre preform which is not wetted with the molten matrix, it is necessary to apply pressure above a pressure threshold a lower bound for which is given by eq. (9.10). Obviously, capillary effect determines, to a first approximation, a maximum infiltrated length of the preform. However, in practice, the infiltration kinetic occurs to be of a great importance because, first, the time of contact between the fibre and molten matrix is

547

548

Ch. XllI, w13.1

Liquid infiltration PREFORM+MA TRIX SOLIDIFIED MEL T / >

PREFORM+MEL T

/

PREFORM /

/

>

Po

, > > >

T~'/////////// x , =i

/

xz

x~

/

x4

(b)

I! ..!

L x

Fig. 13.1. Schematic illustration of uni-dimensional infiltration process.

limited by possible interaction, and second, the matrix viscosity can yield the processing time longer than a permissible limit. 13.1.2.

Infiltration

The infiltration of a fibre preform with a molten matrix can be described in terms of the filtration theory dealing with the flow of liquid through a porous medium. The main equations of this theory are K u = - - - X7p, P div u = 0

(13.1) (13.2)

where u is the velocity vector of the liquid, p the hydrostatic pressure, K the tensor of filtration coefficients, or permeabilities of the porous medium, p viscosity of the liquid. The first equation, which is called the D'Arcy's law, just states the proportionality between the velocity vector and the pressure gradient, Vp. The second one is the continuity equation. Note that D'Arcy's law is applicable to laminar flow only, which means that the Reynolds number Re = p r - ~ / p where p is the density of the melt, r the channel radius, and ~ the average melt velocity, should be less than unity, a condition normally fulfilled for the infiltration processes in composite technology. In the unidirectional case, eqs. (13.1) and (13.2) are written as

Ch. Xlll, w

Infiltration mechanics

Xdp

u-

349

(133)

~dx'

du

~=0.

(13.4)

Here K is a scalar value. D'Arcy's law is often written in terms of the liquid flux, q, so that eq. (13.3) is replaced with k@ q . . . . . /tdx

(13.5)

Here q = A u where A is the cross section of a sample and k a constant with a dimension different from that of constant K. The value of permeability can be either found in a special experiment or derived by considering laminar flow through a system of pores of an idealized geometry. Statistical approaches to model a porous medium more adequately are also known. To give an example of a simple geometrical approach, we remind first the HagenPoiseuille formula that is actually a solution to the general Navier-Stokes equation describing the laminar flow in a single cylindrical channel. It gives the average flow velocity in the channel as r2 @ - ----. 8/tdx

(13.6)

where r is the channel radius. Comparing eqs. (13.6) and (13.3) yields r2 K

__

~ .

8

A simple Kozeny approach to find the value of k is to model the porous medium by a block of solid material intersected by straight cylindrical channels of a constant radius. Then t'/7"oR4 dp

q-

(13.7)

8/t dx

and k-

n rcR 4 8#

where R is the channel radius and n the number of the channels per unit cross-section. Introducing the porosity 4) and the pore surface area in a unit volume S we can write 2

S2

1

550

Ch. XIII. $13.1

C7'

I.iyrric/ itrf~l/rtrtron

CD

m.,,.

.J

c~ 0

0 b,l ('D

c~

9

b-,.

cD

Substituting the above equations into eq. (13.7) we get the Kozeny constant, to be used in eq. (13.5), as

~D

~D

i.. I

,..,.

~D

p-I

Introducing the "specific" pore surface area, i.e. the pore surpdce area per unit volume of the solid,

~...,.

C::~.,O.c~

~..,,.

N

we get the Kozeny-Carman constant

s~

0

c~

~..j.

~

::r"

~.c~

m...

c~

-~.

~..,.

m--

,-1

~

Fukunaga and Chda [ I 801 deriving the eflective channel radius of a characteristic cell in the hexagonal fibre array and following the procedure just described. found permeability of a fibre bundle in the longitudinal direction as

"I

D..,1

J""

~-, "C~

~

0

~

~

r.~

cD

C~

~: ~--~. -Jr-

....I 9 ~

Here r, and t:, are the radius and volume fraction of the fibre plus a matrix layer solidified on the fibre surface. For the liquid flow through the interstices of un~directionallyaligned fibres, Yamauchi and Nishida [7 171 borrowed the following approximate expressions:

~..,.

m-,.

§

I

m

§

c~

I

,,-.-

and

.-..I

'-1

(~'J)

~/~ "1

'-1

,.~ c~ ~

-...I

Jr-

0

c~

Equations (13.1 1 ) and (13.12) give the pcrmcabilities in the longitudinal and transverse directions, respectively. For the same purposes. Mortensen et 31. [476] preferred to use the following couple of expressions:

Ch. XIII, w13.1

ln[iitration mechanics

rs K - 3.192

Vs

551

(13.14)

which are approximations to numerical solutions. Equations (13.13) and (13.14) are valid for 0.5 < Vs < 0.8 and 0.2 < Vs < 0.8, respectively. An equivalent radius of the effective channel for a three-dimensional random fibre packing, to be used in eq. (13.6), can be introduced [373] as 1 -

ri

(13 15)

n

where ri are principal radii of an interspace. Then some statistical considerations together with experimental observation yield an approximation of the probability density function for req as f(r) -Ar -l.

(13.16)

Here and further on, index eq is dropped, R1 _< r _< R2 and constant A is determined by the obvious condition fR R2 f (r) dr - 1. I

This yields the distribution function of the volume fraction of the interspaces with a radius r in a unit preform volume as 4 3(1 --/)f) 2 v(r) -- -5 Tcr3f (r) -R32 r

(13.17)

Here the condition

j~

R2 v(r) dr - 1 - vf I

is taken into account. 13.1.3.

T h e r m a l effects

Two thermal effects are of importance. First, a change in the temperature field as a result of flow of the matrix material with a higher temperature than that of the fibre preform, and second, the influence of the latent heat of the molten component on the temperature field during the matrix solidification. Let us write the general equation of the heat transfer in an isotropic body as

,-.

•

r tr hq

rO

Ch. XI11. $13.1

d~ .~.

+

II

d~

(13.18)

where -

4

II

PC'

PC

9

q = -.

=

II

,I

k a- = - = const,

~9

o

~

h

~~

~4

0a >:., ;:"

._'~:

.

~

o

"6

'-u.-u =

"U = ' -

Xo

~

""'~

~

= ~

9..-~

~

.--

~

0

.. o

=

_= ~ E

~

oo

~

ila=

~

~

8

=

,.~=

@

-u

IIere k , r*, and p are the tliemal conductivity, heat capacity, and density of the body. respectively, and ij = q(.x-,j),z,t) the heat source within the body. Solution of the corresponding Couchy problem is well known [620], it is written via error ti~nction of argutrlerit x / 4 . '1.0 apply eq. ( 13.18) to a problern o r the infiltration depicted in fig. 13.1, we writc it for uni-tiimensional case with the heat source being the flowing molten matrix "~

"u

".~

.-~

~

o

.~

r

o

=

~

~.~

r.~

~" -

,.4.2

~

~v

..

< r < .Q): v

(xl

d~ o.I

t'q

i~,

+

i,

.,-.~

,-4

0',

5"

.,,u

"-~ u-

[c

.,.,i

is the liquid flow velocity, a2 is given by eq. (13.19) with

-

where

-Jr-

~. +

~"

II

d

eO

a=

;>

r.~

r.~

.= .-'-

~.

-~ ~

~

where a letter without a subscribe relates the value t o the composite. and ~ ~

II 9

...0

. ,....~

E

0

~

E

~

.=

~=

.,,-,

o

.xi ~ a ' ~~

V

~ ~

~.

v'~" ~

._

At I: < x < 13. the heat exchange between the fibre : ~ n dmolten matrix has to be included into the aniilysis. so that

~.2

.~

~,o

~ ~

~

~

,4:: o . ~

E:

~

==

......,

~'~.

"-

"--

~ o

where fi is the initial tenlperature of the tibrc pref'orni, 7;:, the melling point of the matrix material (in the case of pure nietal or eutectics). (I:, ry) the volume fraction of the solid~fedmatrix. and AH the enthalpy of firsion of the metal. -

9

"~

~

-.

I,.~ ~

=

....~

"*'~

~=

I1"~

~.~

~

~~.=

I,..,

r.~

o

8 ..

e0

8 8 ..0

E~

.-= --

.

supplying more nucleation sites;

~.g~

~

"~ >,

-u'-

"

=.

=-._•

~o

u'=

--

.~

Let a liquid matrix be in contact with u reinforcement. We can consider a temperature in the system as being uniform since the characteristic time, I = d;'/rr., where a n d al.are the diameter and thermal diKusivity of'the fibre. is at most of the order of I ms [475]. The presence of fibres can influence thc crystallization process of the matrix by a number of ways, those being [475]:

h{t~'ltration mechanics

Ch. XlII, w

553

9 impeding convection in the liquid metal; 9 changing stability of the plane front of the alloy solidification. Increasing the density of the nucleation sites results in reducing grain size of the matrix as compared with the unreinforced matrix. Although this does not occur with pure aluminium matrix, AI-Cu and AI-Si alloys demonstrate the grain refinement due to the catalysis of heterogeneous nucleation of the primary phase. Another way to refine the matrix grain structure is to cast liquid metal into preform initially at a temperature below liquidus. This yields rapid solidification of a layer around the fibre. Unless this layer remelts, it will consist of fine equiaxed grains. Impending the convection in the liquid metal by a fibre system may change the shape of dendritic structures from equiaxed to columnar with corresponding change in the effective matrix properties, possibly, to a better direction. Obviously, crystallization of the molten matrix starts at the matrix/fibre interface. Therefore, the configuration of the solidification front may not be plane. A question arises whether such a configuration tends to make the front unstable, that is to increase the deviations from the plane infinitely, or not. Mortensen (see Ref [475]) has calculated a critical value of the wavelength, 2,, below which an infinite plane front is stable. It happens that the value of 2, is normally smaller than fibre spacing. Hence, the fibre should have a stabilizing effect on the matrix crystallization. Actually, the stability of a bent solidification front (fig. 13.2) is determined by the ratio of the temperature gradient, dT/dy, to the rate, Uy, of the front movement.

13.1.5. Some examples of the analysis The analysis of infiltration process that combines a number of the effects just considered is, in general, a complicated non-linear problem. Hence, most approaches to solve the problem are of an approximate nature. Certainly, a most comprehensive solution of the filtration problem bound to the thermal one, that is a solution of the system given by eqs. (13.3) and (13.20), was presented by Mortensen et al. [476]. The system mentioned was rewritten in a new (single) variable,

Y ,z"

Fig. 13.2. A schematic representation of the solidification front in the fibrous composite.

554

Liquid infiltration MELT

PREFORM +MEL T

Ch. XIII, w

PREFORM

/

t9o

>

;

I

>

>

~

~

~

~

r--

,

"/////////// X l

-',

X 2

X4

......L ......J

Fig. 13.3. Schematic of the infiltration without solidification of the matrix.

(13.22)

Z = x/~hVff

which is chosen in such a way as to have Z = 1 at the position of the infiltration front, x = x 2 (fig. 13.3). Hence, L = Oxfi. Since eq. (13.4) gives a constant infiltration velocity, u, and accounting for a volume filled with melt yields u=

l/tUrn v/,/ .

(13.23)

Equations (13.3) and (13.20) are now written as dp dz

I)m/202 K

(13.24)

and dT ( Z - fl)~_7_~,~a=

a 2 d2T ~t2 dz 2

(13.25)

where fl = b v m . At x > X2 (uninfiltrated preform), instead of eq. (13.25) we write ap2 d2T ~2 dz2

dT _ Zdz -

(13.26)

where 2 ap-

kp ppCp

F r o m eq. (13.24) we have for the case when the initial preform temperature, Tr, is sufficiently high so that the matrix is not solidifying during the infiltration (fig. 13.3)

Ch. XIII, w13.1

lnl~'ltration mechanic,~

K A p I2 "

555

(13.27)

Here Apl2 = Pl - P 2 is the pressure drop on the XlX 2 interval. If solidification of the matrix at point Z - 1 occurs, the fibres become covered with a sheath of the solid matrix at x2 < x < x3 (fig. 13.3). In this zone, the matrix temperature is constant and equal to the matrix melting point. Because of the input of "fresh" liquid matrix into the partially solidified region, remelting of the solid matrix takes place, so remelting front, x = x2, or Z = Zs moves to the right. Assuming a thickness of the solid matrix sheath to be constant, hence the permeability of the xzx3 zone to be constant, and considering the total permeability of the x~x3 zone as that of two zones in a series yield a modification of eq. (13.27) in the form

'/'-

I

(13.28) /t/)m K~2 + K23 J

where K12 and/s are permeabilities of the corresponding parts of the preform given by one of the eqs. (13.10) to (13.14). Note that to calculate KI2 by using the equations mentioned, one has to take Vs = vf and rs = rf. For the calculation of K23, Vs = vr + Vsm and rs = rrv/Vs/Vf where Vsm is the volume fraction of the solid matrix in the xzx3 zone. Now eqs. (13.24) to (13.26) are solved under appropriate boundary conditions. The results of calculations of values L/x/t compare very favourably with experimental data for infiltration of pure aluminium into two-dimensional random preform made of Saffil alumina fibres of a diameter of 3.8 Jam obtained by the same authors [390], fig. 13.4. The infiltration kinetic and further matrix solidification occur to be strongly effected by crystallizing the melt when its temperature reaches the melting point as a result of the heat transfer to the preform and subsequent remelting of the solid matrix sheaths around the fibres due to additional heat carrying by "fresh" portions of the melt which continue to go into the preform [476]. Hence, a composite specimen with a pure metal matrix consists, after finishing the process, of two zones. The first one, "remelting zone", located at the preform end opposite to that contacting the pure matrix, contains a fine-grained matrix, the second one contains a coarser matrix [390]. Also, the fibres in the "remelting zone" are exposed to the molten matrix for a shorter time. The calculations by Mortensen et al. [476] allow to estimate the boundary between the two zones. The crystallization process of alloy matrices differs from that of pure metals. Michaud and Mortensen [406, 407] developed the infiltration theory by Mortensen et al. [476] for such a case and analyzed segregation of alloying elements in the matrix along both the infiltration direction (in the case of adiabatic process) and the transverse direction (in the case of heat exchange between the specimen and the die wall). Together with experimental data by the same authors [407] on infiltrating a

Liquid infiltration

556 0.015

9

|

Ch. XIII, w

0.020

9

oo

0.015

T, = 2 5 5 - 257~

~1.010

Tm~ = 680 - 683~

~

'~Po = 3.38 - 3.52 MPa

o 0.005

0.010

o vf = 0.24

0.005

rf- 284- 290~ T.~ 9esooc

0.000 0.0

'

0'5.

"

110

'

3.0

0.000 0.20

9

,

,

0.22

,

0.24

,

Vr

i

0.26

9

0.28

0.0150 0.03

vr

= 0.235- O.240

~tpo = 3.42 - 3. 53 MPa

0.02

-~.o~2s I&&

l

vr = 0.24

0.01

rm ~ - 6 7 0 . 680~ ~p

0.00

200

,

I

300

i

Tf--/~

93.42 - 3.53 MPa

I

=

500

0.0100

[

" I

600

,

~

I

TmO/Oc

700

Fig. 13.4. The infiltration length divided by square root of the time versus technological parameters of the infiltration of a mat of Saffil alumina fibres with pure alumina. Ap,, is the applied pressure drop, Ap the pressure drop duc to melt viscosity, Tr the initial preform temperature. T,~,I,is the initial melt temperature. Experimental data by Masur ct al. [390].

binary alloy and those by Jerry et al. [282] who studied the infiltration of an industrial alloy, the calculation results give a pretty clear picture of what is going on when an alloy infiltrates the fibre preform with a temperature below the liquidus of the metal. Perhaps, the most important feature of the microstructure of a composite with 6alumina fibre and a matrix of an aluminium-copper alloy of hypoeutectic composition is a non-homogeneous copper distribution along the infiltration direction. The copper concentration increases sharply towards the infiltration front. Again, as in the case of a pure metal matrix, a zone where solid and liquid metal coexisted during infiltration contains a fine-grained matrix. On the other hand, a zone where the preform was exposed to a molten matrix only during infiltration contained large grains in the matrix volume. In the case of an industrial alloy (AI-4.4Cu-0.27M g-0.18Ti) used as a matrix in a squeeze casting experiment, copper segregation, fibre volume fraction, as well as matrix grain structure change along the infiltration direction in a manner predicted by the calculations. With the melt temperature decreasing from 700 to 500~ the non-homogeneity in copper distribution and fibre volume fraction increases.


Ch. XIII, w

557

In a simplified analysis, Yamauchi and Nishida [717] neglected a complicated kinetic of the matrix solidification and estimated approximately the limiting infiltration distance depending on both the pressure and preform temperature. They just averaged the values of permeabilities for parallel and normal melt flow to fibre alignment given by eqs. (13.11) and (13.12), respectively, and used the averaged value in D'Arcy law, eq. (13.5). The volume fraction of the solidified matrix was obtained from the heat balance, eq. (13.21), assuming the instantaneous heat exchange between the molten matrix and preform as soon as they get in contact. Other main assumptions are as follows. First, the compressive stress on the uninfiltrated volume equals the pressure acting on the left surface of the preform (fig. 13.5). Second, deformation of the preform starts when the pressure exceeds a critical stress for the preform. The deformation means that the fibre volume fraction in preforms increases. A dependence between stress and fibre volume fraction is assumed to be known. Third, back pressure of air is neglected. Integrating eq. (13.5) yields the pressure in the molten matrix (Pro in fig. 13.5) P-

j/t/

-~-Tx +P0

(13.29)

where K t is an effective permeability. The velocity of the infiltration front is effected by a partial solidification of the matrix entering the preform, so d x f = u(1 - Vsm) dt 1 - Vs

. . . .

~ XI

(13.30)

i'~T-//'~--' ' / / -- {j

X 3

.')C4

p - -

(b) X

p

!

(c) X

Fig. 13.5. A simple infiltration model by Yamauchi and Nishida [717]. (a) Schematic view; (b) pressure distributions before preform deformation starts (pf and Pm are the pressure values in the preform and molten matrix, respectively); (c) pressure distributions when the preform is deforming.

L~

(ira

i,-.

II

O

i',a

~.

Ca

""

O

O

where xf is the location of infiltration front, xr = .r3 in fig. 13.5. Hence,

~

~

E

.o

Since p = 0 at x = XF,eq. (13.20) yields

L~

__..~_

O

m

I:::

,-,

"~

I ~ io

4 1 ' ~ ;COS ' ~ 0~ dl( I - cr) a-" ~

pc = -

Ca

7,

I

.-96"

Actually, to get the total pressure on the melt, we have to add the capillary component given by eq. (10.9) to the value obtained, so that eq. (13.32) should be rewritten as

~o

~"

~"

~g

=- =- o .8

~i., _~. ~ ~.~

L~

L~

I

,..,,

m

I

+

II

I

I

o

::r.~

o

~

~

~

where df is the fibre diameter and ul. the initial value of the fibre volume fraction. Equations (13.31) and (13.33) together with eq. (13.21) give the dependence of the infiltration length on the pressure applied provided the preform does not deform. Suppose at t = t, the applied pressure reaches a critical value for the preform to start to deform. Then it can be shown [717] that eqs. (13.31) and (13.33) transform into

"~-~- ~.~&~

I

L~

I

I

and

~-~.~ -.

----,-- ~ -,.

~

~_

~~.

I~

~~

~ ~ (1~

--.

~'~ ~.=

~9 o ' ~ _ . ~ ~9 ' = I ~ ' ~ ~

~;~

~m ~

~9

~.

9

B.~~-~~

~.~ o

~

"a

,~.O~

.

~ ~~~=

~m

-.

where values K , I>,,,,.and o, are constant at t 5 I, and they, a s well as ci in eq. (13.34), become to be dependent on />(, at r > t,. The systeni was solved numerically by using ) a preform made of silicon carbide whiskers [717]. experimental dependence I ! ~ ( Pfor In particular, it was found that a limiting infiltration distance existed for each preform temperature as shown schematically in fig. 13.6. Long et al. [373] formulated and solved, in an approximate manner, a nonstationary infiltration problem aiming mainly at the understanding of an effect of air trapped in the preform on the pressure developed in it. They considered infiltration


Ch. XIII, w

559

g p

r,

Fig. 13.6. Schematic dependencies of the infiltration distance on.pressure according to Yamauchi and Nishida [717].

of a random fibre preform and assumed an equivalent radius of the interspaces, r, and corresponding volumes to be distributed according to eqs. (13.16) and (13.17) with R1 and R2 being the smallest and largest values of r, respectively. Firstly, they rewrite eq. (9.9) as YESCOS0 r -- ~

(13.37)

Pcap

where index cap accents the capillary nature of the pressure in the present context and r is the equivalent radius of an interspace given by eq. (13.15). Now at the infiltration front ( x - x2 in fig. 13.3) the local pressure, according to eq. (13.37) is Pex (X) -- 7LS COS 0 Rmin

(13.38)

where Rmin is the smallest interspace penetrated by the melt. Hence, under this pressure the saturation degree of the preform is fR '~2 (1 -- Of) (R 3 -- Rmin 3 (x)) S(x) -- rain v(r)dr -- R32

(13.39)

From eqs. (13.38) and (13.39) we have ]4de

dS

- ~ - C(~(X), dx

where

C

~..

3(1 - vf) YLS COS OR2

is a constant and ~ = Rmin/R2.

(13.40)

560

Liquid infiltration

Ch. XlII, w13.1

Suppose now that a pressure gradient along the x-axis exists in a melt infiltrating a medium containing the cylindrical channels of radii ranging from R I to R2. This means, in particular, that the channels are interconnected. Within each interval of the radii, dr, they are homogeneously oriented in the space. Then applying eq. (13.6) to a channel of radius r inclined at angle cz to the x-axis, yields r 2 dp - (r) -- ~ - - ~ c o s ~.

(13.41)

Averaging over r and 0r we can write

r2['(r),

U - 81----~t-~

dr

d -n/2

cos ~g(~)d~

(13.42)

where g(~) is a distribution density function, Actually, Long et al. [373] calculated U by writing an expression for U(r) as --

Dr r2 d p

U - ~-8~ dx

(13.43)

where

D r - fnCOS n(r)O~idi

(0 0 to v f - 0, we obtain matrix strength equal to about 2 GPa, whereas the unreinforced matrix strength is only about 1.2 GPa. It can also be seen that values of the composite strength up to temperature 1200~ are higher than those reported for composites with A1203 fibres and a Ni3A1 matrix made by hot pressing an assemblage of fibres and foil [588]. Evaluation of the effective fibre strength based on such a set of experimental data cannot be conclusive, but it is clear (fig. 13.30) that the fibre strength reduces at temperatures above 900~ only and at temperature of 1200~ it is about 500 MPa (see fig. 13.29b). > Rupture strength of the composites has not been studied in detail. However, a limited number of the creep tests conducted [428] (the results are presented in fig. 6.9), together with the calculations based on these results and used in

582

Liquid infiltration

Ch. XIII, w

Fig. 13.28. Scanning electron micrograph of a part of a cross-sectioned specimen After Glushko et al. [194].

Section 6.2.2, shows that rupture strength of the composites at a temperature of 1200~ and the time base of about 1000 h can be expected as high as 150 MPa.

13.7. Ceramic-matrix composites An example of composites with ceramic matrix produced by liquid infiltration method is that containing a brittle oxide matrix and relatively tough fibres, namely molybdenum wires [299]. To provide strong enough bond at the interface the composites have been produced by the infiltration of a fibre bundle with a melted oxide. In this process, the temperature of the melt is about 100~ higher than the melting point. The oxides are AI20~ and eutectic compositions AIzO3 + ZrO2 (nonstabilized as well as partially stabilized with MgO or Y203). All the specimens are unidirectionally reinforced. The specimens with the eutectic matrix, when crystallized, were withdrawn from the hot zone of the furnace at a rate of 3.5 ram/rain. The composite specimens intended for bending tests had a diameter of 8 mm. They were tested at room temperature in three-point bending. To obtain values of the stress intensity factor the compact specimens shown in fig. 13.31 were used, of dimensions thickness 4 to 4.5 mm, width 16 to 17 mm, and crack length 6 to 9 mm. The values of strength and critical stress intensity factor obtained are given in Tables 13.1 and 13.2. The bending strength of unreinforced ceramic A1203 + ZrO2

Ch. XIII, w

Ceramic-matrix composites

2.5

t3

1

'

!

,

0 I

583

9

2.0

1.5

1.0 0.0

O.1 I

.2

0.3

Vf

1000 o ,,

13

800

looooc

[]

9 1100~ o

1200~

600 /

.-'"

400

O

b

2~

|

I

|

0:2

Fig. 13.29. Strength of A1203/ZrO2 + Y203 fibre/nickel aluminide-alloy matrix composites versus fibre volume fraction. Open points stand for bending strength, solid point stands for tensile strength. (a) Room temperature, (b) high temperatures. After Glushko et al. [194].

obtained by crystallization of the melt is 188 + 44 MPa (14 specimens were tested). These strength data were used above in Section 5.5 to make a comparison to theoretical predictions. Fracture surfaces of composites given in [299] show little fibre pull-out and no interface delamination, so the strength of the interface is sufficiently high. It should be noted that all specimens must have quite high residual stresses. Indeed, if the difference between the values of Poisson's ratio for the matrix and the fibre is neglected we can write the residual stresses as

Liquid infiltration

584

Ch. XIII, w13.7

2.5

2.0

"~ 1.5 1.0

0.5

0.0

0

,

t

250

~

i

500

,

i

750

,

i

1O00

T~ ~

,

1250

Fig. 13.30. Temperature dependence of the bending strength of AleO3/ZrO2+Y~O3 fibre/nickel aluminide-alloy matrix composites with fibre volume fraction between 0.2 and 0.25. After Glushko et al. [194].

Fig. 13.31. Specimen of the AlzO3/Mo composite after fracture toughness testing.

Ch. X I I I, w13.7

T

a m ~vf

Ceramic-matrZr composites

EmEf

E

58 5

E m E f (~m - c~f)AT

(~xf-~m)ATa/~vm

(13.47)

E

where E is the Young's modulus of the composite, (~lI1 and c~f are the thermal expansion coefficients, AT is the difference between the temperature when the matrix ceases relaxation and the temperature of testing, AT < 0. If the properties of materials are used as they are given in Table 13.3, and it is assumed AT = - 1 2 0 0 to - 1 5 0 0 ~ the initial matrix stresses will appear to be too high to get the matrix without transverse macrocracks. They have to be present in TABLE 13.1 The

bending

strength

a* of oxide matrix/molybdenum fibre composites produced by the liquid infiltration

method Sample

Matrix

d (mm)

vf

Oexp (MPa)

rYcaI (MPa)

62/1 62/2 63/1 63/2

A!203

-

0

198

-

-

0

181

-

-

0

218

-

-

0

206

-

93/1 93/2 149 153/1 153/2 154 160

A1203

0.08 0.08 0.10 0.05 0.05 0.05 0.04

0.41 0.41 0.48 0.35 0.35 0.39 0.33

514 542 596 573 475 502 490

528 528 520 530 530 520 770

5.5

A1203 + ZrO2

0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.10 0.10

0.45 0.49 0.49 0.34 0.34 0.41 0.41 0.475 0.475

670 660 736 567 560 613 620 580 590

610 680 680 565 565 610 610 540 540

7.2

94/1 94/2 101/2 101/2 102/1 102/2

A1203 4- ZrO2 + 6 % M g O

A1203 + ZrO2 + 1.7%Y203

0.42 0.42 0.486 0.486 0.465 0.465 0.465 0.465

590 585 590 530 540 550 580 530

600 600 520 520 550 550 590 590

7.1

1o3/1

0.08 0.08 0.10 0.10 0.10 0.10 0.08 0.08

89/2 90/1 90/2

91/1 91/2 92/1 92/2 100/1 100/2

103/2

A1203 + ZrO2 + 2.5%MGO A1203 + ZrO2 + 6.8%Y203

KI~1 ( M P a - m t/2)

6.8 7.1 6.6

Note: The values of Ac~AT which are necessary to describe the experimental dependencies, are equal to 0.00275 and 0.00400 for the alumina matrix and all the alumina-zirconia matrices respectively. Gxp are the experimental data, a~,l are the calculated data, K~I are the average values of the matrix critical s t r e s s intensity factors.

Liquid infiltration

586

Ch. XIII, w

TABLE 13.2 Fracture toughness of some oxide matrices and oxide-molybdenum composites produced by the liquid infiltration method. Sample

Compositions

K, MPa.ml/2

2 8 13 21 4 14 15 22

A!203

6.07 5.50 4.96 23.7 7.52 7.24 6.73 28.4

A!203 + Mo AI203 + ZrO2

A!203 + MgA1204 + Mo

TABLE 13.3 The properties of the composite's components. Material

~"( 10-6K -I )

E (GPa)

A1203 ZrO2 ZrO2 + 3%MgO A1203 -q- ZrO b A1203 + ZrO2 + 3%MgO Mo

9.1 8.0 11.8 8.7 10.0 5.8

400 350 200 380 340 350

" In the interval 300 .... 1400 K b The eutectic mixture.

accordance with the ACK-theory (Section 4.4) and have not been observed in the experiments. Note that calculations show that the appropriate value of AT for A1203/Mo system is 1400~ (see fig. 5.37).

Chapter XIV INTERNAL CRYSTALLIZATION Most types of fibrous composites are being fabricated by using pre-made fibres and putting them into a matrix. In this chapter, an alternative process will be presented. The process, called the internal crystallization method (ICM), has been invented in Russia and is rather unknown in the West. The method is based on crystallization of fibres from the melt within the volume of a matrix [431,433,434]. In addition to the fabricating of various composites, the method provides a means to produce a variety of substances in the fibrous form. This can be a routine to produce fibres to test their mechanical and physical properties, the latter being of importance due to a possibility to obtain, say, fibres of complex oxides, a class of materials with an enormous spectrum of physical properties. It should be noted that it is possible to reinforce a chosen matrix with the fibres crystallized in an auxiliary matrix, which can be more suitable for using in ICM than the wanted matrix.

14.1. Technique Basically, the method of internal crystallization includes the following exercises: (1) Preparation of the matrix by forming continuous cylindrical channels in it. (2) Infiltration of the channels in the matrix with a melted fibre material. (3) Crystallization of the fibres in the channels.

14.1.1. Preparation of the matrix The first step of the fabrication route is the formation of continuous cylindrical channels in the matrix to be later infiltrated with a melted fibre material. This can be done in a number of ways. In general, that is a matter of inventive work. To illustrate the possibilities, we give here just one example. To prepare the matrix for a metal-matrix composite we can use the diffusion bonding of an assemblage of the foils and wires under special conditions. First, a layered assembly of foils and wires, both normally being of the same material, is prepared as shown in fig. 14.1a. This can be done by winding the ingredients onto a mandrel. The assembly then undergoes diffusion bonding (fig. 14.1b) under such 587

588

In ternal co'stallization

000 O0

(a)

Ch. XIV, w

)o() o) I

J

IIIIIIIIIIIIIIIIIIII

(a)

()

()

.

_

t

TTTTTTTTTTTTTTT'TTTT-t

Fig. 14.1. The fabrication steps of a metal matrix: (a) assembling the foil and wire layers; (b) diffusion bonding, and the final shape of the matrix with channels (c) and the fibre cross-section (d). After Mileiko and Kazmin [433].

temperature/pressure/time conditions as to provide a sufficiently strong bond between the foils and the wires without the gaps between the neighbouring wires being filled with the solid material. This yields a structure, as in fig. 14. l c, which has continuous cylindrical channels. The shape of the transverse section of a single channel is shown in fig. 14.1d, and that will be the shape of a future fibre. A particular set of diffusion bonding conditions depends upon the matrix material and the geometry. Certainly the temperature can be fixed for a particular matrix material. For example, for molybdenum it is 1200~ and a suitable regime might be 1200~ - 5 MPa (average pressure)- 4 h, for nickel matrix: 800~ - 2 M P a - 1.5 h. It is clear that a particular composite part can be shaped easily before the infiltration. For example, a specimen for a tensile test can be made by using EDM.

14.1.2. Infiltration Obviously, the infiltration can be performed if the melting temperature of the fibre is lower than that of the matrix. If chemical interaction between the matrix material and the melt is restricted then the infiltration is clearly easy because no strict limitations are to be imposed. It is very convenient to have a good wetting of the matrix by the melt of a fibre substance. However, the last two demands are usually contradicting each other (Section 9.3.1). Perhaps in this context, combinations of refractory metals as the matrices and metal oxides as the fibres are the only ones which satisfy both requirements entirely. Therefore, such metal matrix composites as sapphire/molybdenum, yttrium-aluminum garnet/molybdenum, etc. can easily be made. In case of the sapphire/molybdenum composites, the melting temperatures of the matrix and the fibre materials are 2610 and 2070~ respectively, and the wetting angle is about 15~ at 2100~ Chemical interaction can hardly be observed at the melting point of the fibre. So the infiltration of a melt into cylindrical channels in a matrix is a self-driven process. It starts when a piece of the matrix contacts the melt.

Technique

Ch. XIV, w

589

The temperature of the A1203 melt should be about 2200~ to provide necessary overheating. The infiltration time is short, of the order of minutes. In the case of a matrix with a low melting temperature such as nickel, one has often to look for a complex oxide with a melting point just below that for the matrix. One can expect, first, a chemical interaction between the components, and secondly, a large enough value of the melt viscosity just above the melting point. Both expectations are attained in the case of 2A1203. M g O . 3CaO-fibre/nickel-matrix composites (see below, Section 14.2.5). In this case, the infiltration time increases by two orders of magnitude, an interaction zone develops. Hence, we should perhaps be looking for fibre/matrix combinations which are not generally considered as proper composites. An example of such combinations is a matrix containing say substance A and a fibre of a eutectic composition in an A - B system. The only precaution to be observed when executing the infiltration, is to keep the infiltration temperature just above the melting point of the fibre. In particular, there have been tested Ti/Ti - TisSi3, N i / N i - Ni3Si, A1203/A1203-A15Y3O12. The results obtained will be discussed below, in sections 14.2.6 and 14.2.7.

14.1.3. Crystallization The final step of the fabrication process is the crystallization of the fibres in the channels. A possible model of the crystallization process is considered qualitatively in [433]. Suppose we have a furnace with two temperature zones " I " and " I I " (fig. 14.2) and there is no direct thermal exchange between them. Immediately after the infiltration, the specimen is located within Zone I and its temperature is Tl which is higher than the melting temperature Tm of the matrix. Now if the specimen is moved instantaneously in the direction of the arrow (fig. 14.2) by AL, the

/

H t-O

zST

_!

f"

>

"-A

Fig. 14.2. Schematicrepresentation of temperature profiles in the crystallization zone. After Mileiko and Kazmin [433].

590

Internal crystallization

Ch. XIV, w14.1

temperature profile along the specimen starts to change, as shown by lines marked with t = 0, tl, t2, . . . . To initiate the crystallization it is necessary to have some value of overcooling, say Air. So at some time, say t3, the temperature at the right-hand end of the specimen is Tm - AT and spontaneous crystallization within the length L* occurs. There is no reason to observe any results of this process other than polycrystalline fibres within this part of the specimen, but it is important to note that as the channel effective diameter decreases, the probability of having a single crystal occupying the whole channel transverse section increases. Now, if the specimen is moved from Zone I to Zone II at a definite rate, u, then, obviously, the length L* of the specimen with spontaneously crystallized fibres will depend on the value of u: the larger the rate the bigger is this length. After the initial stage of crystallization in a moving specimen, crystals at the left-hand boundary of the crystallized zone occur as seeds for fibres growing in channels. Thus the fibre can have single crystalline structure along the whole channel length to the left from the initial crystallization front. A quantitative model of the process, which remains to be performed, should yield a critical effective diameter dc,- of the channel such that at a particular value of u single crystalline seeds should definitely appear. Figure 14.3 presenting the sapphire fibre bundle extracted from the molybdenum matrix gives an impression of a real object. Actually both poly- and single-crystalline parts of a fibre can be seen when looking at a layer of the sapphire fibres extracted

Fig. 14.3. A bundle of sapphire fibres removed from molybdenum matrix after internal crystallization. After Mileiko and Kazmin [434].

Ch. XIV, w

Fibres and composites obtained by I C M

591

Fig. 14.4. A layer of sapphire fibres photographed in polarized light, the polarization planes being perpendicular to each other. Molybdenum wires are located between the sapphire fibres. (a) Crystallization rate 15 mm/min, (b) Crystallization rate 7 mm/min. After Mileiko and Kazmin [433].

from the matrix (fig. 14.4). Note that optically anisotropic sapphire crystals, observed in a polariscope with crossed polarization planes, appear to be transparent only if the light beam direction does not coincide with the c-axis and the apparent color is determined by the phase difference of two partial beams. The latter depends on both crystal thickness and its orientation. Figure 14.4 reveals that the size of the polycrystalline part of a layer of the fibres increases when a pulling rate of a specimen increases, that conforms to the conclusion from the model discussed above. The crystal orientation is not constant within a fibre layer. So a specimen with single crystalline fibres can be considered to be an effective polycrystal. The angle, ~0, between the c-axis of a hexagonal optically transparent crystal, e.g. sapphire, and the fibre axis, can be measured by observing fibre orientations with respect to the polarized light beam direction corresponding to the black appearance of the fibre. Some of the histograms of q~ are shown in fig. 14.5. It can be seen that the values of ~0 lie mainly between 45 ~ and 90 ~ It should be noted that no apparent dependence of the shape of a histogram on the parameters mentioned above has been observed.

14.2. Fibres and composites obtained by ICM We begin with a description of the strength of fibres and composites obtained by the internal crystallization method and then analyze the experimental data.

i

0 0

o

0

o

o

0"1

o

o

o ~

o

o~

,

~ "

o

o

. .;~ . . .=

o

,

o

r.0 o

o

,

!

i

! !

o

,

o

,

,

'

|

Q

.~ i

o

Ch. XIV. $14.2

I ~ i r c , ~ - ~ t(t ~r l. ~ ~ . ~ r r r l l i : t r t i ~ ~ n

.b.~"

O-

=.~.

9

~-~

~...,,

9

To (i.e., the point X1 in fig. 14.48). If it is kept at this temperature for some time followed by slow cooling, there prevails particles of compounds .4xBy in a "ballast" matrix. It is the wellknown flux method (see for example [701]). Suppose that the liquid can be removed from the liquid-solid mixture existing at some temperature T'. Then the compound .4xBy occurs in a pure state without the ballast matrix. This can be done if the liquid could be removed and used for the matrix of a future composite. The forgoing procedure is the basic idea of the "blotting paper" technology. The procedure starts with a layer of powder of composition X1 contacting with a porous layer which is wetted by a liquid of composition X('. When this sample is heated up to temperature T > To, thi~ porous layer starts to act as a blotting paper. That is, the liquid infiltrates this layer and the solid crystals remain after cooling to form a layer of the desired material. In this way, multilayered composites can be formed. The "blotting paper" filled with a crystallized liquid yields a matrix of high mechanical strength bounded strongly to the main layer. If heating goes on slowly then the liquid starts to go into the porous layer at temperature just above To. The composition of the melting layer then changes. The kinetic of the process is determined by the heating rate, wetting properties, temperature dependence of the liquid viscosity, etc. The porous layer must have the appropriate wettability; it should be chemically passive as it is in contact with the melt of the liquid mentioned. The melting temperature of the material of inert layer should be sufficiently high. If the main layer is the superconductive YBazCu307 oxide, possible candidates of the porous layer are the oxides A1203, SrTiO3, ZrO2, and MgO, the last of which is

632


Ch. XIV, w

preferred [82]. Magnesia and barium zirconate BaZrO3 was used in the experiments to be described. 14.4.2. Technique." Main route The actual quasi-binary phase diagram of the Y203-BaO-CuO [377] differs from the simple scheme in fig. 15.48 because really solid and liquid coexist in the more complex combinations. In the experiments [18], it appeared to be necessary to try a variety of initial compositions to arrive at the desired composition. The slurry casting technology is applied to make the pre-layers using either MgO or BaZrO3 and a main layer made of mixture of simple oxides in fine powder form. The slurry is prepared using a polymer binder, blended in benzine-acetone mixture. A stack of 3 to 21 layers is heated to temperature 400~ to evaporate the binder and to attain a preliminary sintering. Then the temperature is elevated up to 950~ for the infiltration of the porous layers and crystallization of YBa2Cu307_ x oxide in the main layers. The specimen is then cooled in the furnace, during which time saturation of YBazCu3Oy_x oxide with oxygen occurs. Failure surface of a specimen in fig. 14.49 shows the microstructure of the layers. The specimen is prepared at 950~ for 1 hour. The main layer has a large degree of the porosity; that is because the liquid has been removed from the layer. The X-ray phase analysis reveals in the main layer the CuO-phase as an addition to the 123 phase; in the matrix layer, copper oxides as well as barium cuprate are present. The specimens obtained are characterized by a very wide transition region to the superconductive state along the temperature axis, which starts at about 90 K and finishes at about 40 K. That is, perhaps a result of non-homogeneous oxygen saturation of the YBazCu307 x oxide in a porous structure containing grains of various sizes. Under these consideration, the highest transition temperature is reached when the initial composition corresponds to Y~Ba29Cu63Ox. 14.4.3. Technique." Remelting If the prepared specimen is reheated up to a temperature above 1000~ at which the 21 l-phase (YBaCuOs) and a liquid co-exist, then the liquid-solid mixture will produce porousless "black" layer (123-phase), because it occupies less volume than the initial state. The above prevails if no significant interaction occurs between the liquid and that in the pores of the blotting paper. Such a process differs from a melt quenching process in two aspects. For a layered composite, an interaction of the melt with the atmosphere is "shielded" by the matrix layers. Second, remelting of the main layer occurred without using a crucible. The remelting procedure alters the composite microstructure as illustrated in fig. 14.50. The specimen is processed at 950~ for 1 hour and the superconductive layer is remelted at 1050~ followed by slow cooling in a furnace. A change in the porosity of system is observed. The pores of different sizes in the main layer have become larger and moved toward the layer interfaces. The local texture emerges in some areas of the specimen.

Ch. XIV, w

"Blotting paper" technology

633

Fig. 14.49. Failure surface of layered composite with the matrix layer containing BaZrO3 as a base. The composition of the raw material for the main layer corresponds to YsBaz9Cu63Ox. The matrix layer is on the bottom part. At the left upper corner a typical pore located within the main layer can be seen. After Aptecar et al. [18].

X-ray microanalysis of the main layer reveals the presence of the "green" phase (211) together with the main "black" phase. In the matrix layer, the BaCuO2 phase is present. Figure 14.51 displays the temperature dependencies of the specific resistance of the YBazCu307_x-MgO composites before and after remelting. A strong dependence of the specific resistance and the transition temperature on the fabrication procedures is obvious. Possibly, an optimal cooling rate after remelting has not been reached in these experiments. The obtained transition temperature is about 10 K lower than that could be achieved physically. Under these fabrication conditions, it is considered high. A texture of the superconductive layer is shown in fig. 14.52. It corresponds to the failure surface of a layered composite with Y9Ba36Cu55Ox as a precursor of the main layer and BaZrO3 as a base of the matrix. The specimen is processed at a temperature of 950~ for one hour ....It is then heat

634


Ch. XIV, w

Fig. 14.50. Failure surface of layered composite of the type shown in fig. 14.49 after remelting at 1050~ The main layers are in the top and bottom of the photograph. The pores are now located in the interlayer zones. After Aptecar et al. [18].

treated in a furnace with a 4.6 K temperature gradient per millimeter for a specimen 50 m m in length. The specimen is then heated up 1120~ at the hot edge and cooled with a temperature gradient of 48 K per hour to 900~ at the same edge location.

14.4.4. Technique." D(ffusional grol~'th of superconductive la),er If the initial mixture of the powders correspond to the stoichiometric "black" phase YBa2Cu3OT_• is heated to 1050~ it will decompose into so-called "green" phase (Y2BaCuOs) and a liquid. Cool the main layer with the " g r e e n " phase only. Now, heat the specimen to 920~ to form a " b l a c k " phase layer as a result of diffusion. Such a scheme can be useful. Data in fig. 14.53 show the dependence of the transition temperature on the regime of diffusion ageing. They correspond to the YBa2Cu3Ox/BaZrO3 composite with a superconductive layer obtained as a result of diffusional restoration. Heat treatment temperature is 900~

"Blotting paper" technolog3'

Ch. XIV, w 600

I

I

w

I

I

I

I

"',1

:

:

:

I

635

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? 400

..... .

\

.

.

.

.

. . . . .

without remelti~g r e m e l t i n g : c o o l i n g r a t e 20 K ~ . m i n remelting: cooling rate ! K/min

~" 2 0 0 9

ooom~O

o oo

oo o

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o

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J

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I

60

o:

o~-J"~__

x

,

100

T/K

x

x

:

~

I

140

i

i

180

Fig. 14.51. Temperature dependence of specific resistance of the layered composites before and after remelting. After Aptecar et al. [18].

Fig. 14.52. Failure surface of a layered composite with Y9Ba36Cu55Ox as a precursor of the main layer and BaZrO3 as a base of the matrix. The specimen was heat-treated and than cooled under a temperature gradient. The texture in the superconductive layer is obvious. After Aptecar et al. [18].

Ch. XIV, w14.5


636

120 o O ~ 1 7 6 1 7 6 1o7o6o1o7o6o o o o o o o o o

100

eo

80920 ~ 920 ~ 920 ~

.....

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60

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3 4 5

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40-

o

aaaaaaaa aaaaaaaaoaaaaaaa

o

20-

o

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0

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~

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g

a

~

a,

a ~ 1 7 6 1 7 6 1 7 r6dmnn~n~ 1 7 6 1 7 *6 1 7 6

60

80

r/K

I

I

100

120

Fig. 14.53. Temperature dependence of specific resistance of YBa2Cu30• layer. After Aptecar et al. [18].

140 with a superconductive

14.4.5. Technique." Fabrication of Mo-A1203 composite Using the "blotting paper" technology it is possible to make composites, whose constituents possess high melting points, at moderate temperatures. For example, when the Al203-melt is infiltrated into the molybdenum matrix by application of the internal crystallization method, the melt is heated up to 100~ higher than the melting point, that is about 2200~ The "blotting paper" technology may produce a similar result with temperature not higher than 1800~ The porous layer initially contains molybdenum only. The initial composition of the main layer is A1203 + 10 m%Y203. The AI203-Y203 system has a eutectic 1.96 parts of A1203 to 1 part of Y203 at about 1760~ Heating the upper layers of the packet to 1800~ the eutectic melt would disassociate to the molybdenum layer leaving the sapphire crystals in the main layer. The dependence of the A1203/Mo composite bending strength on the volume content of sapphire layers is shown in fig. 14.54. The beneficial effect of the composite is clearly revealed by the curves reaching a maximum.

14.5. Fibres produced by ICM

For the present book, this is a final paragraph which cannot be completed yet since it is the author's intention here just to mention on a current research line in his laboratory. It appears now that molybdenum can be used as an auxiliary matrix to produce ceramic fibres in a rather economical way. Such fibre, an example being

Ch. XlV, w 800

Fibres

,

,

,

,

.

.

produced

.

.

637

bl' ICM

.

.

.

~ 600 \ b 40O

6

200

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-

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0 0.4

i

l

i

I

0.6

i

VA~.o.

i

l

I

l

I

I

0.8

1.0

Fig. 14.54. Variations of bending strength of A1203/Mo composites with volume content of sapphire layers: neutral plane parallel to layers. After Aptecar et al. [18].

shown in fig. 14.55, promises a prospect in obtaining composites for elevated and high temperatures. A usage of the fibres produced in such a way, to obtain composites with special physical properties can also be expected.

Fig. 14.55. A batch of A1203/A15Y3012 fibres produced by using ICM.


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AUTHOR INDEX

Numbers refer to pages on which the author (or his work) is mentioned. Numbers between brackets are the reference numbers in the Bibliography. No distinction is made between first author and co-author(s).

Abbaschian, G.J., 428, 567, 653 [364] Abiven, H., 448, 662 [587] Aboudi, J., 115, 639 [1] Achenbach, J.D., 92, 639 [3]; 101,639 [2] Ackermann, L., 579, 639 [4] Adams, D.F., 84-85, 668 [712]; 91,639 [7]; 115, 639 [6] Adams, J.C., 426, 640 [40] Adsit, N.R., 284, 639 [8] Ahmad, I., 244, 639 [9] Akasaka, T., 222, 659 [510] Akiyama, S., 566, 642 [80] Akizuki, T., 421,663 [600] Alfutov, N.A., 122, 639 [10] Alipova, A.A., 488-489, 639 [11] Allen, B.C., 432, 639 [12] Allred, R.E., 457, 649 [256] Alman, D.E., 522, 639 [13] Amateau, M.F., 522, 667 [706] Ananth, C.R., 439, 642 [75] Anderko, K., 616, 618, 648 [230] Anderson, C.H., 466, 666 [683] Anderson, W.P., 335, 358, 660 [526] Andreikiv, A.E., 289, 639 [14] Andrews, R.M., 565, 639 [15] Anishshenkov, V.M., 333-335, 639 [16]; 338, 656 [424]; 346, 348-351,656 [423] Annin, B.D., 218-219, 222-224, 639 [17] Aptecar, I.L., 630, 632-637, 639 [18]; 630, 639 [19] Archangelska, I.N., 295, 639-640 [20] Argon, A.S., 434-435, 643 [101]; 434-435, 647 [213] Arnold, S.M., 105, 479, 509-511,641 [59] Arsenault, R.J., 254, 640 [21]; 419, 653 [366] Asami, Y., 542, 668 [716] Ashby, M.F., 14, 640 [22]; 14, 640 [23]; 369, 646 [182]; 418-419, 662 [575]; 472-473, 667 [703] Aslanova, M.S., 59, 640 [24] Asthana, R., 466, 665 [650] Aswath, P.B., 523, 640 [36]

Aveston, J., 131-132, 177, 640 [25]; 177, 640 [26]; 62, 640 [27] Awerbuch, J., 221,287, 640 [28] Azzi, V.D., 122, 640 [29] Babich, B.N., 456, 466, 470, 618, 661 [545] Babich, I.Yu., 392, 404, 647 [218] Bacack, G.P., 466, 644 [137] Bacon, J.F., 512, 640 [30]; 512, 661 [550] Bader, M.G., 566, 579, 640 [31] Badini, C., 451,467, 640 [32] Bahei-El-Din, Y.A., 112, 644 [145] Bahrani, A.S., 503, 643 [108] Bakarinova, V.I., 448, 640 [33] Baker, S.J., 466, 640 [34] Bakhvalov, N.S., 89, 640 [35] Bandyopadhyay, A., 523, 640 [36] Banichuk, N.V., 14, 640 [37] Bao, G., 435, 643 [119]; 479, 511-512, 652 [345] Baranova, G.K., 630, 639 [19] Barbero, E.J., 91-92, 654 [374] Barglay, R.B., 460, 640 [38] Barlow, C.Y., 254, 640 [39] Barnard, T., 51,653 [370] Barrailer, V., 430, 648 [222] Barranco, J.M., 639 [9] Bashforth, F., 426, 640 [40] Basyleva, O.A., 31,641 [63] Bates, H.E., 57, 640 [41] Baumann, S.F., 452-453, 640 [42] Baxter, W.J., 244, 254, 257, 576, 640 [43] Beakou, A., 100, 102-104, 665 [659] Beaumont, P.W.R., 369, 646 [182] Becher, P.E., 32-33, 304, 640 [44] Bednorz, J.C., 630, 640-641 [45] Bender, B., 458-459, 641 [46]; 523, 641 [47] Bender, B.A., 523, 650 [283] Bengisu, M., 304, 641 [48] Berger, M.H., 50-51, 60-61,641 [62] Bernhart, G.A., 73, 653 [352] 669

670

Author &dex

Berry, D.S., 80, 664 [618] Bert, C.W., 404, 641 [49] Bhagat, R.B., 522, 667 [706] Bhalla, A.K., 503, 641 [50] Bhatt, R.T., 437, 521,645 [153] Bi, J., 318-319, 660 [528] Bibring, H., 68, 661 [556] Birchall, J.D., 61,641 [51]; 62, 641 [52] Bischoff, E., 453, 643 [ 120] Bockstein, S.Z., 47, 641 [53] Bohlen, J.W., 453, 641 [69] Bolotin, V.V., 93, 641 [54] Bonfield, W., 460, 640 [38]; 466, 640 [34] Bordia, R.K., 526, 649 [253] Borovikova, M.S., 428, 662 [580] Bourrat, X., 524, 644 [142] Bowie, O.L., 219, 641 [56]; 284, 641 [55] Bradbury, J.A.A., 61,641 [51] Braddick, D.M., 244, 575, 577, 649-650 [274] Bradley, D.J., 513, 655 [403] Bradley, S.A., 522, 641 [57] Brennan, J.J., 72, 641 [58]; 302, 641 [65]; 513-514, 661 [551] Brentnall, W.D., 486, 665 [658] Brindley, P.K., 105, 479, 509-51 I, 641 [59]; 452453, 640 [42] Brun, M.K., 458, 664 [613] Bujalski, D., 51,653 [370] Bullock, E., 317-318, 641 [60] Bunsell, A.R., 46, 51,641 [61]; 50-51, 60-61, 641 [62]; 365,666 [672]; 497, 648 [240]; 571,645 [164] Buntushkin, y.P., 31,641 [63] Burkland, C.V., 72-73, 668 [720] Bussalov, Yu.E., 71,641 [64]; 295, 570, 646 [176] Butkus, L.M., 369-370, 668 [728] Cao, H.C., 302, 641 [65]; 302, 641 [66]; 435, 643 [119]; 578, 649 [263] Cappleman, G.R., 426, 641 [67]; 566, 579, 640 [31] Caputo, A.J., 524, 664 [625] Carlson, M., 466, 666 [683] Carman, G.P., 100, 641 [68] Caroll, N.T., 427, 649 [252] Carpenter, H.W., 453, 641 [69] Caulfield, T., 443, 642 [70]; 445, 665 [653] Chamis, C.C., 508, 664 [630] Champion, A.R., 430, 562, 577-578, 642 [71] Chan, K.S., 193, 198-199, 642 [74]; 194, 198-199, 642 [73]; 352, 366-367, 642 [72] Chandra, N., 439, 642 [75] Chang, F.K., 223, 642 [77] Chang, R., 600, 602, 642 [76] Charbonier, J., 579, 639 [4]

Charles, J.A., 448, 654 [388] Chatellier, J.Y., 100, 102-104, 665 [659] Chawla, K.K., 464, 642 [78] Chen, J.L., 84, 642 [79]; 114, 664 [632]; 115, 118, 664 [631] Chen, K.C., 63, 658 [474] Chen, K.I., 566-567, 650 [290] Chen, X., 491,509, 667 [693] Cheng, H.M., 566, 642 [80] Cherepanov, G.P. 157, 642 [81] Cherepanov, Yu.G., 122, 639 [10] Cheung, C.T., 632, 642 [82] Chiang, Y.-M., 566, 643 [102] Chin, E.S.C., 104, 106, 244, 368, 580, 659 [498] Cho, K-M., 579, 642 [83] Choi, Y.-S., 579, 642 [83] Chou, T.-W., 77, 95, 98, 123-124, 134, 137, 233, 254, 257, 642 [85]; 181, 514, 650 [296]; 650, [295] Chou, T.C., 452, 642 [84] Choy, K.L., 423, 642 [87]; 424, 466, 642 [86] Christensen, R.M. 77, 89, 95, 99, 642 [88] Christman, T., 467, 642 [89] Chung, K.H., 491,649 [254] Cinibulk, M.K., 544, 642 [90] Clarke, D.R., 437-438, 654 [375] Claussen, N., 32-33, 642 [91]; 72, 642 [92]; 522, 667 [6981 Claveyroals, G., 448, 662 [587] Clegg, W.J., 579, 642 [93] Clougherty, D.P., 419, 645 [149] Clyne, T.W., 424, 469, 651 [311]; 426, 641 [67]; 448, 654 [388]; 469, 476, 651 [312]; 520, 647 [203]; 566, 579, 640 [31] Coleman, B.D., 45, 233, 642 [94] Collins, J.M., 566, 643 [102] Comninou, M., 199, 644 [! 41] Connel, S.J., 491,509, 667 [693] Conti, P., 222, 643 [96] Cook, J., 147, 189, 643 [97] Cooke, C., 544, 648 [238] Cooke, R.G., 281,648 [231]; 513, 648 [232] Cooke, T.F., 53, 61,643 [98] Cooper, G.A., 131-132, 177, 640 [25]; 169, 287288, 643 [99] Cornie, J.A., 426, 659 [51 I]; 431,659 [512]; 434435, 643 [101]; 434-435, 647 [213]; 470, 643 [100]; 550, 553, 555,658 [476]; 555-556, 564, 654 [390]; 566, 643 [102] Cottreli, A.H., 164, 643 [103] Courtright, E.L., 31,643 [104] Cox, B.N., 131-132, 182, 654 [385]; 159, 643 [107]; 285, 643 [105]

67 1

Author index o~

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Ebcrhart. M.E., 419, 645 [I491 Ebcrt, K.J., 115. 667 I7071 Eckel, A.J., 546. 645 [I501 Edwards, L.?579. 642 [93] Efimov. V.B.. 630. 639 [I 91 Eggeler. G., 329, 645 [I511 Egin, P.A., 334. 337. 645 [152]; 338, 656 [424] Eisenmann. J.R.. 219. 666 [677] Eldridgc, J.I., 105. 479, 509-51 1, 641 [59]; 437. 521. 645 [I531 Eliezer, Z., 257-258, 668 I7301 Elzey, D.M., 481482. 484485, 488,493, 645 [154]; 491. 645 [155]; 491. 647 [209] Emiliani, M.L., 453, 643 [I201 Eremichev, A.N.. 381. 383. 645 11471 Eringen, A X . , 138, 645 [i57] Evans. A.G.. 105. 650 12781: 131-1 32. 182,654 [385]; 165. 168, 665 16521: 169. 182. 332, 439, 453,645 11611; 182- 183, 654 13861; 182,457, 513, 645 [159]; 302,641 1651; 305, 645 [ l h 3 ] ;415, 433, 645 [IhO]; 418-419, 662 15751: 434, 645 11621; 435, 643 [I 191; 453. 643 11 201; 510, 667 16941; 567, 577-578. 651 13171; 578. 649 1263) Everett, R..527. 650 [283] Ezis, A , , 522. 663 15981

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Faber, K.T., 305. 645 [I631 Fastnacht. R.A.. 630, 650 I2841 Favre, J.-P., 263, 645 [IQS]:437. 643-644 [123]; 49 1 , 658 [469] Favry, Y., 497, 648 12401: 571. 645 [164] Fcillard. P., 263. 645 [I651 Feingold. E., 432, 665 16371 Feng, C.R., 254, 640 [21] Feodosiev, V.I.. 7+9: 645 [I661 Ferber, M.K., 522. 653 [354] Ferraris. M.. 45 1, 467. 640 [32] Ferris, D.H., 315, 317. 645 11671 Ferro, A.C.. 427. 431, 645 [ I 681 ,...

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Dalgleish, D.J.. 41 5, 433, 645 [I 601 Daniels. H.E.. 233 234, 643 [I 151 Darkcn, L.S.. 505-506, 643 [I 161 Das. G.. 450. 467. 643 [ I 171 Dauchier, M.M., 73. 653 [352] Dauskardt. R.H., 371. 643 [I181 D a u ~ J.-C.. . 571, 666 [675] Davidson. D.L., 352, 366367, 642 [72] Davies. M.E., 630, 650 [284] Davis, J., 434. 645 [I621 Davis, J.B., 435, 643 [I 191; 453, 643 [I201 Dcott, V.D.. 467, 664 [616] Dean, A.V.. 562. 643 [I211 Derby. B., 423, 642 (871; 427, 43 1, 645 [I681 Dergunova, V.S.. 420, 652 [331] Deribas. A.A., 502. 643 [I221 Desarmot. G . , 263, 645 [165]; 437, 643-644 [I231 Deslanches. G.. 579, 639 [4] Dcve. H.E.. 105, 650 [278] Dharan, C.K.H.. 6, 644 [124]; 341, 644 [125]; 339, 668 17291 Dharani, L.R.. 194-197. (161 15441 Dhingra, A.K.. 60. 644 [126]; 430, 562, 577-578, 642 [71] DiCarlo, J.A., 62- 63. 644 [I271 D~cfcndorf.R.J.. 52, 644 [I281 Din~iduk,D.M., 31, 644 [I291 Dinwoodie, J.. 61. 641 [51] Diwanji. A.Y., 445, 648 12731; 574, (44 1 1 301 Dobhs, J.R., 480, 644 [I 311 Doblc. G.S.. 500, 644 [I 321 Dollhoph V.. 428, 466. 668 17231; 562. 652 [3461 Donald. I.W., 51 3. 655 [403] Dorcy. S.F., 281, 648 [231] Dorokhovich. V.P., 476, 496, 650 [297] Dover, B.. 60. 645-646 [I721 Downes, T., 11 1-1 12, 661 (5471 Dragone, T.L.. 317. 644 11331; 644 [I341 Drapcr, S.L.. 105,479, 509-51 1,641 [59]; 45 1 452, 644 [I 351 Dresselhausc. G., 46-49, 644 11361 Dresselhause, M .S.,46--49, 644 [I 361 Dubus. A.. 556, 650 [282] Dudarcv. F..F... 466, 644 [I371

_.~~.~o~~o

Dudck, H.J.. 4484l9. 644 [139]; 470, 479, 644 [I381 Dugdala, D.S.. 157. 644 1140) Dundurs. J., 199. 644 [I411 Dunn. M..321. 323, 665 [647] Dunn, S.A.. 432, 664 [633] Dupel. P 524. 644 11421 Duprce. P.L., 480, 644 [I311 Duva, J.M.. 473, 644 [143]: 491. 644 [I441 Dvorak, G,J., I 12, 644 [I451 Dymkov, I.A., 118. 12& 121, 405, 408409, 645 [146]; 122. 639 [lo]; 38 1. 383, 645 [147]; 405.408, 645 [I481

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Cripin: R.M.. 428, 658 [490] Crossland. B.. 503, 643 [I081 Crossman. F.W., 618, 623, 626, 643 [I091 Crow, P.D.. 473. 644 [143] Currey. J.D., 34. 64, 643 [I 101 Curtin, W.A., 267. 643 [I 1 I]; 267, 643 [I 12); 276. 643 [I 141: 299, 302, 643 [I 1 31; 49 1 , 644 [I 441

~

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Cox. H.L.. 97, 124. 643 [I061

672

Author index

Figueredo, A., 424-425, 662 [576] Finn, R., 424, 645 [169] Fischmeister, F., 460, 650 [293] Fisher, H.E., 423, 645 [170] Fishman, G.S., 542, 666 [680] Flemings, M.C., 550, 553, 555, 658 [476]; 555-556, 564, 654 [390]; 567, 655 [402] Flower, H.M., 551,558, 560-561,653-654 [373] Fomina, G.A., 630, 632-637, 639 [18]; 630, 639 [191 Fortunko, C., 575, 653 [356] Fourmeaux, R., 48-49, 666 [676] Foye, R.L., 115, 645 [171] Fras,zr, H.L., 449, 663 [603] Frechette, F., 60, 645-646 [172] Fredriksson, H., 564, 650 [280] Freese, C.E., 284, 641 [55] Frenkel, Ya.l., 471,646 [173] Freudenthal, A.M., 37, 646 [I 74] Friedel, J., 25, 646 [175] Friedrich, E.M., 295, 570, 646 [176]; 459, 646 [177] Friend, C.M., 579, 646 [! 78] Fujiwara, C., 509, 646 [179] Fukube, Y., 478, 515, 575, 659 [513] Fukunaga, H., 550, 646 [180] Gakhov, F.D., 20 I, 646 [ 181 ] Galasso, F.S., 423, 651 [318] Galkin, Yu.A., 476, 662 [577] Ganczakowski, H.L., 369, 646 [182] Garcia, D.E., 522, 667 [698] Garmong, G., 295, 646 [I 83] Garofalo, F., 329, 472, 483, 646 [184] Gaydosh, D.J., 451-452, 644 [135] Geguzin, J.E., 472, 474, 646 [185] Gelachov, M.V., 107-109, 537, 646 [186]; 255, 527, 533-536, 656 [425]; 384, 656 [426]; 518-520, 538, 540-541,656 [439]; 537, 541,646 [187]; 622, 624-630, 657 [446] Generazio, E.R., 546, 645 [150] Geng, L., 567, 668 [713] George, E.P., 581,662--663 [588] Gerold, V, 352, 662 [572] Gigerenzer, H., 501, 570, 646 [189] Girot, F.A., 531-532, 567, 646 [190] Givargizov, E.L., 62, 646 [19 I] Glatter, I.Y., 424, 648 [227] Glushko, V.I., 58, 646 [193]; 58-59, 265, 277, 328, 581,656 [428]; 270, 275, 646 [192]; 330-331,656 [427]; 563, 581-584, 646 [194] Goda, K., 550, 646 [180] Goddard, D.M., 421,646 [195]; 421,647 [196] Goetz, R.L., 486, 647 [197]

Goland, M., 228, 647 [198] Goldberg, H.A., 46--49, 6441136] Goldsmith, W., 339, 668 [729] Golofast, E.G., 339-340, 343, 656 [442] Gomez, M.P., 335, 358, 660 [526] Goo, G.K., 452, 647 [199] Gooch, D.J., 600-601,647 [200] Goodier, J.N., 154, 368, 647 [201] Gordon, F.H., 520, 647 [203] Gordon, J.E., 147, 189, 643 [97]; 3-4, 14, 647 [202] Gotman, I., 454, 647 [204] Goto, S., 315-317, 647 [206]; 315, 647 [205] Gotoh, A., 521,658 [468] Grag III, G.T., 467, 649 [255] Graves, J.A., 452, 647 [199]; 480, 644 [131] Greil, P., 526, 647 [207] Griffith, A.A., 23, 647 [208] Groves, G.W., 600-601,647 [200] Groves, J.F., 491,647 [209] Groves, M.T., 422-423, 653 [350] Gryaznov, V.P., 479, 656 [429] Guermazi, M., 454, 647 [210] Gukassyan, L.E., 490, 647 [211] Gundel, D.B., 450, 647 [212] Gupta, V., 434-435, 647 [213] Gupta, V.J., 434-435, 643 [101] Gurland, J., 235, 647 [214] Giiselsu, A.N., 100, 665 [645] Gfiscr, D.E., 235, 647 [214] Gusev, E.D., 500, 647 [215] Gutans, Ju.A., 44, 647 [216] Guth, J., 543, 654 [379] Gutmanas, E.Y., 454, 647 [204] Guz', A.N., 373, 647 [217]; 373, 647 [219]; 392, 404, 647 [218] Guz', I.A., 373, 647 [219] Guzei, L.S., 441,443, 664 [621] Gvozdeva, S.I., 61 !-614, 647 [220] Gycnkcycsi, J.Z., 546, 645 [150] Ha, J-S., 464, 642 [78] Haasen, P., 254, 294-295, 658 [488] Habib, F.A., 513, 648 [232] Haggcrty, J.S., 57, 647-648 [221]; 57, 663 [605] Hahn, H.T., 221,287, 640 [28] Hait, E.B., 124, 665 [649] Hall, !., 445, 648 [223] Hall, l.W., 367-368, 654 [378]; 430, 648 [222]; 450, 648 [224]; 574, 644 [I 30] Halpin, J.C., 98, 648 [226]; 100, 648 [225] Halzlebeck, D.A., 424, 648 [227] Hancock, J.R., 284, 648 [228] Hanigovsky, J.A., 422-423, 653 [350]

Author index

Hannant, D.J., 177, 181,648 [229] Hansen, M., 616, 618, 648 [230] Hanser, F.E., 341,644 [125] Harris, B., 281,648 [231]; 513, 648 [232] Hartley, M.V., 466, 468, 648 [233] Hartman, G.A., 369-370, 668 [728] Hartman, H.S., 430, 562, 577-578, 642 [71] Hasegawa, H., 456, 567, 662 [578] Hashin, Z., 77, 89, 94-95, 99, 648 [234]; 89, 648 [236]; 94, 648 [235] Hay, R.S., 424, 648 [237]; 544, 648 [238] Hayami, T., 468, 664 [628] He, M.Y., 191-192, 194, 648 [239]; 567, 577-578, 651 [317] Heaney, J.A., 422-423, 653 [350] Hearn, D., 497, 648 [240] Hedgepeth, J.M., 134, 648 [241]; 141-142, 648 [2421 Hegemier, G.A., 92, 100-101,658 [477] Hejazi, M., 91-92, 658 [487] Henmann, J.H., 544, 668 [722] Henstenburg, R.B., 276, 648 [243] Herbell, T.P., 546, 645 [150] Herring, H.W., 442, 648 [244] Herrmann, G., 101,639 [2] Herron, M.A., 454, 648 [245] Heuer, A.H., 418-419, 662 [575] Heyliger, P., 575, 653 [356] Hill, R., 84, 648 [246]; 89, 648 [2471 Hillig, W.B., 562, 649 [248] Himbeault, D.D., 420-421,649 [251]; 420, 649 [249]; 421,649 [250] Hink, R.C., 42-43, 652 [334] Hitchcock, S.J., 427, 649 [252] Hojo, M., 145, 659 [500] Hollenback, S.A., 526, 649 [253] Hong, S.H., 491,649 [254] Hong, S.I., 467, 649 [255] Hoover, W.R., 284, 286-287, 649 [257]; 457, 649 [256] Horsfall, I., 579, 642 [93] Houpert, J.-L., 276, 279, 649 [258] Hsueh, Ch.-H., 437, 649 [259]; 437, 649 [260]; 437, 649 [261]; 437, 649 [262] Hu, M.-J., 578, 649 [263] Hubert, P.A., 566, 579, 640 [31] Hughes, D.C., 177, 181,648 [229] Hui, Ch.-Y., 143, 653 [351] Hull, D., 25, 649 [264]; 95, 649 [265] Humphreys, F.J., 254, 649 [266] Hurley, G.F., 598, 649 [267] Hutchinson, J.W., 191-192, 194, 648 [239]; 437438, 654 [375]

673

lasonna, A., 527, 649 [268] Iijma, S., 521,658 [468] Inal, O.T., 304, 641 [48] Interrante, L.V., 423, 645 [170] Irwin, G.R., 156, 649 [269] Ishikawa, T., 50-51,649 [270] Issupov, L.P., 112, 649 [271] Ivanov, V.G., 488-489, 639 [1 l] Ivanov, V.V., 396-399, 656 [440]; 488-489, 639 [11]; 500, 647 [215] Iwakuma, T., 91-92, 658 [487] Iwamoto, N., 428, 649 [272] Jackson, P.W., 244, 575, 577, 649-650 [274]; 420, 460, 466, 649 [273]; 460, 650 [276] Jacobson, N.S., 454, 653 [358] James, M.R., 371,643 [118] Jang, Ch.W., 517, 519, 650 [277] Jang, H.M., 517, 519, 650 [277] Jangg, G., 520, 664 [612] Janssen, R., 522, 667 [698] Jansson, S., 105, 650 [278]; 513, 650 [279] Jarfors, A.E.W., 564, 650 [280] Jarvis, C.V., 503, 650 [281] Jena, P., 419, 653 [366] Jeng, S.M., 72-73, 668 [720]; 491,668 [718] Jerine, K., 100, 648 [225] Jero, P.D., 416, 651 [309] Jerry, P., 556, 650 [282] Jessen, T.L., 458-459, 641 [46]; 523, 641 [47]; 523, 650 [283] Jin, I., 424, 426, 429, 431, 552-553, 658 [475] Jin, S., 630, 650 [284] Johnson, B., 369-370, 661 [552] Johnson, W., 48-49, 650 [285]; 339, 650 [287] Johnson, W.S., 353, 369, 650 [286]; 457, 658 [481] Jones, C., 449, 650 [288] Jones, L.M., 428, 658 [490] Jones, R.M., 77, 89, 95, 650 [289] Ju, C.P., 566-567, 650 [290] Kablov, E.N., 31,641 [63] Kadoi, M., 632, 654 [377] Kadyrov, V.Kh., 476, 496, 650 [297] Kalita, V.I., 490, 647 [211] Kalnin, I.L., 430, 650 [291] Kaminski, B.E., 219, 666 [677] Kammloth, G.M., 630, 650 [284] Kang, Ch., 567-568, 650 [292] Kannappan, A., 460, 650 [293] Kao, W.H., 453, 668 [719] Karakozov, E.S., 481,650 [294] Karandikar, P., 181, 514, 650 [296]; 650 [295]

674

Author index

Karasek, K.R., 522, 641 [57] Karpinos, D.M., 476, 496, 650 [297] Katinova, L.V., 476, 662 [577] Katzman, H.A., 422, 650 [298] Kaysser, W.A., 470, 479, 644 [138] Kazmin, V.I., 184, 562, 582-583, 650 [299]; 244, 656 [430]; 587-589, 591-592, 595, 597, 599-602, 656 [433]; 587, 590, 597-598, 603-610, 656 [434]; 587, 656 [431]; 611,656 [432]; 619, 621-623, 656 [435] Keda, T., 632, 654 [377] Keith, H.D., 630, 650 [284] Keller, K., 543, 654 [379] Kelly, A., 21, 23, 25, 48, 77, 122, 124, 651 [301]; 131-132, 177, 640 [25]; 133, 263, 317, 319, 651 [306]; 164, 169, 650 [300]; 169, 287-288,643 [99]: 177, 181,648 [229]; 177, 640 [261; 294, 651 [302]; 315,651 [305]; 317, 319, 651 [304]; 651,651 [303] Kendall, E.G., 421,646 [195]; 428, 432, 651 [307] Kendall, M.G., 40, 651 [308] Kent, E., 62, 664 [623] Kerans, R.J., 416--418, 660 [527]; 416, 651 [309] Kerr, W.R., 486, 647 [197] Khan, T., 31,658 [482]; 68, 661 [556] Khanin, E.I., 570, 663 [593] Khanov, A.M., 503, 668 [715] Khokhlov, V.C., 170, 220, 657 [455] Khvostunkov, A.A., 107-109, 537, 646 [186]; 255, 527, 533-536, 656 [425]; 373-374, 376, 378, 656 [436]; 373, 377, 379, 656 [437]; 396-398, 401, 498-499, 656 [438]; 396-399, 656 [440]; 488-489, 639 [11]; 497, 508-509, 662 [582]; 501,657 [462]; 518-520, 538, 540-541,656 [439]; 527, 662 [583]; 537, 541,646 [187]; 564-565, 651 [310] Kiely, CJ., 449, 650 [288] Kieschke, R.R., 424, 469, 651 [311]; 469, 476, 651 [312] Kiiko, V.M., 100, 102-103, 651 [313]; 107-109, 537, 646 [186]; 255, 527, 533-536, 656 [425]; 518-520, 538, 540-541,656 [439]; 537, 541,646 [187]; 619, 621-623, 656 [435] Kikoin, I.K., 506, 651 [314] Kilin, V.S., 420, 652 [331] Kim, B.S., 138, 645 [157] Kim, J., 60, 645-646 [ 172] Kim, R.Y., 181,651 [316] Kim, W.H., 445-446, 651 [315] Kim, Y., 567-568, 650 [292] Kimura, S., 542, 668 [716] Kimura, Y., 52, 658 [472] Kingery, W.D., 432, 639 [12]; 432, 652 [347] Kiser, J.D., 437, 521,645 [153] Kishkin, S.T., 47, 641 [53]

Kitahara, A., 566, 642 [80] Kitao, T., 52, 658 [472] Kizer, D., 423, 570, 655 [405] Klein, M.G., 461,466, 655 [404] Klipfel, Y.L., 567, 577-578, 651 [317] Kmetz, M.A., 423, 651 [318] Kobayashi, K., 566, 642 [80] Koczak, M.J., 287, 664 [614]; 445-446, 651 [315] Kodama, H., 72, 651 [319]; 521,658 [468] Koechendorfer, R., 428, 466, 668 [723]; 562, 652 [346] Kohara, S., 447, 651 [320] Kolesnichenko, G.A., 428, 651 [321] Komai, K., 369, 651 [322] Komura, O., 422, 521,542-543, 654 [391] Kondakov, S.F., 121-122, 652 [325]; 325, 651 [324]; 339-340, 343, 656 [442]; 341-346, 657 [450]; 394--396, 651 [323] Kondrashova, N.V., 563, 581-584, 646 [194] Konkin, A.A., 48, 624, 626-627, 652 [327] Koop W.E., 114, 664 [632] Kopecky, Ch.V., 500, 652 [328] Kopjev, I.M., 71,641 [64]; 244, 652 [348]; 247, 250, 652 [329]; 295, 570, 646 [176]; 459, 646 [177]; 490, 647 [211] Koslowski, H., 579, 639 [4] Kosolapova, T.Ya., 110, 652 [330] Kostikov, V.I., 420, 652 [331]; 424, 652 [332] Kostrov, B.V., 210, 652 [333] Kotchick, D.M., 42-43, 652 [334] Koutsky, J.A., 432, 664 [633] Kovalenko, V.P., 270, 275, 646 [192]; 381,384, 386-388, 391-394, 652 [335] Kovchik, S.E., 28-29, 652 [336]; 281-282, 652 [337] Kowbel, W., 418, 459, 652 [338] Kozhevnikov, L.S., 58, 646 [193] Kozlov, A.N., 244, 652 [348] Kreher, W., 33, 652 [339] Kreider, K.G., 105, 244, 661 [553]; 475, 652 [340] Krison, M.E., 464, 652 [341]; 469, 653 [371] Krucinska, I., 49, 652 [342] Krueger, W.H., 430, 562, 577-578, 642 [71] Krukonis, V.J., 52, 652 [343] Kryssan, V.A., 229, 231-232, 652 [344] Kuchkin, V.V., 396-399, 656 [440]; 488-489, 639 [11]; 500, 647 [215] Kudinov, V.V., 476, 662 [577] Kumar, K.S., 479, 511-512, 652 [345] Kun, Y., 562, 652 [346] Kurkjian, C.R., 432, 652 [347] Kuzmin, A.M., 244, 652 [348] Kyono, T., 445, 648 [223]

Author index

Labelle, H.E., 56, 652 [349] Lachman, W.L., 73, 524-525, 655 [395] Lackey, W.J., 422-423, 653 [350] Lackman, W.L., 501,570, 646 [189] Lagoudas, D.C., 143, 653 [351] Lahaye, M., 55-56, 448, 450, 466, 653 [355] Laizet, J.-C., 571,666 [675] Laliberte, J.M., 423, 651 [318] Lamicq, P.J., 73, 653 [352] Langdon, T.G., 32, 304, 668 [711] Lange, F.F., 33, 653 [353] Lara-Curzio, E., 522, 653 [354] Larkin, D.J., 423, 645 [170] Lawley, A., 287, 664 [614]; 445-446, 651 [315] Layard, M., 62, 664 [623] Layden, G.K., 513-514, 661 [551] Le Petitcorps, Y., 55-56, 448, 450, 466, 653 [355]; 421,662 [586] Leckie, F.A., 513, 650 [279] Ledbetter, H., 575, 653 [356] Lee, J.D., 218, 653 [357] Lee, K.N., 454, 653 [358] Leis, H.O., 366, 368, 653 [359] Lekhnitsky, S.G., 95, 653 [360]; 155, 220, 229, 653 [361] Leonov, M. Ya., 157, 653 [362]; 653 [363] Leucht, R., 448-449, 644 [139]; 470, 479, 644 [138] Levi, C.G., 428, 567, 653 [364]; 510, 667 [694] Lewis III, D., 458-459, 641 [46]; 523, 641 [47] Lewis, M.H., 415, 653 [365]; 477, 661 [565] Li, S., 419, 653 [366] Liang, F.L., 453, 659 [494]; 564, 659 [493] Liang, L.C., 437-438, 654 [375] Liebowitz, H., 156, 186, 663 [610] Lienkamp, M., 44, 653 [367] Lilholt, H., 294, 651 [302]; 321,323, 665 [647] Lim, S.-W., 567, 653 [368] Lin, J.H.Ch., 566-567, 650 [290] Lin, K.Y., 222, 654 [383] Lin, R.Y., 569, 653 [369]; 569, 667 [686]; 569, 667 [687]; 569, 667 [685] Lin, T.S., 419, 668 [724] Lin, W., 72-73, 668 [720] Lingle, R., 100, 665 [636] Lipowitz, J., 51,653 [370] Lirn, J.-L., 450, 648 [224] Listovnichaya, S.P., 464, 652 [341]; 469, 653 [371] Liu, C.H., 112, 644 [145] Liu, C.T., 453, 668 [719] Liu, H., 448, 461,466, 653 [372] Liu, H.L.~ 418, 459, 652 [338] Loefvander, J.P.A., 453, 643 [120]; 510, 667 [694] Long, S., 551,558, 560-561,653-654 [373]

675

Louwen, J.N., 419, ,645 [149] Lowden, R.A., 522, 653 [354]; 524, 664 [625] Lowengrub, M., 155, 664 [619] Luciano, R., 91-92, 654 [374] Luty, E.M., 466, 654-655 [394] Ma, Q., 437-438, 654 [375] Ma, Zh., 318-319, 660 [528] Mace, J.G., 73, 653 [352] Macheret, Y., 112, 644 [145] Macmillan, N., 21,654 [376] Madaleno, U., 448, 461,466, 653 [372] Maeda, M., 632, 654 [377] Magata, A., 367-368, 654 [378] Magini, M., 527, 649 [268] Mah, T., 543, 654 [379]; 544, 648 [238] Majumdar, A.J. 611,654 [380] Majumdar, B.S., 470, 654 [381] Makarov, S.A., 476, 662 [577] Mall, S., 115, 118, 122, 662 [568]; 369-370, 654 [382] Malyarenko, A.A., 250, 252, 668 [726] Manthiram, A., 257-258, 668 [730] Mar, J.W., 222, 654 [383] Marchetti, F., 451,467, 640 [32] Mareck, E.V., 428, 660 [524] Margolin, H., 452, 460, 465,-467, 470, 659 [495]; 453, 659 [494]; 460, 659 [496]; 460, 659 [497]; 564, 659 [493] Marjoram, J.R., 460, 650 [276] Markevich, Yu. E., 570, 663 [593] Markov, A.M., 500, 652 [328] Marshall, D.B., 131-132, 182, 654 [385]; 159, 643 [107]; 182-183,654 [386]; 302, 641 [65]; 416-418, 660 [527]; 436, 654 [384] Martin, M.R., 522, 641 [57] Maslennikov, M.M., 11,654 [387] Maslennikova, V.R., 428, 660 [524] Mason, J.F., 448, 654 [388]; 579, 642 [93] Masson, J.J., 515, 654 [389] Masur, L.J., 550, 553, 555, 658 [476]; 555-556, 564, 654 [390] Matsuhama, M., 509, 646 [179] Matsui, T., 422, 521,542-543, 654 [391] Mattheck, C., 64, 654 [392] Matveev, V.V., 338, 661 [541] Maximenko, V.N., 218-219, 222-224, 639117] 223, 654 [393] Maximovich, G.G., 466, 654-655 [394] May, M., 62, 664 [623] Mazdiyasni, K.S., 63, 658 [474] Mazlout, L., 52, 644 [128] McAllister, L.E., 73, 524-525, 655 [395]

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Naidlch. Yu.V.. 424. 42X. 6.55 14801 Naik, R.A.. 457, 658 (4811 Nak;~.M.. 428. 666 [667] Naka. S 31, 658 14821 Nakamura. K.. 478. 515. 575, 659 [513] Nardonc. V.C.. 5 13-5 14. (158 148?] Nartova. T.T., 448. 640 [33] .~

~. ~

603-610. 656 (4341; 587, 656 [431]: 61 1 6 1 4 , 647 [220]; 61 1, 656 [432]: 615-620. 657 [445]; 619. 621-623.656 [435]: 622,624 630, 657 [44h]; 630. 639 1191: 651 [303]; 657 14541: 630. 632 637. 639 [IX] Miles, D.E.. 3 17. 3 18, 641 [60]; 329, 657 [46.3] Millcr, E., 62, 664 [623] Milstcin. F. 21. 657 [464] Minoda. Y.. 456. 567, 662 [578] Minoshima. K., 369, 65 1 [322] Miraclc. L).B., 31. 644 [129]; 470, 654 [SX I ] Mishima. Y.. 448. 461. 46(1. 653 [I721 Misra. A.K.. 451 452. 644 [I 351; 45 I ,452. 657 14651: 454. 521. 657 [466]; 470. 657-658 14671 Miyakc, M.. 422. 521. 4 2 543, 654 [391] Miyoshi. T.. 72. 651 13\91: 521. 658 I4681 Mockford, M.J.. 62. 641 1521 Mollicx, I... 491, 658 [469] Moon. J.H.. 517. 519, 650 [277] Moran. P.A.P.. 40. 651 [30X] Morcton. R..49. 658 [470] Mori. T., 87. 658 [4?1] Murita, ti., 52, 658 [472] Morot, V.O., 476. 496. 650 [297] Morozov. E.M.. 28 29, 652 [336]: ZX 1-282. 652 [337] Morris. A . W . H . . 564. 658 [473] Morschcr. G.N.. 63. 658 14741 Mortcnscn. A.. 424. 426. 429. 431. 552-553, 658 [475]: 550. 553. 555.658 14761; 555 556. 564.654 [390]: 555. 655 [406]: 555. 655 14071; 556. 650 [282]; 565, 639 (151; 566. 643 [I021 Mortimcr. D.A.. 428. 432. 658 [489]; 418. 658 [490] Morton. J.. 228, 666 [66 I] Moschcllc. W.R.. 369 370. 654 [382] Motzfcldt, K.. 59. 666 16791 Mucllcr. K.A.. 630. 640-641 [45J Mura, T.. 86. 658 14791, 87. 665 16481 Muriikami. 11.. 92. 100-101. 658 14771 Muri~kami.Y . . 294. 659 [SO?]: 295. 461. 659 [501]: 46 1. 059 [503] Murali. K . . 331, 658 14781 Murthy. V.S.R., 415. 653 13651 Muto. N.. 447, 651 [320]

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McCartncy. L.N., 123. 12.5..126, 130. 655 [397]; 131, 178. 180. 655 [396] McDanels. D.. 324. 655 13981 McKamey. C.G., 581. 662-663 [588] McLean. M.. 3 15-3 17. 647 12061; 3 15. 647 12051; 3 17-31 8, 641 1601: 329. 655 13993: 329, 657 14631 McMeeking, R.M ..567, 577-578. 65 1 [3 171 Mecartncy, M.L.. 452. 647 11991 Meetham, G . W . . 10. 655 [401] Mehrabian. R., 428. 567. 653 13641: 567. 577 578. 65 1 13 171: 567. 655 [402]; 578. 649 [263] Mt:ltlichuk. O.Yi1.. 396TY9. 656 14401 Mcnke. G.D.. 486. 665 16581 Meshshcrjakov. V . N . . 448. 640 1331 Mctci~llc,A G . 461. 466. 655 14041 Metcalfc. B.L..513. 655 [403] Mcvrel. K.,57 1 , 666 16751 Mcycrcr. W . . 423. 570. 655 14051 Michaud. V.J.. 5 5 , 655 [406]: 555, 655 [407]: 556. 650 [282] Miketta. M., 515. 654 [I891 Mikhailov, S.E.. 229. 655 14081: 229, 655 [409] Mikhccv. V l . 225-~228.657 [444] Milciko, S.T.. 57. 656 14201: 58--59. 265. 277. 328. 58 1. 656 [42X]: 58. 646 11931: 58, 656 [4? I]; 7 1 . 235, 245. 250. 252. 657 14521: 107- 109. 537. 646 [I86]; 121-122, 652 [325]: 1?0. 172. L74. 220. 222. 359-360. 362, 657 [457]; 170. 220. 657 14551: 184. 562. 582 583. 650 [299]: 184. 186. 204. 657 [460]; 200, 657 [46l]: 204. 2117 209. 657 14591: 21 5. 2 17. 657 [45X]; 275 22K. 657 14441: 235. 239 243. 505-507. 657 144x1: 235. 239. 506 507. 657 14561: 235. 241--234, 246. 201. 334. 48 I . 489. 493494. 496. 502. 655 -656 [419]. 244.656 [430]. 255, 527, 533-536. 656 14251: 270. 275,646 11921: 280. 283 284.286 287. 657 14531: 288.662 15841; 292. 655 [412]: 295. 639 640 [ZO]: 307. 3 1 1. 31 3 3 14. 655 [415];30;. 3 I I. 655 [4I 31; 1 I n. 65s [410]; 320. 652 [4 141: 324. 329. 484. 66 l [559]; 324. 655 141 I]; 325, 651 I.1241: 325. 655 [417]; 330-331. h5h [427]. 333 135. 039 [I(]]; 333. 655 [418]: 338. 656 [424]: 339- 340. 343. 650 14421: 339-7.30. 657 [44Y]: 341- 346. 657 [450]: 346. 348-351. 656 [423]: 373 374. 376. 378. 656 14361: 373. 377. 379. 656 [437]: 377. 380. 657 [451]: 381. 384, 386 3x8. 391 394. 652 13351; 384. 656 [426]; 394 390. 051 13231: 396 198. 401, 498 499. 656 [43X]; 396-199. 656 [440j. 397. 4 10. 656 14221; 479. 656 [429]; 497. 508 509. 662 [582]: 501. 657 [462]; 5 18-520. 538. 540- 541. 656 14391: 527. 662 [583]; 537. 5 4 . 646 [ I 871: 563. 58 1 -584. 646 11941: 56&565. 651 [310]: 5 8 7 589. 591 592, 595, 507. 599-602. 656 [43?]: 587. 590. 597598.

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Pagallo. N.J.. 98. M8 12261: 100. 660 152 I]; 100, 660 [522]: I X I. 65 I [3 1 h] Page, R.A.. Lcvcrant. G . R . . 330. 660 15231 Pailler. R.. 55-56. 448. 450. 466. 653 [355]: 524. 644 [I 421 Palmquist, R.W.. 470. 643 [I001 Panascnko. G.P., 89. 640 [lS] p.' ~ r. ~. ~ ~ s A y uD.. k . 428. 660 15241; 428. 662 [580] Pannsyuk. V . V . . 157. 653 [362j: 289. 630 [I41 Pnnkov, G . A . . 630. 632 h3?, 639 (IX]: 630, 639 [I91 Papkovich, P.F.. 394. 660 1.5251 pap rock^, S., 423. 570, 655 1.1051 Pillis. I1.(.' 335. 758. Ah0 [52h] Park. I . - M . . 579. 642 [83] Park, J.M., 276. 643 11 141 Parthasaruthy. T.A , 4 1 6 418, 660 [527]; 416. 651 [309]: 548. 654 [179] Partridge. P.G., 470. 666 [6X?) Pattnaik. A.. 523. 650 12831 Paul, H., 423, 570. 655 [405] Peltier, J.F.. 448. 662 [587] Pcng. L., 318-319. 660 [52X] Penty. R.i\.. 57.5 576, 660 [579] Pcppcr, R.T.. 422, 660 [530]: 50 1 . 570, 646 [I 891: 575-576. 660 (5291 Peregudov;~.G . Yu.. 725 228. 657 I4441 Pcrnot. J.J.. 369-370. 654 [382j Perov, B . V . , 667 [692] Peters. P.. 366. 368. 653 17591 Pclcrs. P.W.M.. 138. 14(&141. 650 [509] Petrov. Y u.N.. 244. 652 [34X] Pctrov. Yu.M.. 420. 652 13311 Pezzotli. G 34. 660 [531] Phillipovsky. A . V . . 466. 6 5 4 6 5 5 [394] Phillips, D.C., 71. 181. 289. 477, hhO 15331; 72, 060 [535]: 420, hGD 15341 Phillips, M.C.. 467. 6G4 [6l6] Phoenix. S.L., 44. 660 [537]; 45.666 [b78]: 143. 653 [351]: 165166. 299, 302. 660 15361; 233. 660 [538]; 276, 279, 649 [258]: 276, 648 [243] Piehler, H.R.. 492. 658 [492] Piekarski, K., 420- 421. 649 [2Slj; 420. 649 [249]; 42 1 . 049 (2 501 Piggot. G.H.. 62. 641 [52] Piggott, M.R.. 263. 435: 660 [539] Pilipovsky. Yu.L.. 464. 652 [341]; 469. 653 [371] Pines, R. Ya., 477. 660 66 l [540] Pinto, l'.J.. 62, 641 [52]

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Och~ai.Sh.. 138. 140-141. 659 [509]; 143 144, 659 [SOS]; 145. 659 [500]; 152. 659 [504]: 252, 659 [506]; 252. 659 [SO?]; 294. 659 [502]; 295. 461. 659 [SO I]; 46 I, 659 [503]; 464, 650 [SO81 Og;~sa.T..222. 659 [ 5 l 0] Oh. S-L.. 424 425. 662 [576]; 426. 659 151 I]: 431. 659 [5 121 Ohama. S.. 509. 646 [I791 Ohrincr. E.K.. 581. 662 603 [588] Ohsaki. T.. 478, 51 5. 575. 659 [S 131 Okamolo. 1.. 428. 666 (6671 Okan~uto.T.. 468. (164[628] Okamura. K., 51, 059 [514]; 421, 063 [600] Oku. Y . , 468. 664 [628] Okura, A,, 447. 659 [5151 Olifcrcnko. V.I., 461. 663 [602] Omaletc. 0.0..522. 659-660 [Sib] Orowan. E.. 25. 157. 660 [SIX]; 660 15171 Orzhckhovrky. V . L . , 500. 652 [328] Osamura. K.. 143-144. 659 [SOS]; 145, 659 15001; 252. 659 15041: 252. 659 15061; 252, 650 [507]; 464. 659 [508] Ovcharenko. V.E., 466. 644 11371

Ovchlnshy, A S .247. 250. 652 [729]. 247.250, 660 [519]. 247. 662 [579] Own, S -ti 42. 660 [520]

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Naslain. R..4X49.666 [670]: 55-56.448.450. 466, 653 [-755];421. 662 [560]: 42 I . 662 [586]: 524. 5444546. h5K [484]: 524. 644 [142]: 525. 545 546. 658 [485]: 531-532. 567. 646 [I901 Nathal. M . V . . 105. 479. 509 51 1. 641 1591: 451452. 644 11 351 Naych-Hashemi. H.. 352 353. 658 14861 Nazarova. M.P.. 47. 641 1531 Nefcdov. A.S.. 501. 657 [462] Ncmat-Nasscr. S., 91-92. 658 [487] Nctri~viili.A.N., 276. 643 [I 14) NCIIIII;~II. P . 254. 204 295, 658 [488] Nicholas. M.G., 427. 649 [252]; 428, 432. 658 [4X9]; 428, h5H [490j; 431. 658 [40lj: 432. 664 [62?] Nieulaou. P D.. 492, 658 14921 Nich. -1'.Ci.. 452. 642 [841 N ~ k ~ t l nI..V.. . 220. 211-132, 652 [744] Niqh~Ja.Y.. 550. 557 550. 668 17171; 567. 653 1.7681 Nix, W.D.. 717. 644 [1?3]; 644 [I341 Nixon. A.C.. 579. 646 [I781 Nocblc. R D.. 466. 665 [65O] Nourbakhsh. S.. 452.460.465.467.470.659 14951; 453. 629 14941; 4WI. 659 [4Y6]: 460. 659 [497]: 564. 659 (-1931 Nuismcr. R.J.. 219. 667 [699] Nuncs. J.. 104. 106. 244. 368. 580. 659 [498] Nutt. S.R..55. 659 14991

678

Author &dex

Pisarenko, G.S., 338, 661 [541] Polilov, A.N., 189, 191,661 [542]; 381,661 [543] Pollock, W.D., 457, 658 [481] Polyakov, V.A., 100, 665 [644] Pompe, W., 33, 652 [339]; 459, 646 [177] Popejoy, D.B., 194-197, 661 [544] Porter, J.l~., 371,643 [118] Portnoi, K.I., 456, 466, 470, 618, 661 [545]; 603, 661 [546] Prangnel|, P.B., 111-112, 661 [547] Prewo, K.M., 72, 641 [58]; 105, 244, 661 [553]; 369-370, 661 [552]; 369, 661 [549]; 476, 496, 661 [548]; 512, 640 [30]; 512, 661 [550]; 513-514, 658 [483]; 513-514, 661 [551] Pryce, A., 514, 661 [555] Quenisset, J.M., 421,662 [569]; 531-532, 567, 646 [190] Rabe, J., 51,653 [370] Rabinovitch, M., 68,661 [556]; 491,658 [469]; 571, 666 [675] Rabotnov, Yu.N., 25, 148, 155, 376, 386, 661 [560]; 112, 661 [557]; 311,324, 487, 661 [558]; 324, 329, 484, 661 [559]; 381,661 [543] Raj, R., 276, 279, 649 [258] Rand, B., 48, 661 [561] Rashid, M., 420, 661 [562] Ratke, L., 523, 661 [563]; 523, 661 [564] Raviart, J.-L., 571,666 [675] Ravikovich, A.I., 223, 654 [393] Razzel, A.G., 477, 661 [565] Reifsneider, K.L., 100, 641 [68] Reissner, E., 228, 647 [198] Rhee, W.H., 460, 659 [496]; 460, 659 [497] Rhodes, J.F., 541,667 [708] Rice, J.R., 157, 661-662 [567]; 213, 661 [566] Riek, R.C., 567, 655 [402] Risbud, S.H., 454, 648 [245] Ritchie, R.O., 368, 666 [674]; 371,643 [118] Ritter, A.M., 480, 644 [I 31] Rizza, J., 450, 648 [224] Robertson, D.D., 115, 118, 122, 662 [568] Robinson, H.H., 44, 660 [537] Rocher, J.P., 421,662 [569] Rogers, W.M., 541,667 [708] Rosen, B.W., 42, 373,662 [571]; 235,662 [570]; 89, 648 [236] Rosenkranz, G., 352, 662 [572] Rowlands, R.E., 122, 662 [573] Ruckenstein, E.J., 632, 642 [82] Rudnev, A.M., 611-614, 647 [220]; 615-620, 657 [445]; 615, 662 [574]; 619, 621-623, 656 [435]; 622, 624-630, 657 [446]

Ruehle, M., 302, 641 [65]; 418-419, 662 [575] Russel, K.C., 424-425, 662 [576]; 426, 659 [511]; 431,659 [512] Rykalin, N.N., 476, 662 [577] Ryoson, H., 369, 651 [322] Sahin, O., 452, 460, 465, 467, 470, 659 [495]; 460, 659 [496]; 460, 659 [497] Sakamoto, A., 456, 567, 662 [578] Sakamoto, H., 72, 651 [319]; 521,658 [468] Sakharova, E.N., 247, 662 [579] Samsonov, G.V., 428, 662 [580] Sandifer, J.D., 544, 668 [722] Santella, M.L., 581,662-663 [588] Sara, R.V., 420, 662 [581] Sarkissyan, N.S., 58,646 [193]; 235, 239-243, 505507,657 [448]; 235,239, 506-507, 657 [456]; 288, 662 [584]; 497, 508-509, 662 [582]; 527, 662 [583] Sarkissyan, O.A., 339-340, 657 [449]; 341-346, 657 [450] Saunders, S.C., 42, 660 [520] Savruck, M.P., 153, 662 [585] Sbaizero, O., 302, 641 [65] Scattergood, R.O., 72, 664 [617] Schamm, S., 421,662 [586] Scheed, L., 448, 662 [587] Schienle, J.L., 522, 641 [57] Schlautmann, J.J., 644 [I 34] Schmucker, M., 464, 642 [78] Schneibel, J.H., 581,662-663 [588] Schneider, H., 464, 642 [78] Schreurs, J.J., 470, 643 [100] Schueller, R.D., 531,663 [589] Schulte, K., 138, 140-141,659 [509]; 515, 654 [389] Schwartz, P., 44, 653 [367]; 44, 660 [537]; 45, 664 [627]; 45, 666 [678] Schwartzfager, D.G., 59, 666 [679] Scott, R.A., 223, 642 [77] Sedov, L.I., 5, 288, 663 [590]; 10-11,663 [591] Seibold, M., 526, 647 [207] Seith, W., 441,663 [592] Semenov, B.I., 570, 663 [593] Semiatin, S.L., 486, 647 [197]; 492, 658 [492] Semirchan, A.Kh., 501,657 [462] Seo, Y., 567-568, 650 [292] Serebryakov, A.V., 235, 239-243, 505-507, 657 [448] Scvely, J., 48-49, 666 [676] Seyyedi, J., 352-353, 658 [486] Sha, G.T., 114, 664 [632] Shahinian, P., 598, 663 [594] Shalman, Yu.l., 11,654 [387] Shanley, F.R., 14, 663 [595] Shaskolska, M.P., 605, 639 [5]

Author index

Sheehan, J.E., 57, 647-648 [221]; 57, 663 [605] Shepard, L.A., 295, 646 [183] Shermergor, T.D., 89, 663 [596] Sherwood, R.C., 630, 650 [284] Shesterin, Yu.A., 476, 662 [577] Shetty, D.K., 436, 663 [597] Shih, C.J., 522, 663 [598]; 72-73, 668 [720] Shimoo, T., 421,663 [600] Shin, H.H., 513, 663 [601] Shinora, T., 448, 461,466, 653 [372] Shorshorov, M.Kh., 461,663 [602]; 476, 662 [577] Shtinov, E.D., 384, 656 [426]; 501,657 [462] Shyue, J., 449, 663 [603] Sierakowski, R.L., 339, 663 [604] Sigalovsky, J., 57, 663 [605] Signorelli, R.A., 324, 655 [398] Sih, G.C., 153-154, 663 [607]; 156, 186, 663 [610]; 162, 164, 663 [611]; 162, 281,663 [608]; 162, 663 [606]; 162,,663 [609] Simancik, F., 520, 664 [612] Singh, J.P., 72, 664 [617] Singh, R.N., 458, 664 [613] Sirotenko, L.D., 503, 668 [715] Siu, S.C., 368, 666 [674] Skinner, A., 287, 664 [614] Skvortsov, D.B., 255, 527, 533-536, 656 [425] Slate, P.M.B., 503, 650 [281]; 503, 664 [615] Slepetz, J.M., 104, 106, 244, 368, 580, 659 [498] Smith, J.E., 467, 664 [616]; 469, 667 [688] Smith, P.A., 369, 646 [182]; 514, 661 [555] Smith, P.J., 448, 654 [388] Smith, P.R., 480, 644 [131] Smith, R.L., 44, 667 [690] Smith, S.D., 452-453, 640 [42] Smith, S.M., 72, 664 [617] Sneddon, I.N., 80, 664 [618]; 155, 664 [619] Sobolev, S.L., 552, 664 [620] Soboyejo, W.O., 449, 663 [603] Sokolovskaya, E.M., 441,443, 664 [621] Somekh, R.E., 424, 469, 651 [311] Sorokin, N.M., 71,235, 245, 250, 252, 657 [452]; 280, 283-284, 286-287, 657 [453]; 377, 380, 657 [4511 Spain, I.L., 46-49, 644 [136] Speyer, R.F., 513, 663 [601] Spiridonov, L.S., 100, 102-103, 651 [313] Springer, C.C., 223, 642 [77] Stanking, R., 432, 664 [622] Stanley, D.R., 62, 641 [52] Stanton, M.F., 62, 664 [623] Stark, J., 523, 661 [563] Starostin, M.Yu., 58, 646 [193] Starrett, S., 369-370, 661 [552]

679

Stepanov, A.V., 189, 664 [624] Stinton, D.P., 524, 664 [625] Stobbs, W.M., 111-112, 661 [547] Stock, A.T., 339, 664 [626] Stohr, J.F., 68, 661 [556] Stoloff, N.S., 522, 639 [13] Streckert, H.H., 424, 648 [227] Street, K.N., 315, 651 [305]; 317, 319, 651 [304] Stumpf, H., 45, 664 [627] Styrka, T., 49, 652 [342] Subramanian, K.N., 420, 666 [670] Subramanian, R.V., 42, 660 [520] Suganuma, K., 468, 664 [628] Sugihara, K., 46-49, 644 [136] Suib, S.L., 423, 651 [318] Suleimanov, F.Kh., 170, 220, 657 [455]; 657 [454]; 170, 172, 174, 220, 222, 359-360, 362, 657 [457]; 174-176, 664 [629]; 235, 239, 506-507, 657 [456] Sullivan, T.L., 508, 664 [630] Sun, C.T., 84, 642 [79]; 114, 664 [632]; 115, 118, 664 [631] Sung, Y.-M., 432, 664 [633] Suresh, S., 333, 364-365, 664 [634]; 467, 642 [89] Sutcu, M., 40, 165, 665 [635] Sutherland, H.J., 100, 665 [636] Sutton, W.H., 432, 665 [637] Suzuki, N., 468, 664 [628] Suzuki, T., 72, 651 [319]; 448, 461,466, 653 [372] Svendsen, L., 564, 650 [280] Sveshnikov, A.G., 149, 665 [638] Svetlov, I.L., 47, 641 [53]; 456, 466, 470, 618, 661 [5451 Swanson, G.D., 284, 648 [228] Takahashi, S., 246, 665 [639] Takeda, N., 339, 663 [604] Takemura, M., 421,663 [600] Talley, C.P., 52, 665 [640] Talreja, R., 363, 665 [641] Tamuzh, V.P., 44, 647 [216] Tan, S.C., 376, 665 [643] Tanabe, Y., 542, 668 [716] Tanaka, K., 87, 658 [471] Tandon, G.P., 100, 660 [521]; 100, 660 [522] Tarakanova, T.T., 500, 647 [215] Tarasova, O.B., 448, 640 [33] Tarnopolskii, Yu.M., 100, 665 [644] Tauchert, T.R., 100, 665 [645] Taya, M., 86, 665 [646]; 87, 665 [648]; 321,323, 665 [647] Taylor, H.M., 233, 660 [538] Tegeris, A., 62, 664 [623] Ten, V.P., 124, 665 [649]

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Vaidya, K.U.. 420. 666 16701 Valentin, I ) . , 448. 062 15871 Valentine, T.M., 432, 658 149 I] van Dovcr, R.B., 630, 650 [284] Van Dykc. 141 112. 648 [242]

Waddoups, M.E.. 219, 666 [677] Wadlcy. H.N.G.. 48 1 -482. 484 485.488.493, 045 [I 541; 49 1. 644 [I 441; 49 1. 045 [I 551; 49 1 . 647 [209) Wagrtcr. 11.D., 45. 666 (6781 Waitc. M.J.. 432. 058 14911 Walkcr. P.S., 244, 575. 577. 049- 650 [I741 W:tllcnbcrg. F.T.. 59. 666 [h79] Wnllindcr. M., 564. 650 [2XO] Wang. U.-J.. 427. 666 [6Xlj W ; I I I ~F.. . 318-319. 660 [52X] Wang. H.. 542. 666 [hXO] Wang. J.Y.. 569, 653 [369] Wang. L..254. 640 [21] Wang. S.S.. 449. 650 [28X] Wang, Zh., 3 18 319, 060 [528] Ward. C.H., 31. 644 [I291 Ward-Close, C.M.. 479. 666 [682] Warrcn. R.. 466. 666 [683] Warricr, S.G.. 569. 653 [369]; 569. 667 [686]: 569. 667 [687]: 569, 667 ((1851 Warwick. C.M., 448. 654 [38X]: 469. 476. 651 13121: 469. 667 [hXX] W:tsscrmi~n.(;.. 523. 661 [563]; 523. 661 15641 Wa(i~rl;~lx. 0..420. 466, 667 [6X9] W;~rson.A.S . 44, 607 [ti901 Wall. W.W., 4b 47. 667 16911: h67 [hY2] Walls. .I.F.,426, 641 167) W:~wncr. F.E.. 55. 659 [490]: 450. 647 12 121: 53 1. 663 15891 Wchcr. C.H.. 49 1 . 509. 667 16931; 5 10. 667 16941 Wcl>cl. K.. 51 5. 654 [3X1)] Wcclon. J.W.. 324. 655 13981 Wcihull. W., 37. 667 [695] Wc~singcr.M.D.. 496, 667 [696] -,1

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[Jcki. M..428. 666 [667] Clhlmilnn. D.K , 566. (A3 [l02] I l s l l i ~ k ~M.U.. . 4x1. hhh (hi581 Clslinov. L . M . , 401. 663 [602]; 461. 666 166l)j

Vanin, G.A. (Van F o Fy). 89, 666 Ihll] Varenkov. A.N.. 424. 652 13311 Varin. R . A . . 420-421, 649 [251]: 420, 649 12.191; 42 1 , 649 [250] Vasilenkov, Yu.M., 464, 652 [341]; 469. 653 13711 Vl~ssel.A,, 491. 658 [469] Vauterin, 1.. 365. 666 16721 Vcga-Boggio. J.. 53. 55. 666 [h73] Veltry, R.D., 5 12, 640 [30] Vcnkateswara Rao. K.T.. 368. 666 16741 Vcnkateswaran. V., 60. 645-646 11721 Viala, J.C.. 448, 662 15871 Vidal-Sctif. M.-II.. 57 1. 666 [675] Villeneuvc, J.F., 4 8 4 9 , 666 [676] Vingsbo. 0.. 53. 55. 666 [673] Vinogradov. L.V., 461. 663 16021 Vuba. K.T.. 435. 666 [663]

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Tcnncy. D.R.. 442, 648 [244] Tcrcntjcva. L.M.. 244. 652 134x1 Tewari. S.N.. 466. 665 [hSO] Thomas. M.. 31. 658 [482] Thomason. P.F., 169, 665 [65 I] Thompson, R.R.L., 339. 664 [626j Thouless, M.D.. 165. 168. 665 16521; 302. 641 [66] Tiefel. T.H., 630, 650 [284] Ticgs. T.N.. 522, 659-660 [516] Tien, J.K., 443. 642 [70]; 445. 665 I6531 Tikhonov, A . N . , 149. 665 [h38] Timofccva. N.I., 603. 661 [546] Timoshenko. S.P.. 13. 355 356,665 [654]; 373.665 [655] Ting, J.-M., 543. 665 16561 Tirard-Collet. R.. 556. 6.50 [2H2] Toloui 8.. 580-58 1. 623. 665 [657] Torovec. L.A., 466, 644 [I371 Tosyali, 0.. 304. 641 (481 T o ~ h .I.J., 486. 665 16581: 500. 644 [[32] Touraticr. M., 100. 102 104, 605 [h59] Trcgubov. V.F.. 466. 644 [I371 Trcsslzr. R.E.. 424.1, 652 13343; 466. 470. 665-066 [6601 Trtrnnov. S.V.. 235. 239 -243. 505 5117. (157 144x1 TI-ofimov, V.V., 405, 408, (145 {14X] Tsai. M.Y.. 228. 606 [Ohl] Tsai. S.W.. 91. 639 (71; 122. 640 [ZOj Tsang:trakis. N.. 104. 106. 244. 368. 580. 659 [49XJ Tsirlin. A.M.. 52 54. 666 [662]: 71. 2.75. 245, 250. 252. 657 [452]; 280. 283 284. 286 287. 657 [453]; 377. 3x0. 657 (4511; 461. 663 [002J Tsou, H.T.. 418. 459. 652 [3!8] Turusov. R.A., 435. 666 [663] Tvardovsky. V.V., 174 176. 664 16291; 184. 562. 582 583. 650 [299]: 184, 186. 204.657 [460]: 200. 657 [461]: 200. 202. 666 [604]: 204. 206. 606 [665]; 204. 207 209. 657 [459]: 204. 207. 209. 666 [66G]; 2 15. 2 17. (157 [45X]; 270. 275. 646 (1921: 381, 384, 386 388. 391 394, 652 [735] Tyson. W.K.. 133, 263. 317. 310. 651 [306] I'zou. I1.Y.. 162. 164, 667 161 1 1

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