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
Structural materials
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
Effectiveness of materials and structures
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
Structural materials
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)
Effectiveness of materials and structures
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.
Structural materials
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
Effectiveness of materials and structures
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
Structural materials
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
(<0
~
(3)
(b)
Fig. 1.5. Failures of engineering structures. (a) Tensile fracture of a turbine blade. Such occasion yields catastrophic consequences through disbalance of the jet engine rotor as well as penetration of pieces of the blade into the fuselage. (b) The aeroplane wing on the surface (1), at a normal flight (2), and at an abnormal flight when change in the shape of the wing can change the aerodynamic performance of the aeroplane. (c) Buckling of a shell under external pressure that destroys completely an underwater vehicle.
Consider now a rotating element, for example, the drive shaft of an automobile (fig. 1.7). At a critical rotation speed, oJcr, equal to the natural frequency of the transverse vibrations of the shaft [654], the amplitude of the vibrations grows indefinitely. The natural frequency is
f ~ ~---~(E) l/21I) l/2
(1.14)
where E is the Young's modulus of the material, A the cross-sectional area and I the second moment of the cross-sectional area. Therefore a critical material parameter for vibration of various beams is the specific modulus, E/p, which is the square of the longitudinal wave speed in the material.
14
Structural materials
Ch. I, w
A Fig. 1.6. A body stressed by centrifugal forces.
In general, a design philosophy based on the science of strength of materials includes measures of efficiency of various kinds of structural elements [22, 23, 37, 202, 595]. Performance, p, of a structural element, which can be for example its mass or cost, depends on (i) the functional requirements (the speed of rotation of the turbine blade or drive shaft, rigidity to the bending and twisting of the aeroplane wing, the maximum depth of a submarine), R, (ii) the sizes, S, and (iii) the material/ shape properties, Q. Following Ashby [23], let us assume the three groups of the parameters, R, S, and Q be separable in a function determining p, that is (1.15)
p -fl(R)fz(S)f3(Q).
This assumption allows us to analyze dependence of the performance on the three groups of the parameters separably. Therefore, ifp has been maximized with respect
j,L
I
l~\\\\Xl
l~\\\\~l
_
-~-
i
IZ/I//A
r a Fig. 1.7. Transverse vibrations of the draft shaft.
Effectiveness of materials and structures
Ch. I, w
15
F
Fig. 1.8. A beam loaded in bending. to f3, it will work for all the functional requirements and global sizes of the structural element. 2 For the case of bending (fig. 1.8), the functional requirement may be a specified stiffness of a beam, C, and the performance we are looking for is the mass,
m --Alp
(1.16)
where 1 is the beam length. The stiffness, which is the ratio of generalized load F to generalized displacement 6 is F
E/
C - -~ -- K l--T
(1.17)
where K is a coefficient depending on a design of the beam and loading configuration. Define now the dimensionless shape parameter
~ stiff B
4M A2 ,
(1.18)
such that ~stiff = 1 for a solid circular cross-section. Combining eqs. (1.16)-(1.18) '+'B
yields
m--(4rc)l/2(4KC)l/2 15/2
P
( E ~ i f f ) 1/2 .
(1.19)
The relationship obtained is written in the form of eq. (1.15). The material/shape performance index for bending, o~iff
( E ~ i f f ) 1/2 -
P
,
(1.20)
should be maximized to supply a m i n i m u m weight to a performance parameter, which is the weight of the beam.
2 This is an obvious simplification, but at this stage, a useful one.
16
Structural materials
Ch. I, w
When we are looking for a strong lightest beam loaded by bending moment M(x) where x is the coordinate along the beam axis we should minimize the integral m -- f0 l #(x) dx where d m = 2(2rt)1/3 (M(x))2/3
P
(O., ~Brength) 2/3
(1.21)
and
~ Bstrength
_ 4rtl/2 W A3/2
(1.22)
.
thstrength , is chosen such Here W is moment of resistance. Again the shape parameter, ~B that "WB hstrength - - 1 for a solid circular cross-section. The material/shape performance u index is strength- (~ ~)Brength) 2/3
QB
(1.23)
P
Obviously, in the case of an element subjected to a tensile load (fig. 1.5a), the material strength performance index Q~iff
--,
E P
t,.)strength ~v
---
O',
P
(1.24)
where cr, is the material strength or an admissible value of the stress under the service conditions. Note that the stiffness under tension and compression load are usually the same if buckling is avoided. For the case of a cylindrical shell (fig. 1.5c) with a solid wall loaded by hydrostatic pressure, the buckling pressure (see below, Section 8.3), is p , ~ k E ( R / L ) ( h / R ) 5/2
and E2/5
Qbuckl SH-hydro -- ~ "
P
(1.25)
Therefore, we can introduce two series of the material/shape performance indexes. The first one portrays stiffness, buckling and vibration characteristics of a structure
Ch. I, w
QE --
Effectiveness of materials and structures
(E~bE) ~
17
(1.26)
P
a n d s e c o n d o n e is r e l a t e d to t h e s t r e n g t h o f a s t r u c t u r e
Q~ -
(1.27)
p
where ~ and//are parameters depending on a particular performance of an element, _< 1,/~ _< 1, ~bz, a n d qS~ a r e t h e s h a p e f a c t o r s . T o c o m p a r e j u s t a p e r f o r m a n c e o f v a r i o u s m a t e r i a l s in o n e t y p e o f t h e s t r u c t u r e w i t h e q u a l p e r f o r m a n c e r e q u i r e m e n t s , it is n e c e s s a r y t o t a k e q5z - 1, a n d qS~ = 1. T a b l e 1.2 s u m m a r i z e s b a s i c c h a r a c t e r i s t i c s o f v a r i o u s m a t e r i a l s t h a t c a n be a b a s e f o r t h e i r c o m p a r i s o n . L e t us p l o t d a t a f o r s o m e m e t a l s a n d c e r a m i c s o n t h e Y o u n g ' s m o d u l u s / d e n s i t y p l a n e (fig. 1.9). W e p l o t a l s o lines o f the c o n s t a n t v e l o c i t i e s o f
TABLE 1.2 Melting temperature, density, Young's modulus and specific Young's modulus of structural materials. The data have been compiled from various sources. Material Metals Fe Al Ti Ni Mg W Mo Nb Be Ceramics TaC HfC NbC TiC TaB2 ~-BN TiB2 MgO SiC
B4C B A1203 MoSiz Si3N4
Melting/sublimation temperature ~
Young's modulus, E GPa
Density, p kg/m 3 910-3
E/p (m/s) 2
1536 660 1665 1455 650 3400 2620 2415 1287
200 70 100 200 45 410 330 82 240
7.87 2.7 4.5 8.9 1.74 19.3 10.2 8.57 1.85
25.4 26.0 22.2 22.5 25.9 21.2 32.4 9.6 130.0
4255 3890 3615 3260 3040 3000 2850 2825 2600 2470 2300 2050 2020 1877
550 450 500 490 690 86 540 290 460 450 400 400 440 385
14.3 12.6 7.56 4.92 11.70 2.29 4.45 3.65 3.2 2.5 2.7 3.97 6.1 3.2
38.5 35.7 66.1 100.0 59.0 37.5 121.3 9.4 143.8 180.0 148.1 100.8 72.1 120.3
18
I000
'
~r~
'
-
'
*
9
'
~
'
/
'
'
HS
M(
.. "
9
Fe ~ ~ ~
~CF
f
_-
~ Ti~ ~ J 0
~
Nb
/
All
_
"~"
~)~
9
Be~)
100
'~
9
-
\
Ch. I, w1.1
Structural mater&ls
-
0
O)
/
~oooo
Metals Ceramics ***** High-modulus sbrengfh (HS)
ooooo
10
I
1
(HM) and
high-
carbon fibres (CF) I
p /
I
I
g/ m
I
I
I
I
10
Fig. 1.9. Young's modulus versus density for some metals and ceramics used in engineering. Young's modulus for carbon fibres are in the fibre longitudinal direction. Also plotted are the dotted lines corresponding to fixed velocities of two longitudinal wave (10000 and 5,000 m/s). Note the two exceptions in the metal row, beryllium and niobium.
longitudinal wave in solids which is just the square root of the specific Young's modulus of isotropic solid, VL- ~/~,
(1.28)
being a measure of the efficiency of materials in vibration environment (see eq. (1.14)). Obviously, all the commonly used metals (iron, aluminium, titanium) are equally good or equally bad from this point of view, the latter being true if to compare the metal behaviour with that of commonly used ceramics. We already mentioned ceramics as superior materials when compared with metals at high temperature environment. Now we see another advantage of a possible use of highmodulus/low-density materials. To emphasize an approximate equivalency of metals in applications where the specific Young's modulus determines the structure performance, the value of QE normalized by that for iron is shown in fig. 1.10. We see that the ratio is nearly 1 for all metal alloys. At the same time, it is clearly advantageous to use lighter metals in structural elements for which ~ < 1 (for example, beams under bending (~ = 1/2) or cylindrical shell under hydrostatic pressure (~ = 2/5)). Usage of ceramics and graphite, certainly in combination with metals as matrices, yields an essential improvement in the performance. Figure 1.11 compares the
Ch. I, w
2.5
Effectiveness of materials and structures
'
J
'~I
'
i
,
I
'
'
'
I
'
n
,
I
0(--
00oo0
'
1
l
,
I
'
'
19
'
ol= J / / 2
*****
o~=2/5
*****
2.0
1.5
Co
9
~
.~
~ 9
o
1.0
0 , 5 0--1
i
I ~
~
o
I
9
I
I
41
I
I
I 61
I
# /
!
I 8l
_
.
I
I
i
101 9 I J12
g/om3
Fig. 1.10. Material stiffness index for metal alloys for tension and vibration (c< - 1), bending (~ -- 1/2), buckling of cylindrical shell under hydrostatic pressure (a -- 2/5) normalized by that of iron.
12.5
r
I0.0
~. "-"
,,~
I:
I: 5.0
~
~
***** ~=I/2
* * * * * ~=2"/5
2
F ~
o.q.5
o
o
~
2.0
2.~
~
~
"~
:
~
:
o
a.o .3.5 p / g/~
: -
4.0
Fig. 1.11. Material stiffness index for ceramics used in the fibrous form for tension and vibration (~ = 1), bending (~ - 1/2), buckling of cylindrical shell under hydrostatic pressure (~ = 2/5) normalized by that of aluminium.
20
Structural materials
Ch. I, w
performance indexes of some ceramics used in the fibrous form with that of aluminium, which is a usual matrix for light structural composites. We see again, on a quantitative base, that including light substances into a class of structural material leads to a pronounced increase in the performance. On the other hand, the improvement decreases as value of ~ decreases. This calls for more precise design of structures with low ~. Do we need non-metallic materials?
Table 1.2 gives the answer to a first approximation. Yes, we need ceramic materials at least because of their high specific stiffness. We will come back to this question later on and give a more complete answer.
1.2. Stiffness and strength of materials
Now, having a general impression of what kind of material properties effects the effectiveness of engineering structures, we shall discuss physical and technological limits imposed on the properties of various solids, the former being determined by the nature and the latter being a result of physical and engineering development. From the physical point of view, a solid can be either crystalline or amorphous. A variety of crystals are well known. Atoms in solids under given external conditions (temperature, pressure) tend to form a stable configuration corresponding to a minimum of the free energy. All possible configurations are completely determined by seven syngonies. A solid arises either from vapour or melts by a jump, called phase transition, if a necessary kinetic is permissible. The crystallization process of a melt can be arrested and then a supercooled substance inheriting a structure of the liquid is obtained. Such a solid is called amorphous, it has no usual crystal structure. A well known example of such a solid is glass, which is either pure silica or a mixture of silica and various oxides. During the last decade, amorphous polymers have become widely used in modern technology. Note that under extremely rapid cooling, metals can also become amorphous in the solid state and properties of such metals are very interesting. In addition to the structural state, the type of interatomic bond is an important indication in the physical classification of solids. This may be of Vander-Waals, or metallic, or ionic, or covalent type. The type of interatomic bond determines its stiffness and strength, and therefore potential physical properties of a structural material, namely elastic moduli, ideal strength, coefficients of thermal expansion. Solid state theory provides means to establish relationships between micro- and macrocharacteristics of a solid, the former being potentials of the atomic interaction, the latter are named above and such properties as melting temperature, surface energy, sublimation energy etc. are to be added to the above list. On the other hand, microcharacteristics, parameters of the potential of atomic interaction
Stiffness and strength of materials
Ch. I, w
21
being an example, can be determined via macroproperties only. They can be therefore considered first of all as a tool for understanding the relationships between various macroproperties of solids. We shall now present, very briefly, an outline the fracture behaviour of materials and some conclusions of the strength theory. 1.2.1. Strength o f an ideal solid body We call the ideal solid such a solid body that does not contain defects which effect its strength. The strength of the ideal solid is certainly a maximum strength attainable for a particular solid. Although publications of the last two decades or so have revealed a great deal of details of the interpretation of the ideal strength of solids and despite the composite technology is certainly the first practical application of such theoretical knowledge, we do not need to discuss here the subject in detail referring a reader to comprehensive reviews by Kelly [301], Macmillan [376], and Milstein [464]. The simplest way to evaluate the upper limit of attainable strength is, in fact, to build up a model to correlate the ideal strength which cannot be actually measured, to characteristics of a solid which are obtainable, the latter being the surface energy, elastic moduli, characteristic size of the crystal lattice, and so on. Let us follow a usual procedure. Dependence a(x) of the cohesion stress upon the distance between two adjacent atomic planes has to be of such a type that a(x) = 0 at the equilibrium point, say Xo, then function a(x) should have a maximum at some point x, > 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
Structural materials
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 << 1. Here fl is a constant. Taking into account that the value of 2 has an order of 1, we introduce ~ -- 7/2 and obtain a simple relationship, that is a* = 2~/Xo, ~ ,~ 7,
(1.34)
which obviously can be predicted beforehand, at least on the qualitative level. Note that a usual form of eq. (1.34), which is
Stiffness and strength of materials
Ch. I, w
a* = fiE
23
(1.35)
where fl is a constant, follows from the same initial assumptions except for that dependence a(x), eq. (1.30). It is not necessary to compare eq. (1.34) with similar results obtained by Orowan (see [301]) simply because a main parameter to compare, that is a value of the ideal strength, cannot be obtained in a direct experiment. It is also necessary to analyze the upper limit of shear strength of crystals. Frenkel was the first who did it. Obviously a periodical dependence of shear resistance upon displacement u should be observed, so that z - z, sin(2nu/Ax),
(1.36)
where Ax is the period determined by both the crystal structure and orientation of the shear system. A linear size y is determined by the same factors. Hence, dT
dutu=0
= 2nz,/Ax = I~/Ay
where # is the macroscopical shear modulus and Ay a characteristic size if the structure is in the y direction. Finally, "c, = ( 1 / 2 n ) ( A x / A y ) p = ~la,
(1.37)
with ct ~ 0.1. The ideal shear strength given by eq. (1.37) is much higher than it has been ever observed in experiments. Comparing eqs. (1.35) and (1.37) we see that the ratio z*/a* = # / E depends on the Poisson's ratio of the material only (see below eq. (3.16) for the well known relation between # and E). 1.2.2. Strength o f solids containing defects
When a defect making the stress state in a body be nonhomogeneous occurs, the analysis of the previous section based on an assumption of the homogeneous stress field is no longer valid. Such a defect can be either two-dimensional like the crack, or uni-dimensional, e.g. the dislocation. Grifiith's cracks
Griffith [208] was the first to make the situation clear in the physical terms. He applied the energy balance to the analysis of the equilibrium of a crack in an elastic plate (fig. 1.13). If AW is the change of the elastic energy of the plate due to the occurrence of the crack, then an increase in the crack length will be possible if 0AW _> 47. OL
(1.38)
24
Structural materials
Ch. I, w
(7
TTTTTTTTTTTT
Fig. 1.13. A crack in a plate.
This means that the crack can propagate when the corresponding increment in the free elastic energy cannot be compensated by the energy to be absorbed by a newly formed free surfaces. Obviously, 0-2
AW o~ L 2 x -~-,
(1.39)
and the coefficient in eq. (1.39) depends on a particular geometry. Combining eqs. (1.38) and (1.39) yields the Griffith's ultimate stress for a solid containing a crack, that is 0-,
-aV
/'~E Z
(1.4o)
where ~ is a constant of the order of 1. The stress field at the tip of a crack in the elastic continuum is known (see below, Section 4.1) to be singular so that the dependence of the normal stress on the distance, r, from the crack tip is K
0-(r) = x/-~-r
(1.41)
where K is a value called the stress intensity factor having a dimension (stress). (length) 1/2, K cx 0-x/-L where 0- is the applied stress. In the force terms, the fracture criterion can be written as K = K*
(1.42)
where K* is the critical stress intensity factor. Criteria given by eqs. (1.40) and (1.42) are identical as
Ch. I, w
Stiffness and strength of materials
25
K* ~ x / ~ E . Substituting real values of the crack length and corresponding values of the surface energy into eq. (1.40) to estimate the ultimate strength of a specimen containing a notch, yields strength values much lower than those measured in an experiment. In the case of metals, a reason for such a deviation from the Griffith's strength is plasticity caused by a drastic decrease of the ideal shear strength given by eq. (1.37) due to the formation and motion of linear defects called dislocations.
Dislocations and plasticity [175, 264, 301, 560] Dislocations produce elastic distortions of the crystal lattice. Formulating the corresponding plane and antiplane elastic problems yields simple solutions that give the stress/strain fields around a dislocation line and elastic energy stored in the material per unit length of the dislocation, that is wd e< l t b 2 / ( 1 - v) for the edge dislocation and wd e
26
Structural materials
Ch. I, w
The size of the plastic zone at the crack tip in metals can be approximately evaluated from eqs. (1.41) and (1.42), that is r* ~
(1.43)
where 0~y is the yield strength of the metal. 1.3. Structural materials As mentioned above, structural materials are usually divided into metals, glasses and ceramics, and polymers. Carbon materials being very like ceramics in many appearances, are classified as a special category because of their specific properties and applications. We see that such a classification corresponds to the difference in the behaviour of dislocation in the crystal lattice, except for polymers 3 which are the only organics widely used as structural materials. In the present context, there is no need (neither possibility) to recount structural materials as they are. We shall just compare these classes to establish a base to compare composites with conventional structural materials. What are most important properties of a material from the engineering point of view? Some of them, like the stiffness to density ratio, melting temperature and temperature interval of practical use, effective surface energy, availability of the raw materials, etc. are either strictly defined by the nature or placed into a definite interval. Other properties, examples being the strength, cost, resistance to severe environmental conditions, machinability, weldability, etc. depend on a level of the technology. 1.3.1. Metals and alloys
Really plastic metals such as chemically pure single crystals, exhibit plastic yielding at very low stresses, so that engineering metals to meet demands on the stiffness and strength of structures should be strengthened in some way, normally related to some restrictions of dislocation motion. The problem is really very complicated because the strengthening yields some degree of the brittleness. Besides, when we turn to heat resistant alloys, stability of a strong material microstructure is to be preserved and a degree of the strengthening decreases because new possibilities to overcome obstacles for dislocation come about.
Strengthening of metals The yield strength of metals can be enhanced in a number of ways; some of them are as follows.
3Wood is made of natural polymers.
Ch. I, w
Structural materials
27
9 Making polycrystals and decreasing the grain size. Actually, structural metals have always been in polycrystalline form and the Hall - Petch relationship between the yield stress, Cry, and grain size, d, that is r
-- r o + kd -1/2
where r and k are constants, is held down to grain sizes characteristic to so-called nanocrystalline materials in which crystal grains are normally free of dislocation and quasi-amorphous grain boundaries occupy volume of the same order of magnitude as the grains. So deformation mechanisms are different from those based on the dislocation motion. 9 Solution-hardening. Alloying elements can occupy either a regular position in the host lattice substituting the host element or some point inside the lattice skeleton. In both cases, elastic fields are produced that interact with those of dislocations making their motion more difficult. 9 Particle-strengthening. Introducing small particles into a metal matrix provides a variety of mechanisms to restrict dislocation motion. A dislocation moving under a shear stress in its slip plane meets a system of the obstacles and first imposes a force on them. Then the dislocation can either break a particle in which case the effective resistance to dislocation motion will be determined by the particle strength or pass through the obstacle system leaving on the particles loops in which case the smaller the interparticle distance, the larger an increase in the yield stress. So in reality, a sufficiently large hardening can be accomplished if the particle size is sufficiently small. The particles can be introduced by various ways. In particular, it should be mentioned: Precipitation-hardening. This is a very effective method of constructing a hardened microstructure. Perhaps it was originated in aluminium alloys called Dural and then spreads over other metallic systems. The essence of the method is aging a supersaturated solid solution that is obtained by homogenization of it at a high temperature with subsequent quenching to the ambient temperature. - Internal oxidation. This method expands a family of alloys with dispersed obstacles to dislocation motion. The internal oxidation method is essentially the dissolution in a metal matrix of small quantities of an element that forms oxide more easily than the matrix element. The oxygen is made to diffuse from an atmosphere that leads to formation of small dispersed particles of oxide. Similarly, carbides or nitrides can be produced in the form of small particles.
-
9 Intermetallics. A metallic system, say A - B, can contain intermetallic compounds of a A m B n form where m and n are natural numbers. The crystallography of such a compound can be rather complicated, and the dislocation motion in it occurs to be more difficult than in the pure base metals, so values of the ideal strength are expected to be sufficiently high for an intermetallic. This yields, however, to the low temperature brittleness. Special alloying decreasesbrittleness and makes these alloys to be potential candidates for heat resistant materials. 9 Particulate composites. Relatively large particles can be introduced into a metal matrix directly. Such materials are called composites. Actually they are somewhat
28
Ch. I, w
Structural materials
between short-fibres composites and dispersion hardened materials containing disperse obstacles to dislocation motion. The dislocation/particle interaction ceases to be a main mechanism of hardening, the particles start to play their own role that appears in increasing the elastic modulus of the material. At the same time, the size of particles is small enough for an interface zone of an appreciable volume to arise, whose microstructure is greatly effected by the particle. So some mechanisms just mentioned are still working, and new combinations of them can occur. These composites can be produced by various methods. Some of them are - Powder metallurgy; - Squeeze or stir casting; - Spray casting; We do not mention wellknown quenching of steels to produce metastable martensite at room temperatures, work-hardening and some other processes. It should be finally pointed out that a combination of hardening mechanisms usually occur in strong metal alloys. Strength/fracture-toughness
balance
Actually, physical metallurgy of metal alloys aims at such microstructures of the materials that would prevent a global dislocation motion at small shear stresses, but at the same time, provides local plasticity at the crack tip to enhance the effective surface energy. This leads to a balance between the strength, a*, measured in testing a smooth specimen, and on the other hand, the fracture toughness, K*, measured in testing a specimen containing a pre-made notch. Any general requirement to an 300
' 'o' 200
:\' o
. . . . o
'
. . . . . . . . .
:
. 0
\
0
0
0
100
0
0
500
l
!
t
I
J
I000
I
I
i
!
I
I
1500 c~.
I
/
!
0
0 I
I
2000
MPa
I
I
_
'
J
2500
Fig. 1.14. Fracture toughness versus strength for steels. After compilation given in [336].
Ch. I, {}1.3 160
29
Structural materials
I
,
,
,
,
I
,
,
,
r
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I
I
I
I
l
I000
o
o
o
]
o
i
1250
o. /
I
i
_
I
1500
MPa
Fig. 1.15. Fracture toughness versus strength for titanium alloys. After compilation given in [336].
optimal microstructure except those of a qualitative nature cannot be formulated as yet. To illustrate a usual trend for the fracture toughness to go down with the strength going up, we present a plot of strength versus fracture toughness data for steels and titanium alloys (figs. 1.14 and 1.15).
Heat-resistant alloys With temperature increase, the elastic constants of a solid decrease, thermal fluctuations increase; hence, values of the ideal strength go down. More importantly, diffusion processes accelerate enormously, time dependent strains occur as a result on mechanisms of dislocation motion through obstacles by climb, pure stress aided diffusional flow (Nabarro-Herring creep), grain boundary sliding, etc. These processes cause creep and creep rupture of materials. The time base for creep processes of practical interest can be seconds (missiles), minutes (rocket engines), hundredths hours (aviation engines), tens of thousandths hours (stationary gas and steam turbines). Correspondingly, the creep rate can range from 10 -2 S- 1 to 10-11 S- 1. Normally metals are subjected to creep at T > ~ 0.4 Tin. Therefore, if an alloy is able to resist creep at a given time base at temperatures above ~ 0.4 Tm such an alloy can be called heat resistant. For instance, some aluminium alloy can operate at temperatures about 250-300~ They are heat-resistant aluminium alloys. The same is true with regard to titanium alloys operating at temperatures 550-600~ Such aluminium alloys are used in the airframe structures of an aircraft, titanium alloys are used in the airframe structures of supersonic aircraft and in the compressor of a jet engine. Still, increase in the speed of the aircraft and pressure
30
Ch. I, w
Structural materials
ratio of the compressor leads to a necessity to enhance the operating temperature of low density heat resistant materials. The development of heat-resistant nickel superalloys has aimed at the usage of high temperature oxidation resistance of nickel-chromium alloys and organization of precipitation-hardening in the structure that has to be stable at very high temperatures for sufficiently long time. This development was accompanied by additional alloying to enhance the oxidation resistance, improving methods of coating in order to, first, separate oxygen from the metal and, second, make a thermal barrier to decrease the temperature on the metal surface. The maximum operating temperature of various nickel alloys versus time of their invention is shown in fig. 1.16. Perhaps, the ultimate temperature of nickel superalloys was reached by 80s by using directionally-solidified and single-crystal materials. This limits an increase in efficiency of gas turbine and calls for the development of alternative materials. Intermetallic compounds or ordered alloys mentioned above look like a possible alternative. They have been intensively studied during the last decade because they promise to be a solid base for creep resistant materials since the atomic diffusion in them is more difficult than in the basic metals. Main characteristics of some intermetallic compounds are shown in Table 1.3. Titanium aluminide TiaAl (~2-phase) is normally considered at present in combination with a titanium fl-phase (an alloy content can be Ti-25Al-10Nb-3V-
1100
-
1000
o
\
900
800
700 1940
i 1950
~ 1960
J 1970
i 1980
I 1990
Year
Fig. 1.16. Maximumoperating temperature of heat resistant nickel alloys, the creep rupture time is 1000 h under a stress of 150 MPa. D.S. & S.C. marks the band for directionally-solidified and single-crystal alloys.
Ch. I, w
Structural mater&&
31
TABLE 1.3 Some intermetallic compounds. Compound
Melting point ~
Density g/cm3
Young's modulus GPa
Ti3AI TiA1 Ni3A1 NiA1
1600 1460 1390 1638
4.6 3.8 7.65 6.0
150 176 214 200
1Mo) which provides ductility and enhance fracture toughness at r o o m temperature. The upper limit for working temperature for the alloy is set not by creep but oxidation, embrittlement, and susceptibility to burning when it is used for high pressure stages of the compressor [129]. Titanium aluminide TiA1 (~-phase) looks also more attractive from the point of view of ductility when it is used as a main component of a 7 + 0~2 duplex alloy [482]. The creep resistance of alloys based on nickel aluminide Ni3A1 at temperatures around 1100~ can be higher than that of any nickel based superalloy including those in single crystalline or directionally solidified state [63]. Nickel aluminide NiA1 with high melting point is perhaps a most attractive candidate for heat-resistant materials, however, this c o m p o u n d will have a future perhaps as a constituent of a composite. We did not mention here refractory metals, Mo, W, etc. because of an obvious reason: they are hopelessly weak for oxidation at rather low temperatures. 1.3.21 Ceramics
A majority of ceramics (see Section 1.2.2) belongs to inherently strong solids according to the Kelly's terminology. Dislocation motions in them are limited and correspondingly the energy dissipation at the crack tip due to plastic deformation is small. At the same time, ceramics are characterized by very high average r o o m temperature strength (up to 1500 MPa), high melting points, oxidation resistance 4 at high temperatures. Ceramics are becoming of increasing practical importance also because of their versatility. Ceramics have various important physical characteristics which include electrical, ferro-electrical, magnetic, superconductive ones. So ceramics can be a basis for a future generation of the materials that are to combine the load carrying function with other performances.
4 Oxide ceramics, at least those composed of simple Oxides, are really highly resistant to oxidation. With regard to non-oxide ceramics, the situation is not so smooth. Suppose, an acceptable oxidation rate corresponds to a constant in parabolic kinetics equal to about 10 gm/h. Then, experimental data compiled in [104] show that at temperatures higher than 1300~ most carbides, borides, and nitrides, such as HfC, HfB2, ZrB2, A1N, etc., oxidize at rates that are several orders of magnitude higher. Only silicon carbide and silicon nitride that form a silica protective layer in oxidation environments meet the above requirements at 1600~
32
Structural materials
Ch. I, w
However, being inherently strong solids, ceramics are inherently brittle. The critical values of stress intensity factor for pure ceramic materials do not exceed 5 M P a . m 1/2. This makes a material very susceptible to minor defects that arise both during the fabrication and service. Initial defects lead to a very large strength scatter of the materials, so an average value of the strength, which can be very high as mentioned above, is just a half of the truth. The lowest strength of a specimen is normally only a fraction of the average value. A high initial value of the strength of a structural element made of a ceramic is degrading during the service because of a number of factors that cause the growth of initial defects or occurrence of new ones. Such factors are cycling stressing (fatigue), thermal shocks, impacts of hard particles from the environments and so on. All these factors are inherent to the working conditions of such machines as gas turbine, re-entry vehicle, etc. Therefore, a problem of improving the crack resistance of ceramics looks as one of the major problems of the science of structural materials. There are known some ways to enhance ceramic fracture toughness without using fibrous reinforcements [44, 711]. These are: 9 Transformation toughening; 9 Crack pinning by particles of the second phase; 9 Microcracking; 9 Crack deflection and branching; 9 Toughening by matrix grain morphology. All these ways are realized by introducing second phase particles in the ceramic matrix.
Transformation toughening This way of toughening ceramics [44, 91] is based mainly on the martensitic transformation of zirconia (ZrO2) which has tetragonal (t-ZrO2) structure at high temperatures and monoclinic (m-ZrOz) at low temperatures. For bulk zirconia, the transformation starts at point Ms--~ 850~ during cooling and at As ~ 1150~ during heating. 5 The transformation being of the martensitic type is diffusionless and athermal. It occurs with a change in the volume and/or shape of the particle (the t-ZrOz ~ m-ZrO2 transformation corresponds to an increase in volume characterized by volumetric strain eT ~ 0.04 and to shear distortion of about 0.07). When the ZrO2 particles are contained in a stable matrix the transformation during cooling of the material from a temperature of its fabrication may be prevented due to the matrix constrain. So conditions for a stressinduced transformation are formed. Such a transformation takes place when a crack generates stress field in front of it. The stress field occurred is relieved as a result of the phase transformation of metastable particles, so the effective stress intensity
50xide's phase diagrams reveal a possibility to lower the transition temperature by solution of some oxides in the oxide under consideration. For example, yttria and magnesia stabilize zirconia. This makes possible to obtain a partially stabilized zirconia (PSZ).
Ch. I, w
Structural materials
33
factor decreases which corresponds to an increase in the effective critical stress intensity factor. The room temperature fracture toughness increase can be high. Actually, it has been shown (see [44]) that the enhancement of the critical stress intensity factor due to the transformation toughening is l ~ T O( v/EvtcyTo'Tr T
(1.44)
where E is the Young's modulus of the material, Vt is the volume fraction of the tetragonal phase which transforms, r T the width of the transformed zone, ~rT the transformation stress. Similarly to eq. (1.43), we have r T (3((K/oT) 2
(1.45)
where K is the applied stress intensity factor. Considering the balance of energy which should be supplied to transform the metastable tetragonal phase at temperature T and the elastic energy related to work of the transformation stress on the volumetric strain, yields 0"T C( A S ( M s - T ) / 6 T
(1.46)
where AS is the transformation entropy change. Equation (1.44) determines now the characteristic parameters of the material to control the transformation toughening contribution to the ceramic fracture toughness. We see that AK decreases with applied stress intensity factor K increasing and, obviously, the toughening effect is decreasing with test temperature increasing. The room temperature fracture toughness of zirconia toughened ceramics can reach values 8 to 15 M P a . m 1/2 and values 10 to 20 M P a - m 1/2 for PSZ and tetragonal zirconia polycrystals [44, 91]. However, the toughening effect disappears at the temperature at which tetragonal phase becomes stable and this is a main drawback of the transformation toughening mechanism.
Crack pinning by particles of the second phase Similar to the dislocation pinning by small particles which leads to an increase in the yield stress of metals (Section 1.3.1), the crack can be pinned by larger particles of the second phase in the ceramic matrix [353] provided the particles are sufficiently strong and they are strongly bonded to the matrix. Microcracking If the second phase particles are bonded to the matrix weakly they can debond under stressing. This is a possible energy sink and to push ahead the crack through a material with debonding particles requires larger applied stress intensity factor than in the case of an "ideal" material. Calculation of energy dissipated and stored in the fracture process zone characterized by stable microcracking was carried out assuming a number of simplifications of a real situation, shows [339] that if the size of the fracture process zone is known, then the
34
Structural materials
Ch. I, w
effective surface energy for a material depends o n the microcrack density in an extremely non-linear fashion.
Crack deflection and branching A particle of the second phase can deviate the crack and make it to go around the particle. This enlarges the fracture surface and so increases the effective surface energy. Recent analysis of a possible increase in the fracture toughness [531] shows that just a small contribution to a total fracture toughness of a ceramic can be provided by such a mechanism. Toughening by matrix grain morphology Texturing the ceramic structure can perhaps yield an increase in fracture toughness values. Actually, this way of toughening is close to fibrous that, and in some future can emerge as a very effective method. 1.4. Concluding remarks We see that further progress in transportation systems and technique of energy transformations, 6 depends strongly on availability of structural materials with higher strength and fracture toughness values, higher rigidity, higher working temperatures, and lower density. If we look around, we shall see that technical potential of metal alloys that are the main class of the structural materials, are about exhausted at present. The technical history of the Iron Age which is full of brilliant innovations in metallurgy, paradoxically, has led to a search for non-metallic substances to be used in structural materials. Despite ceramics having been known since the Stone Age, they are really a new class of the substances for structural materials. We reviewed briefly the ways of bringing ceramics closer to borders at which they will meet the requirements to structural materials. However, these ways do not yield universal solutions, improvements in fracture toughness are not satisfactory for an overwhelming majority of the applications. Therefore, the advent of glass fibre reinforced polymers ( G F R P ) and then carbon fibre reinforced polymers (CFRP) is an enormous step in the right direction. Brittle substance, glass or carbon, has become a part of structural materials. High modulus and high strength graphite fibres being glued together with a weak polymer have spread over a large area of various application that demonstrate a possibility of overcoming the inherent brittleness of ceramic-like substances by combining their fibrous form and weak interfaces between the fibres! However, a question arises when we look at a combination of an obviously hightemperature material like carbon and an obviously low-temperature organic. Is it a natural combination? The answer is not obvious. Look at the structure of natural materials: they are composed of ceramics and organics [110]. Perhaps, Nature has
6That are two of four main fields of human technological activity, informatics and production of food and consumer goods being two others.
Ch. I, w1.4
Concluding remarks
35
been optimizing the composition of biological structures limiting Itself with both available mineral components and a necessity to use organics. Certainly, the availability of both carbon fibres and a G F R P experience in 60s yields the idea of CFRP. This was a great idea. However, the time space in which Nature has acted, and for people to develop new structural materials, are drastically different. So the great idea will show and has already revealed drawbacks. Low modulus polymers provide weak interfibre bonds easily and this makes the crack to delaminate the interface and to change its configuration. This is good, but at the same time, it can yield uncontrolled delamination and change a configuration of the whole structural element. This is a failure. Hence, one needs to design a composite element very carefully, and this good for.., mainly, for mechanical engineers involved in the process. Thus, fibre reinforced polymers have shown both potential of fibrous structure and restrictions imposed by low modulus organics used as a binder. This has borne an idea to combine ceramic fibres with metal or ceramic matrix. With this, we drop the limitations of polymers but bring new problems. Not to say about technological difficulties as compared with polymer technology, an important problem is how to make fibre a n d m a t r i x materials compatible, how to adjust the microstructure of a composite to a loading pattern, etc. However, it becomes clear, at least on a qualitative level, that combining ceramics and metals, ceramics and ceramics, we broaden enormously the usage of ceramics as structural materials provided we have succeeded in preventing brittle fracture. How to reach this goal, is a subject of Part III. To conclude, we have to point out one more advantage of a wider usage of nonmetallic substances. John Bernal was the first to emphasize that when we are using nitrides, carbides, etc. we are consuming chemical elements which are not so rare on Earth as metals used for structural alloys. Actually, changes in environmental problems which would arise as a result of a wide use of composites based on ceramics, have not been discussed yet. The time for such a discussion is still remaining.
This Page Intentionally Left Blank
Chapter II FIBRES AND FIBROUS COMPOSITES
Before going into the discussion of mechanical behaviour of fibrous composites which shall be done in Part II, we give here a short description of the materials to be considered in detail further on. This is necessary in order to relate mechanical models of Part II to real materials of engineering interest. However, we start with an introduction to structural fibres, because composite properties are primarily determined by inherent properties of fibrous forms of solids.
2.1. Fibre strength An analysis of the failure behaviour of a composite needs a description of the fibre strength. Because a variety of failure models of composites involves a variety of the fibre strength descriptions, the problems do not look simple. Actually, mechanical properties of fibres have to be obtained by measurements of corresponding properties of composites. However, in the present section, before elaborating on models of composite failure, we can present classical approach to the description of fibres strength as a first approximation only. We shall come back to this problem, in Chapter 5, after having presented mechanical models of composite failure. 2.1.1. Weibull statistics
The strength of individual fibres being of a statistical nature is usually described by an exponential distribution function generally called as the Weibull distribution [695]. There can be found a formal basis to use such a distribution. Let us consider such a basis following Freudenthal [174]. We need just two assumptions. The first is obvious, namely the appearance of a single defect corresponding to strength a within length 1 of a fibre means the failure of the fibre of such a length if the applied stress reaches the value of a. Secondly, the statistical homogeneity of the fibre has to be assumed, it means that any part of the fibre does not differ statistically from any other part of the fibre. Let l and l' be two parts of the fibre, and PL (l) the probability of the absence of a critical defect for stress a within length I. Then the probability of the fibre failure is
37
Fibres and fibrous composites
38
Ch. II, w
PF (l) - 1 - PL (l). Connecting in series both parts of the fibre and using the theorem on multiplying probabilities yields
PF(l + 1') -- PF (l)Pi:(l').
(2.1)
Taking logarithmic derivatives with respect to l from both sides of eq. (2.1) gives d InPF(I + 1') d In PF(I) = = -c dl dl
(2.2)
where c is a constant because l' is an arbitrary value. Obviously, PL(0) = 1 and PL(OO) = 0, hence
Pv - exp(-cl),
(2.3)
and, correspondingly, (2.4)
PL(1, a) -- 1 - exp(-c/).
The value of c is to depend on a. Usually when describing the strength, the cumulative distribution function of a Weibull type is taken in the form
P(a,l)-
1-exp
-To
(2.5)
~oo
containing two constants,/3 and a0, as well as a scale parameter 10. The distribution density is
dP p(a)
--
-~
--
lfl(a) loao
~-l
exp -To
~00
"
(2.6)
Therefore, the mean strength of a fibre of length l is
(a) - fo~aP(a)da - ao (~q)'//~ ~ (1) F+
(2.7)
where F denotes the gamma function. The standard deviation is (2.8) and the variation coefficient D
/ F ( 1 + 2//3) _ 1.
= V
ii
(2.9)
Fibrestrength
Ch. II, w
39
Note that x depends on the value of/3 only. It is useful to remember a simple approximation, x = 1.2///. The values of/3 from ~ 2.5 to ~ 5 are characteristic for brittle materials, with a large strength scatter, those from ,-.,8 to ~ 12 are observed usually while testing tough materials with a small strength scatter. It is worth to note that the ratio of the mean strength average values of fibres of lengths l' and l" is given by the following expression (o'(l')) (a(/")) --
(l') -1/~ ~7
9
(2.10)
A scale dependence of the fibre strength is presented schematically in fig. 2.1. Because slope of the line is equal t o , 1//3, it provides a possibility to obtain the scale dependence of a fibre just having obtained the strength scatter, characterized by x, of fibre samples of a fixed length 10. Usually the fibre specimen length is between 10 and 30 mm. To evaluate the strength of a composite to some approximation, one needs the fibre strength on lengths between some fibre diameters and hundredths diameters. The Weibull distribution provides a means of extrapolating the fibre strength to a necessary length according to the procedure which is clear from fig. 2.1. It should be noted now that in the original application of the Weibull distribution to the statistical characterizing of the strength of materials, the volume of a solid was considered as the scale parameter. Obviously, in the case of a batch of fibres of a constant diameter, using both dimensions yields identical results. It is preferable, therefore, to consider the fibre length as a parameter for majority of applications. _
o
v
o O O
0
0
,
I
1
,
I
2
,
....
log (Illo)
I
3
Fig. 2.1. A typical scale dependence of the strength of a fibre. Weibull shape parameter, fl, is determined through the strength scatter for the fibre length corresponding to log(l/lo) = 1 and then used to plot the linear dependence in log-log coordinates.
40
Fibres andfibrous composites
Ch. II, w
If stress 6 is not constant along the fibre then eq. (2.5) has to be replaced with
P(l)
-
1-exp
(
-
To
\ 60 J
dx
)
.
(2.11)
In some cases, it is important to have probability of fibre failure at a particular point along the fibre. To obtain it, let us give a small variation 66 to the stress and find the corresponding incremental number of failures by using eq. (2.11) [635]
6P-(1-P)-g
\ 6o /
66dx.
(2.12)
A pointwise failure probability per unit length for the incremental number of fibres that fail because of the change in the stress, 66, the probability of failure of the fibre within the interval x and dx at the stress between 6 and 66, is
q(x)
fo(6(xt))fl_l(~6dxt
(2.13)
If 66 does not depend on the location, x, then
q(x)
f~ (6(x,))~_ i dx '
(2.14)
Obviously, the fraction of the fibres failed at a current stress within a finite fibre sublength f is obtained by integrating q(x) over the sublength. The cumulative failure distribution function Q(x) up to failure probability P* is
Q(x) - fo P* dP.
(2.15)
It is often necessary to connect the strength distribution to the distribution of defects along the fibre. For such case, the Poisson model [308] that describes a random distribution of points on a line is used. According to the Poisson distribution, the probability of N defects occurring with strength less or equal to stress, 6, and length, l, is
P(N) - [.N(6, l)] u e x p ( - N ( 6 , / ) ) N!
(2.16)
where N(6, l) is the mean number of defects of a kind just portrayed. If we assume
/V(6, l) - To ~00
(2.17)
Fibre strength
Ch. II, w 800
",
,',
, , , , , , i ,
, , , , , ,
~
g. b
600
o
,I
, , , ,,
~ r
~ o o
o
400
o
t~
0
i |o
i
l
o
o
o
o~
o
o
! i , ~I ~t
o
40
oo~
i
~t/ ol ~I |~
Iii ol / II'ool I14 I
o
oO// oOoO/41/ oOo //)OOoOoo ll /
ro o
o
i
o!
o~ o o
o
~176
t~
, , , , ,,
o oo
o
'1 ~ ~
, , , , I , ,",
~~ o
o
f:o,.oO~ rOoooO O 200
, , ~ , , , ,
41
!
i
~
~
o
o ~ o o o o ~ ~
o
%~
oo
o o
o
oO o
oo
. . . . . . .
~ 1 7 6 1 7 6 1 7 6 1 7o6 o
o
o
OD
oo
o
0
o
o
~
t
,
80
l
.........
120
'
X~
.........
I~0
l~
200
Fig. 2.2. Strength of boron-aluminium specimens with fibre volume fraction 21% versus coordinate of a fibre piece, used to make the specimen, normalized by the total fibre length in the batch. The fibre characteristics supplied by a fibre producer are (~) = 3080 MPa, K = 21.7%. The vertical lines mark the points of gluing together the fibre pieces. Experimental data by A.A. Khvostunkov and V.G. Ivanov. then the strength distribution function given by eq. (2.16) for the event to find one or m o r e c o r r e s p o n d i n g defects on the c o r r e s p o n d i n g length, is
P(N(~, l) >_ 1) - 1 - P(N(~, l) = 0) -- 1 - exp - T o
(2.18)
which is, actually, the Weibull distribution.
2.1.2. Some experiments Results of the extrapolating p r o c e d u r e based on the Weibull distribution m a y not be always satisfactory. There can be various reasons for this, an example being a doubtful validity of the a s s u m p t i o n of the statistical h o m o g e n e i t y of a fibre. Such an a s s u m p t i o n can idealize real fabrication conditions. A n example is p r o v i d e d by testing b o r o n - a l u m i n i u m specimens p r e p a r e d in a special way 1. A single spool of the b o r o n fibres, used in this experiment, was characterized by a Russian p r o d u c e r with a pair of strength p a r a m e t e r s , n a m e l y the m e a n fibre strength and v a r i a t i o n coefficient. In the experiment, each specimen c o n t a i n e d a p a r t i c u l a r c o n t i n u o u s p a r t of the spool. The fibre v o l u m e fraction in all the specimens was nearly constant. The results of tensile tests shown in fig. 2.2 reveal the j u m p s of the actual fibre strength
1Unpublished results obtained by A.A. Khvostunkov and V.G. Ivanov.
42
Fibres andfibrous composites
Ch. II, w
characteristics which determine the composite strength. The jumps take place at some points of the fibre. Hence, the fibre batch, the producer thought about as containing statistically homogeneous material (and it was apparently homogeneous when one relays on the results of testing standard fibre specimens of 25 mm length), occurred to be full of statistically different pieces when one looks at the fibre contribution into the composite strength. There are numerous evidences of the effects of different flaw populations on the fibre strength in differem fibre length intervals (see, for example [334, 520, 571]). An example of the normalized scale dependencies for two kinds of fibres those being single crystalline sapphire filaments and silicon carbide fibres on a carbon substrate is shown in fig. 2.3. The fibres were tested as received and after inducing a surface damage which presumably added a new population of flaws. One can see that in the case of sapphire filaments, the additional flaw population predominates over a part of fibre length corresponding to sufficiently long fibres. On the other hand, the additional flaw population on silicon carbide fibres effects the fibre strength on the whole fibre length interval tested. The difference can be attributed to a different response to a particular damage of two kinds of fibres. 2.1.3. Corrections to the fibre strength statistics Such particulars make some authors to follow Rosen [571] introducing either a simple double box distribution shown schematically in fig. 2.4 or more sophisticated bi-modal distribution. An illustration of using a bi-modal distribution to describe the strength characteristics of carbon fibres was given in [520]. If p(x,~,s) is the density of strength distribution, tr and s the expectation and standard deviation, respectively, then the density of the strength distribution of a fibre is taken as p(x) = 21pl(x,#l,Sl) +
~,2P2(X, 6"2,$2)
(2.19)
where 21 and 22 are the corresponding weights, 21 4 - 2 2 - - 1 . The behaviour of carbon fibres with and without surface plating by a polymer are described by eq. (2.19) with lognormal function
pi(x.~,s)-
1
((lnx-6"2)) 2s 2
v / ~ s x exp -
,
x > 0.
(2.20)
The experiments show that the values of '~1 and ~.2 related to the low and high strength branches of the distribution function, correspondingly, are not practically changing with fibre length decreasing from ~ 4 0 mm down to ,,~ 5 mm. At the same time, the values of both 81 and o'2 tend to increase with a decrease in the fibre length. It is interesting to note that thin surface polymer plating does essentially change the original strength distribution making the value of ,t2 to grow up and the value of 21 to go down, respectively. It may be interpreted as healing rough surface defects due
Fibre strength
Ch. II, w l
O V
m
l
I
I
I
b 0
0
KI
n
O " K!
I
I
,I
,
,
,
i
9
0
As received
n
Damaged
I
Sapphire fibres
,
43 i
i
i
i
0
I
~,,
.
V
V
,
.',
,
..==
l
I
i
'
I
i
i
_0
l
.1
9
I
i)
I
10 l
I
I
!
l
i
As received Damaged
Silicon carbide fibres
b
I
O
1 0
i
9 O
=,
i
I
I
.
I
I
I
I
i
'
,
,
l
1
,
I
-
fog(/i/o)10 l
l
'
'
'
'
l
i
Fig. 2.3. Normalized scale dependencies of strength of sapphire and silicon carbide fibres in as received and damaged states. ~0 = 2.5 MPa for sapphire and 4.8 MPa for silicon carbide fibres, l = 25.4 mm. The experimental data after Kotchick et al. [334].
Fibres and fibrous composites
44
Ch. II, w
p(e)
I
! o
b)
cr
l Fig. 2.4. (a) Double box strength distribution; (b) corresponding dependence of the fibre strength on its length.
to polymer coating and that is an indication of the existence of two different flaw populations on the original fibres. Straightforward attempts to adjust the scale effect of the Weibull type and, therefore, the shape of the Weibull distribution itself, are also known. Some authors [216, 690] did it by fitting experimental dependencies of the fibre strength upon fibre length to the expression
al - ao
(2.21)
which differs from eq. (2.10) by constant c~. Substituting eq. (2.21) into eq. (2.5) yields
P - 1 - exp -
~oo
"
(2.22)
That is the Weibull strength distribution corrected by the experimental scale effect. For graphite fibres, the value of e can vary from ~ 1 [367, 537] to ~ 0 . 5 [216].
Fibre strength
Ch. II, w
45
Another example of the adjustment of a distribution function to experimental data is given in [158]. Assuming a statistical procedure, the author came to the following distribution function for dimensionless strength s of a fibre of length l
P(s,l) -- 1 - e x p
((- s ~ + g's ~/2))
,
s>0
(2.23)
where ~ and l0 are the parameters. Standard deviation of such a distribution depends on s and that fits experimental data better than the original Weibull distribution. Finally, we are to mention the procedure of using the Weibull statistics for characterizing stress-rupture properties of fibres, despite being used up to now for describing the behaviour of polymer fibres only. In [678], the cumulative distribution function for lifetime of a fibre experienced load history h(t) is suggested in the form
F(t; l) - I - exp ( - W ( fo' C(h(x)dx) )
(2.24)
where W and c are the functions which can be taken as [94]
c(x) = ax ~
(2.25)
and [678]
tP(x) = bx ~
(2.26)
where a, b, 2 and/~ are positive constants. In a stress-rupture experiment, at a constant stress a - - a 0 , h(t) = a0, t > 0. After integrating eq. (2.24) we obtain
F(t) = 1 - exp(-(t/to)~).
a stress history
(2.27)
That is the Weibull distribution for the lifetime. The exponent,/t, can be considered as the shape parameter, and its values are estimated to be between 0.1 and 0.4 [627] which implies a very large scatter of the lifetime values. The scale parameter for lifetime t~ corresponding to stress a can be obtained as -2
Obviously the statistical characteristics of the lifetime can be obtained from eqs. (2.24) to (2.26) for an arbitrary load history. For example, for h(t) = st, t > O,
Fibres and fibrous composites
46
Ch. II, w
where s is a constant loading rate, a value of the strength will be, say, a = st*. Hence, eq. (2.24) can be rewritten in the form of eq. (2.5) with s2' "] 1/4' O-0 =
abl/,J
and /3 = #2' where 2 ' = 2 + 1.
2.2. Some structural fibres
We do not pretend to present here a full report on strong fibres. It is impossible and not necessary. We can refer to recent books [61, 136, 691]. However, some knowledge of fibre properties should certainly be given. The properties can be understood if the information on the fibre structure is available, the latter being determined by a fabrication method of the fibre. Therefore, this section contains information concerning fabrication schemes and typical structures of fibres for metal- and ceramic matrix composites as well as typical properties of the fibres. This means that we are not dealing with glass and organic fibres which are used mainly as a reinforcement for polymer matrix composites. The description of mechanical properties of structural fibres in this context is necessarily approximate. Some reasons for this were actually discussed above. Besides, microstructure and properties of a fibre are strongly influenced by the fabrication parameters, so that some fluctuations of important parameters of the fibre can change along the fibre even despite usual statistical characteristics do not change (see Section 2.1.2). Fibre characteristics can vary from batch to batch for commercially available fibres. Fibres on the development stage are obviously characterized by larger scatter of the properties. Also it should be noted that the strength of a majority of fibres is strongly dependent on the fibre diameter, an example is shown in fig. 2.5.
2.2.1. Carbon fibres Carbon fibres occupy a special position in the field of structural fibres. This is because, first, the total production of these fibres (more than 10 t per year at present) is far beyond that of all other fibres suitable for reinforcing metal and ceramic matrices. Then, the invention by Watt and Johnson in 60s of a process of the conversion of polyacrilonitrile filaments to high-strength carbon fibres (see [691]) was a starting point of the development of modern metal and ceramic based composites. That position of carbon fibres is mirrored in numerous publications,
Ch. II, w
47
Some structural fibres 9 ,
9, "
"91 0
(9 0 0
0 0
0 0
0
0 0
0 0
0 0 0
00.. ~
1
,
,
, . ,,,,i
10 s
00
0 0
,
9l
0
9,,,.110 4
0
0
l
tl
a l , i,,llo ~
1
s ~ram 2
II
i
i
9, , , , , 0
1
.2
Fig. 2.5. An illustration of the scale dependence of strength of single crystalline sapphire fibres. Open points correspond to whiskers [53], dark points correspond to an approximation to the strength of fibres obtained by the internal crystallization method (see below, Chapter 14).
including special volumes like a fundamental one by Dresselhaus et al. [136]. So we shall give here just a very brief account of the fibres. Yajima [714] reminded that a process of the conversion of organics to inorganics by heating in various atmospheres had been used from ancient times. The heating of wood under airtight cover yields charcoal 2. Actually, pyrolysis of polymers under airtight cover is a process for producing carbon fibres. Despite a great variety of organics which can be converted into carbon, there are just two precursors that are used in the present carbon fibre production: polyacrilonitrile (PAN) and pitch. Chemistry and physics of production carbon fibre from P A N precursor are described in detail by Watt [691]. Acrilonitryl, C H 2 - - C H - - C N , after polymerization and spinning to the fibre form is stretched and transformed into a ladder polymer with a greatly increased thermal stability of structure containing long polymer chains oriented along the fibre axis. Then oxidative stabilization process follows during which the fibre is normally restrained from shrinking. These steps are conducted at low temperatures, around 200~ The result is a highly oriented polymer backbone. The next step is pyrolysis (carbonization) at a temperature of about 1000~ which yields removing from the fibre most elements other than carbon. During pyrolysis, the density of the fibre
2This book is being written in a town called Chernogolovka (Black Head) in a forest area near Moscow. A version of the toponymy of that name refers to the occupation of the local people in former times. They produced charcoal and always had black heads.
Fibres and fibrous composites
48
Ch. II, w
increases, carbon crystallites arise being oriented along the fibre axis, the fibre becomes thinner reaching the final diameter of 6 to 10 ~tm. The final step is high temperature graphitization which determines degree of the crystallite orientation, crystallite size, porosity and consequently, mechanical and physical properties. The fabrication route of the fibre from a pitch precursor, which is petroleum or coal tar, differs from what was just described by melt spinning of the pitch as a first step of the process. The most reliable model of crystal structure of the fibres is due to Ruland (see [136, 285,301]). Graphite has a layered structure, the atoms in a layer form a hexagonal pattern with a characteristic length of 0.142 nm. In a normal graphite structure, neighbour layers are displaced by one bond characteristic length; in the turbostratic form of graphite, the layers are rotated individually around the axis normal to the layer plane, so in such a crystal, the ordering of the layers with respect to each other is lost. In the Ruland's model, crystallites of turbostratic graphite are packed in such a manner as to form a ribbon-like layer plane of a length of some thousands of nm and only some nm wide. A number of the ribbons form a wrinkled microfibril. The microfibrils form branches and elongated needle-shaped voids. The voids lower the fibre density as compare to that of single crystal graphite. Actually, the turbostratic structure of graphite which is characteristic for the fibre, has also somewhat lower density than normal graphite due to a larger spacing between the layers. Still, enhancing degree of the preferred orientation of the turbostratic crystallites yields decreasing void content and increasing both density and the axial Young's modulus (fig. 2.6).
750
i
9
i
9
i
9
i 0
a.
L9 600
0
LU
0
450
0 0
0
300
150 0
I
1.6
0
0
0
0
0 0
0
0 0
0
I
1.8
I
2.0
|
2.2
T / g/cm3 Fig. 2.6. Axial Young's modulus of carbon fibres versus density. The data are compiled from [327, 561, 676].
Ch. II, w
Some structural fibres
49
The layered structure of the fibre provides highly anisotropic elastic properties. Components of the tensor of elastic constants 3 are [285] (in GPa): Cll = 1060, C12 = 180, C13 = 15, C33 = 36.5, C44--4. Here the x3-axis is directed along the normal to the graphite layer. Having known the component of the elastic tensor, one can estimate a range of the effective elastic characteristics of fibres as a function of the misorientation angle. However, to be used in evaluation of elastic properties of composites, fibre elastic characteristics should be obtained by direct measurements which are quite a problem for objects like thin anisotropic fibres. The corresponding methods for obtaining characteristics when measurements of radial fibre strains are involved, include laser diffractographic technique [342] and loading in situ in TEM [676]. Experimental data for axial Poisson ratios lie in a rather wide interval, from 0.1 for pitch fibre P55S with the axial Young's modulus equal to 380 GPa [676] to 0.27-0.28 for those with Ef = 380GPa (fibre similar to PAN T300) [342]. Unlike the axial Young's modulus which increasesmonotonically with the heattreatment temperature, the tensile strength of fibres normally reaches a maximum at a temperature in the range 1200-1500~ [470]. Hence, the fibre strength is determined by the intrinsic structure of a fibre and is also influenced by flaws which can be characterized by various populations of the surface and volume types (see [470] and also Section 2.1.3). If we plot strength and modulus values for various carbon fibres, we will see a region shown in fig. 2.7, an extreme value of one characteristic is reached at the expense of a loss in the value of the other one. Possible control of the fibre structure provides a variety of mechanical and physical properties of the product. Some of them should be mentioned in the present context. Thermal expansion coefficient, ~, is frequently reported as being zero or negative. However, direct measurements made recently by Villeneuve et al. [676] show that both axial and radial values of ~ can vary in a broad interval depending on a fibre type and temperature interval. Still, negative values of ~ can be observed for the axial thermal expansion coefficient. At the same time, Villeneuve et al. [676] found quite large, about 10.10 -6 grad -1, values of the radial thermal expansion coefficient. Thermal conductivity varies within two orders of magnitude with changing precursor material and heat treatment, from ~10 to ~103 Wm -1 K -1 at room temperature [136]. Electrical resistivity is also dependent on the precursor type and the heattreatment temperature [136, 285]. Resistivity values vary from ~10 -6 to ~10 -4 at room temperature. Perhaps the only serious shortcoming of carbon fibre is its natural affinity to oxygen. The fibre oxidizes in oxygen atmosphere at temperatures above 600~ This sets limits for time/temperature conditions of their use in an unprotected state.
3see below Section 3.1.1.
50
Fibres and fibrous composites
Ch. II, w
10.0 13.. (.9
7.5
5.0
2.5
0.0
0
,
I
200
=
I
400
=
I
600
=
I
800
=
I
1000
Ef / GPa Fig. 2.7. A field of the Young's modulus and tensile strength values for carbon fibres commercially available at present.
2.2.2. Fibres produced by chemical conversion of a precursor Carbon fibres were the first to be made by conversion of a polymer precursor to a stiff and strong filament with a microcrystalline structure. This was a start for another fibre to be obtained in such a way. Perhaps, the most important invention was made by Yajima and his colleagues [714] who developed in 70s a series of the polycarbosilane precursors and converted them into SiC fibres. The fabrication process includes four main steps. Firstly, a precursor is distilled to remove a lowmolecular-weight component to provide a better spinnability. Then polymer is spun into fibre which is subjected to curing at low temperature. Finally, the cured fibre is heated up to 1200-1300~ in a nitrogen atmosphere or vacuum, to convert polymer into silicon carbide. The fibre obtained contains, in the addition to SiC, free carbon and excess of silica combined with oxygen, with homogeneous distribution of the elements across the fibre diameter [62]. Such type of the silicon-carbide based fibres is produced by Nippon Carbon under the trade name Nicalon. The structure of Nicalon fibres is that of an amorphous matrix of silica with fl-SiC microcrystals of size of about l0 A embedded in it together with agglomerations of free carbon with sizes of order of nanometers [62, 270]. The structure provides, as shown in Table 2.1 just a half of the value of the Young's modulus inherent to silicon carbide (see Table 1.2) and a sufficiently high strength level.
Some structural fibres
Ch. II, w
51
TABLE 2.1 Some characteristics of the fibres produced from polymer precursors. Fibre type or authors' name
Main substance
Diameter (ave.) gm
Density g/cm 3
Young's modulus GPa
Approx. strength GPa
Source
Nicalon Hi-Nicalon Tyranno Tyranno (6% 02) Lipowitz et al. Tonen HPZ
SiC SiC SiC SiC SiC Si3N4 Si3N4
14 14 10 11 8-9 10 10-12
2.3-2.55 2.7 2.4 2.4 3.1 2.5 2.4
180-220 270 200 200 up to 450 250-300 180
2.8-3.0 2.8 3 3.5 2.6 2.5 2.8
[270] [270] [62] [514] [370] [62, 514] [514]
The strength and Young's modulus of Nicalon fibres tested in air and in inert atmosphere do not decrease as the testing temperature increases up to 1000~ At higher testing temperatures, a decrease in the strength is essential [61]. Nicalon fibres retain their strength after exposure in air at 1000~ for about 100 h, but after the exposure at 1200~ for 100 h only a half of the initial strength value is retained. Even short time exposures at 1400~ lead to a drop in the strength [270]. The reason for the loss of the strength is certainly a high content of oxygen and excess carbon in the fibre. A modified curing process of the precursor fibre by using electron beam irradiation lowers the oxygen content in the fibre and allows to obtain a fibre with high heat resistance [270]. That fibre is marked as HiNicalon (Table 2.1), it has a tensile strength of 1.9 GPa and the Young's modulus of 300 GPa after exposure at 1550~ for 10 h and the strength of 1.2 GPa after exposure at 1800~ fir 1 h in Ar atmosphere. There have been reports on other methods of decreasing oxygen content in silicon carbide fibres, which are applied to the polymer precursors. The procedures yield an increase in thermal stability of Nicalon type fibres. Another way of the retention of the fibre strength is to make SiC crystallites larger to suppress recrystallization, however remaining the crystallite size sufficiently small to provide reasonable initial strength. The corresponding procedure which includes the pyrolysis of the precursor at a temperature higher than 1600~ is described by Lipowitz et al. [370]. The crystallite size is said to be about 30-40 nm, fibre properties are presented in Table 2.1. The fibre without excess carbon retains 87% of its initial strength after exposure in Ar atmosphere at 1800~ for 12 h and 66% of the strength after the same exposure in air. A neighbour to the Nicalon family of fibres is the Tyranno family produced by Ube Industries. These fibres are known to contain a small amount of titanium which does perhaps inhibit recrystallization. The fibre retains its strength up to about 1200~ for short exposure times [62]. Fibres which contain silicon nitride as a main substance are also produced by converting a polymer precursor in inorganic material. Properties of two of them are
Fibres andfibrous composites
52
Ch. II, w
presented in Table 2.1. Also there are possibilities to use a polymer precursor to obtain alumina fibres (see for example, [472]).
2.2.3. Fibres produced by CVD-methods For the first time, the chemical vapour deposition process for making structural fibres was used by Talley [640] who made thin boron rods by the reduction of boron tribromide by hydrogen and discovered high modulus and high strength of the specimens. Then it was developed into an industrial technology in both USA and USSR [343, 662]. The schematic draw of the apparatus for producing fibres by CVD-method is shown in fig, 2.8. It consists of glass tubular chamber with mercury seals that, at the same time, serve as electric contacts for heating the substrate drawn through the reactor by a mechanism comprising let off and take up spools. The mechanism provides uniform unwinding of the substrate and its necessary tension as it passes through the reactor. At the present time, boron, silicon carbide and boron carbide fibres have been produced according to this scheme. CVD-process is very flexible. In particular, it can be used to make fibres with changing radial chemical composition either in a continuous or discontinuous manner. Obviously it is very convenient to introduce into the technological scheme a stage of fibre coating by the same method. On the other hand, the flexibility means that a large number of parameters effects the result. There are known attempts to model the process to make the experimental work shorter. For example, Diefendorf and Mazlout [128] tried a computer simulation of the CVD of TiB2 fibres in a vertical cold wall reactor based on finite-element, dynamic analysis scheme to describe qualitatively the rate limiting mechanisms at different fabrication conditions. However, the results of such attempts are still far from direct practical usage.
Inlet gaseous mixture
L e t o f f s"9
~j
E x h a u s t gas
seals Ua~d electric co
Fig. 2.8. Schematic draw of a reactor for continuous production of fibres by CVD-process.
Ch. II, w
53
Some structural fibres
425 I~. 400 Ua 375 350
3 2 5
300
0
'
'
'
100
'
200
'
'
'
300
'
'
400
'
500
TIOC
Fig. 2.9. Temperature dependence of axial Young's modulus of boron fibre measured by E.G. Golofast. Boron fibres
Boron fibres [662] are produced by using either tungsten- or carbon core in the temperature range from 1000 to 1300~ The fibre is passing through the reactor for 30 to 60 s, when the following reaction occurs 2 BCI3 + 2 H2 = 2 B + 6 HC1. The boron deposit, being of fl-rhombohedral structure with a rather small grain size (around 30 A) possesses a very attractive set of mechanical and physical properties. First, very high specific stiffness at room temperature (Table 1.24) which retains at elevated temperatures (fig. 2.9). The transverse modulus of the fibre has not been reported, but some degree of the anisotropy can be expected: the texture of the boron deposit determined by growth conditions is obvious. In the case of tungsten substrate, a nonhomogeneous profile of the wire causes a distribution of nucleation sites on the surface. This, in turn, yields non-homogeneous growth of the domains revealed by a corn-like external surface of the fibre (fig. 2.10). The surface of a fibre on the carbon substrate is much smoother [98]. The shear modulus of boron fibres in the plane normal to the fibre axis is 177 GPa [673] which yields the first approximation for the Poisson ratio (assuming the elastic isotropy) equal to 0.13.
4The fibre density depends on the substrate material and the ratio of the fibre and substrate diameter.
r~ I:T
-I
Fig. 2.10. Surk~cu\lrLlcrure o f a boron 1ibl.s
.~ ~
p ~~o -" <
~::J" Cr 9 ~
~':~ ~
~:r :::r
,,
~.
~
-'~
p
t~
~-.
="="=
,~" "~
::r'~ ~
E.~
~ ~ ::~-. ~ ~.
=
'-'9
=
q" o
:~..
---~
.~ o r~ a"
~.
ze:g -
~
~o=
r
~ = . ~
~
=.=
;a-~
"-I
c~--',, ~
I
'-I
<
~ ~
~
--" r ~ . "
.~
'-"
~- ~__.,.
t'~
9
0
~
(j~
.-F, "~
~
~
,--*
~
~.~
-.
~. -.-. 9
9-~-o
2~~,y~
9
9
9
~
-
9
"d
=r"
~.
9
~.~
~
9
el.
Fracturc caused by a rough dclkct in the boron deposit such as for example. a macrocrystnllinc domuin. This corresponds to a low-strength part of the fibre strength distl-ibution. That caused by a11 inclusion in thc boron dcposit. Corresponding strength values lie in the interval 1.1-om 0.6 to 4.5 GPa. F r a c t ~ ~ at r e a site of tlic radial crack. Sue11 defects cot-respond to the strcngtli of 0 . 2 3.6 (;Pa. Fracture because o f dclkcts it1 the s~lbstr;ltc.111 the case of tungsten substrate. the defects irrclude delaminations, voids. and inclusions at the interfi~ceoTsuch a size that they are resolved by optical microscope. Corresponding strength values are between 0.6 and 3.0 GPu. ,
9
~ "~
~ rt~~
~:~, ,,,,,-~ ~,,, "~ ~..r '--"
2=N
~_.
Tensile strength of boron fibres, being stutisticnl by nature, is detertnined by a distribution of f a w s which arise at various stages of the fitbrication process. Tsirlin [662] s t ~ ~ d i e nature d of the Raws, and rclutionships between the flaw appeal-wnce and size. their locations and strength characteristics. Comparing his results with those published by other authors, he distinguished five types of the fi-acturc:
Ch. II, w
Some structural fibres
55
9 High-strength mode corresponding to defects which are beyond the optical microscope resolution. It should be noted that correlation between the fibre strength and size of a defect yields the evaluation of the effective surface energy between 2.8 and 6.4 J/m 2 [673]. The nano-crystalline structure of the fibre is certainly preserved up to a temperature of about 700-800~ This sets a limit for the fibre use at elevated temperatures. The cost of raw materials, their purification as well as that of the CVD-process make the price of boron fibre rather high for their use in most of civil applications. Certainly, boron fibres are more suitable for using with metal, mainly aluminium, matrix. Silicon carbide fibres
Silicon carbide fibres are produced also on carbon- or tungsten-core in the temperature range between 1200 and 1400~ [355, 499]. Various chlorosilanes are used as reactants. The following reaction normally occurs: CH3SiC13(g) ~ S i C ( s ) + 3 HCI(g) . There is a number of particular structural SiC fibre types [355]. Pure silicon carbide fibres have diameter either 100 jam on tungsten core (produced by SIGMA, Germany) or 140 jam on carbon core of a diameter of 32 jam (produced by Textron Speciality Materials, USA). In the case of tungsten core, a layer of about 1 jam thick composed of tungsten silicide, WsSi3, and tungsten carbide, WC, occurs between the core and silicon carbide during the fibre synthesis. The fibre is almost pure silicon carbide near the core but becomes a SiC/Si mixture close to the outer surface. In the vicinity of the surface the content of free silicon can reach 20 wt% . All other known fibres are produced on carbon core coated by a thin layer (from 1 to 3 jam thick) of pyrocarbon to enhance the electrical conductivity of the substrate as well as to ease the effect of the thermal mismatch between the substrate and the deposit due to low shear modulus of the coating. The latter prevents the breakage of the carbon filament when the deposit is expanding and, hence, the catastrophic local overheating of the growing fibre. Silicon carbon fibres being a cheaper product than boron fibres, have been developed to be mainly used with metal matrices. To make them more compatible with aluminium and titanium matrices, fibres with carbon-rich outer layer (SCS-2 and SCS-6) are produced by Textron Specialty Materials. SCS-2 fibres are studied carefully by Nutt and Wawner [499]. They have discovered that in the layer from the pyrocarbon coated core to a radius of about 40 jam columnar subgrains of/3-SIC of the radial orientation have 40 to 50 nm in width, outside of the radius mentioned the subgrain width becomes 90 to 100nm. The subgrain length can be up to a few micrometers, The structure of the carbon-rich zone arises as a result of both a specially organized deposition process and coating with a thin carbon layer
Fibres andfibrous composites
56
Ch. II, w
TABLE 2.2 Some characteristics of CVD SiC fibres. Fibre material
Trade mark or technicalname
Diameter Density ~tm g/cm3
Young's modulus GPa
(a) MPa
/3
Source
sic/w SiC/C SiC/C SiC/C SiC/C
SIGMA SCS-0 SCS-2 SCS-6 SCS-0
100 140 142 146 140
390 420 385 390 420
3340 4950 3270 4670 4950
8 10 7 14.5 10
[355] [355] [355] [355] [355]
3.32 3.32
consisting of several sublayers. The first sublayer consists of small crystallites of fl-SiC in a carbon matrix, the second sublayer is mainly carbon with SiC decreasing to zero, the third one is carbon crystallites more randomly oriented, and the fourth outer sublayer contains carbon with the trace of silicon. The total thickness of the carbon-rich outer surface is about 1 ~tm. In the case of SCS-6 fibre this layer has thickness of about 3 to 3.5 ~tm [355]. Strength characteristics of silicon carbide fibres are given in Table 2.2.
2.2.4. Fibres crystallized from the melt Fibres can be produced from the melt by a variety of methods. Oxide fibres are known to be obtained in single crystalline form by a modification of the Czochralsky-Stepanov's method called the edge defined, film-fed growth (EFG). To speed up the crystallization rate, a melt can be crystallized on a subtrate (microrope). Also oxide fibres can be obtained by crystallizing the melt in pre-made channels in the matrix (the internal crystallization method described in details in Chapter 14). Melt with rather large viscosity can be spun by inviscid melt spinning that yields normally a glassy state of the fibre. It should be also noted that unidirectional crystallization of eutectics (Sections 2.3.1 and 2.3.2) is also a way of making fibres from the melt. Such fibres normally aim at using at high temperatures, so they should have either single crystalline or composite structure.
Czochralsky-Stepanov's-EFG method Stepanov's modification of the well known Czochralsky's method introduces a shaper to pre-determine a shape and size of the capillary column at the top of which the liquid/solid interface arises (fig. 2.11). The technical adjustment of Stepanov's method to the growth of fibres is due to Labelle [349] who has lifted the growth zone above the melt surface with a capillary tube in the crucible. The lower end of the tube is located near the bottom of the crucible, and the growth zone is now fixed relative to the heater independent of the level of the melt surface which goes down with time.
Ch. II, w
57
Some structural fibres
Crystallization
Shaper,,,,
capillary
front
tube
Melt
Fig. 2.11. Capillary shaping-crystallization zone, schematic view.
Both a review of the corresponding techniques and a discussion of the fibre growth parameters, structure and mechanical properties of single crystalline sapphire fibres are presented in a paper dated by 1985 [420]. It seems that the macrostructure and strength of single crystalline sapphire fibres depended strongly on the crystallization rate; it appears that a stable growth takes place at rates no more than 0.5-1.0 mm/s. Crystallographic orientation of a fibre is determined by that of the seed. But as growth rate increases the orientation can change. As a rule, the fibre axis tends to coincide with c-axis of sapphire lattice. The strength of sapphire fibres shows a clear scale dependence [420]. At temperatures higher than 1200~ creep becomes essential. Attempts to improve creep properties by doping single crystalline fibres with Ti +3, Ti +4, and Cr +3 do not seem to yield a definite result [221]. Creep testing of single crystal A1203- MgO spinel fibres [605] does not reveal any advantage of the fibres. Creep properties of eutectic A1203/ZrO2 fibres obtained also from the melt [41] have not been reported yet, although this is certainly a case of enhanced creep resistance (see Section 14.2.4). During the last decade, the technique of growing fibres from the melt has been improved. In particular, the laser heated floating zone process should be mentioned [221]. In this case, there is neither crucible nor shaper. The melt is nested by a solid "feed rod" heated locally by a CO2 laser. The process needs laser power of some hundredths watt, the scheme looks to be very flexible permitting to pull the fibre in any direction. A lower limit of about 10 ~tm for the fibre diameter is set by an optic device. An upper limit is determined by the laser power, material melting point, etc. Unfortunately, a permissible growth rate is also low, of the same order of magnitude as in the case of the modified Stepanov's method (EFG).
58
Fibres and fibrous composites
Ch. II, w
Crystallization in micro-rope (MIGL-fibres) To increase the drawing rate, by at least an order of magnitude as compared to the E F G method, it was suggested [421] to use a metal micro-rope as a carrier. The carrier or moving substrate, which is a molybdenum micro-rope containing some tens of molybdenum wires of a diameter of 50 to 100 lam, moves along an edge of a feeder which supplies the oxide melt from a crucible. It is infiltrated with an oxide melt when it passes the feeder (fig. 2.12). Then, on its way to a take up spool, the melt inside of the micro-rope crystallizes. If no special precautions are taken, the fibre shape occurs to get an irregular shape shown in fig. 2.13. The drawing rates in experiments conducted changes from 0.25 mm/s to 85 mm/s. Appropriate oxides can be eutectics containing A1203 as a component, for example A1203 - ZrOz(Y203). Microstructure of a fibre occurs to be of a complicated nature due to crystallization under rather complicated conditions [193]. Testing single fibres in a copper matrix, Section 5.3.2, fig. 5.28, reveals very low strength at large fibre length and sufficiently high strength at the lengths close to critical value. High temperature strength of the fibres and composites reinforced with them shall be discussed in Section 13.6.
Inviscid melt spinning Advanced fabrication methods of glass fibre are based on the utilization of high viscosity of oxide mixtures containing silica (the corresponding values reach
Fibre
\
(3ruclDle Micro-rope
Heater Fig. 2.12. A scheme of fabrication of MIGL oxide-based fibres. After Mileiko and Glushko [428].
Ch. II, w
Some structural.)qbres
59
Fig. 2.13. A possible cross-section of the M I G L fibres. The molybdenum wire diameter is 0.05 ram. After Mileiko and Glushko [428].
100 P a . s for non-alkaline alumoborosilicate glass known as E-glass [24]) that provides a possibility to draw amorphous fibres from the melt with fairly high speeds. On the other hand, the crystallization process of oxides described in the previous section is possible because of very low viscosity of the melts. It is impossible to use directly the high speed production methods used for silica containing fibres for making a variety of oxide fibres, perhaps in noncrystalline state, as a jet of a liquid with low viscosity happens to be unstable. Inviscid melt spinning of oxide fibres [679], involves jet stabilization by an increase in surface layer viscosity through suspension of solid carbon in the molten jet surface. This yields an increase in surface melt viscosity and formation of a carbon-rich skin in the fibre. A schematic drawing of the process is shown in fig. 2.14. Wallenberg et al. [679] demonstrated the possibility of the method by making alumina and various alumina-calcia fibres of a diameter of 100 to 400 ~tm. The jet of a melt with viscosity between 0.04 Pa 9s (alumina) and ~ 0.34 Pa 9s (a mixture of about equal masses of alumina and calcia) comes from the orifice due to applied pressure of inert gas and meets the propane gas jet immediately. Propane decomposes at the surface of the jet and carbon enters into the surface layer of the liquid jet. A part of carbon is used in reduction of aluminium oxide which yields formation of aluminium carbide particles. Together with remaining carbon particles they form a
Fibres and fibrous composites
60
INERT
Ch. II, w
_ _ ~ _ _ _
GAS
111111
A
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. ..
=...% ~c_.=
_= P~oP~
A
~
Iil
FIBRil
Fig. 2.14. Schematic drawing of inviscid melt spinning process.
suspension with a viscosity value comparable with that of E-glass. It is sufficient to prevent the growth of Rayleigh waves on the jet surface and stabilize the fibre diameter. The strength level of the fibres obtained in this way is rather moderate, however elevated temperature strength seems to be higher than that of E-glass. Still, no data have been published on using these fibres in structural composites, but the method looks promising because of a principal possibility to produce crystalline fibres at high speed.
2.2.5. Sintered fibres This is a family of methods which follow the typical powder metallurgy routes. They consist of the slurry- or sol-gel spinning stage, and then drying and sintering resulting fibre. The output is a polycrystalline ceramic fibre with sufficiently small crystalline size. The first industrial process for producing fibres by extruding a slurry containing the powdered fibre material was developed by Du Pont in late 70s [126]. It was applied to manufacturing of nearly pure ~-A1203 fibres (FP in Table 2.3). The grain size in this fibre is about 0.5 lam, mechanical properties are shown in Table 2.3. The next step by Du Pont in this direction was the development of alumina based fibre strengthened by partially stabilized zirconia. The fibre called PRD-166 reveals better strength properties which are retained sufficiently well at temperatures up to at least 1000~ [62]. Nearly the same fabrication method can be applied to silicon carbide fibres. Frechette et al. [172] demonstrated it, and provide preliminary experimental data to show an excellent strength retention of the fibre up to about 1600~ and good creep resistance at 1400~ On the other hand, the initial strength level does not look very attractive.
Some structural fibres
Ch. II, w
61
TABLE 2.3 Some characteristice of sintered fibres. Fibre type or authors' names
Main substance
Diam. (ave.) ~tm
Density g/cm 3
Young's modulus GPa
Approx. strength GPa
Source
FP PRD- 166 Frechette et al. Saffil Saffil Safimax Nextel 312 Nextel 440 Nextel 480 Nextel 610
A1203 A1203 SiC r/-A1203 + 4%SIO2 6-A1203 + 4%SIO2 A1203 + 4% SiO2 A1203 -+- SiO2 q- B203 A1203 q- SiO2 q- B203 A1203 + SiO2 + B203 A1203
20 18 25-50 3.5 3.5 3 11 11 11 12
3.9 4.2 3.3 3.4 3.3 2.7 3.05 3.05 -
380 380 300 300 300 154 189 224 390
1.4 1.9 1 1.8 1.5 2 1.75 2.1 2.3 2.4
[62] [62] [172] [51] [51] [62] [62] [62] [684]
A green fibre can be composed of a precursor to the metal oxide to be converted to the oxide at a relatively low temperature. The corresponding process is described by Birchall et al. [51] who disclosed a method developed by ICI to fabricate "Saffil" fibre. In this case, the precursor is aluminium oxychloride; other salts also containing aluminium can be used. An aqueous solution of a precursor contains also an organic polymer to control rheology in the spinning process. During the post-spinning treatment, the fibre is dried without distortion and internal voids formation and a hydrated material (solid or gelled) is obtained in the form which permits to handle the fibre further on. Then, at a higher temperature, dehydration takes place and organic matter is burnt out. The decomposition of the precursor salt at the next stage of the low-temperature heat treatment produces amorphous alumina. It seems to be difficult to perform crystallization process in a controllable manner to achieve a combination of high density, small grain size, and high purity to obtain a high-strength fibre. Generally, a compromise can be reached by adding into the fibre acidic oxides of Si, P,B or Zr which lower the maximum working temperature of the fibre. In "Saffil", silica is a chosen doping and a variety of alumina phases (q, 7, 6, 0, ~) is obtained in the fibre depending on the time/temperature conditions of the crystallization. In a commercial product, all these phases, as well as mullite can be present in some combinations. The phase composition and the fibre diameter determine the fibre strength (Table 2.3). Note also that "Saffil" was developed originally as a refractory felt and produced in a variety of forms, some of them are convenient for particular composite fabrication techniques; for example, mat is suited for squeeze casting in making metal matrix composites. "Safimax" fibre produced also by ICI is long staple fibre. The solution or- sol-gel spinning process is used by 3M to produce a family of continuous fibres under trade name "Nextel" that contain A1203 (about 65 wt %) as well as SiO2 and B203 [98]. The fibres can be either crystalline mixture of 7-A1203 and mullite or pure mullite. Grain size is of order of tens nm [62].
62
Fibres andfibrous composites
Ch. II, w
Fibre5 with high alumina content are also produced along the route which include spinning of a solution of an organometallic compound, followed by calcining the resulting precursor fibre.
2.2.6. Whiskers Whiskers also occupy a special position in the field of composites. They were the first solids to reveal the tensile strength values comparable with those of the ideal strength. This gave birth to an intensive study of mechanisms of the whisker growth, their structure and mechanical properties. There are known, at present, a few of the process to grow whiskers. Most important mechanisms involved are called "Vapour-Liquid-Solid" and are based on solid state chemical reaction. The first one was normally observed in most of the experiments performed in 50s-60s [191]. The second one became especially important since patenting in 1973 by Cutler a process of production of silicon carbide whiskers from rice hulls. In that process, silica and carbon, presenting in rice hulls (the former used to make the problem of utilizing the by-product of the rice processing difficult), react at a rather high temperature to form silicon carbide. Whiskers grow at nucleating sites formed by impurities in the hull. Whisker diameter is usually between 0.1 and 1 ~tm and this makes them dangerous for the human health. Studies on experimental animals [623] and special cell culture [52] have shown that carcinogenicity of thin (< 1 lam in diameter) fibres and whiskers is a real danger that comes from inhalation of the fibres.
2.2.7. Concluding remarks At present, a variety of structural fibres is known. Some of them are commercially available, some are at development stage. Comparing various types of fibres, one bears in mind mechanical properties, workability, availability, and cost. The strength and Young's modulus are most important mechanical properties of fibres. These points were discussed above in detail. Also we discussed briefly the stability of the fibre microstructure at high temperatures. This is important since metal and ceramic based composites are mainly aimed at service at elevated and high temperatures. The same reason calls for knowledge of creep properties of fibres. DiCarlo [127] has been studying creep of various fibres for many years. Some of his results are presented in fig. 2.15. Let us mention also a problem of the fibre cost. Normally, with increasing production, the cost decreases. However, to start the increasing production, it is necessary to have a reasonable initial price of the product to start application. In the case of oxide fibres growing from the melt, which promise a real jump in creep resistant characteristics of composites, the initial price announced in late 60s [27] occurred to be too high to start any structural applications. Hence, up to now the price of such fibres is so high that only optical applications of them seem to be reasonable. In such cases, a solution is certainly to be found if non-traditional ways
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64
Fibres andfibrous composites
Ch. II, w
9 Metal matrix composites. 9 Ceramic matrix composites. Each group, especially the first one can be subdivided into a number of subgroups, considering aspect ratio of reinforcing fibres, fibre morphology, fibre spatial orientation, etc. Such parameters determine possible fabrication routes and mechanical behaviour of a material. Structure and mechanical properties of passive sufficiently rigid biological composites are reviewed by Currey [110]. In the present context, it is important to note that 9 The effective surface energy of biological composites composed mainly of brittle substances is normally 103-104 J/m 2 which is about two orders of magnitude higher than that for high-modulus engineering ceramics. This is a result of various mechanisms acting to arrest a crack in the material. In particular, there should be mentioned an experiment by Zioupos and Currey [732] carried out by using a modern technique. They have actually shown that the stress concentration in bone is relaxed via microcracking of a more highly mineralized component. 9 The structure of a biological composite has a fibrous feature on different levels of the scale. It is easily visible on the wood cell structure which includes a number of layers in a tubular cell wall, some of them composed of cellulose fibres oriented at the angle specific for each layer. This certainly gives an appropriate balance of the effective values of stiffness, strength, and fracture toughness of wood. 9 Biological structures are adjustable to the load conditions. That feature was made clear in 30s by a Russian called V.F. Razdorsky. He studied behaviour of the stem of sunflower under compression and tension loads applied during their growth. He found that the sunflower's stem loaded by tension occurred to have a tensile strength higher than that for a normally grown plant and, at the same time, a very low compressive strength. And visa versa, the stem loaded by compression was strong in compression and weak in tension. Using an optical microscope with a small magnification, Razdorsky found that changes in the strength characteristics were accompanied by changes in thickness of the cell wall. Mattheck [392] has observed adaptive growth of trees that changes the macrostructure around a site of mechanical damage and so makes the tree safe in carrying its biological aim to maximize light reception by means of a large crown. 9 All load-carrying structures of animals and plants carry, at the same time, some purely biological functions. The peculiarities of biological composites just listed as well as those remaining outside of the above list cannot be found in modern engineering materials. But they, no doubt, look attractive due to a perfect adjustment of a material and its properties to its function in the organism. Of course, engineering materials serve under different conditions, temperature being a most important one. But maybe we need to consider more effective ways of energy transformation which do not involve high temperature cycle. It does not mean a call for a wider use of the horse in
Ch. II, w
Composites
65
transportation systems, but maybe we need to look more carefully at the methods that living creatures use to accumulate, transform, and spend energy. Still, at the present time and for the next century we definitely need composites with polymer, metal, and ceramic matrices. So, avoiding a discussion of polymer matrix composites, we come back to the main subject of the book.
2.3.1. Metal matrix composites If the origin and subsequent development of advanced fibre reinforced polymers were based on the experience in glass fibre-reinforced-rubber technology, metal matrix composites (MMCs) did naturally find their roots in metallurgy. Therefore, casting, powder metallurgy, diffusion bonding, etc. have come to be traditional technological processes for making MMCs. Examples of M M C s are boron/aluminium, carbon/aluminium, short-fibre/aluminium, silicon-carbide/titanium.
Boron/aluminium composites are normally produced by hot-pressing a blank composed of plasma-sprayed mono-layers. These processes shall be described in Chapter 11, here we just present a micrograph (fig. 2.16) which gives an impression of a typical composite microstructure and a photograph of typical boron/aluminium structural elements, such as tubes, shells, rings (fig. 2.17), which portrays possibil-
Fig. 2.16. A fracture surface of a boron/aluminium composite. Scale bar = 100 ~tm.
66
Fibres and.~'brous composites
Ch. II, w
Fig. 2.17. Examples o1" boron/aluminium structural dements produced by Laboratory of Reinforced Systems of Solid State Physics Institute of Russian Academy ot" Sciences. The length of the longest tubes shown is about Im.
ities to use boron/aluminium in structural components. Typical properties of boron/ aluminium composites are as follows:
Longitudinal Young's modulus = 220-250 GPa, Transversal Young ~' modulus - 100-120 GPa, Shear modulus in a reinforcement p l a n e - ~ 48 GPa, Poisson ratio-- ~ 0.2 (measured when loading a specimen in the fibre direction), Density = ~ 2.7 g/cm3 Longitudinal tensile strength = 1.5-2.0 GPa, Transversal tensile strength = 100--200 MPa. The properties are very good, most of them retain up to about 400~ However, because of high cost of boron fibre, the usage of boron/aluminium composites is rather limited. Still, in the former Soviet Union, they were successfully used in both space and aviation applications.
Carbon/aluminium composites are produced by various liquid infiltration methods (see Chapter 13). If problems of the wetting, carbide formation on the interface, and some others are overcome, the material of a high longitudinal strength (around 1000 MPa), high stiffness (about the same as that for boron/aluminium) and low density
Ch. II, w
Composites
67
(~2.2 g/cm 3) is obtained. Figure 2.18 pictures failure surfaces of a graphite/aluminium composite to give an impression of its microstructure. The composites are usually considered to be used in more applications than boron/aluminium. However, a number of problems should be solved to make them available and reliable. No doubt, the problems mentioned are soluble.
Short-B'bre/aluminium-matrix composites can be produced by using either powder metallurgy methods (see Chapter 12), or pressure infiltration, or squeeze casting fabrication routes (see Chapter 13). The fibres can be either whiskers or short alumina and silicon carbide fibres. Typical microstructure of a composite is illustrated in fig. 2.19. Cheaper fibres and cheaper fabrication processes yield composites that can be widely used now. The range of possible applications is really very broad, from pistons of the engine to airframes. Silicon-carbide/titanium composites are very similar in many respects to boron/ aluminium composites, just an upper limit for the service temperature is essentially
Fig. 2.18. Failure surface, of a graphite/aluminium composite. Mean fibre diameter is 8 gm. Courtesy to A.A. Zaboltsky.
Fibres andfibrous composites
68
Ch. II, w
Fig. 2.19. Failure surface of a silicon-carbide-whisker/aluminium-matrix composite. Courtesy to A.P. Gribkov.
higher, up to 600-700~ Therefore, their usage in structures of the transatmospheric plane seems to be inevitable if such planes are to be advantageous. A special group of metal matrix composites is materials obtained by direetionally solidification of eutectics. Because we all avoid a description of the corresponding technological procedures in Part III, a very brief account of the fabrication process (see, for example, [556] for details) will be given here. Consider system AB which is characterized by a eutectic-type phase diagram (fig. 2.20). This means that there can be found a unique concentration, CE, corresponding to the lowest temperature, TE, at which the liquid phase exists. At point (C~, TE) of the phase diagram two solids (~- and fl-phases) and the liquid are in
r~ TA
L LIQUID /
(sozzp)
cA
c~
Fig. 2.20. A binary eutectic. L stands for liquid phase.
G
Ch. II, w
Composites
69
quasi-equilibrium. At that point, the two phases are crystallizing separately and the growth direction is perpendicular to the solidification front. One of the phases, namely that of lower content, is crystallized in a fibre or lamella form. Thus, a typical structure for fibrous composites, arises as shown in fig. 2.21 provided the crystallization front is plane. The fibres occurring in such composites have all the features characteristic to whisker. They are thin, the diameter being dependent on the crystallization rate, they are normally single crystals, their strength, evaluated via the strength of the composite, is very high. They are much better than whiskers in one respect: the fibres do not exist outside the composite, their toxicity is not important. Heat resistant composites of Co-TaC and Ni-TaC systems obtained in this way can have very good properties. The specific rupture strength, necessary for turbine blades to serve for 100-1000 h, can retain up to temperatures of about 1070-1100~ that is nearly 50~ higher than for convential nickel superalloys. A drawback, that is structural instability, revealed especially when a thermal gradient in the reinforcement direction exists, can be certainly overcome. However, another drawback, which is a very low crystallization rate, of order of 1 cm/h, prevents these composites from a practical usage. The need to enhance stiffness, strength, fracture toughness, working temperature of metal alloys was the driving force in development of metal-like composites. "Metal-like" means that a composite can be jointed to a metal element by using routine and reliable }nethods such as welding of some kinds, bolting, etc. This means
Fig. 2.21. Scanning electron micrograph of a Ni-TaC eutectic composite. Courtesy of I.L. Svetlow.
70
Fibres and fibrous composites
Ch. II, w
also that a user of the material will not meet problems of extremely high anisotropy, necessity for lightning protection, enhanced rain erosion, and so on. In that respect, MMCs are "Metal-like" materials. However, in many respects, MMCs are "composite-like" materials. This is because: 9 Any composite does not exist, as a rule, separately from a structural element. The material itself arises in a process of making a structural component or, at least, an element shaped very close to the component to be used in a structure. In particular, this leads to a necessity to study properties of a composite connecting them to a fabrication technology of the structural element. 9 The microstructure of a composite for a particular structural element should be optimized with regard to a loading pattern of the element. In mechanical engineering, the optimization means usually a solution of the problem of fibre orientation. We will see below that actually the optimization should include a choice of the fibre and matrix materials, their volume fractions, the interface structure, and so on. We should also remember that a particular composite is a less universal material than a metal alloy. For example, optimum structures of one composite system, e.g. B/AI or SiC/Ti, can be different for different applications. Ideally, a composite should be designed together with designing a structural element. 9 As mentioned in final remarks to the previous chapter, brittle fibres introduced in a composite structure do not cause the brittle behaviour of a composite. This can be true under some conditions, but untrue under other ones. Actually, brittle/ tough (brittle/ductile) transitions are known from a "metallic" experience (temperature, strain rate, a type of the stress state), but when we are dealing with composites, the number of variables becomes much larger. On the other hand, it is important to point out that similar to stable microcracking of a brittle component in bone, stable brittle fibre breakage in a composite provides relaxation of the stresses so that fracture toughness increases. In addition, this process in a homogeneously loaded structural part (fig. 2.22) enhances the effective fibre strength. The field of MMCs is now expanding towards using intermetallic alloys as a matrix. Because of room temperature brittleness of intermetallics, such composites are really making a bridge between MMCs and ceramic matrix composites. 2.3.2. Ceramic matrix composites Governmental and industrial institutions in developed countries, including USA, France, Japan, and former Soviet Union, have conducted an intensive R&D work by ceramic matrix composites (CMCs) due to clear understanding of the importance of structural materials, first of all, for high and very high temperature applications, in the general technological advance. Besides that, CMCs provide the opportunity to follow biological structure in combining load carrying and functional actions in one element, examples being catalytic convertors, various sensors, fuel cells, etc. Actually, the area of applications of CMCs can be extremely
Ch. 11, w
Composites
71
Fig. 2.22. A longitudinal section of a boron/aluminium-composite specimen after tensile testing. Fibres were breaking and a portion of weak points of brittle fibres were eliminated in this process. The fibres became effectively stronger. (After Mileiko et al. {452].)
large, from heart valves and various implantations in the human body to thermal shields for missiles moving with highest speeds. Cutting tools, armour plates, bearings, elements of piston engine and gas turbines are in that series. Therefore, a variety of possible microstructures of CMCs is too large to be illuminated in a short series of figures. Ceramic matrix composites are known as the oldest composite materials; the Bible refers to this particular type of materials when it tells a story of the fabrication of bricks with and without straw. Actually, bricks made of muck and clay and fired by sunlight (in South Russian regions) are also ceramic matrix composites. Phillips [533] discusses the idea that carbon fibre reinforced ceramics may have an ancient history as goat hair added to clay during manufacturing pottery jars and vessels, being converted to a form of carbon fibre during firing the product. The pre-history of modern fibrous CMCs had a start in the early 60s when the first attempt to increase fracture toughness of ceramic materials by introducing refractory metal wires in a ceramic matrix were carried out [64]. Those materials could hardly have any commercial future: despite the fact that fracture toughness and their strength were better than those of homogeneous ceramics, oxidation of the reinforcement prevented their practical usage. So, that period of the CMC history
72
Fibres and fibrous composites
Ch. II, w
coincides, at least partly, with the period of an intensive development of alternative methods of toughening ceramics discussed above, Section 1.3.2. Technically, those methods were based on introducing second phase particles into a ceramic matrix, so that it was obviously advantageous to replace an equiaxed particle with a particle of the aspect ratio larger than 1. This brings additional possibilities to arrest the crack, such as energy dissipation due to fibre pullout, crack bridging, etc. These mechanisms are subject to a detailed analysis in Chapters 4 and 5. Fabrication methods such as diffusion bonding in hot pressing or powder metallurgy routes, liquid infiltration, directional solidification of eutectics, etc. which are commonly used for production of MMCs, are also applied to fabricate CMCs. Normal ceramic procedures such as reaction-bonding, chemical vapour deposition and impregnation are also used in composite technology. The real progress of this type of materials started with the advent of a variety of modern structural fibres. The present scenario of ceramic composite field is rather rich and does not have a definite boundary. Some materials that are very close to CMC may be or may not be, in a strict sense, called CMCs. Carbon/carbon composites, fibre reinforced concrete are examples of such materials. Table 2.4 presents typical ceramic matrix composites. Whisker-reinforced ceramics are normally produced by routine powder metallurgy methods that can vary to adjust them to a particular condition of dealing with mixtures of the powder and thin whiskers which can possess the toxicity/ carcinogenic properties. The composites so far obtained can have fracture toughness two to three times higher than that for the matrix materials, but usually is accompanied by a decrease in the strength. Note that a combination of various ways of toughening ceramics as that given by an example of SiCw-reinforced AlzO3matrix toughened by introducing particles of tetrogonal ZrO2 (Table 2.4) yields a combined effect. Glass and glass ceramic matrix composites produced by a powder metallurgy route based on using a slurry to deposit a matrix powder precursor on the tow of carbon or silicon carbide fibres, yields a family of composites to work in an oxidizing atmosphere up to either about 450~ (with carbon fibres) or about 1100~ (with SiC fibres) [535]. CVI method is used to produce Nicalon SiC-fibre/SiC-matrix laminate composite with the largest value of fracture toughness ever reached in non-ductile materials, TABLE 2.4 Strength and fracture toughness of some ceramic matrix composites (approximate values). Composite
Fracture toughness MPa
Strength MPa 9m 1/2
Source
SiCw/Si3N4 SiCw/A1203 SiCw/A1203 + t-ZrOz SiC/LAS glass ceramics SiC/SiC composites
10 11 14 17 30
500 600 700 830 400
[319] [617] [92] [58] [720]
Ch. II, w
Composites
73
that is 25-30 MPa 9m 1/2 [352, 720], and the strength value was at a level of 400 MPa. These properties retain up to a temperature of about 1400~
2.3.3. Carbon/carbon composites CVI as well as methods based on the pyrolysis of a liquid precursor impregnated into a fibrous pre-form are used to produce carbon/carbon composites [395]. These composites provide a possibility to use excellent high temperature properties of graphite in structures subjected to mechanical loading: strength and fracture toughness of the composites can be higher than those of polycrystalline graphite. Initially, carbon/carbon composites were developed to be used in purely military objects such as heat shields for re-entry vehicles, rocket nozzles, disk-brakes for military aircraft, etc. Now, with the technology advance it becomes possible to use these composites in other disk-brake systems, medical applications, and so on. Carbon/carbon composites are also fabricated by using either thermosetting resins with high carbon yield or pitches as a matrix precursor. Both species are carbonized in a process organized in such a way as to reduce carbon matrix porosity.
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Part I1 FAILURE MECHANICS OF COMPOSITES
In this part, we shall consider failure mechanics of composites as a base for understanding the behaviour of metal and ceramic matrix composites. If a real understanding is achieved then we can formulate the requirements of composite microstructure to provide a composite with necessary properties. We restrict ourselves to specific questions, these being the crack interaction in non-homogeneous solids, tensile and compression strength, creep and creep-rupture, and some topics in fatigue and impact behaviour. We start, however, with rather known problems of deformation of composites and description of failure behaviour in terms of effective characteristics, otherwise, the part would not be complete.
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Chapter III DEFORMATION AND FAILURE OF COMPOSITES
Understanding of the elastic-plastic behaviour of fibrous composites is important from the standpoint of composite technology. We are normally looking for stiffer materials, and non-metallic fibres provide a possibility of finding out what we are looking for. However, to make the possibility a reality, we need to know how the macroresponse of a composite depends on microcharacteristics of the composite structure. In this respect, metal- and ceramic-matrix composites are different from polymer matrix composites which have been a main subject in the study of the elastic behaviour of non-homogeneous materials. Metal and ceramics are much stiffer than polymers used as a matrix. Hence, corresponding composites are less anisotropic than fibre reinforced polymers. When we are dealing with polymer matrix composites we see, roughly speaking, fibres put together with a glue. (Surprisingly, such a structure produces a lot of interesting behavioural features!) Obviously, the stiffness of that class of non-homogeneous materials has to be carefully analyzed and this has been done. When we are looking at metal- and ceramic-matrix composites, we see fibres that strengthen or toughen a matrix. At the same time, incorporating fibres into metal or ceramic matrix yields a material with new elastic-plastic properties. Therefore, we can gain an additional and important benefit. Failure and fracture behaviour of non-homogeneous solids are greatly effected by features of the stress/strain fields in such solids. The fibre/matrix interactions, especially near the fibre end, failure of the interface, etc. make an important contribution to macro-deformation and failure processes in composites. This chapter contains limited information on general points of mechanics of metal- and ceramic-matrix composites. The reason for the limitation is availability of good books containing many details on the subject (see, for example [85, 88,234, 289, 301, 400]). We start with a brief outline of elastic-plastic behaviour of anisotropic solids not to refer a reader to corresponding textbooks.
3.1. Elastic and plastic behaviour of anisotropic materials This is a subject of the classical elasticity and plasticity theories.
77
78
Deformation and failure of composites
Ch. III, w
3.1.1. Elastic behaviour
If a body of an arbitrary shape is stressed and strained, its stress state is described by a second-order tensor, r and the strains are given by tensor eij. These tensors are symmetrical, that is aij = aji, cij = cji. A body is called elastic if the work done on the body does not depend on the way of loading but is determined only by the initial and final states of the body. In that case, there should exist the elastic potential such that , ij =
i~eij
(3.1)
For a linear elastic solid: 1 r - ~ EijkleijCkl
(3.2)
O'ij = Eijklekl.
(3.3)
and
This is a general formulation of the Hook's law which allows inversion: ~ij = Sijklokl.
(3.4)
Components of the stiffness tensor, Eijkl, and those of the related compliance tensor, Smnpq, provide linear relationships between six independent stress components r and six strain components Ckl. To illustrate the compactness of the tensorial expression accompanied with the summation on repeating indexes, we write down just one relationship of those expressed by eq. (3.3): all = Ellllell + El122e22 + El133e33 -+- El123e23 -k- El131e31 + El112~12. Note that tensor components are transforming with changing the coordinate system according to the rule: ! Eijkl -- Emnpq aim OCjnOCkp0Clq
(3.5)
where 0tim are the direction cosines of the x 'i axis with respect to the Xm axes. The number of 0~im in the compact form of eq. (3.5) is equal to an order of the tensors. The number of mutually independent components Eijkl is 21 for a most general case because the symmetry of tensors oij and eij yields the symmetry of tensor Eijkl: Eijkl = Eklij = Ejikl = EijJk.
Ch. III, w
79
Elastic and plastic behaviour of anisotropic rnaterials
Before reducing the number of independent constants for particular cases of the elastic symmetry, we note that in m a n y operations with matrices Eijkl and Sijk~, it is convenient to replace the corresponding four-indexes matrices with the two-indexes matrices, C ~ and S~13, respectively. The rule of the conversion is clear from the comparison of the following equivalent matrices:
Ellll
El122 El133 El123 E2222 E2233 E2223 E3333 E3323 E2323
Ell31 E2231 E3331 E2331 E3131
EllI2 E2212 E3312 _ E2312 E3112 E1212
Cll
C12 C13 C14 C22 C23 C24 C33 C34 C44
C15 C25 C35 C45 C55
C16 C26 C36 C46 C56 C66 (3.6)
These matrices are symmetrical with respect to the main diagonal. When we are using matrices C ~ and S~13, it is convenient to represent symmetrical tensors 0"ij and eij as vectors 0"~ and ci3 with six components. Then the H o o k ' s law can be written as 0-1 0"2
0-3 0"4 0"5 0"6
Cll _
C12 C22
C13 C23
C14 C24
C15 C25
C16 C26
C33
C34
C35
C36
C44
C45 C55
C46 C56 C66
x
~1 f?2 ~3 (~4 ~5 ~6
(3.7)
Suppose that a body has a plane of the elastic symmetry, say XlX2. This means that the elastic potential, eq. (3.2), does not change when the coordinate system changes: x 1 - Xl, x~ - x2, x 3 -x3. After such a change, displacement components ul and u2 do not change, u3 changes the sign, i.e. u~ - -u3. Because cij - l(ui,j + uj,i), elj - eij if there is an even number (including 0) of indexes 1 and 2 among ij , for example, s -- ~33, ~t12 -- ~12- In the opposite case, (~ij --6ij, for example, 613 --el3. In the sum determining the elastic potential, eq. (3.2), only terms, containing just one component eij changing a sign, do change their own signs. Therefore, if we are to keep the potential of the same value, we should have coefficients Eijkl in these terms equal zero. This means that Eijkl = 0 if the total number of 1 and 2 indexes in this coefficient is an odd one. The number of non-zero components appears to be equal to 13. Note that this number is observed only when the symmetry axis (normal to the symmetry plane) coincides with a coordinate axis. Suppose now that there exist two planes of the elastic symmetry, say x lx2 and x2x3. This an important case of orthotropy often observed in fibrous structures. Applying the above considerations to a matrix with 13 independent constants already, we find that the number of independent constants decreases to 9. The matrix of the stiffness tensor takes a form: I
I
__
I
__
"
Deformation and failure of composites
80 Ellll El122 El133
El122 El133 E2222 E2233 E2233 E3333 0 0 0 0
i
0
0
0 0 0 0 0 0 E2323 0 0 E3131
0
0
0 0 0 0 0
Ch. III, w
(3.8)
"
E1212
It can also be seen that the existence of two planes of the symmetry (or two axes of the symmetry) means that the third plane (the third axis) orthogonal to the considered ones is also a plane (an axis) of the symmetry. A case of the plane stress state is of practical importance. Let 0"33 - 0"31 = 0 " 3 2 - - 0. Then the Hook's law with stiffness matrix given by eq. (3.8) yields El133 -- E2233 -- E3333 = E2323 = E3131 = 0
and the Hook's law can be written as
0"2 0"3
--
(c,, el2 C12 0
C22 0
0 C66
>
s s
(3.9)
9
Note that the number of independent constants for the plane case in an orthotropic material reduces to 4. Another important case of the elastic symmetry of fibrous structures is transversal isotropy which means the existence of a plane of isotropy. Let the Xl axis be normal to the plane of isotropy, i.e. the components of the stiffness matrix, eq. (3.6), does not change when the coordinate axes transform as I
I
X1 -- Xl , X2 -- X2 COS 0 't- X3 sin
I
0, X3 -- --X2 sin 0 + X3 COS 0 .
Applying the condition just formulated to components Eijkl (see for example [618]) we obtain the following matrix Ellll El122 El122
0
0
0
0
0
0 0 0 1/2(E2222 -E2233) 0
0
0
0
0
0
El122 El122 E2222 E2233 E2233 E2222
0 0 0 0 E1212
0 0 0 0 0
0
E1212
(3.10)
which contains just 5 independent constants. A special case of the elastic isotropy can be deduced from the above considered case of one plane of the elastic isotropy. It is sufficient to assume existence of the second plane of the isotropy orthogonal to the first and perform the corresponding procedures with the matrix given by eq. (3.10). Let the x2 axis be normal to the
Elastic and plastic behaviour of anisotropic materials
Ch. III, w
81
second plane of isotropy. Then the additional relationships between components Eijkl follow: E1212 = 1 / / 2 ( E 2 2 2 2 - E 2 2 3 3 ) ,
Ellll
=E2222,
El122 = E 2 2 3 3 -
and just two independent constants remain. Introducing the two constants as # = 1 / 2 ( E 2 2 2 2 - E2233) , 2 --- El122
transforms matrix Eijki to the following form: 2+2p 2 /l 0 0 0
2 2+2p 2 0 0 0
2 2 2+p 0 0 0
0 0 0 # 0 0
0 0 0 0 # 0
0 0 0 0 0 #
"
(3.11)
Constants 2 and p are Lame's elasticity coefficients which are normally used in the equations of elasticity problems for isotropic solids. In practice, it is often convenient to use engineering constants instead of components of tensors Eijkl (C~) or Sijkl (S~). We illustrate a procedure of introducing them by considering a case of unidirectional loading of an isotropic body. Let the only non-zero component of the stress tensor be all. In such case, the Hook's law with matrix Eijkl given by eq. (3.11) yields all = 20 + 2#ell,
(3.12)
0 = 20-Jr- 2#e22, 0 = 20-+- 2#e33
where 0 = e l l -t- s eq. (3.12) yields
nt- s
is the volumetric deformation. Summing three equalities in
fill - - (32 + 2#) 0 and all --
#(3 2 + 2#)
•22 -- s
,~ _+_ #
~11~
2 -- - - ~ 0 - Z#
2 -- 2(2 + #) e11.
Now the Young's modulus is
Deformation and failure of composites
82
e --
0-11
ell
]2(3/~,
=
+ 2]2)
Ch. III, w
(3.13)
/~ -~- ]2
a n d the Poisson's ratio is e22
v-
(~11
~33
=
(~11
=
2 . 2(2 @/~)
(3.14)
U n d e r the stress state o f pure shear, for example, ) , 1 2 - 2 e 1 2 - ),21 with all other stress c o m p o n e n t s being equal to zero, we have the H o o k ' s law in shear 2712 - o-12 - ]2),12 . Therefore, L a m e coefficient ]2 is the shear m o d u l u s in a system o f the engineering c o n s t a n t s 1. U n d e r hydrostatic stress state, all = 0 2 2 = 0 3 3 = 0- and shear stresses being, obviously, zero, the first three lines of the H o o k ' s law, eq. (3.1 1), yield -
(2) ,~+~
0
a n d the bulk m o d u l u s is K -
0-
2
-O -
(3.15)
2 + -j ]2.
The following useful relationships between engineering constants can now be obtained: K E m
E 1 --2Y
9K]2 3K+]2
E ]2 -
2(1 + v) '
3 K - 2]2 v - 2(3K + ]2) "
(3.16)
(3.17)
F o r transversely isotropic solids, the engineering constants are the Y o u n g ' s m o d u l i in the direction o f the s y m m e t r y axis and in the plane of isotropy, Ell and E 2 2 = E 3 3 , respectively, the shear moduli in the plane of isotropy and the o r t h o g o n a l plane, ]223 and ]212 = ]213, the Poisson's ratio connecting the longitudinal and transversal d e f o r m a t i o n s , v j 2 - - v ~ 3 , and that connecting the d e f o r m a t i o n s in the plane o f isotropy, v23, the plane strain bulk m o d u l u s for loading in the plane of
1 Usually the shear modulus in the engineering system is denoted as G. However, we will keep # for this value remaining G for the energy release rate in fracture mechanics (see the next chapter).
Elastic and plastic behaviour of anisotropic materials
Ch. III, w
83
isotropy with zero strain in the direction of the symmetry axis, k - k23. We are to remind that only five constants are independent in this case. Relationships between the engineering constants and components of matrix Cij, eq. (3.10) with the two-indexes system, and the inverse matrix, Sij, are C l l - E l l nt- 4v22 k , C12 - - 2kv12 , C22 - / * 2 3 -+- k, C23 = -/123 q-
k,
C66 - - / 1 1 2 "
(3.18)
In the plane of isotropy, the relationship similar to eq. (3.16) in the isotropic case is held and E22 /-/23 --
2(1 + V23)"
The relationships inverse to eq. (3.18) are
(3.19)
2C22
E l l - - C l l - C22 __1_C 2 3 '
E22 -
V12 - -
V23 ~
2 ( C 2 2 -k- C23 - 2 C 2 2 / C l l ) ( C 2 2 - C23 --I- 2C44) 3C22 -+- C23 + 2C44 - 4 C 2 2 / C l l C12 C22 nt- C23
(3.21)
,
C22 -+- 3C23 - 2 C 4 4 -
4C22/Cll
(3.22)
3C22 q- C23 -t- 2C44 - C 2 2 / C l l
k = (1/2)(C22 +
(3.20)
(3.23)
C23).
For an important case of the plane stress state of an orthotropic solid with the Hook's law given by eq. (3.9) we have $11 = 1/Ell , 822 = 833 = 1/E22 ,
S12 = S13 = - V l 2 / E l l = - v 2 1 / E 2 2 ,
866 = 1//.12.
(3.24)
and
Qll
Q12
---
Ell E22 , Q22 - , 1 - VleV21 1 - - V12V21 v12E22
1 - v12v21
=
VzlEll
1 - v12v21
, Q66 -
~12-
(3.25)
84
Ch. III, w
Deformation and failure of composites
3.1.2. Plastic behaviour
The von Mises yield criterion for an isotropic solid is well known: 2f(0-ij)
-- (0"22 -- 0"33)2+(0-33
-- 0"11)2+(0-11
-- 0"22)24-20"23 -t'- 20"~1 -t- 20"22 9
(3.26)
Hill [246] generalized the von Mises criterion for an orthotropic solid as 2 f (0"ij) -- F (0"22 - 0-33)2+ G (0-33 - fill
+ 2L~3 + 2M ~2 + 2N ~122
)2
-+- H (0-11 - 0"22
)2
(3.27)
where F, G , H , L , M , and N are constants. Obviously, one constant can be taken as equal to 1. Note that eq. (3.27) is written in the principal directions of the orthotropy. For a transversely isotropic solid, with x2,x3 being the isotropy plane, M=N.
We can now use all the methods of plasticity theories forgetting about the anisotropic nature of function f . In particular, assuming function f to be the plastic potential and using the associated flow rule yields the plastic strains increments as
ef d2
de p - ~
(3.28)
where d2 is a non-negative proportionality factor to be determined considering particular flow hypotheses. Assuming the existence of a unified dependence between the equivalent stress and strain (that is valid for proportional loading) one can obtain
3
(d 0)
d ~ - ~ \d~o) \ ~--T/
(3.29)
where 0-o and e0 are the equivalent stress and strain. In the case of an isotropic solid, 0-o and e0 are proportional to the second invariant of the corresponding tensors. Note that the equivalent-stress/equivalent-strain curve can be approximated by a power function
_
( o)n
/ 30/
\o-n/ An anisotropic behaviour is obviously relevant to that of unidirectional fibrous composites. In this case, the plastic deformation in the fibre direction, the x~ axis, is usually neglected [79, 712]
Ch. III, w
85
Elastic behaviour of composites
delPl -- 0.
(3.31)
Substituting eq. (3.27) into eq. (3.28) yields [712] d~Pl = ( G ( a l l -- 0"33) + g ( a l l -- 0"22))dR
(3.32)
and
G=H =0. Therefore, for the transversely isotropic composite, only three parameters in eq. (3.27) remain. Obviously, one of them can be set equal to unity and eq. (3.27) reduces to 2f(aij) - - F ( a z 2 - a33)+ 2a23 + 2M(a21 + a22).
(3.33)
3.2. Elastic behaviour of composites
When we are testing an isotropic specimen to measure its elastic moduli we normally believe that the meaning of the moduli has been defined. We do not think of the difference between the moduli of, say, metal polycrystalline specimen we are dealing with, and local elastic moduli within a grain. Therefore, nearly the whole technique of engineering continuum mechanics is based on semi-intuitively defined effective properties of non-homogeneous materials. A strict definition of the effective properties is far beyond the scope of the present book, so we give just a sufficiently strict definition.
3.2.1. General remarks and definitions Let Hook's law be valid at any point of a non-homogeneous body, that is b-,(r) 0"ij (r) -- *-,ijkl~kl 9
(3.34)
where E (r) is the tensor of local elastic coefficients. Define the averaged values of the stresses and strains as -
ij(r) d r ,
=
ij(r) d r .
(3.35)
Suppose, volume V is sufficiently large to represent the material structure as a whole and sufficiently small not to expect changes in values of ~ij and ~ij with changing V, then the averaged values of the stresses and strains can be expressed by the Hook's law
86
Deformation and failure of composites
6ij - E~kl~kl
Ch. III, w (3.36)
where E* is the effective stiffness tensor, and ~ij - Si]klakl.
(3.37)
where S* is the effective compliance tensor. To prove a possibility to represent relationships between the averaged stresses and strain in the form given by eqs. (3.36) and (3.37), it is essential to assume the continuity of the tractions and displacements on the fibre/matrix interface. It can be shown that E* and S* are the fourth-order tensors with the symmetry inherent to the elastic characteristics tensors. Moreover, for statistically homogeneous composites, we have
E~rsS~skl- Iijkl. A most complete and general way of treating the problem of elastic behaviour of non-homogeneous solids is due to Eshelby. Further on, some results obtained by using Eshelby's method will be used. So we give here a general scheme of the method following Taya's presentation [646]. Let an infinite elastic matrix characterized by the elastic moduli tensor E (m) to contain an ellipsoidal inclusion (fibre) characterized by tensor E (f~. Then the uniform stress field ~r~ (fig. 3. la) is disturbed by the fibre. If the stress disturbance is denoted by #, then the total stress is ~r~ + #. The problem in fig. 3.1 is reduced now to finding in the equivalent inclusion problem. This problem is shown in fig. 3.1 (on the righthand side) where the original fibre is replaced by the inclusion with the stiffness tensor being that of the matrix, E(m); the new inclusion is given 'transformation strain' or 'eigenstrain' according to Mura's terminology [479]. To ensure the equivalency, the following equation must be satisfied:
O"0 -+-r7- E (f). (~o q_ ~) _ E(m). (~o _f_~_ (,)
(3.38)
where co is the uniform strain defined by o"0 -- E (m) 9co
(3.39)
and ~ is the strain disturbance caused by the fibre. Eshelby showed that g - S. ~o
(3.40)
where S is the fourth-order tensor depending on E (m) and the aspect ratio of the fibre. Equations (3.38) and (3.39) yield ~ Y - E ( m ) ' ( ~ - e *)
(3.41)
Ch. III, w
87
Elastic behaviour of composites
0'-0
IIIIITTTTIIIIIT
IIITTITITITIIII eigenstrains
s E(/)
x3
\i i
i 9
.
-
9
i
9
-
i
i -
272
X2
.
. .
.
.
.
....
llllllllllllllt
l lllllltlllltll
Fig. 3.1. Eshelby's model: an ellipsoidal fibre in an infinite matrix. The original fibre (on the left hand) and equivalent inclusion (on the right hand).
Unknown e* can be found from eq. (3.38) where ~ is eliminated by using eq. (3.40). Here the interaction between fibres was not considered. If the fibre volume fraction is not small the above model should be modified. According to [471,648], the Eshelby formulation is modified by replacing g with g + ~ in eqs. (3.38) and (3.41). Here ~ is the volume average of strain g over the matrix domain, which is related to the volume average of the stress in the matrix, (a)m, by
(O')m -- E (m). ~
(3.42)
Note that
1/v
(O')m =-Vmm
~dV
(3.43)
m
The volume integral of ~ over the entire composite should vanish, this leads to" + vf(S - I ) - e * - 0
(3.44)
where I is the identity tensor. Eliminating now g by using eq. (3.42) we obtain e*. Introducing now the effective stiffness tensor of the composite, E*, we equate the elastic energy of the effective (homogeneous) body to that of the equivalent body"
Deformation and failure of composites
88
1 ~(~o~o).(E,
)-1 _ 1
(,~o~o)
(E(m) ) -1 -]-VfO"0" s
Ch. III, w (3.45)
Because e* is now determined, eq. (3.45) gives tensor E*.
3.2.2. Unidirectional composites Obviously, a regular structure, for example, a doubly-periodical that in a plane normal to the fibre direction in a fibrous composite, should be of a definite elastic symmetry (fig. 3.2). Considering the structure of a body yields a further reduction of the number of mutually independent moduli as compared with the general case of anisotropy discussed in Section 3.1.1. For example, let the fibre and matrix in a model shown in fig. 3.2a to have elastic characteristics E ft) and E (m). The
V-_J
k:h/
<.Vd
----
()
a;
---"
0
.2
Fig. 3.2. Two regular fibre packing leading to transversely isotropic (at the top) and orthotropic symmetry of composites. The fibre and matrix are isotropic.
Ch. III, w
Elastic behaviour o f composites
89
orientations of all the main axes of the symmetry of all fibres are the same. Then we obviously have
, I = Eijk , I (Eft) Eijk p(m) \ rspq, L'ghmn,
d/a)
(3.46)
Therefore, if such functions are known, then having obtained one component of tensor E* we can deduce all others just excluding from above relationships the dimensionless parameter, d/a [234]. This finding has not so much practical importance, since to use it, one needs to obtain expressions written in the structural form of eq. (3.46). Actually this means finding a solution of the problem. Still, the relationships between engineering constants of a composite with transversal isotropy obtained by Hill [247] are of some use. In particular, we have
11
4 ( V f - Vm)2 [ ( ~ ) Ell --/~ + ( i ~ f -- 1/km) 2 - ~ *
* Vf--Vm [~-2 ( ~ ) ] V12 -- 9n t- l / k f - 1 / k m -
"
'
(3.47) (3.48)
Here and further on, the bar over a symbol means the averaging over the fibre and matrix. For example,
( k ) =kf-t-~mm'Vf Vm
(3.49)
The actual evaluation of the elastic characteristics of composite materials is a subject of a special research line in composite mechanics which includes a variety of approaches ranging from exact solutions of the corresponding plane and antiplane elasticity problems to rough estimations of the Voigt-Reuss boundaries (see for example [35, 88, 234, 289, 400, 596, 671]). Exact solutions do normally lead to very complicated final formulae, so they can now be readily replaced by numerical procedures which allow to consider arbitrary geometries of the structure. On the other hand, if the geometry is not known in detail, an exact solution has not so much practical sense and an evaluation of the upper and lower bounds for the effective moduli remains to be the only way to estimate the elastic response of a composite. There are few known approaches to make a gap between the bounds smaller, the Hashin and Shtrikman [234] bounds produce the smallest gap. The most widely used presentation of engineering elastic constants for transversely isotropic material composed of isotropic fibres and matrix is that obtained by Hashin and Rosen who introduced the composite cylinder assemblage model [236]. Direct solutions of the corresponding problems of the theory of elasticity yields exact values for four moduli:
90
Deformation and failure of composites
Ell -- E + 4 ( v f - Vm)2 (.o, V12
~-}-(Vf
Vm) (~m
(3.50)
(3.51)
~----~)
]A12 -- ]Am ]Af -[--------~ ]Am -+- ~ '
(3.52)
kfkm
{ + ]Am. k+]Am
k-/~
Ch. III, @3.2
(3.53)
Here the bar over a symbol again means averaging as J~ = Ef/Af q- gm/Am
and the hat means a pseudo-averaging according to rule fi -- ]Af/Am -~- Pm/Af"
Also -1 O)
--
/Af/Am
-t- ~mm-1-
We remind that for isotropic materials: k f - Ef/2(1 - v f - v2), Am -- Em/2(1 - v f - v~). The shear modules in the plane of isotropy,/-/23, and consequently moduli E22 and are bracketed by bounds. For
V23
kf > 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)
Elastic behaviour of composites
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
Elastic behaviour of composites
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
Deformation and failure of composites
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
Elastic behaviour of composites
(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
Elastic behaviour of composites
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
Deformation and failure of composites
]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 << 2 << S
(3.97)
where s is the characteristic size of the material structure and S is the cross-sectional size of the specimen. If the first inequality is satisfied, the ultrasonic wave 'sees' the material as homogeneous. With the ultrasonic frequency increasing, the dispersion becomes noticeable. The wave-velocity/frequency dependence can be evaluated theoretically for a regular structure [2, 477]. However, a specimen may contain non-homogeneities of different sizes, irregularities in fibre packing, fibre waveness, fibre/matrix delaminations being examples. This makes the situation complicated, and it would be better not to rely either on the inequalities, eq. (3.97), or the calculation of dispersion characteristics but to find a non-dispersive frequency band in a direct experiment. A typical setup to measure wave velocities is shown schematically in fig. 3.5. The ultrasonic pulses with the frequency of about 5 kHz are produced by a generator and put into a piezoelectric transducer. The pulses are passed through the experimental setup twice, with the specimen and without it. A signal from a receiving transducer of the same type as the input one is observed on the screen of an oscilloscope and the time difference, At, between them is given by a time indicator. Obviously, velocity V = L/At, where L is the pathway for the wave in a specimen. To input the
Deformation and failure of composites
102
Ch. III, w
+
+ Generator of synchronizing pulses
Oscillograph
High frequency enerator
V-~
~
T.r a n. s d u. c e .r .c o u. p l e ]__] [__ ~ wlLnouL s p e c i m e n --~: ~]]: ~1
Transducer
couple
w It h sD e c l m" e n "
Fig. 3.5. The experimental setup for measuring velocities of ultrasonic waves.
transverse wave into the specimen, the specimen/transducer configuration shown in fig. 3.6 is used. Attention has to be focused on the transducer/specimen interface to ensure appropriate bonding which is especially important for measuring transverse wave velocities. A particular solution depends on specimen properties.
Metal matrix composites Detailed studies were performed by Kiiko and Spiridonov [313] who tested boron/ aluminium composites with v f ~ 0 . 4 5 by (i) ultrasonic pulse technique, (ii) measuring the natural frequency of a specimen, and (iii) static loading, as well as by Touratier et al. [659] who tested a variety of metal matrix composites and analyzed carefully the dispersion behaviour of the ultrasonic waves in nonhomogeneous specimens. Longitudinal
wave
Transducer matic
t
anSverse
interface
wave
pecimen Fig. 3.6. The piezoelectric transducer to input the transverse wave into the specimen along the normal to its surface.
Elastic behaviour of composites
Ch. III, w ,
10000
w
,
V[,22
ooooo
,
_
_
.
.
.
.
4O00
V/,33
***** ~
E
E
V~oI
_
~/
7000
~
9 9 9 9 9Y
7000
0
2
4
6
f / MHz
(o)
'~.
(b)
7
'7
8000
5000
9000
,
(=)
ooooo
103
6000
8
0
f /
n
ooooo
Vs, t 3
.....
v~,~
3500
6
un~
3000
0
2
f / MHz
4
Fig. 3.7. Ultrasonic pulse velocities versus the frequency in the pulse for boron/aluminium composite, vf ~ 0.45. Experimental data by Kiiko and Spiridonov [313].
A specimen for measuring longitudinal and transverse waves is the cube with an edge of 10 mm in the experiments by Kiiko and Spiridonov and a parallelepiped with the cross-section of 17 • 17 mm in those by Touratier et al. For unidirectional specimens, the Xl axis is directed along the fibre. Dependencies of the wave velocities presented in fig. 3.7 illustrate peculiarities of the dispersion in boron/aluminium composites. A part of them can be certainly prescribed to various non-homogeneities in the composite volume in addition to the main one with a characteristic dimension equal to the mean distance between the fibre axes. The dependencies for longitudinal waves may be used to calculate moduli, namely Young's moduli. The other ones are obviously obtained in too narrow bands of the frequencies to make one sure about being in a non-dispersive frequency band. So just experimental values of the Young's moduli of boron/aluminium composites are shown in fig. 3.8 together with calculated values of the moduli.
300
,
|
........
,
!
,
|
200
,
,
f
Ultrasonic Sta~c Vibration
,
,
,
Ultrasonic Stafic Vibra$ion
a,
\
150
250
200
\
~
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
U.~" 100
. . . . . 150 0.35
,
, 0.40
,
, 0.45
vl
,
t 0.50
,
0.55
50
0.35
0.40
L.B.
0.45
0.50
0.55
'VI
Fig. 3.8. The longitudinal (on the left-hand side) and transverse (on the right-hand side) Young's moduli of a boron/aluminium composite with a fixed fibre volume fraction, vf ~ 0.45, obtained by various
techniques. Also shown are the calculated dependencies of the moduli on fibre volume fraction. The experimental values of wave velocities are after Kiiko and Spiridonov [313]. The calculations were made by using eqs. (3.50) a n d (3.55), Ef -- 400 G P a , Em = 70 G P a , vf = 0.20, Vm = 0.35.
104
Deformation and failure of composites
Ch. III, w
We see, first, that the values obtained by ultrasonic technique is always higher than those obtained by measuring the natural frequency and static stress/strain curve. Secondly, unlike the longitudinal Young's modulus for which the difference between various evaluations does not exceed 12%, the difference between the largest and smallest value of the transverse Young's modulus is as large as 30%. A possible reason for such a big scatter is a larger influence of the fibre packing on the transverse modulus as was pointed out above, Section 3.2.2. Touratier et al. [659] did the experiments in much larger frequency interval, up to 20 MHz, and were able to access all the elastic constants, every time relying on the non-dispersive wave length interval. Their results for a unidirectional AlzO3/A1 composite are given in Table 3.1. The comparison of experimental data, obtained in static tensile loading, with various theoretical evaluations of the Young's moduli is shown in figs. 3.9 and 3.10. We see that the transverse modulus data are again with a large scatter. This is especially true for the titanium matrix composites and because some values are below that for the pure matrix, it can be assumed that at least some of the fibres are delaminated from the matrix. The gap between the upper and lower bounds for E22 given by eq. (3.55) is large especially for the large difference between the Poisson's ratios of the constituents. So the approximate estimation of the transverse modulus by using eq. (3.66) seems to be justified. The usage of exact expressions as those given by eqs. (3.57)-(3.62) can lead to a fairly good description of the real behaviour, however some deviations of a real structure from a model yield rough deviations of calculated data from experimental ones (see fig. 3.10). Note also that there can be detected a trend for the values of E22 to increase slightly with increasing the oxygen content in the titanium matrix. We show also experimental data by Nunes et al. [498] obtained by static testing of AlzO3FP/Mg-composites (fig. 3.11). The data illustrate, first, a usual scatter of the experimental results and, second, a possibility to evaluate the Young's modulus of the fibre by the linear interpolation of the experimental data. The next figure, fig. 3.12, presents the same series of the experimental data, just illustrating the behaviour of +0 laminate. The experimental values of the Young's modulus along the x-axis, which is the direction of the symmetry, are higher than those calculated by using eq. (3.79).
TABLE 3.1 Engineering constants of elasticity of unidirectional Al203FP/Al-composite (vf = 0.5) obtained by Touratier et al. [659]. Method of determination
E~l GPa
E22 GPa
Vl2
~//23 GPa
~12 GPa
Ultrasonic experiment Static experiment Calculation, eqs. (3.50)-(3.51) Calculation, eqs. (3.55)-(3.54)
218 207 226
152 144
0.22 0.24 0.29
57.0 48.0
61.0 48.0 56.3
140-149
52.9-58.1
Ch. III, w
Elastic behaviour o f composites
300
,
,
,
,
,
,
,
,
.
.
.
.
oooooAs oeee|
received Heat-treated
9
0
.
.
105
y q
~
~ 9
~
250
\z00 0
150
100
50 0.3
0.4
0.5
0.6
0.7
v l
Fig. 3.9. The longitudinal and transverse Young's moduli of boron/aluminium composites. Experimental data by Prewo and Kreider [553]. The solid lines are calculated from eqs. (3.50) and (3.55), the shortdotted line corresponds to an approximate estimation according to eq. (3.66), the long-dotted line is calculated from eqs. (3.57)-(3.62) and (3.20). Ef = 400 GPa, Em = 70 GPa, vr = 0.20, Vm = 0.35.
250
200
'
I
ooooo Oxygen
'
t
I
'
'
tO00 p p m ooooo Oxygen > !000 ppm o )o000 B r i n d l e y et al O0 ) 0 0 0 0 B r i n d l e y et al ~~)Jansson et al 0 ~ O 9 <
%
Eft
~
\
/ /
-~
~.~ 1 5 0 /
9 E22
/ -
kJ
0
100 0
50
i
0.20
0~
0
0
0
o
I
0.26
I
,1,,
0.30 Vy
,z.
I
0.35
...,
0.40
Fig. 3.10. The longitudinal and transverse Young's moduli of SiC-fibre/titanium-matrix composites. Experimental data by Brindley et al. [59] and Jansson et al. [278]. The solid lines are calculated from eqs. (3.50) and (3.55), the short-dotted line corresponds to an approximate estimation according to eq. (3.66), the long-dotted line is calculated from eqs. (3.57)-(3.62) and (3.20). E f = 4 0 0 G P a , E m = 108GPa, vf = 0.25, Vm = 0.26 (the data are from [59]).
Deformation andfailure of composites
106
Ch. III, w
250 0
J
200
~.~ 1 5 0
\ ..100
24~ 200~
ooooo 9o , , ,
50
0
~
i
.~,,
0.0
I
..1
,~
.~
0.2
I
~
~
0.4
0.6
"Of Fig. 3.11. The longitudinal Young's moduli of Al203FP/Mg-composites versus fibre volume fraction with the corresponding linear approximations leading to values of the Young's modulus of the fibre equal to 343 GPa at room temperature and 337 GPa at 200~ The experimental data after Nunes et al. [498].
250
[ L
, , ooooo
--~
ooooo
200
~
....
,
,
vi=0.536 vi=0.536 vi=0.35 vi=0.35
,
,
,
(experiment) (calculation) (experimer~t) (calculation)
~_.. 1 5 0 ---
0
~100
\
50
0
0.0
i
i
22.5
i
I
i
45.0
/
i
67.5
grad
i
90.0
Fig. 3.12. The experimental and calculated values of the Young's modulus in the direction of the symmetry axis of AlzO3FP/Mg-laminates (+0). The experimental data after Nunes et al. [498], the calculations were made by using eq. (3.79), the values of Ell0 are shown in fig. 3.11, Vm - 0 . 3 5 , vf --0.2.
Elastic behaviour of composites
Ch. III, w
107
Ceramic matrix composites Ceramic matrix composites can be composed of constituents with very different elastic properties since the variety of ceramics is enormous. Still, the elastic behaviour of such kind of composites has not attracted so much attention as that of polymer-matrix- and even metal-matrix composites. We shall here illustrate problems which can arise when we are dealing with ceramic based composites by considering the elastic behaviour of graphite-fibre/carbide-matrix composites with three-dimensional random structure [186]. These composites are produced by powder metallurgy methods that are routine ways of making ceramics (see Chapter 12), so they are characterized by some matrix porosity which effects the elastic constants. Another special point of these materials is usual just for composites reinforced with three-dimensional graphite fibres. High values of the Young's modulus in the axial fibre direction are accompanied by correspondingly low values of the transverse moduli. So the effective Young's modulus of randomly distributed three-dimensional fibre skeleton is approximately equal to the Young's modulus of isotropic polycrystalline graphite. Gelachov et al. [186] measured values of the Young's modulus of composites with boron-, niobium- and tantalum-carbide matrices by using static loading in compression and the ultrasonic technique. In the latter case reasonably low frequency, 0.6 MHz, was used to ensure a low effect of the wave dispersion. The stress/strain curve of the composite occurs to be non-linear, the larger the fibre volume fraction, the larger is the non-linearity. Examples of the behaviour of unreinforced boron carbide matrix and graphite-fibre/boron-carbide-matrix composites are shown in figs. 3.13 and 3.14. A speculative reason for the non-linearity is porosity of the matrix. However, strict evidences to support such a conclusion were not provided. To interpret original experimental data presented in Table 3.2 we need to obtain values of the Young's modulus of the void-free matrices using the data on porosity of the composites (see Section 12.4.3). We accept the dependence of the Young's modulus, E0, on the porosity according to the Hashin and Shtrikman estimates of the upper and lower bounds on the elastic moduli, eqs. (3.94) and (3.95). Being dependent on the volume fraction, vf, of randomly oriented fibres, the two bounds being coincident at vf = 0. Hence, assuming fibres and matrices to be isotropic and to have the same Poisson's ratios, vf = Vm = v, we have the values of the bulk modulus, K0, and shear modulus, #0, of the void-free matrix expressed via the corresponding values of Ep and ]2p of a solid with porosity p: -1
Ko=Kp
1-
(3k+22)(1-p) | --
3k+62
,0=,p(, ' 7
Deformation and failure of composites
108 120
Ch. III, w
, l l l l l V l l i l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l V l l l l l l
4
--
5
-
100 8O
,y .j.-"
../ .k" ..7 .j
eo
-
b
=
/
i i i
0.0000
0
|
i |
|
,..?"
.y .j..-J"
J
i
l
0.0001
0.0002
0.0003
0.0004
V V V V l V | I W I | W I I | | V V I I | | | I l l I V W I | | V |
60
/
40
/ //"
/
1
I
|
|
I
0.0005
V
I
|
I
V
l
I
I
I
2
/
/
/
I
0.0006
V
I
|
|
|
/
I
|
I
I
I
|
3
20
/,,,//
0 0.0000
0.0002
0.0004
. . . . .
0.0006 E
0.0008
0.0010
0.0012
Fig. 3.13. (a) Strain/stress curves of B4C-matrix. (b) Those for C/B4C composite with 20% of Kulon fibres. Loading-unloading circle numbers are shown. Solid lines are used for loading, dashed lines for unloading. After Gelachov et al. [186]. where v k - - 1 - v2 '
2
-
v ~ 2(v + 2 ) "
E q u a t i o n (3.17) yields t h e Y o u n g ' s m o d u l u s
2.5E+005
3.0E+005
=r'
2
2.3E+005
F
2.5E+005
t,J
~ 2.1E+005 ~ 2.0E+005 \
1.5E+005
\ ~1.9E+005 5
1.7E+005
~) 1.OE+O05
!
Illlll,||llll,ll|llllll,lllllllpl,llllllll,,,,|lllllllllllll
1.5E+005 0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006 r
2.5E+005
1.4E+005
..........................................................
,,, ..... , .........
, .........
, ...................
, .... , ....
The 4 t h a n d 5 t h l o a d i n g v i r v l e s
"x
1.2E+005
2.0E+005
~....
\ 1.5E+005
1.OE+O05
//
d) IIIIlllllflllllllllllllllllllllllllllll[lllil[lllll[I 1.OE+O05 0.0002 0.0000
IIII1
0.0004
0.0006
IIIIIIIllllllllll[llllll
8.0E+004 0.0000
....
II . . . .
Illl|llllllll[lllllll'lll
0.0002
0.0004
0.0006
O.O00B
0.0010
0.0012
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0000
0.0020
0.0040
0.0060
3
i, . . . . . .
[,IIIILi*IIIIIILII[*IIIIIII
.....
I
0.0080
Fig. 3.14. T a n g e n t m o d u l u s , o b t a i n e d by differentiating the strain/stress curves of B 4 C - m a t r i x (a) a n d C / B 4 C c o m p o s i t e s with 2 % (b), 4 % (c) a n d 2 0 % (d) K u l o n fibres, versus strain. L o a d i n g - u n l o a d i n g circle n u m b e r s are s h o w n . Solid lines are used for loading, d a s h e d lines for u n l o a d i n g . A f t e r G e l a c h o v et al. [186].
Deformation and failure of composites
ll0
Ch. III, w
T A B L E 3.2 Young's moduli of short-carbon-fibre/carbide-matrix composites: experimental data
Matrix
Fibre
Kulon Kulon Kulon VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 VMN-4 Kulon Kulon Kulon Kulon Kulon Kulon Kulon
B4C B4C B4C B4C NbC NbC NbC NbC NbC NbC NbC NbC NbC NbC NbC NbC NbC TaC TaC TaC TaC TaC TaC TaC TaC
vf
0 0.02 0.04 0.2 0 0 0.02 0.04 0.06 0.08 0.10 0.12 0.15 0.20 0.20 0.30 0.30 0 0.02 0.04 0.06 0.10 0.20 0.30 0.40
Es (GPa)
Eu
286 190 228 136
249 196 156 96.2 282 283 342 246 346 229 267 235 272 284 221 121 153 388 401 342 402 244 196 148 121
(GPa)
Es is the static modulus, taken as a maximum value of tangent modulus, Eu is the modulus obtained by ultrasonic method. Kulon and VMN-4 are the Russian trade marks for graphite fibres (see Section 2.2)
9K0/t0 E0 - 3K0 + it o
N o w we may proceed by taking the value of p either as an average value of the experimental data obtained as a result of testing the matrix, or as a value of the porosity of the composites at vf = 0, obtained after approximating experimental dependence p(vf). The same is true with regard to the value of the Young's modulus at vf = 0. It can be taken either as an average value obtained by testing the matrix, or as the corresponding result of approximation of all the experimental points E0(vf) corrected by the porosity p(vo). More experimental points are involved in the calculation, and so a more reliable result can be expected to obtain. We therefore give in Table 3.3 the values of E0 calculated by the three procedures described in the table. Comparing the data obtained with the corresponding data published previously [330] we have to conclude that the averaging of all of the data
Ch. III, w
Non-elastic behaviour
111
T A B L E 3.3 Y o u n g ' s modulus of carbides corrected to zero porosity Carbide----~
Young's modulus- l l / GPa Y o u n g ' s m o d u l u s - l0 / G P a Y o u n g ' s m o d u l u s - 00 / G P a
B4C static
B4C ultrasonic
NbC ultrasonic
TaC ultrasonic
412 400
401 349
407 389 349
562 570 484
11 - Both the Y o u n g ' s m o d u l u s and porosity values at vf = 0 . a r e taken as a p p r o x i m a t i o n s of their dependencies u p o n the fibre volume fraction (vf ~ 0). 10 - The porosity value at vf = 0 is taken as an a p p r o x i m a t i o n of its dependence u p o n the fibre volume fraction (vf ~ 0), but the Y o u n g ' s m o d u l u s at vf - 0 is taken as an average value of the experimental d a t a obtained at vf = 0. 00 - Both the Y o u n g ' s m o d u l u s and porosity values at v f - 0 are taken as the average values of the experimental data obtained at vf - 0.
independent of fibre volume fraction gives a more reliable result. Such results are presented in Table 3.3 as bold characters. Hence, we shall use these data later. With the values of Young's modulus of void-free matrices just obtained, and the approximations of the dependencies of the porosity on fibre volume fraction presented in Section 12.4.3 we can evaluate the upper and lower limits of the Young's modulus of the composites by using as the fibre Young's modulus the Young's modulus of polycrystalline graphite, that is Em = 7 GPa. The results are presented in figs. 3.15-3.17. Again, as in the case of metal matrix composites, we see a large gap between the upper and lower limits of the Hashin-Shtrikman evaluation for the system with components with very different elastic characteristics.
3.3. Non-elastic behaviour
Because of the inherent plasticity of metal matrix, composites with such a matrix can reveal non-recoverable strain at low applied stresses. Plastic behaviour of metals, unlike elastic, is sensitive to the microstructure. The microstructure of a metal matrix can change as a result of mechanical or chemical interaction with the fibre. Also we have just seen that some elastic characteristics of fibrous composites are greatly effected by the composite structure. Therefore, the plastic behaviour is expected to be influenced by structural factors on two levels. It should be noted that the very definition of plastic characteristics of composites used in engineering mechanics, such as the yield stress, becomes a problem if this demarcation point is to be chosen traditionally, for example, by introducing a "proof stress" corresponding to an arbitrary chosen plastic strain [547]. For example, in composites with whisker reinforcement, the material structure being highly inhomogeneous on the dislocation level, reveals progressive transition from the local plastic flow to the bulk one. This makes some authors search for a new definition of the yield point. Prangnell
Deformation and failure of composites
112 300
,
,
,
Ch. III, w
(
200
\ ~1oo
o 0.0
0.1
0.2
300
300
\ I00
0
0.0
I
|
0.1
v;
I
0.2
Fig. 3.15. Young's modulus of C/B4C composites with Kulon fibres versus fibre volume fraction. Experimental points and lower and upper limits according to Hashin-Shtrikman theory. (a) Static moduli, (b) Ultrasonic moduli.
et al. [547] suggested a definition based on the analysis of the stress-strain curve up to the third derivative: the minimum in d~r/de corresponds to a m a x i m u m in the rate of change in the local slope of the stress-strain curve. This point is manifested as the yield point. Despite such a controversial situation, or perhaps because of it, an analysis of the yield surfaces of metal matrix composites can reveal a number of interesting features [145, 271,557].
Ch. III, w
113
Non-elastic behaviour
400
,
_o_ \\
,
,
0
o
o "
)
o
/\,
o
\200 r~
100
Calczela~ed f o r closed p o r o s i t y
0
0.0
0.1
0.2
vl
Fig. 3.16. Young's modulus of C/NbC composites with VMN-4 fibres versus fibre volume fraction. Experimental points and lower and upper limits according to Hashin-Shtrikman theory.
600
!
'-, ' "
.
.
;
.
~oo , , o ~ 400
d
,
.
m
,
Calculated for ~o~al p o r o s i t y
~
o
[., ,, o \300
[- ~l
r~
l-t,.
200
o .g
o
o
o
~
~z
loo O
0.0
"~"~~
.....|
0.1
0.2
vl
|
0.3
I
0.4
Figure 3.17 Young's modulus of C/TaC composites with Kulon fibres versus fibre volume fraction. Experimental points and lower and upper limits according to Hashin-Shtrikman theory.
Deformation and failure of composites
114
Ch. III, w
The inherent complicity of the elastic-plastic behaviour of non-homogeneous solids justifies the existence of two approaches for modelling the plastic behaviour of composites and a number of versions of both approaches. The macromechanical approach is just analyzing ways of appropriate determination of the parameters involved in the plastic potential of an anisotropic solid and constitutive equation, eqs. (3.27) and (3.28), respectively. The micromechanical approach is based on a treatment of the material as a non-homogeneous solid, and that is similar to the approach to evaluating the elastic characteristics considered in the previous section. There are also some combined approaches when determination of the parameters mentioned is based on some structural considerations. The macromechanical approach was effectively used by Sun et al. [632] in analyzing the elastic-plastic behaviour of silicon-carbide/titanium-matrix composite (vf = 0.35). Assuming the orthotropic symmetry and considering the material under the plane stress condition, the authors reduced the plastic potential, eqs. (3.27) and (3.33) to a form containing just one parameter: 1
f(oij) - ~ (022 + 2a66o'~2) .
(3.98)
The equivalent stress, a0, and the equivalent plastic strain increment, de0, are ao = x/~f
(3.99)
'
and de p -
(deP2)2+ 2--~66(d),~'2)
')"2
.
(3.100)
It can be shown that for off-axis specimens under uniaxial loading in the x axis direction: -
,p
-
(3.1Ol)
where 1/2
c o ( 0 ) - (3(sin40 + 2a66 sin 2 0cos 2 0)
(3.102)
An appropriate choice of the value of a66 forms a unique master curve ~r0(e0) approximated by a power function, eq. (3.30), with n = 7 from the room temperature up to 316~ So an off-axis stress/strain curve can be constructed by adding the elastic strains O"X
ex - Fxx + exP
(3.103)
where Ex is the Young's modulus given by eq. (3.79). The comparison of the experimental data and the approximation just outlined seems to be fairly good.
Non-elastic behaviour
Ch. III, w
115
In the framework of a micromechanical model, the geometry is to be first defined and then simplified to account just for main features of the composite behaviour and, possibly, to rely on the introduction of some adjusting parameters by comparing the model and real composite behaviour. Secondly, the conditions on the fibre/matrix interface should be formulated. Thirdly, a particular plasticity theory is to be chosen. Finally, a numerical procedure is performed to obtain a macroresponse of the model. The first attempts to analyze the plastic behaviour of metal matrix composites [6, 171,707] were performed on the plane models with regular arrays of elastic fibres in elastic-plastic matrix. As a rule, a finite element method was used for the numerical calculations. To analyze completely the elastic-plastic response on the basis of this approach, one would need enormous computational work. It is also important that qualitative conclusions would be hardly possible to draw. So simplifications of the real geometry occurred to be wanted. A simplified geometry, is often constructed [631,568] following the Aboudi's rectangular array of representative-volume elements (fig. 3.18). The fibre of a square cross-section with an area equal to that of the actual fibre occupies subregion AF of region A which includes also the matrix subregion, AM. Part B is a pure matrix region. The stress and strain fields in each region are assumed to be uniform. A choice of linear sizes h allows to adjust the model elastic-plastic response to a real one. Sun and Chen [631] adopted the following stress and strain region/region relationships for plane stress state, 0-33 = 0-13 --- 0"23 = 0: _
~IA1
_
cB1 - - ~ 1 1
_
,
~A2 - - cB2 - - ~22,
_
,1
),A
1
_
B
12 - - 3~12 - - 3~12"
_
(3.104) (3.105)
Averaging the stresses and strains over regions and using eqs. (3.104) and (3.105) yields (3.106) 0-11 - - 0-11 - - VA0-A1 -~- VB0-B1, 0"22 - - 0"22, 0-12 - - 0-12
Fig. 3.18. A b o u d i ' s r e p r e s e n t a t i v e - v o l u m e
e l e m e n t [1].
(3.107)
116
Deformation and failure of composites
Ch. III, w
where
hi hi + h2
UAF
/)AM
h2 hi + h2
h3 h3 + h4
h4 h3 + h4
Here values h are obvious sizes of the subelements in fig. 3.18. The fibre is considered to follow the Hook's law for the orthotropic material, eq. (3.9), with the elastic constants EI~ , E ~ , p(f) 12 a n d ,V(f) l2, so in region AF we have just Hook's law, that is in the incremental form
{de AF } -- [SAF] {do"AF }.
(3.108)
The matrix is an isotropic elastic-plastic material and its plastic behaviour obeys the constitutive equation, eq. (3.30), with potential function
f (00ij)
1[
112
+
II2
l/
II
+ 3<)
(3.109)
where the double prime refers a value to the matrix. By using the power approximation, eq. (3.30), for the equivalent-stress/equivalent-plastic-strain curve, parameter d2 and, therefore, the increments of matrix plastic strain are expressed in terms of stress components 001~and the equivalent stress 00o. " Adding the elastic strain increments, determined by the Hook's law for the matrix, we obtain the following relationships for subregion AM:
{de AM} = [SAM] {do"AM}
(3.110)
where
1
9 (00AM]n-3sAMsAM
sAM -- Em + 4 o~k,, 00-----7,] sAM
s1A2M_
003
'
Vm 9 (00AM'/n-3s1AMs2AM -E--~ + 4 ~ 2 0"----7-/ 003
9 (000AM']n-3sAMsAM -
1 9 (000AM)n-3 AM AM sAM --Em q-~o~ ~ $2 $2 k, 00n J 0"3 '
9 (000AM'~ n-3 $2AMSAM 3
sAM--~0~ ~ \ an I sAM
= ~ + 9~ tim \ 00n J
0"3
(3.111)
'
0~
where e, an and n are constants in the approximation given by eq. (3.30) and
117
al
I
i,,o
~_,
Non-elustic hehuriour ~l --
II
Ch. 111, 43 3
o"
~-,.
9
IxO
-,
~
, ~ ~j.
>.~
"> 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
Deformation and failure of composites
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
Deformation and failure of composites
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
Deformation and failure of composites
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
Deformation and failure of composites
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, ~ << 1. The behaviour of function f(z) at z ~ +1 is most interesting. It occurs
f (z) -
4rt#e0z - I n a + 2 In 2 + (2 + / t ) / 2 ( 2 + 2/~)
+
(3.134)
Therefore, the ratio of the interface shear stress, r, at z - + 1 to the average stress o 0 - e0E will be r --~ o0
/~2 1 Ee-Inc~+21n2+(2+/0/2(2+2/0
(3.135)
Equation (3.135) unveils that the ratio r/a0 depends weakly on the elastic characteristics of the matrix. On the other hand, when ~ varies from 1/10 to 1/1000 this ratio changes between 1 and 100. It follows immediately that the matrix behaviour in the vicinity of the interface may not be elastic unless the interface fails. Therefore, in the case of metal matrix the latter deforms plastically, in the case of ceramic matrix the interface can fail. Hence, results of the elastic stress analysis may be used for very rough estimations only, and there is no practical demand to build up exact solution of the elastic problem for the intact interface. It is sufficient to rely on approximate solutions, one by Cox [106] is well known. A simple analysis of the shear-lag type of this problem was performed by Cox (see for example [85, 301]), we give just the final result, the fibre axial stress, d, and interfacial shear stress, r, as are functions of longitudinal coordinate x. We have
o(x) - gfs 1 -
cosh( (//2 - x))) cosh( //2)
(3 136) '
Fibre~matrix stress transfer
Ch. III, w
"c(x) -- Efeq sinh(a'l'2 ( / - x)) cosh(~l/2)
125
(3.137)
where e is the composite strain, and
q
H
In(R/r))
~
2rc/.t m
In(R/r)'
r and l are the radius and length of the fibre, respectively, and R is the radius of a cylinder to model the matrix, or the distance from the fibre axis at which the matrix strain is unaffected by the fibre.
3.5.2. Interface partially debonded Following McCartney [397], we consider an axi-symmetrical problem (fig. 3.22), in which the non-zero stress components are ar, a0, az, and r = arz, and the displacement components Ur and Uz. The matrix crack is located in the plane z = 0. The inner radius of the single-fibre model, Ri, and the outer radius, Re, are such that V f - (Ri/ee) 2. The equilibrium equations to be satisfied everywhere in both components are ~O'r
~'C
O'r -- 0"0
~O"z
9"17
T
0---r--t-~zz + ~ r
= 0
~---Z+ ~r + -r - 0
(3.138) (3.139)
The Hook's law with thermal stresses are for each component O'r
V
E (az + ao)+ aAT,
(3.140)
o"0 G0 -- E
v /~ (O'r -+- O'z) nt- ~ A T ,
(3.141)
(7z s = E
v E (O'0 q- fir) -t- o~AT,
(3.142)
(r--
E
? = z/2#,
E = 2#(1 + v)
where strain components
(3.143)
Deformation and failure of composites
126
Ch. III, w
TTTTTTTTTTTTTTTTTTTTTTTTTTT N O SLIP REGION I
DE BONDING
AND
FR ICTIONAL LIf REGION
II
J
f
L Re Fig. 3.22. A matrix crack in a concentric model of a composite. (After McCartney [397].)
s
=
~Ur Or '
Ur e0---,r
~Uz ez =~i~z '
OUr ~Uz 2 7 - -~z +~'0r
(3.144)
Here E, v, p, and ~ are Young's modulus, Poisson's ratio, shear modulus and the coefficient of thermal expansion; AT is the change in the temperature relative to that temperature at which the strain is zero provided the model is not loaded or stretched on the external boundary. The conditions at the fibre/matrix interface are for all z _> 0: Ulr(Ri, z) :
) u tt(gi,z r
t7rt( e i , z) -- t7rtt(g i , z),
(3.145)
$' (Ri, z) -- "f (Ri, z).
Single prime refers a component to the fibre, double prime refers a component to the matrix. On the external cylindrical surface for all z _> 0: o'er(Re, z) - 0,
atz~
~:
"~tr(Re, z) = 0.
(3.146)
Fibre~matrix stress transfer
Ch. III, w O'zt (r~ (x)) - - o - ~t
"ct(r~ oo) -- 0,
tr"(r, oo) - tr"~
Z,"(r, 00) -- O,
t
127
(3.147)
tt
0"oo Uf -~- 0"~ Um ~ O"
and there should be O" t -- Ef(~ - ~fAT) - 2VfVmf~/e 2,
ffoott - E m ( e -
(3.148)
~mAT)-+- 2VmVff~/g 2
where (1 -+- Vm)o~mATR2
~b --- (Vm -- Vf)e -+- (1 + v f ) ~ f A T 1
F 1--Vm
2,Um ~
1--vr
(3.149)
Vf -'['- - ~ f Vm
On the plane z --0:
U'z(r , 0) = 0,
z'(r, 0) -- 0,
z"(r, 0) -- 0,
a~(r, 0) = 0.
(3.150)
In region 1" Uzt(R i, Z) -- Uztt(Ri,Z ) ,
l <_ z < ~
(3.151)
In region 2 of frictional slip, the Coulomb friction law sets the interface condition: "/:'(Ri,2) --- -rltTtr(Ri,z),
0< z< l
(3.152)
where r/is the interfacial coefficient of friction. The exact analytical solution of the problem just formulated would lead to an extremely complicated analysis, as one can guess, having the structure of the corresponding formula for the case of a rigid fibre perfectly bounded to plastic matrix, eqs. (3.133)-(3.135), To obtain sensible expressions, McCartney built an approximate solution: .gt -- ~1)m Ct(z)r,
t ~ t -- o-0~
(3 153)
"c" = R2/r -- vfr R~ C'(z), 1( re (1) o'~t - - 2 / ) f R T - 1 + In ~f
(3.154)
-In
(r~))
1 (R 2 )A(z) C"(z) + ~ \ r 2 - vf ~ ,
(3.155)
-Deformation andfailure of composites
128
,, 1 ( r 2 ao-~ vf~2i2-1+ln
tt
(1)
-
In ~
Ch. III, w
(z)-5\r 2 +vf )~ ,
tt +2v OfC(z),
(3.157)
, 1((1) O'tr -- O"0 -- ~ In ~
r2 ) A(z) - ~ Vm C tt(Z) -'F 1)m 2R 2 ,
'
' - 2 ~t)m C(z)
,
1(
(3.158)
(3.159)
G z ~ Gcx~
Ur -- ~ m
t
uz - ~
A(z)
(3 160)
G(z) - 2R~ } r,
1 (Umct(z)_ R2 (G t ~f
(3.156)
~
At(z)~) r2
( z ) - 2R2 j
(3.161)
~+Hf(z).
Primes at functions A, C, G and H mean differentiating with respect to z. This solution satisfies exactly eqs. (3.138), (3.139), (3.143), and (3.145)-(3.147). The constitutive equations, eqs. (3.140)-(3.142) are satisfied averaging stress and strain components over interval (0, Ri) for the fibre and (Ri,Re) for the matrix. The conditions on the interface, eqs. (3.150) and (3.151) and that on the crack plane are satisfied approximately. This yields A (z) - -2c~ - p4fl C(z) + ogR~C" (z)
(3.162)
where 1 -- Vf p - ~1 +- Vm 1 -- Vm /)m, Em ~- Em vf + Ef O)
m
pEm'
21
( 1
1L~m~pEm jI,
fl -I-
Vm/)f Vf/)m + ~, -~m Ef
1 ( 1~) ~lnvm
-I.
(1) m
vf
(3.163) (3.164) '
also
H;(z) =
2vm C(z) Ef R~
VfVmA(z) a~ Ef R~ ~ Ef
-+-o~f A T -+-~ m 2vf C(z)
H'(Z)-Em
Gtt (Z)
VmVfA(z)
En
Art(z)) 2R2
o.tt
O(3
Em
-
(1))Ct t
~ff + ~f In ~
(z),
(3.165)
Fibre/matrixstresstransfer
Ch. III, w -~-~ +
m
(1--I-- Uf
2Em q
(z)
1 --1-~m Em
2R2 j - 8 ~
1 --t-~'mUmin ( 1 ) )
Emvm
129
~
m
C"(z).
-2 (1-+-l) )Att (Z) (3.166)
Equation (3.151) yields the following fourth-order differential equation for the function C(z) in region 1" R 4C"" (z) - 2al R 2C" (z) + b 2C (z) - 0
(3.167)
where G 2F'
al
bl -
(3.168)
and constants F, G, and H defined by
Vm ( ( l + v m ) I + l )
F=l-vf
8pEr Emvf
Um~
l+vmfl(I-
Em
/~ I,
('+Vm)
G -- flco + 2vfEf-I -
H = 2E*
EfEm
Em p
l+vm( Vm) 4Em I - ~ 09, v ~ f ) 1m Om + (E 2
li) (3.170)
4 ~2 --,
(3.171)
p
E* -- Ef/)f -Jr-Emvm,
(3.169)
It
~ -- v r - (1 - vf) ~ . P
(3.172)
Equation (3.152) yields the following second-order differential equation for the function C(z) in region 2: 2~b
RZc"(z) + 2azRiC'(z) - b~C(z) - (o9 + 1) where
a2-- q(o91+ I ) '
b2 - 2
~fp (co//+ I)"
(3.173)
(3 174)
Note that it follows from eqs. (3.154) and (3.157) that C ( ~ ) - 0,
C'(c~) - 0.
(3.175)
Now for the case of a perfectly bonded composite (1 - 0), when the matrix cracks on the plane z - 0 leaving the fibre intact, it follows from eqs. (3.153), (3.154) and (3.157) that eq. (3.150) yields
-Deformation and failure of composites
130
c'(o) -o,
R 2~rIto~
c(o) = - ~ = 2vf
Ch. III, w
(3.176)
C0,
and the solution of eq. (3.167)is for
bl > al 9 C(z) --- C o ( c o s ( q z / e i ) --[- ( p / q ) s i n ( q z / R i ) ) e x p ( - p z / R i ) ,
for
bl - al 9 C(z) = C0(1 + pz/Ri) e x p ( - p z / R i ) ,
for
bl < al 9 C(z) = ( C o / 2 q ) ( ( p -4- q ) e x p ( q z / R i ) - (l9 - q ) e x p ( - q z / R i ) )
(3.177) exp(-pz/Ri)
where parameters p > 0 and q > 0 are defined by
P-
a l -'[- bl ~ ,
q =
/vial 2 bl[
Therefore, eqs. (3.153)-(3.157), (3.158), and (3.159) determine now the stress fields in both components. We do not write here the approximate expressions for the strain fields referring to the original McCartney's paper [397], that lead to the expressions for the additional displacement at infinity, Aor arising from the presence of the matrix crack and the average opening of the matrix crack, A0, namely
Ace=
2Vm Ef (1
A0--2
-
E~-~m
2vffl/p) C'(0) R2
(3.178)
2
(3.179)
where c(o) =
C(x)~
(3.180)
In the key case of non-zero length l, when slipping occurs over frictionally constrained part of the interface, function C(z) for region 2 is given by the solution of eq. (3.173) which has the form
C(z) -
(pdp )( ~-~ + Co
kz a2
kz)
(a2z)
cosh Ri + --ffsinh Ri exp - --~-iJ
Pq~ 2fl '
O
where k -
C a 2 + b 2.
This function and that for region 1 given by the solution of eq. (3.167) should yield the stress and displacement fields continuous at z - l.
Fibre~matrixstresstransfer
Ch. III, w
131
It can be shown that these continuity conditions for the stresses are satisfied whenever the function C(z) and its first derivative are continuous at z = 0. The continuity conditions for the displacements u r and u tt r can be easily satisfied only for averaged values of the displacements. For the case bl < al the required solution of eq. (3.167) is
C(z)-
B
(A c o s h ( ~ i ) + - s ~i n h i ( ~i ) ) ) expq ( -
pz
l < z < c~
(3.182)
where
q(-Rii)- ~V+PUsinh(ql)) q Rii
A-(Ucosh
B-- ( (V+pUlcosh (ql) Rii
-Uqsinh
kl
( p C ) (
U - C(I-) -
-~ + Co
t pl)
( qRiil ) ) t pexp l) a2
(3.183)
exp Ri '
kl)
Rii '
( a21~ pC
cosh Ri + --ffsinh Ri exp - Ri ]
V_RiC,(I_)_ (pflp co) k2-a 2sinh Rii kl exp ( - ~a21~ -~ + k ].
(3.184)
(3.185) (3.186)
To glue the functions C(z) and C'(z) at z - l it is necessary to find the value of I as the solution of the following transcendental equation (o9 + I)((q 2 -
pZ) U - 2pV) + -2 V - 4- U - 2dp q P
(3.187)
where U and V are functions of l written above. Equation (3.187) must be solved numerically. When bl = al, q = 0 and eqs. (3.182)-(3.187) are simplified, so the corresponding relations are easily obtained. When bl > 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
-
(3.188)
132
Deformation and failure of composites
2-c Vf -z, o it (z) -- --Ri- -l)m l
0_
Ch. III, w (3.189)
Ri Vm .. ~ Vf
(3.190)
F o r the plane strain:
E,c du(K)(z)
a (~) (z) - 1 - v2
(3.191)
dz
where subscript • is either f or m, and superscript (x) is either single or double prime. In particular, a ('~) - a('C)(l) -- E K e / ( 1
- v2).
(3.192)
Hence, z (l--z) 2 Ri
(3.193)
u' (z) -- e( l -- z) + ( 1 - - v 2)~ff
"C V m ( l - - z )
u"(z) -- e(l - z) - (1 - v2) gm vf
2
where the displacements are selected to be zero at z The crack opening is Ri 1 - v 2 vmEm [0.,(0)]2
2Ao - ~
E*
(3.194)
Ri l.
(3.195)
Ef
Obviously, the additional displacement between the ends of the debonding zone is
A, - 2 v ~
Ef
1:12 . Ri
(3.196)
Note that in [25, 385], the formula corresponding to eq. (3.195) is written for the plane stress state, so that the factor (1 - v2) is replaced by 1. Using the equations obtained above, the case of a cracked fibre in a composite with perfect bonding on the interface can now be considered so that r
o) - o,
This yields
r
z) - o,
-'z(r,z) - o
Fibre~matrix stress transfer
Ch. III, w C(0) - 0,
C(0) - RZcr~ = C1. 2Vm
133 (3.197)
Equation (3.177), with Co replaced by Cl, defines the appropriate function C(z), so eqs. (3.162), (3.165), and (3.166) can now be used to determine functions A (z), G(z), H~(z) and H~m(z). Thus, the stress field is defined.
3.5.3. Critical fibre length The shear stress concentration at the vicinity of the fibre end causes plastic deformation of a metal matrix. In this case shear stress may be assumed to be constant, say ~*, on some length near the fibre end and zero in the center zone if the fibre length is sufficiently large (fig. 3.23a). This corresponds to the linear increase of the tensile stress in the fibre up to a maximum value, a ~, determined by the applied load and value of 7*. When the fibre length decreases the situation changes qualitatively. The end zones link each other (fig. 3.23b) and we see the only point along the fibre where the fibre stress reaches a maximum value. With the fibre length decreasing further, the maximum value of the axial fibre stress decreases (fig. 3.23c). The fibre length can decrease as a result of breaks occurred when the fibre stress, d, reaches the fibre strength, a~. The last fibre break occurs under the condition depicted in (fig. 3.23b) when the fibre reaches the length l*. Kelly and Tyson [306] were first to analyze this process, they called the corresponding fibre length as the critical length. Obviously, the critical aspect ratio of the fibre is l--
*
= ~f df 2"c*
(3.198)
The definition of critical length of a fibre is implicated into the analysis of either mechanical behaviour of short-fibre composites or the situation in the vicinity of a fibre break. Note also that a part of fibre loaded not to a maximum possible stress, O'max, is often called 'ineffective length'. In case of the ideal plastic behaviour of the
I--!
F[ I
Q
-,/
Fig. 3.23. Variations of the interface shear stresses and tensile fibre stresses along the fibre length according to Kelly and Tyson.
Deformation and failure of composites
134
Ch. III, w
matrix, this is exactly half of the critical length. On the other hand, in a purely elastic case, it remains to introduce a conditional value, say 0.95trmax, to be treated as a transition point from 'ineffective' to 'effective' fibre length. The average fibre stress is ~, - I 1 fot a'(x)dx
(3.199)
Equation (3.199) for a constant shear stress at fibre/matrix interface yields a~
l < l*
(3.200)
3.6. Shear-lag analysis Stress/strain fields in a composite around some kind of irregularity, a broken fibre being an example, are to be evaluated for many purposes, for example, to follow the failure processes in the composite. Normally rigidities of the components in a fibrous composite differ from each other significantly. On the other hand, values of the ultimate strain of the components are also very different, usually these values are much lower for a more rigid component. So the problem of stress redistribution around a broken fibre arisen because of the latter argument becomes sufficiently simple due to the former one. The situation gave birth to the shear-lag method by Hedgepeth [241] that is based on the assumption that the fibres carry only axial stresses and the matrix between the fibres transmits only the shear stresses. Despite the existence of more rigorous solutions of the theory of elasticity problem, this simple method is used nearly always when the corresponding solution is to be looking for. Solution of the problem is important for the analysis of failure processes in composites. So we need to present the method here, but because it is now well known (see, for example, the book by Chou [85]) we shall describe the method very briefly.
3.6.1. Fundamentals Consider a regular plane fibre/matrix array of the Hedgepeth type as shown in fig. 3.24. Let element Ay of a fibre be loaded only by axial stresses try and shear stresses z on the fibre/matrix interface. Writing the equilibrium conditions, excluding transverse forces Q, neglecting the transverse normal stresses, and making Ay ~ 0 yields
ht day + ~(n) _ .~(n-1) _ 0
(3.201)
Shear-lag analysis
Ch. III, w
135
Yi _#_nnu_l . ,
(
. 1
"
_t
.
. 1
1
. 1
1
. 1
1
1
1
1
i
#
1
1
1
1
1
1
1
1
.
.
.
.
/!!!!. i////,
t / i l l ,
/////,
1 1 i / i .
y
r i l i l l I i 1 1 1
"///// t1111,
I/ll/IA
g////A
v
X
/ / ! 1
i
//t"///
~//,~
n
,n,- I
(Q+AQJh ~
m- ~~lXu ~
n+
l
n+2
(a (n) + A cr (n) h~)
I rCn) A u
Qh'J-i o(")h' Fig. 3.24. Plane model of a composite considered in the simplest version of the shear-lag analysis.
The equation just written implies that the increment of the normal stress carried by fibres in a raw is equilibrated by a difference in the shear stresses in the matrix on both sides of the fibre. The shear stresses in the matrix, ~, are expressed via the corresponding shears
" Vn+l (Y) - Vn(Y) 7(n) -h"
(3.202)
where dn is the displacement along the y-axis of a fibre/matrix interface point. So
136
Deformation and failure of composites .c(n)
/'/m t = ~h" [/)n+l
t (y)].c(n-1)
(Y) - Vn
]Am __ _h" _ [Urn(y)
,
- Vn_ 1 (Y)]
Ch. III, w
(3.203)
where p is the shear modulus of the matrix. Substituting eq. (3.203) and the Hook's law for the fibre dv~n dy
if(n)--Ef
(3.204)
into eq. (3.201) gives d2vn - -Jr- C 2 (Un+l -- 2Vn + Vn-1 ) - - 0
(3.205)
dy2
where C 2 = ~m
1
(3.206)
Ef h'h"
Suppose a k-fibre is broken at y = 0 (k = n + 1 in fig. 3.24). If the array is loaded by a uniform tension in the y-direction at infinity, the problem can be reduced to that of an array with no load at infinity but a uniform compressive load, cr at the crack surfaces. So to obtain the solution of the original problem it is necessary to add a uniform tension of the same magnitude as the compression applied to the solution of the reformulated problem. Thus the boundary conditions are written as at y - c~" aty-O:
Vn - - 0 r (n)-Od-d&-
dy --
vn--O dvn - - 0
n#k n - k --, cx~
dy
~ 1
vf Ef
n-k
Applying the transformation v(Y)--
Z
Vn(y)exp(-inO),
(3.207)
n----(x3
whose inverse is Vn(y) - ~
~n(Y, O)exp(inO)dO,
(3.208)
7r
to eq. (3.205) transforms this equation into d2~
dy 2
0 4c 2 sin 2 ~ v -- 0
(3.209)
with corresponding boundary conditions. Omitting the obvious mathematics we write the solution in final form for k - O, that is the stress in a fibre with index + r
Shear-lag analysis
Ch. III, w
a(yr)_av~ [1 - ~l f o ~ cosr0sin~exp 0 ( -2cysin ~) dO] .
137
(3.210)
Here the r-counting starts from the broken fibre ( r - 0). If y - 0 we have
_a [1- lI(r)] -- K ( r ) v5 L
5
a v--f
(3.211)
- - ~
(3.212)
I
where I ( r ) -- f0 ~ c o s r O s i n ~ d O 0
4 r 2 -- 1
We see that a fibre closest to the broken one undergoes the overload equal to 4/3, for the next fibre this value is just 16/15 and so on. The fibre overload decreases very fast with the distance from the broken fibre, so simple analytical solutions for a three fibre configuration obtained by Fukuda and Kawata [85] can be seen as an approximation to a real behaviour. In this case the set of equations given by eq. (3.205) is replaced with two equations: d2v0 2 ~ § d{ 2 O~
-- VO) -- 0
d 4v0
3 d 2v0
d~ 4
~ d~ 2
d2vl -~
§
1 O~
=0
-- /21) -- 0
(3.213) (3.214)
Integrating the equation obtained and using the approximate boundary conditions, we obtain the solution as (~2 + ~ + ~ 1 exp(-2~)) '
v0 - a' v0 - ~'
+ ~
1
(3.215)
ao - a'(1 - exp(-2~)), (71 --
~'(
1 +~ 1e x p ( - ~ / ) ,
For the case of 2s + 1 fibres broken in a row, all the breaks laying at y - 0, the problem is also solved explicitly. For r > 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
Deformation and failure of composites
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
Deformation and failure of composites
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<- a.
(3.234)
Outside of this volume the situation is as in the simplest model, and the equilibrium equations are given by eq. (3.205).
Shear-lag analysis
Ch. III, w
143
The problem reduces to an integral equation with respect to function f(~) - vo - Vl - -
(3.235)
I "0
where ~ is the dimensionless y-coordinate. A numerical procedure is applied to solve the integral equation and the results obtained show that stress concentration factor in the nearest fibre to the broken one decreases from value 4/3 for applied load P* to cause the beginning of the matrix yielding down to ~ 1.14 when the applied load equals 10P*. Within the interval (P*, 10P*) the plastic zone expands in the ydirection linearly. A problem for viscoelastic matrix, which is rather relevant to the behaviour of polymer matrix composites, is obtained in [351]. The integral operator acting on (Vn+l --2Vn 4-2Vn+-l) replaced C2 in the second term of eq. (3.205). The equations obtained is solved by using the method of Laplace transform, which is a usual strategy for solving problems of viscoelasticity, to reduce the problem to the elastic one. The solution provides a means for analyzing the time evolution of the overstress of fibre due to fibre breaks. This allows also to observe sequential fibrebreaks as a result of stress redistribution with time. An important case of nonelasticity is that of delamination on the fibre/matrix interface.
3.6.5. Non-uniformfibre packing The fourth direction of the study is important because often the failure processes in composites dependent on non-homogeneity of fibre packing essentially (see Section 5.2). Still, the only attempt by Ochiai and Osamura is known [505]. Three main assumptions are made by the authors. First, fibres cut by a crack are considered as a single element, 'core', and therefore, the displacements of the fibres within the core do not depend on x. Secondly, only the closest neighbours are subjected to stress concentration. Thirdly, a bi-linear approximation of the stress/ strain curve of the matrix is valid, with #m as the shear modulus, #~ as the slope of the shear-stress/shear-strain curve in the plastic region, and z0 being the yield shear stress. Under these assumptions the equilibrium equations for the purely elastic case (fig. 3.27a)will be d2~l d~ 2
d2v2 --
m
d{ 2
d233 d~ 2
(3.236)
= --a2(v2 -- Vl) 4- al (Vl -- ~),
a2 (v2 - 31) + - a3 (32 - ~)
n
n
-- --a3(v2 -- 33) 4- a4(v3 -- ~).
'
(3.237)
(3.238)
144
Deformation and failure o f composites
0
1
4
3
2
Ch. III, {}3.6
..11 I I I I i I I
I I I I i I I
,,//
/ I I I i I I
I I I I I I I I I
,._t_l I// /.,,z //z r r /.i/ i// ,,.//
It
II f-
0
1
2
3
4
(b)
0
1
2
3
4
(c)
Fig. 3.27. Ochiai and Osamura's scheme of non-uniform packing of fibres in an elastic-plastic matrix [505]. A group of the fibres (together with the matrix in between) is labelled by '2'. The whole overload is taken by fibres '1' and '3'. (a) The matrix is deformed elastically. (b) Plastic yielding in the matrix between the closest neighbours. (c) Plastic yielding in the matrix on both sides of the group of broken fibres.
Here Vn ~ Vn
v/Ef/Am IA fh~' h a' #m h
- *VuT-rha an
,, ,, h o, , 1 -~ (h'( + h ,,2 + h ,,3 + h~), -- 1 / f n , f n = hn/ho,
(3.239) (3.240) (3.241)
Shear-lag analysis
Ch. III, w
145
n is the number of fibres in the core, h is the thickness of the plate under consideration. For the partly elastic case (fig. 3.27b): d2~1 d~ 2
d2v2 z
d~ 2
d2f3 d~ 2
(3.242)
--- --[fla2(v2 -- Vl) q- (1 - fl)Z'o] + al (Vl -- ~),
_
fla___22 (V2 -n
Vl) q- (1 m fl~)70 nt- -a3- ( V 2 -- ~)
n
n
'
(3.243)
(3.244)
= --a3(v2 -- v3) -Jr- a4(v3 -- ~).
Here
"CO /gfh~h
(3.245)
7V-j
and fl- p~/mUm. For the elastic-plastic case (fig. 3.27c)" d2fl d~ 2 d2~2 d~ 2 d2~3 d~ 2
= -[fia2(v2 - vl) + (1 - ~)~0] + al (fl - ~), _ fla__~2(v2 - vl) + 2 (1 - fl)70 + fla___~3(v2 - ~), n
n
n
= --[fla3(v2 -- v3) q- (1 - fl)go] 4- a4(v3 -- ~).
(3.246) (3.247) (3.248)
A solution of each set of the equations can be easily obtained in an explicit form, the procedure is similar to that for the three-fibres configuration considered above. The qualitative conclusions drawn by the authors are rather predictable. In particular, they state, that (i) the larger the non-uniformity of fibre spacing, the larger the deviation of the stress concentration factors from those for uniform fibre spacing; (ii) the wider spacing between the closest to the core and the next fibres, the higher is the stress concentrations in the closest fibres; (iii) when the number of broken fibres increases, the above tendencies enhance. The real importance of the model considered is in using it in a computer simulation to analyze quantitatively the influence of non-uniform fibre packing on the strength of metal-matrix composites. The results will be described in Section 5.2.3. Note that in a recent paper, Ochiai et al. [500] combined the approach just discussed with the model in which both fibre and the matrix transfer load by shear and carry axial stresses (see Section 3.6.2).
This Page Intentionally Left Blank
Chapter IV MACRO- AND MICROCRACKS IN N O N - H O M O G E N E O U S MATERIALS
Cracks in non-homogeneous materials have much more opportunities to be arrested than in homogeneous materials. At least, a designer of a composite should organize the structure of the composite in such a manner as to provide the crack with a chance mentioned. If the designer goes over, the results discussed above can be achieved. This means that brittle substances in the form of fibres or matrix, or even in both components, can be effectively loaded in the composite. Therefore, grasping the behaviour of cracks in non-homogeneous bodies is of a major importance for a composite designer. Perhaps, a most significant virtue of the crack behaviour in composites is a variety of the interactions of macro- and microcracks. An analysis of the interactions of macro- and microcracks is important for all types of composites, fibrous and particulate, metal- and ceramic-matrix composites, fibre-reinforced polymers, carbon/carbon composite, and reinforced cements. A macrocrack in any solid under load gives birth to microdefects in front of its tip. The scale level of the microdefects can be very different, from the atomic or dislocation level, in which case plasticity of some physical nature will occur, to the rather macroscopic level, microcracks and macrovoids being examples. The creation of such microcracks has been observed experimentally in metals, ceramics, polymers and fibrous composites. The fracture toughness of the solid is determined by a complicated interaction of defects on different levels. It is often difficult to choose a predominant type of interaction. The study of the variety of appearances of a system of microcracks leads to an obvious assumption. Namely, a microdefect such as a microcrack, flaw, etc. should arise not at an arbitrary point but at such a point where an imperfection exists in the structure. The scale level of such imperfections varies from the atomic (vacancies being an example) to quite macroscopic in the case of brittle fibre breaks in composites. The problem of an interaction of the original (macro-) crack with the secondary (micro-) crack was formulated Cook and Gordon [97] who considered the situation of only one or two secondary cracks and showed clearly a possibility of crack arrest by microcracks occurring ahead of its tip. The C o o k - G o r d o n microcracks occur as a result of the failure of a weak interface. Both the fibre and matrix can carry defects which can transform into microcracks under appropriate circumstances.
147
148
Macro- and microcracks in non-homogeneous materials
Ch. IV, w
Hence, it is convenient to consider all these situations as particular cases. We shall do it restricting ourselves by some approaches to the problem choosing such results which illustrate the behaviour of fibrous composites and can be used as a base for the evaluation of fracture characteristics of composites. However, we start with a brief outline of the results of the general crack theory which will be used further on.
4.1. Cracks in homogeneous solids The energy approach of Griffith (Section 1.2.2) is based on the analysis of the stress state in the vicinity of an elliptical hole assumed to be a model of the crack. An extreme case of the Griffith crack is a cut with zero radius that occurs to be a convenient model of the crack in a continuum. To illustrate the convenience of such an ideal model we consider a standard procedure of stress analysis following Rabotnov [560] in the case of isotropic body. Then we discuss some known approaches to formulate fracture criteria. 4.1.1.
Stress analysis
Isotropic case We start with the simplest case of an a n t i - p l a n e p r o b l e m , that is a crack under longitudinal shearing (fig. 4.1). The general solution of the anti-plane problem is to satisfy the equilibrium equations by assuming displacements as u, -- u2 -
1
O,
u3 -- - u ( x , , x 2 )
P
where p is the shear modulus. Then the Hook's law gives
| ee|
xe
I
| | |174174 Fig. 4.1. The shear crack.
(4.1)
Cracks in homogeneous solids
C h . IV, w ~)u "/21 - - 0"31 - -
149
i~u "/22 - - 0"32
Oxl
Ox2
0"11 - - 0"22 - - 0"33 - - 0"12 - - O.
(4.2)
The non-zero components of the stress field, shear stresses, should obey the equilibrium equation ~'gl
8"62
Ox---~+ ~
- 0.
(4.3)
Differentiating the first expression in eq. (4.2) with respect to one with respect to Xl we may write O'Cl
~'C2
~X2
~Xl
X2
=0.
and the second
(4.4)
We see that the equations obtained are the Cauchy-Riemann relationships between derivatives of the real and imaginary parts of an arbitrary function of a complex argument [638]. Take z - x l + ix2, the function, w ( z ) , can be expressed in the form W(Z) -- U(Xl,X2) nt- iV(Xl,X2)
where u and v are real functions. Therefore, an arbitrary function w'(z)
-- Zl - i z 2
(4.5)
satisfies eqs. (4.3) and (4.4). Hence, this is the general solution of the problem of the elasticity theory under consideration. Boundary conditions are known to determine the function completely. It is also known that u - Re w ( z ) .
(4.6)
To obtain the special solution we take function w(z) -
-Ki
-
-g
-
sin ~ + i cos
,
0 - arctan-x1
(4.7)
which corresponds to a solution of the anti-plane problem. Following the usual procedure we may write R e w ( z ) = K V /2~r s i n ~0 .
(4.8)
At 0 = ~z" Re w ( z ) - x/7, at 0 - - ~ " Re w ( z ) - - x / 7 . This means that at Xl < 0 the displacement, u3, is discontinuous, that is shearing of any two points of opposite
Macro- and microcracks in non-homogeneous materials
150
Ch. IV, w
surfaces of the crack which coincide initially, in opposite directions along the XlX3plane is such that the j u m p occurred, u~- - u 3, proportional to x/7. According to eq. (4.5) Zl -
z2 -- - I m w' (z).
Re w'(z),
We have also w'(z) =
2~
sin ~ + i cos
.
At 0 = +~z, w' is real, this means that "[3 is zero; hence, no forces are actually applied on the surfaces of the cut. Finally we have K 2/~r 0 V~-sin # 2'
U3 - - - -
K zl - - ~ s i n ~
0 " 2'
z2 =
K ~
0 cos-. 2
(4.9)
The stresses tend to infinity with r goes to the crack tip (r = 0). Consider now a more real situation when the crack in an infinite body has a finite length, so that it lies within the interval [-g, +g] on the xl-axis. At infinity, Zl = 0 and ~2 • "CO. We may take function w ( z ) in the form w ( z ) -- A i x / z 2 - g2.
(4.10)
F o r points on xl-axis, v/z 2 - g z _ x / x 2 _ g2 is real at ]Xl 1> 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 << g, the following approximate formulae are obvious u2 -
x ( 1 - v) #
,
x
0"22 - ~ / ~ r '
K -- 0"v/-~.
(4.21)
The energy release rate is given by Irwin's relationship in the same way as in the case of a longitudinal shear crack, eq. (4.16), that is G-
(1 -
v ) K 2 _-- 1 -
2p
V2K2.
(4.22)
E
F o r plane stress s t a t e (0"33 - - 0 ) : K 2
G - --~-.
(4.23)
The stress/strain field at the vicinity of the crack tip is found if the stress intensity factor is determined. This calls for the determination of stress intensity factors for various configurations of the crack/load systems. The corresponding results are well known, they are compiled in numerous textbooks (see, for instance, Refs [585, 607]). We just give some formulae which are to be used further on.
Macro- and microcracksin non-homogeneousmater&&
154
-
Ch. IV, w
L
)
37
%(*) Fig. 4.3. A crack with the normal symmetrical load on its surfaces.
If an arbitrary load fly(X) is applied on the crack surfaces (fig. 4.3) then the stress intensity factor can be written as [607]
K -- ~zv'~
L
fly(X)
--X
dx
(4.24)
for plane strain conditions, and
K -- 2(a:R) -1/2
fo
fly(r)
2 _ r2
xdr
(4.25)
for a penny-shaped crack with axisymmetrical loading of the crack surfaces. If a body loaded at infinity by homogeneous stress fly contains a crack loaded on its surface by tractions fly(X) = p(x) then we can write
x/1 _ ~2 d~
(4.26)
for plane strain conditions, and
K -2(L/lz) 1/2 foI(fly --P(~))~ d~ x/1 - ~
(4.27)
for a penny-shaped crack. Here ( = x/L. Goodier [201] showed that a crack under tensile load could be treated as a thin ellipse. Strictly speaking, such a treatment needs to be considered in the framework
Cracks in homogeneoussolids
Ch. IV, w
155
of non-linear theory of elasticity. Still we can use an approximate expression for the displacement of the crack surfaces: X ~
u(x) ~ ~
(1 - (x/L)
2) 1/2
(4.28)
An exact expression for the crack profile u(~) via the stress at infinity, a y and the traction, p(t), distributed on the crack surface is given by [619]
u(~) -- 4(1-v2)L f l fiE
l V/s 2 __ ~2
foS (a~176 Y -P(t)) tdtds V/S2 __ t 2
(4.29)
for a penny-shaped crack. Finally we note that the plane configuration just considered is called mode I. The transverse shear configuration is called mode II, and longitudinal shear crack belongs to mode III. The corresponding stress intensity factors are denoted as KI, KII, and KIII, respectively. If the stress intensity factors are obtained then the stressstrain state in the vicinity of the crack tip, where singular terms dominate, are expressed by simple formulas. For example, in the case of mode I, stress components at a point with polar coordinates (r, 0), originating at the crack tip, are ax - ~ c o s
1 - sin sin
,
ay-~rrCOS
l+sin
,
K
0.
0
sin--
(4.30)
30
"~xy = ~ r r COS ~ Sln ~ COS-~-.
Anisotropic case Equations (4.17) are replaced with the following relationships [361]: fill
--
2Re [p2(I)t(Zl)-+-q2q/t(Z2) ]
0"22 = 2Re [(I)'(Zl) q- tFt(z2)] ,
,
(4.31)
0"12 = - 2 I m [pO'(Zl) + qW(z2)] where functions (I) and W are complex potentials, p and q complex roots of a characteristic equation of the corresponding plane problem for an anisotropic elastic solid. For the case of crack (fig. 4.2) in an orthotropic plate, the application of the routine procedure yields [560]
Macro- and microcracks & non-homogeneous mater&&
156
K ~
pqK
- 2e2e~'
~,l
K-
2v%7'
0"x/~-l
Ch. IV, w
(4.32)
Note that the stress intensity factor determining 0"22 in the vicinity of the crack tip is the same as in the isotropic case, but that for 0"11 differs from the former by a factor pq. The energy release rate for a crack in an orthotropic plate is given by the following expression [610]: {allaZ2) 1/2
G-
\
2
J
a22
1/2 + 2a12 q-- a66
\a117
K2
2all
(A.~'~] \--
.-,.-,]
where aij are expressed via technical constants
all
--
l(
a12 =
1
1 -
vl2 ~ ~
El 1 + v23
)
,
:
1
-
v
3),
1 a66 = ~ .
/tl2
Here E1 > E2 and the crack coincides with one of the principal axes of material symmetry. 4.1.2. Fracture criteria
The formulation of a fracture criterion is either based on some kind of the hypothesis or being just a postulate. The boundary between two interpretations is not always clear. Linear elastic fracture mechanics
If linear elastic idealization of a real fracturing solid is assumed, the critical value of the stress intensity factor, Kc or K,, can be introduced [269]. When under monotonic loading the stress intensity factor reaches the critical value, K = K,,
(4.34)
the crack is assumed to start to advance. As a result of the drastic idealization, the value of Kc occurs to be not a unique characteristic of a material but a property to be dependent on the mode of loading (tension or shear), the stress state (for example, plane stress or plane strain), strain rate, etc. This makes experimental determination of Kc be a problem that should be solved each time under well defined conditions. On the other hand, the very problem of the evaluation of this parameter from the first principles seems to have no meaning.
Ch. IV, w
Cracks in homogeneous solids
157
Plastic deformation ahead of the crack front developing under the loading changes the material in the process zone and this makes the concept very questionable. So when one uses the concept one should be aware of the limitations and consider results as approximate. The error can be sometimes unpredictable. An alternative formulation of the fracture criterion is based on the relationships between the stress intensity factor, K, and energy release rate, G, eqs. (4.16), (4.22), and (4.23). Equating the critical value of G to double surface energy, G = 27
(4.35)
we rewrite in fact the fracture criterion in the energy terms. In the original Griffith's treatment, 7 is the surface energy in normal physical meaning. For non-elastic materials, Orowan [518] extended the Griffith criterion written by just replacing the meaning of 7 with the effective value that included all the energy dissipation accompanied by the crack propagation. In the case of metal materials, the main contribution to the dissipation is due to plastic deformation. If the deformation theory of plasticity, which is, actually, a non-linear elastic model, is assumed then deformation energy U(eij) is the potential and integral J =
Udy
--
O'ij
~
njds
(4.36)
occurs to be independent of the integration path (this integral was introduced in fracture mechanics in [81, 567], although the idea is going back to earlier Eshelby's works.) Calculating J-integral along a definite path helps to apply fracture criterion to a particular situation. Note that J - -G.
(4.37)
Thereupon, the J-integral approach is really another alternative formulation of the linear fracture mechanics concept. Plastic and cohesive zones
The stress singularity at the crack tip is obviously a mathematical abstraction that emerges as a result of linear elastic idealization of real behaviour of a material. Dugdale [140] and Leonov and Panasyuk [362] suggested models of a crack with plastic yielding in front of its tip. In fact, both models can be analyzed by applying constant stresses to the crack surfaces at the vicinity of its tip as shown in fig. 4.4 which is a particular case of a crack shown in fig. 4.3. With "pure" crack length 2/ and the length d - L - l of the crack surfaces near the crack tip being pulled together by a yield stress t~*, eq. (4.26) modified for the plane stress condition leads to
2 ( arcsin )
158
Ch. IV, w
Macro- and microcracks in non-homogeneous materials
-L
L X
21 Fig. 4.4. A crack with its ends pulled to each other by a constant stress equal to the yield stress.
If stress a ~ is applied at infinity it generates stress field c o r r e s p o n d i n g to the stress intensity factor K~ - tr~v/-~.
(4.39)
The singularity at x = + L will disappear, that is KId + K ~ = 0
(4.40)
- = cos
(4.41)
if
t
-b;-
At d << l, that is a condition for validity of linear fracture mechanics, we have
d _ 1 - cos
7-
(4.42)
E q u a t i o n (4.42) is valid for the plane strain condition as well, just the coefficient changes. Analyzing small plastic zone, it is useful to apply the J-integral, eq. (4.36). F o r example, if we choose the integration c o n t o u r as shown in fig. 4.5 then along the horizontal parts of it dy = 0, trijni = if*n2, n z d s = dx, hence J-
(u A - u~)o'*
(4.43)
Ch. IV, w
Cracks in homogeneous solids
159
O'*
> 37
Fig. 4.5. A contour around the Dugdale zone to calculate the J-integral.
where u A and uy are displacements at points A and B, respectively. Because of eq. (4.37) and due to the symmetry with respect to the x-axis we can write the crack opening as G 5 = 2--.
0"*
(4.44)
The crack begins to propagate, this means that the leading and trailing edges of the plastic zone start to advance simultaneously, when G reaches a critical value, G,. Equation (4.44) shows that crack opening displacement has also a critical value. In the above considerations, the idea of plasticity was involved in the interpretation of a constant value of stress cr*. Nearly the same technique can be used for the process zone of any kind the latter being expressed by a particular form of the dependence of pulling stress on displacement of a surface point, ap(U). The process zone in which the material changes, and strictly speaking, this makes the whole problem to be non-linear, can involve atomic cohesion as in Barenblatt's model or polymer crazing. There can be observed a variety of the physical processes involved in fracture of non-homogeneous materials, the crack bridging by intact fibres is perhaps the most important mechanism (see below, Section 4.4). It is important to note that independent of a particular nature of the cohesion forces, a part of the J-integral value associated with the cohesion stress will be for a critical state J, - 2 fo ll* ap(U) du
(4.45)
where u, is the displacement value at which the cohesion disappears. Bridged zones
In a fibrous composite, a part of the crack can be bridged by intact fibres. Cox and Marshall [107] summarized the behaviour of bridged cracks by introducing the properties of a material in the bridged zone of a length d (fig. 4.4) in a phenomenological manner, just prescribing a particular dependence of the bridging forces, p, on the displacement, u, of the crack surface from the middle plane of the crack.
160
Ch. IV, {}4.1
Macro- and microcracks in non-homogeneous materials
Let us drop the requirement to have no singularity at the cohesive zone tip expressed by eq. (4.40) which is the basic condition for the plastic zone configuration. Then the critical stress for crack growth, a,, is defined by 0", X / / ~ -- K ~ -I- K d .
(4.46)
To investigate stability of the initial crack growth with respect to the bridging length d - L - l, find the derivative da,
dd
1 K~ + Kd
---
2L
x/~
+
-
-
1 dK d
(4.47)
v / ~ dd"
For small d, assuming for a m o m e n t that l and d are independent variables, there should be
(
0Ka'~ ~ Od J! \ ~l ] l + d
(4.48)
Hence, eqs. (4.26) and (4.28) lead to +
dd
p(~)
.
(4.49)
Assuming now the dependence p(u) for u < u0 (fig. 4.6) as p -
-
P0
(=0)" u
=
~> 0
(4.50)
with p0 and u0 being constants, we rewrite eq. (4.47) as
29
(a)
P
Po
Uo
(b)
Po
U
Uo
U
Fig. 4.6. Possible constitutive relationships for a material in the bridged crack zone.
Ch. IV, w
Cracks in homogeneous solids
f t
do-, dd
K~
+
2po [2K~ V~] ~ ( ~ ) ~
[uoeo
}
161
(4.51)
It follows from the equation obtained: < 1 9
da, dd~C~
= 1 9
da,dd_ > 0 if _< am/2 and ~da, < 0 if l > am /2,
c~ > 1 9
for alll,
da, dd < 0 .
Here the characteristic length of the bridging zone is
l+a am---~uo
(4.52)
2~ uoE2
For ~ < 1, the extend of the bridging zone for the stable growth estimated from the extremum condition, da,/dd = 0, is
amdm~2(l+~176176
~m
(4.53)
which is valid provided K d << K ~. For example, for the case e = 1/2, which is to be discussed further in this chapter, eq. (4.53) yields dm cx l 4. The magnitude of a, for the value of d = dm at which crack starts to propagate at a decreasing load, is also estimated from eqs. (4.46), (4.51) and (4.53). For the case e = 1/2, this is O',max ~
0",
[ (,/31
(1) 1 + 8 ~mm
"
(4.54)
Thus the zone of initially stable growth depends on the initial crack length expressed in the characteristic length of the bridging zone, called as the non-catastrophic bridging length scale. One can see that for relatively small l/am, the value of a, rises only slightly above its value at the onset of the crack growth. If the dependence p(u) is assumed to be a decreasing function (fig. 4.6b), for example, linear that
p(u) = Po - 2u
(4.55)
then eq. (4.49) yields
8, (~oo0)2 I1 q- g~0]
dm~"
where a0 = K ~ v / - ~ is the ultimate stress in the absence of bridging.
(4.56)
Macro--and microcracks in non-homogeneous materials
162
Ch. IV, w
Sih's criterion As mentioned above a macrocrack propagates by forming voids and microcracks ahead of its tip. This is characteristic for cracks in metals considered to be homogeneous on the macroscopic level. So in any case the crack propagates through a material which is essentially different from the original one, no linearity which is a basis for Kc concept can be observed. A basic assumption of the S-criterion suggested by Sih [606, 608] is that fracture initiates from an interior element located at a finite distance r from the crack front. Detailing the idea yields the following working assumptions: 9 Crack initiation takes place in a direction of an extremum of the strain energy density factor, S, that is, OS/OO = O, OS/Oc~ = 0 at 0 = 00 and ~b = 4~0. Here r0, 00, and 4~0 are coordinates of a fracturing element in the local coordinate system with the origin at a point belonging to the crack front, 0 is located in the plane normal to the crack front. 9 Crack advances when the minimum strain energy factor, Smin, reaches a critical value, S*. 9 The radius of the process zone region, r, locating the points of initial fracture is assumed to be proportional to Smin. The strain energy density factor for an element (r, 0, 4~) near a crack of arbitrary shape under arbitrary external load can be expressed as
S - al,K 2 + 2alzKIKII h-az2K?l -+-a33K?lI
(4.57)
where
all
--
a22 -
(2 - 4v - cos 0)(1 + cos 0) 16p cos~b
,
4(1 - v)(1 - cos0) + ( 3 c o s 0 16/t cos ~b
a12 --
2 sin 0(cos 0 - 1 + 2v) 16p COS~b
1)(1 + cos0) ,
a33
'
1 4/~ cos ~b'
and # and v are the elastic constant. The stress energy density criterion allows an essential generalization [609, 611]. Take the strain energy density function, that is derivative of the energy with respect to volume, dW/dV, as a measure of the material state. We calculate this function according to dW S = -. dV r
(4.58)
Here S is the strain energy density factor, and r the radial distance measured from the location under investigation. In the case of a crack problem, it is the crack tip. The value of S can be found either by the elastic solution (eq. (4.57)) or by an appropriate elastic-plastic solution. In any case eq. (4.58) is valid. The total energy density d W / d V can be divided into two components:
Ch. IV, w
Cracks in homogeneous solids
dW = ( d W ) + ~ - ( d W ) dV ~ p c
163
(4.59)
(dW/dV)pis the portion of the energy density used in plastic deformation and (dW/dV)c the portion available to drive the crack whose critical value is
where
characteristic of the material and can change for each increment of crack growth. Figure 4.7 is a graphical interpretation of two components of the energy density referred to the uniaxial case. As a given state p moves and reaches f, (dW/dV) that is the total area under the stress/strain curve, becomes critical or (dW/dV)*. So the energy to drive the crack (dW/dV)c is distinguished from that dissipated by plastic deformation (dW/dV)p. Coming back to the basic assumption of S-criterion we state that crack initiation occurs when the portion of the volume energy density (dW/dV)c in a crack tip element reaches the critical value (dW/dV)*c;this is a condition of local failure. Crack grows in a step-by-step manner when the following relationships are fulfilled: dW)* c
S1 rl
$2 r2
S~ rj
S* r* - const
(4.60)
If material properties in a crack tip element, the stress/strain curve being an example, change for each step of the crack growth, then (dW/dV)s must also be adjusted for each j. A useful assumption is that the rate of change of S with respect to crack length is constant: AS = const Al
(4.61)
The general fracture criterion needs the accuracy in applying it to a particular case, and this is a kind of disadvantage as compared with simple fracture criteria
O-
/
P
d
Fig. 4.7. The graphical interpretation of the strain energy density.
Macro- and microcracks in non-homogeneous materials
164
Ch. IV, w
considered above. However, it has been applied in an appropriate way, the result will describe many fracture events such as changes in crack growth due to differences in specimen size and loading rates, peculiarities of fatigue crack growth, etc [611].
4.2. Energy dissipation The first approach to a description of the mechanical behaviour of composites as compared with homogeneous materials (Chapter 2) has shown that non-homogeneity as well as interfaces make fracture processes in composites much more diversified than those in traditional materials. Hence, before starting to discuss the behaviour of cracks in non-homogeneous solids we shall present a concise portraiture of some possible energy dissipation ways in composites.
4.2.1. Fibre pull-out Fibre pulling out of the matrix can be looked at as a special case of cracking in non-homogeneous solids. Because pullout is a result of the fibre/matrix interface fracture and slippage on the interface overcoming frictio'n, the process is accompanied by energy dissipating. If such a process takes place at the vicinity of the macrocrack tip the energy dissipated contributes to the effective surface energy [103]; therefore fibre pullout can be an important potential to enhance crack resistance of materials. Suppose a crack propagating in a matrix makes fibres to break not on its plane but somewhere outside due to both non-homogeneous fibre strength and the fibre stress concentration over some length around the crack plane. The length on which the fibre stress decreases from a maximum value at the crack surface, say a/vf, down to a regular value, a', at some distance from the crack surface depends on the shear stress on the fibre/matrix interface. The fibre break is followed by pulling-out the fibre over length l. The work done during this process is 1
W (1) -
f0 z(x)sxdx.
(4.62)
where v(x) is the shear stress at the interface and s is the perimeter of the interface. If the fibre has a circular cross-section and ~ is constant, then (4.63)
W (1) -- z n r l 2.
In the case of a discontinuous fibrous composite with unidirectional reinforcement, all the fibres with their ends within a distance l,/2 from the crack surface, are pulled out of the matrix. The portion of the fibres pulling out is l,/l, where I in the fibre length. Thus the total work done per unit area of body cross-section is [300] Vf g* 12
W-12-
r
for l < l,
(4.64)
Energy dissipation
Ch. IV, w
165
and Vf l, z* l 2
W--121--
r
for l > l,.
(4.65)
Continuous fibres (l >> l,) can be pulled out if they are characterized by strength scatter. In such a case the fibre stress which reaches a m a x i m u m value on the free surface of the matrix can be lower than the local strength of the fibre. The failure condition can be fulfilled at some point away from the crack plane. The situation was considered in [536, 635, 652], we will here follow Sutcu [635] in main detail. As usual, constant shear stress on the fibre/matrix interface is assumed, so the axial fibre stress as a function of the distance from the free matrix surface (fig. 4.8) is
a(x) - 2__f([_ x)
(4.66)
r
where f is the length of the fibre to decrease the fibre stress from a m a x i m u m value, a ( 0 ) - 2z rf to the background stress which is assumed to be zero. Note that f depends on the external load. The fraction of fibres, P(f), that have failed on both sides from the matrix crack is given by eq. (2.11) with the corresponding change in the integration limits, that is
P([)-
1-exp
~ o J - f \ ao }
fl+llo
~oo
"
(4.67) This means that prior to the present stress, which determines length f, has been reached, the fraction P([) of the fibres failed. To obtain the average fibre length
FIBRE
STRESS
FIBRE
Fig. 4.8. The axial fibre strength along the fibre length at the vicinity of the matrix crack which leaves the fibre intact.
166
Ch. IV, w
Macro- and microcracks in non-homogeneous materials
between the free matrix surface and a break point for the present stress state, we use the pointwise probability given by eq. (2.14), which yields
q(x) -- ~ (1- l) ~-1
(4.68)
per unit length of the fibre. Hence, the average fibre distance for the failure event within P and P + dP is l" (/'p) - f0 ~xq(x)dx - fl +------~.
(4.69)
Integrating the incremental pullout length over a probability interval [0, P] yields the cumulative average pullout length at a given probability P (4.70)
(/p(P)) -- j~0 P (/'p) dP After complete fracture, P = 1, the total average length will be (Lp) -- (lp(1)) --
/0'
(lPp}d P - A
/0
x~-~-r+,exp(-x) dx = AF
+
(4.71)
where F(-) is the gamma function and !
Here ~o r
(4.72)
lo
It is convenient to introduce the characteristic length ] as [536]
a~
(4.73)
and to rewrite eq. (4.71) in the form !
+i" l Note that at fl ~ cx~, 60 ~ l, where l, is the critical length defined above.
(4.74)
Ch. IV, w
Energy dissipation
167
The cumulative average pullout length, (/p(P)), can be expressed now in the normalized form"
(lp (P))
1 ~ ~,fl+1'
(L p )
{fl+2)] P I-" \_~_g
(4.75)
where numerator is the incomplete gamma function
~(t, ~) --
Xt-1 exp(--x) dx
~0~
and e-ln
1 ) 1-P "
(4.76)
A rough estimation of the pullout work will be
W - ~zr2V-J-f (Lp)2- vf(2(fl + 1))~-~F2(~ 0-k-_~) + _b~
(4.77)
Here a ~ - aoco ~+'.
(4.78)
Quantity a ~ can be called the characteristic fibre strength. Obviously, eq. (4.77) yields a sufficiently exact result only at large enough values of ft. Sutcu constructed the expression for the work of pullout as
W - ~7~r Vf2 fo ~ where z Q(z) -
~gr(z6) 2Q(z) dz
(4.79)
x / 6 ~ and Q(z) is obtained from eq. (4.71) rewritten as
f
~
dP
q(x, l)-d-[ d l
(4.80)
Making use of eqs. (4.67) and (4.68) and replacing the argument with dimensionless value z, we write finally eq. (4.79) in the form W - 2(fl)vfa~ ~
r
8+,
(4.81)
168
Macro- and microcracks in non-homogeneous materials
Ch. IV, {}4.2
where
2(/3) -
F {fl+3"~ \?-#-r] 2(2rc)~+L}(fl+ 2)(//+ 1)~+L} "
(4.82)
For the case of multiply matrix cracking eq. (4.81) is valid with 2 replaced with
{e+3~
2'(//) --
F\/~+l,]
12(2r0 -~r2-}"
(4.83)
Thouless and Evans [652] solved the same problem by using a slightly different statistical procedure. Adopting weakest-link statistics and assuming c in eqs. (2.2)(2.4) to be a power function, c - (cr/So)~, they did naturally arrive at two-parameter Weibull distribution for the probability that the flaws in a surface area of fibres, A0, have a strength less than ~r, in the form [ 2rtrZcr~+l ](2rcrfl(~_2vx/r)~-l/AoS~o). p(o-, x) - exp - v(~ + i)--~oSgJ
(4.84)
The mean pullout length for fibres that failed at a stress less than a = s is given by
fioxp(~,x)dxd~ A0 (So)/~ ( ; + 2 (s']/~+l) =41tr K 7 + I ' \ S , J o0 /f0 2 f0 P(~, x) dxd~
{lp(S)) -- fg 2
(4.85)
where S,
_
k,
+ 1)),/(t,+,l
~r 2
After complete fracture, the average pullout length will be (4.86) The mean strength of all fibres is
< >-2f0 f '-~P(~,x)dxd~- S*F(~+ 21)
(4.87)
Fibre cracking
Ch. IV, w
169
4.2.2. Interface &bonding In a brittle-matrix composite, the interface debonds before pullout occurs so that debonding can contribute to the energy dissipation. Kelly [300] discussed a possible value of the total debond energy basing on Outwater and Murphy's assumption on equating the energy of debonding, /4~, with the elastic energy, ~ , stored in the fibre as a result of debonding, that is Wdmax
=
g~r~r2 f - (22fr ~ ; ) ~drtr u~r* f r 9 -1--27 21: 32Ef ao of
2
2 ~r~ *l, = Wf d x - rcr 96 Ef O'f
(4.88)
where 7d is the specific debond energy. Comparing this value with work of pullout Wp of one fibre, which is given by eq. (4.63), yields Wd = 1 (o-~'~
Wp
(4.89)
3 ~Ef]
Obviously, the debond energy is negligible as compared with the pullout energy. Absolute values of 7d are at most 100 J/m 2 for polymer matrix composites [300] and 5 J/m 2 for ceramic matrix composites [161]. In Section 4.5.2 we shall continue the discussion of the interface debonding.
4.2.3. Plastic deformation Cooper and Kelly [99] observed plastic flow of metal matrix at the fibre break. The matrix deforms nearly as it follows from elastic-rigid-plastic analysis performed later [651]. So the semi-empirical expression for the energy dissipation in a composite due to matrix plastic flow derived by Cooper and Kelly, is
2~
Gpl-
Vm G ~ vf h--m
(4.90)
where G~ is the double effective surface energy of the matrix material, may be used, at least for the semi-quantitative models.
4.3. Fibre cracking The fibre cracking occurs ahead of the macrocrack tip under the loading as shown in fig. 4.9. The microcracking zone can change the stress state at the crack tip. The material in front of the crack also changes with the increasing external load. Thus, the crack propagation in a fibrous composite in such situations needs to be analyzed very carefully. This is a subject of this section. We shall see that the occurrence of microcracking zone either shield the crack tip or make the crack to be 'sucked' into the zone.
170
Macro- and rnicrocracks in non-homogeneous materials
Ch. IV, w
Fig. 4.9. Fibre breakage near crack surface in boron-aluminium composite. The original macrocrack was located on the right-hand side of the specimen. (Courtesy of V.M. Kiiko.)
We start with a simple plane problem which throws light at the physical situation, then proceed with results of a more complicated consideration based on a computer simulation approach. 4.3.1. A simple nlodel
Following [455, 457] let us consider a plane model of the non-homogeneous media containing alternating layers of two kinds, fig. 4.10. One layer carries regularly spaced defects such that a microcrack arises when the stress at any point of the defect reaches the ultimate value, a*. Let the microcrack length be d. This means that the microcrack cuts a layer of the first kind but does not penetrate neighbouring layers of the other kind. There can be various reasons for the prevention of such a penetration, for example, the local plastic deformation of one kind of the layers or the local interface delamination. For simplicity, the case of isotropic layers with equal elastic moduli will be considered. The defects are numbered by two indices (fig. 4.10), the first one shows a defect number along the layer and the second shows a layer number in the series of defected layers. Let the even layers bear only the even defects and vice versa. We assume also that all the defects are contained within a finite region which surely includes the region of microcracking. Each microcrack generates an elastic field of its own, and each microcrack is assumed to be influenced by the macrocrack field and the fields of only the direct neighbour rnicrocracks. So the stress at point (q, m) of a layer is determined as:
Fibre cracking
Ch. IV, w
171
J
i
(q,
/
Fig. 4.10. A plane model of a non-homogeneous solid.
0"~q'm) -- a(r, O) + Aa(q, m) + ao ,
(4.91)
where a0 is the applied stress at the infinity,
a(r, O ) - aox/7 ~(O)r
2424 :
-
v/m2b 2 + q2h20
arctan
(qh)
and Aa(q,m) the stress generated by the presence of microcracks at points (q + 1, m + 1). Index y on a here and further on is dropped. Introducing dimensionless values ~ - a / a * , g-d/b, l-lib we rewrite eq. (4.91), dropping the bar notation,
a(r, 0) -- a(q, m) --
0vq V/2~z(/+/t)
~b(q, m),
(4.92)
where
dp(q,m) - (l + ~)3/2(3~- 2) 2 v/-~, , - (1 + q2)1/2 q _ q h = q--)/vf, m ' mb+d d z=h/d. Vf-b+ d ,
(4.93)
The condition for defect (k, s) to be located within the region of microcracking is"
172
Macro- and microcracks in non-homogeneous materials
Ch. IV, w
(4.94) The equality in eq. (4.94) means the defect lies at the boundary of the microcracking region. The value of a(k, s) is given by eq. (4.92), and this is slightly higher than should be within the microcracking region and lower outside this region because it does not take into account a decrease in the elastic moduli of the media due to microcracking. The value of ~r0 should be included in eq. (4.94) because the microcracking zone may be large enough. For an arbitrary defect, the value of Aa is given by sum Aa(q, m) - tcijAo'q+i,m+j(q, m) where i , j = + 1, ~cij - 1 if point (q + i, m + j) lies within the microcracking zone, and ~cij - 0 if the point lies outside this zone. The value of Aoq+i,m+j (q, m) can be obtained by taking into consideration the non-homogeneous stress field of the macrocrack. But it will lead to complications, so we will assume these fields to be locally homogeneous. The mutual influence of only the direct neighbour microcracks, which has been already assumed, helps in the assumption mentioned above. Usually, d <_ l and it also supports the assumption. So"
Aaq+i,m+j(q,m) - cI~2~zV/1V/ 1 -~p !f k ( q
+ i, m + j)qb(i,j)
where i,j - 4- 1. If a point lies within the microcracking region, then eq. (4.94) gives a series of inequalities 0 t (k, s) _> 1,
(4.95)
and each one is determined by its own set of values. This is because the position of the boundary of the cracking zone is not known. Determining the boundary demands the consideration of all the possible sets of tci~ - 0, Gst - I. The calculation of variants with increasing value of a0 [457] brings a picture of microcracking in front of the macrocrack. Two examples are shown in fig. 4.11. Note that the assumption of the mutual influence of neighbouring microcracks yields the essential enlargement of the region of cracking comparatively with the region obtained in the case Aa = 0 (no mutual microcracks influence at all). The calculation also gives the dependence of the largest number n of cracks along one "fibre" upon the value of c~0. These dependencies are presented in [457]. Now we can find a limit size of the microcracking region, namely such a size when the region stops to grow up in y-direction and goes on to grow in x-direction only.
Fibre cracking
Ch. IV, w
m
m
173
m
i m
m
m
,--
m m
mm
mu
m
m m
m
U
m
mm
m m m
m m
m
m m
m
m
m
m
m
I mmm m -. mmm mm
mmm m mmmmm
nmm
mm
mm
m m m m
m
m m m m
co
=
m
mmm m n u
m m m
0.20
m
mmmmm
m
mm Um mmm
co=
m m
m m m
m m m
m m m
m m
m m m
u
0.25
Fig. 4.11. Microcracking in the model composite ahead of the crack (~ - 6, l = 250, vf = 0.1). The thick bars are obtained neglecting mutual interaction of microcracks.
This means the start of macrocrack propagation or K reaching the value of K*. Assuming eq. (4.35) is valid we write the energy release rate as G - GO+ AG(a0)
(4.96)
Here, Go is determined by linear summation of the values of G of the composite components, and AG(a0) is determined by the microcrack's density and depends on applied stress o0. Note that AG is analogous to Orowan's plastic correction to the surface energy by Griffith. It should be likewise noted that, generally, the value of AG can be effected by other processes of local failure (delamination, pull out and so on).
Therefore, if AG(a0) can be obtained, then the value of K* will be determined from the solution, with respect to a of the equation:
(G(ao)C) '/2- 2o--x/1
(4.97)
where l is the crack length, 2 a factor depending on the body shape and C an appropriate combination of the elastic characteristics of the composite. A diagram in fig. 4.12 illustrates the situation. An obvious complication of the situation due to the dependence of the stress intensity factor K on microcracking is also traced. A possible method of solution to this problem is to prescribe some value 7 of dissipated energy to each microcrack. There is no strict way to apply this procedure. For example, the strain distribution in the cracking region (without taking into account the cracking itself)can be found, and then assume: 7 --
~y
-~y~/m,
(4.98)
where 7m is the effective surface energy of the matrix, and ey0 the largest matrix strain in the cracking region at the critical condition ( K - K*). Such a procedure is therefore iterative in nature.
Macro- and microcracks in non-homogeneous materials
174
Ch. IV, w
K 7 /
'
"1" ~"
o-o o-o
Fig. 4.12. The graphical illustration of the solution of eq. (4.97). Curve K(ao) can deviate from a linear dependence because of microcracking in front of the crack and a corresponding decrease in the effective elastic moduli of the material within the process zone.
Additional rough estimations are obtained in [457], where 7 is taken in form: 7 = 7m( 1 + vn)
(4.99)
where v _< 1 is a constant. energy dissipation takes propagating macrocrack, intensity given by v. Then (Gin(1 + vn(ao))) I/2=
Equation (4.99) means that ~'r = 0, and a fully plastic place at microcracks closest to the surface of the while the other cracks dissipate the energy with the eq. (4.97) is reduced to:
ao x/~ El~2
(4.100)
where E is Young's modulus of the isotropic composite. This equation gives the values of K* and a~), fig. 4.12.
4.3.2. Computer simulation The simple model considered above was analyzed more carefully in [629]. For the purpose of stress analysis, the solid is assumed to be homogeneous and anisotropic, and microcracks at the vicinity of the crack tip may appear at definite points regularly distributed in the solid as shown in fig. 4.10. The criterion for a microcrack formation in the above model is the normal stress at a corresponding point reaches a critical value. The microcrack being formed causes the energy dissipation through plastic deformation of the matrix near the microcrack or fibre pullout or some other way (see Section 4.2.1). Unlike this, in a computer simulation procedure performed
Ch. IV, w
Fibre cracking
175
in [629], the criterion of transformation of a defect into a microcrack is written in the form
F(ax,
fly, O'xy) - -
(4.101)
)'fd -I
where 7f is the energy dissipation prescribed to a fibre break, function F depends also on combinations of effective elastic moduli of the composite. The criterion for the macrocrack to start to grow is K~ - Ki~, and the value of K~ is calculated taking into account the cracks interaction, the value of Ki~ is determined by the material structure. In the stress analysis of the system, a combination of elastic fields of N microcracks is considered. The normal, Pk, and shear, qk, components of the surface load are applied to the surfaces of k-microcrack. The macrocrack is endured to the external load as well as to the stress field, ay, developed by all microcracks. Unknown value pk, qk are the sums of loads occurred in the case of an isolated microcrack in a homogeneous elastic field and those contributed from the macrocrack and the existing microcracks. These values are expressed as the power series with respect to the x-coordinate. Because usually h _> 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
Macro- and microcracks in non-homogeneous materials
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
Macro- and microcracks in non-homogeneous materials
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)
Macro- and microcracks in non-homogeneous materials
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
Macro- and microcracks in non-homogeneous materials
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<
E~
f ..................................................... ! l
ol
: l i
/
E~
< Ei
< EI
/ /
Es < Ern
1
I
o
1
l
_.2.
(t/,O
I
9i 0 - z
Fig. 4.24. Critical value of the Gd/Gp ratio for crack impinging on coating/fibre interface (crack 2 in fig. 4.22c) versus normalized thickness of the coating. Schematic representation of the data by Popejoy and Dharani [544] for the model characteristics shown in fig. 4.23.
Interface cracking
Ch. IV, {}4.5
197
.-.:,, 3
1
I
o
I
2
to
Fig. 4.25. Critical Gd/Gf ratio versus normalized initial debond length (fig. 4.22d). Schematic representation of the data calculated by Popejoy and Dharani [544].
problem via an integral equation which is solved numerically. So the plastic correction to eq. (4.30), o.d, is calculated to write down the stress components in the matrix in vicinity of the crack as gI
- 2eze
..
(0) +
d(r, 0).
(4.153)
Obviously, the correction obtained lowers the normal stresses in the matrix in a close vicinity of the crack tip relative to that given by the elastic solution and, correspondingly, enhances those stresses away of the tip. Under the assumption of a particular slip plane configuration mentioned above, the transition "lowers/ enhances" takes place at about 1/10 of the semi-height of the plastic zone. Hence, for the small-scale yielding and sufficiently low fibre volume fractions, the stress in the fibre ahead of the crack tip can be estimated assuming the plastic correction in eq. (4.153) is negligible. For the crack with its tip at the interface, the stress in the center of the fibre is o.i=
KI
(4.154)
where df is the fibre diameter. If the von Mises criterion is assumed, then one can write "Crn ~_= 0-'
0-m v ~ KI
(4.155)
Macro- and microcracks & non-homogeneous mater&&
198
Ch. IV, w
where a m is the yield stress of the matrix. Equating a' to the effective fibre strength, 4 say a~, eq. (4.155) yields the boundary line between fibre fracture and matrix yielding, that is Tm-- V ~ a * v ~
(4.156)
Assuming the interface shear stress is equal to Tm (see fig. 4.26a), the condition for fibre fracture and matrix yielding to occur prior to interface debonding is obtained, namely ~--~-~> Trn 0"f
(4.157)
fff
where T~ is the ultimate stress for interface debonding. Within the same approach as for the problem depicted in fig. 4.26a, Chan [73] solved the problem of interface debonding or occurring a doubly deflected crack tip (fig. 4.26b). He showed that the stress ahead of the debonded crack could be expressed as
Cry- g(/)KI
(4 158)
v7
for y = 0. Here l is the debond length, and g(.) a non-linear function calculated numerically. This yields the criterion for fibre fracture to occur rather than debonding as *
1
T! > ~ . (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
Macro- and microcracks in non-homogeneous materials
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
Macro- and microcracks in non-homogeneous materials
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,
K,
(4.178)
where Km and K~ are the critical stress intensity factors for the matrix and inclusion, respectively. The crack will propagate along the tangential line into the matrix if K, _>Kin,
K,
(4.179)
which is valid for quasistatic crack propagation only.
4.6. Cracks in dually non-homogeneous solids Introducing two or more levels of non-homogeneity can be considered as an old technical idea since Nature has been normally designing leaving creatures in this very way. So we consider here just two cases of dually non-homogeneous solids to illustrate possibilities provided by such types of structures. First, we will analyze the crack propagation in a solid with elastic modulus changing along a coordinate axis periodically and show that the crack can propagate in such a composite in a nonsteady fashion. Secondly, we consider the cracking of a solid containing inclusions of two kinds. It will be shown that the dispersion of either the interface energy or the size of an inclusion can yield an enhancement of the effective energy release rate of the composite.
4.6.1. Periodically changing elastic modulus A quasistatic formulation of the problem for a longitudinal shear crack and its solution is presented in [665], that for crack mode I is in [459], the concise discussion of the problem as a whole is presented in [460]. A relatively simple dynamic problem for anti-plane shear crack growth in a periodically non-homogeneous solid was solved in [666].
A quasistatic approach Let shear modulus /~(X l ) be dependent on a coordinate axis and change along a crack propagation direction. Average shear stresses Iz23) are applied at infinity. Because of the symmetry, the problem should be solved for x2 > 0 only. Performing now the Fourier transformation, with respect to the xl-coordinate, to b o t h equations of equilibrium ~Z31 ~- ~ 3 2 -- 0
and the Hook's law
(4.180)
Ch. IV, {}4.6
Cracks in dually non-homogeneous solids
OU3 "/5"31 -- ~ ( X l ) ~ x l ,
OU3 "/732 -- ]-/(Xl) ~X---7'
205
(4 181)
we obtain integral-differential equation ~2 -- Y/ T'(q,X2) --
1 2re
a ( r / - 2)'c(2, x2)d2
(4.182)
where (Xl) ei2Xdx a (/],) -- - i [or ec /2(Xl )
j_
and function T'(g/,X2) --~(r/,X2)
,
in which the bar denotes Fourier transform, to be searched for. The boundary conditions are
o) -
~'(q, x2) Ix2--+oo--0.
The solution is taken in the form OO
(4.183)
g(g/,X2) -- l:O(t/,X2) --1--Z "t'k(rl'X2) k=l
where "CO(rl,X2) -- ~ O ( t / ) e x p ( -
I ~ I x2)
is the solution to the problem for a homogeneous solid with the same boundary conditions, functions "Ck(q,x2), k _> 1, satisfy the ordinary differential equation d2
- r/] "Ck(r/,x2) --
if
2It
a(r/-- ~)'Ck-1(2,x2)d2
(4.184)
(X3
with the known right-hand value and zero boundary conditions
r
O)-
k('7,x2)lx2
-o.
Now return to a line crack located at X 2 - 0 and - l _< Xl ~ l in a nonhomogeneous elastic body of infinite extent. If the crack is absent, the anti-plane shear stresses are
206
Ch. IV, w
Macro- and microcracks in non-homogeneous materials
#
q702 - (-~ (~).
~'01 - - 0 ,
(4.185)
Subtracting the above solution from the general problem just considered, it suffices to solve the problem of traction applied to the crack surfaces with vanishing stresses at infinity and consider only the upper half plane because of symmetry. The boundary conditions for the latter are
u3(xl,0)
]Xl I~ l,
]2 (~'32) (]2)
"~32(Xl, 0) --
Ou3 ~31 ~ ~xl ~
0,
0
[xl [> I.
(4.186)
Equation (4.186) presents the mixed boundary conditions that are to be enforced on the governing eqs. (4.180) and (4.181). Let the unknown function v0(r/) be expanded in series OG
9o(~1) - To(q)+ Z
k=l
Tk(r/)
(4.187)
in which To(q) is the corresponding Fourier transform of the stresses 1732(Xl, 0) for the crack problem of a homogeneous body. Satisfaction of eq. (4.186) yields a system of recursive dual integral equations [665] for Tk(r/) expressed through Tk-1(r/). Once Tk(r/) are known, the stress intensity factor KIII follows Kill(l)
--
lim ~
r/---,~ k=0
v/~
sin(q/
I
(4.188)
~/4)"
An analysis of the corresponding problem of in-plane extension can be done in a similar way. Here we just present the final results, that is a set of approximate expressions for the stress intensity factors for cracks in non-homogeneous bodies. Suppose a crack occupies the line [Xl [< l at X2 = 0 in a solid with the Young's modulus changing periodically according to (4.189)
g ( x l ) -- g 0 exp(~c cos(~xl ))
where • determines the amplitude of the function and 2n/~ is the period of the function. Let the average stresses at infinity be ( a 2 ) = a ~, ( a l ) = (vl2)= 0. The crack surfaces are free of tractions. Then the reduced boundary conditions at x2 = 0, which are similar to those expressed by eq. (4.186), are ~12(xl,0) = 0 ,
a2(xl,0)-- --(a~/Io(K))exp(Kcos(~xl)),
"~12(Xl, 0) -- 0,
UZ(X1,0) -- 0,
0U2(Xl, 0) aX 1 -- 0,
I Xl l> l.
Ix1 l< l,
Ch. IV, w
Cracks in dually non-homogeneous solids
207
Here I0(x) is a modified Bessel's function. The solution to the plane problem just formulated yields the stress intensity factor [459]
2 ~k=l fo~X3/2fk(x) dx V/1 -- X2
g i -- K 0
v/~
where the functions equations, and
K~
fk(x)
(4.190)
are the solution of the corresponding dual integral
2a~~foteXp(xc~
Io(tC) v/12 - x 2
(4.191)
the stress intensity factor for the crack in a homogeneous body and the boundary conditions given by eq. (4.190). Fracture criterion, eq. (4.34) with eq. (4.16), must be applied locally, so that the crack starts to propagate when
KI -- K, -- v/GE(xl)/(1 -
(4.192)
v:).
Here G does not depend on the coordinate. The following approximate formulas of stress intensity factor may be written [666] for crack length much smaller than the characteristic dimension inhomogeneity (~l << 1) KI ~
{
E(l)
v
(0"2) ~ 5
[ E ( / ) 1 ] (0.1)} X ~ '
1 -- V [(E) --
gii ('t12)~ ,
(4.193)
gii I -- (-c32)-~,(t) ~ 47i. If the crack length is much larger than the characteristic dimension of inhomogeneity, then
(E),/gi gII
l
E(x)
+ x -x
dx,
~ (~12)V~,
Km ~ ~
(~3~) f '
(4.194) U(x)
~/~t +x
dx.
Using eq. (4.192)-(4.194) yields asymptotic formulas for the critical stress a* for a crack of a length l in a composite with the Young's modulus given by eq. (4.189).
Macro- and microcracks in non-homogeneous materials
208
Ch. IV, w
For small cracks: (4.195)
1(1 - v2) e x p ( K ) '
and for long cracks:
tr*
~/(E}Io(~c)G V ~l exp(•
k
cos(•/))
k =0
n =0
( k - 1n)!n' Jo(~l(k-
2n)
(4.196)
"
where J0 is the Bessel's function. Some results of the calculation of the critical stress on the crack length can be found in [459]. One can obtain spatial modulation of the material stiffness presented by eq. (4.189) by distributing fibres in a composite, for example, in a fashion illustrated schematically in fig. 4.31. Then using eqs. (4.191) and (4.192) yields dependence of the critical stress on the crack length shown schematically in fig. 4.32. If the initial crack length, 10, is smaller than lb the crack is of a Griffith's type, this means that when the applied stress reaches a critical value the crack becomes unstable and propagates. If lc < 10 < ld then it is necessary to increase the applied load to make the crack to propagate, so in the crack length interval (/c, ld) the crack propagates in a stable fashion until the stress reaches the critical value ~ . Then the crack propagation is unbounded. If lb < 10 < lc, the crack tip, being unstable, jumps to the corresponding point of portion cd of the curve which represents a stable growth
I]ltl
'"'i
11111
III1H .....
11111
3., Fig. 4.31. Schematic of a non-homogeneous composite with periodic variation in the modulus of elasticity due to a corresponding fibre distribution.
Ch. IV, {}4.6
Cracks in dually non-homogeneous solids
209
__o ....
f
Fig. 4.32. Critical stress versus crack length in a non-homogeneous material. Schematic representation after Mileiko and Tvardovsky [459].
of the crack. The events within portion d e f of the curve are the same as those within portion abcd. Therefore, the ultimate stress is presented by the line abdefgh . . . . Hence, for the crack in a doubly nonhomogeneous material (as contrasted from Griflith's crack) there exists the possibility of stable growth and this yields the critical-stress/crack-length curve that, on average, goes above the Griffith's curve for a material with the same mean characteristics. An effect of the fluctuation of the specific surface energy, 7 ( x l ) = 7o exp(ecos(~x)), with a coordinate upon the crack propagation was analyzed in [666] by considering longitudinal shear crack in material with both 7 and # changing with the same period. It was shown that values of the ultimate shear stress are affected more by changing in ~c(associated with shear modulus) than by changing in (associated with specific surface energy).
A dynamic approach Strictly speaking, the analysis of crack growth that includes both unstable and stable regimes, as for cracks with initial length lb < 10 < lc in fig. 4,32, needs to be performed in the framework of a dynamic approach. In Ref [666], non-homogeneity of the solid is assumed to be reflected by the specific surface energy in the form '~(X) = ~0 "~ '~1 s i n ( 2 n x / T ) .
(4.197)
The problem is formulated for a pair of equal and opposite concentrated forces P0 applied to the crack surfaces and moving at the same velocity v0. There are results
Macro- and microcracks& non-homogeneousmaterials
210
Ch. IV, w
used by Kostrov [333] giving the relationships between the stress intensity factor and crack tip velocity, i, normalized by shear wave speed c2 - v@/Pwhere p is the mass density, for a current configuration of the load/crack system as well as the modified Kostrov's relationships replacing that by Irwin, eq. (4.16), namely
KIII -- 1 4 ] A ] ; ( x ) v / l -
(]/r
2.
(4.198)
If a distance ~0 between the load application point and the crack tip is much larger than T, then the following solution of the corresponding differential equation is written
1 -- A~2(x)
(4.199)
i(t) - c2 1 + A~2(x) where
471;2//2 2 2
A=-~o
?o~o, ~ ( x ) -
'},'(X)
7--~'
(4.200)
and the average normalized velocity is vo -- =
C2
T 1+A~2(x) 9 f : l_d~2(x)dx
(4.201)
The calculated dependencies of the normalized average velocity, Vo/C2, o n degree of inhomogeneity, 71/70, are plotted in fig. 4.33 for various values of A. One can see that the oscillation of the crack tip velocity in accordance with oscillation of the specific surface energy lowers the average crack velocity as compared with that in a homogeneous material with the same mean specific surface energy. Let us introduce the effective value, 7eff, of the specific surface energy of nonhomogeneous material to characterize the enhanced crack resistance of such a material with respect to that of the material with the same mean value of 7- The stress intensity factor for the crack under consideration is [333]
KIII--- ~ p o i l
- (v0/c2)2~0
(4.202)
If 7(x) and i in eq. (4.198) are now replaced, respectively, by 7eft and v0 and the result equated with that in eq. (4.202), we obtain
eff= ,/1 --( 01C2) V A
?o
(4.2O3)
Cracks in dually non-homogeneous solids
Ch. IV, w
1.0
o"
0.8
'
I
'
I
'
I
'
I
-
211
'
A=0.1
0.6 0.4 0.2
-
0.0 0.0
05
0.3
!
0.2
0.4
0.6
0.8
7'1/7o
1.0
Fig. 4.33. Normalized average crack tip velocity versus degree of inhomogeneity for various value of A.
Comparing the result obtained with the plot shown in fig. 4.33 we see that )~eff/70 > 1.
4.6.2. Non-homogeneous interfaces Consider now a composite in which interface debonding is the only way to enhance fracture toughness of the original matrix. A pure example of such materials can be a ceramic filled with ceramic particles. For fibrous composites, a contribution of the interface debonding is less than that of other mechanisms. Yet, it is interesting to note some possibilities to control energy dissipation due to debonding via a structure of the reinforcement system. Let a model composite be under longitudinal shear (fig. 4.34). The effective interface energy, 7fro, is a random variable. The distribution density function for 7 is O(co). If the total fibre volume fraction is vf and V~ is the volume fraction of fibres which may be delaminated, then the volume fraction of fibres delaminated at the applied stress r is "~ Vfd(T) -- rOd
O(co)dco.
(4.204)
The complete delamination of each fibre can be regarded as the creation of a cylindrical hole. Hence, if the initial effective shear modulus is ~0 ~ ~/fVf-~ ~/mVm
Macro- and microcracks in non-homogeneous materials
212
Ch. IV, w
0 0 0 0 ~
0000 OO %0 Fig. 4.34. A composite model.
then at the applied stress T the current modulus is
~(r
- ~o + ( ~
- l,o)~fd(r ~~d
(4.205)
with the final value of the modulus
I L l - flf(Vf -- POld) Jr-~mVm . The delamination criterion of a single fibre can be obtained by comparing the change in elastic energy of the system due to the bounded fibre, 00 = 0 occurring in eq. (4.169), that is AUo -
gR 2
/l m - / i f ,~2 /Am (/Am -~/If)
(4.206)
with that due to complete delamination, 00 = ~z in eq. (4.169), that is ,c2
AU0 - ~zR2 ~
tim
,
(4.207)
and equating the difference of energy in two states to 2~zRTfm. This yields 272 -- /Am ('tim -+- ~f) 7fm
#f
(4.208)
R
We can write the delamination criterion obtained for sufficiently dilute reinforcement as
Ch. IV, w
Cracks in dually non-homogeneous solids
172 __ //(1:)(//m + ]gf)(_O /~f R
213 (4.209)
to use it together with the dependence of the current shear modulus of the solid with delaminating inclusions, given by eq. (4.205). Equations (4.205) and (4.209) give the stress/strain curve for the material with delaminating inclusions. Let the length of a macrocrack in the composite be 2l (fig. 4.35). At infinity, uniform stresses 17/713- - 0 , 1723 --17 are applied. Fibre/matrix delamination can occur near the crack tips; this reduces the effective modulus. Assume that l >> R,
1 ) ) A 1/2,
7 << ~00
(4.210)
where A is the area of the delamination zone, 7o the effective surface energy of a composite without delamination, and COo the threshold value of co. The configuration described by the first two inequalities of eq. (4.210) means, in particular, that a large number of fibres exists in the delamination zones so that the material behaviour may be presented in terms of the effective properties. The third inequality implies that
17,~
o
where ~,0 is the critical stress for the crack in a composite with no delamination, that is 170
=
~/2/t070
(4.211)
and 170 is the stress at which delamination first occurs, that is
//Z0(]'tm +/gf)
v
(4.212)
Making use of the solution for the elastic-plastic problem of a solid with strain hardening [566], the radius of a quasiplastic zone or the zone of delamination as shown in fig. 4.35 is found:
Fig. 4.35. Delamination zones around the crack tips.
214
Macro- and microcracks in non-homogeneous materials
p -~ll
.
Ch. IV, w
(4.213)
The centers of these circular zones are located at (/1,0) such that ll-I
1-e
(4.214)
where dx
1~o
/~o x2~(x)
1
2
The stress/strain state in an elastic region can be found via the function w' (z) (see eqs. (4.5) and (4.6)). The boundary conditions at infinity and at the boundary of quasiplastic zone suggest that i'cz
w'(z) = -
(4.215)
V/Z which is analogous to eq. (4.11). Since we neglect energy dissipation due to friction, unloading in ( r - 7) domain5 will take place along a straight line towards the origin as shown in fig. 4.14c; the behaviour will be quasielastic. With the change in elastic energy due to the cut given by Woo = rt/2r2/2#0, the total change in elastic energy including the two quasiplastic zones is given by
W0-
l 'r2
9
(4.216)
The zones of delamination are much smaller than the crack length so that no interaction occurs. In other words, the radius of the delamination zone is assumed to be independent of the crack length and it can be regarded as a material parameter. Assuming that the size of the delamination zone is constant during crack growth, the change in elastic energy due to unloading of the material behind the crack front can be calculated and it can be shown that this effect does not essentially contribute to the total energy balance. So it will be neglected. Let p* be the critical radius of the quasiplastic zone referred to fibre delamination with the average interface energy (oo). Thus, the energy dissipated due to creation of the quasiplastic zone is GA. A* where A* - ripe, and GA -- 2V0d(o9)/R. Shear strain along the line corresponding to p = p* is given by
5Here and further on, y with neither super- nor subscript means average strain.
Cracks in dually non-homogeneous solids
Ch. IV, {}4.6
1
(~)2 __ ~m + ~f ((O) /if g p((o9)) "
215
(4.217)
Hence, eq. (4.213) yields
,/2__p__*
(4.218)
z , = ZOv tell with K
"CO
"CO
The Griffith's energy balance may be written in the form"
6 W - 27061 + 2GA6A.
(4.219)
Making use of eqs. (4.214), critical values of the radius of delamination zone, the effective surface energy of the composite and the ultimate stress, are finally obtained as
4_~m)--+m~4eR'~-1
(11 m -Jr-~f p*=~cR k ~ -~
~'o --if/
7,=7o(1-R(/zm4v0m/zf+/ (1/f) Vfmo17~ '
(4.220)
2#0m0 11m + / I f p*
"c, -- V
~---K-I
].if
R
The above results are valid for a broad class of function O(~o) provided that the delamination zone is small. Note that the distribution function determines the function z(7). To illustrate the dependence of fracture characteristic of the material on distribution function of the interface energy and. to show the importance, in this respect, of the second and third moments of the distribution, the curves ~(l) are calculated [458] for the distribution function O(m) shown in fig. 4.36. Here ~ - z,/z0, [ - / ( 2 # 0 7 0 / m 2 ) -1. The distribution function is related to the stress/strain curve (fig. 4.37): q,(o~) = o
0 < 03 < gOlO
--g-
gOll < f_,O<' 0)20
0,)10 < 03 < 0311 ,
03 > 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
Macro- and microcracks in non-homogeneous materials
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 <_ d _< 2r*, c a n be r e w r i t t e n as
a- -- a*[l+c~P+qv/d/2r*1-1 Pq where ~ d e p e n d s o n d/w; ~ < 1
(4.230)
at d/w>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
Macro- and microcracks in non-homogeneous materials
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
Macro- and microcracks in non-homogeneous materials
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)
Strength and fracture toughness
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
<0"~> : O~
(5.8)
where (a~(l*)) is the mean fibre strength at a length 1', 1" the average distance between fibre breaks, a m the matrix strength and ~ a constant.
236
Strength and fracture toughness
Ch. V, w
At large values of the fibre volume fraction fibre/fibre interaction should not be neglected. This results in the need to take account of the non-homogeneity of fibre packing. Suppose that the non-homogeneity may be depicted as a combination of fibre bundles. Let q~n be the fraction of fibres combined in a bundle of n fibres. Such a bundle has the characteristic size nd, d being the fibre diameter, n - 1 , 2 , . . . , k, and k ~-]~n=l (On -- 1. A crack of length e _> 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
--
(5.9)
where L is the total fibre length in the composite and//n a constant. A crack arising in this way is stable in the composite if < ~r~n)
(5.10)
where critical stress 0"~n) for the composite containing a characteristic microcrack cutting n fibres, is
(5.11)
_
Here C can be taken for an orthotropic material according to eq. (4.33) G the energy release rate of the composite, and df is the fibre diameter. If to express the critical energy release rate of the matrix material, G~ as G~ - O'ms
m
(5.12)
where hm is a characteristic size, then, using eq. (4.90) we can write G--{
~G~ f ) (1 v'~-dfvf 2-Ghm -~
atat V vf ~_> <, vv~. f
(5.13)
Here vf ---
1 q- h m / d f
Let us consider graphically the case of two bundles, with n equal to either p or p + q. So (.tip --]-q)p+q -- 1. Let ~0p > q)p+q. At vf < vA (fig. 5.2) the strength of a composite does obviously follow the line OA (eq. (5.2)).
Ch. V, { } 5 . 2
237
Brittle-fibre~ductile-matrix. strength
<(Y g>
OA
C
<(7 9 p+q> < O- P+q ep >
>
I I I
B
D
vf Fig. 5.2. Schematic presentation of the composite strength versus fibre volume fraction dependence.
At v~ < vf < vfB microcracks of characteristic sizes pd and (p + q)d appear when the load path on the ~r- vf-plane crosses the (%) and (o-;+q) lines, respectively. Cracks cutting (p + q) bundles become unstable when the stress reaches the line AB. This line corresponds to the composite strength. At v~ < vf < vfc, when the stress is to reach a value corresponding to the line BC and the first microcrack of length (p + q)d appears, the stress is already higher than the critical value for such cracks (the line AF has already been crossed by the load path). So the composite strength is represented by the line BC. At vc < vf < v~ the previous statement is true for cracks of length pd. At vf > vp, any characteristic microcracks to appear immediately become unstable. Therefore, the gross dependence of the average composite strength on the fibre volume fraction follows the line OABCDE and can be calculated provided the parameters involved in eqs. (5.8), (5.9) and (5.11) are known. It is easy to show that when (pp < (pp+q this dependence corresponds to the line
OABFDE. 5.2.2. Experiments To couple experimental data and the model is important. This is necessary to support the model as well as to reveal structural feature of composite that effect parameters to be used in the micromechanical model. Before doing the comparison we should make clear a semi-qualitative nature of the model. Actually this means that to do calculations we need to obtain the necessary quantities, like (cry) and G~ from a direct experiment on composites using the same fibre or matrix. This is not only because it is difficult to measure the values
Strength and fracture toughness
238
Ch. V, w
mentioned in an experiment on constituents, but what is more important, the properties of the fibre and matrix can differ from those being measured separately. Therefore, the models like that under consideration, aim at the predictions of the strength of particular composites using the values obtained in the experiments with composites containing the necessary constituent. This determines the rules for comparing the model and experimental data. In particular, if we are dealing with the only couple of the fibre and matrix we should be looking at characteristic points of the strength/fibre-volume-fraction dependence, such as points of maximum and minimum, how they are moving along the vf-axis with the fibre and matrix properties changing. A systematic study of the model is performed testing boron/aluminium composites. In such composites, the combining of fibres into bundles can be a result of the occurrence of relatively brittle zones around fibres due to formation of borides on the base of elements, such as magnesium soluted in the aluminium matrix. Boron, with a very low solubility in aluminium, diffuses into the matrix from the fibre surface and finds magnesium to form small magnesium boride particles that strengthen the matrix around the fibre. The size of hardened zone can be comparable with the fibre diameter (see Section 11.6.1). This means that the bundle structure of composites can change during the stage of fabrication or heattreatment, the latter can be either a predestinated procedure or a result of the service at sufficiently high temperature. Now let us trace a change in the composite strength with the bundle structure changing. Suppose we start with the structure characterized by no fibres in the bundle and consider fig. 5.3. Curves corresponding to eqs. (5.8) and (5.9) are labelled by (a*) and (a~~ The strength versus fibre volume fraction dependence no
<(Y no > o
B
b
<(Y n l >
< (YGO
,Of(l) ,Of(2)
< (YGO
no nt
> >
Vf Fig. 5.3. Schematic representation of a sharp decrease in composite strength when the structure with no bundles converts into that containing n~ bundles.
Ch. V, w
Brittle-fibre~ductile-matrix." strength
239
for such a composite follows the line ABDF. Suppose now that at some time during the heat treatment the number of fibres in a bundle becomes equal to n l through the joining new fibres to the bundles as a result of the growth of the brittle zones. The corresponding pair of new lines is (an,) and ( ~ o ) " This means that point 1~ on the original curve ABDF is moving to position 11 point 20 is taking position 2 ~, so that the whole characteristic curve moves to position A CDF. Obviously, there exists a large variety of curve shapes if bundles of various sizes exist in a composite. Note that the ABDF----~ACDF change takes place by a jump corresponding to a sharp drop in the strength of a composite when new fibres link a fibre bundle during heat treatment. The experiments [448, 456] were carried out with two aluminium alloy matrices, AMg6 and AD33, according to the Russian specifications. The first alloy, similar to the 5056 alloy, contains 6% Mg; the second, similar to the 6061 alloy, contains 1% Mg. All specimens were obtained by hot pressing either in vacuum (AMg6 matrix) or in air (AD-33 matrix) of plasma-sprayed precursors under temperature/pressure/ time conditions which ensure increasing the strength of a specimen after subsequent heat treatment begins. In the case of AI-6Mg matrix composites, the number of fibres in each specimen with the AMg6 matrix was counted after specimen testing, to determine an exact value of the fibre volume fraction since the properties to be evaluated may be sensitive to small variations of the fibre volume fraction, so the strength values should be related to more exact values of vf rather than to the average value. This was done by taking into account possible shapes of the strength versus fibre volume fraction curve according to the failure model under consideration. The procedure is illustrated in fig. 5.4. 1100
'"'""'""""'"'"'"'"t 1
\ 900 b
""
A
""~..-~,
/
".
, 700
f
\
\
~-
/
... "-..,i / ~ \ / "~.. * \\ /
/
/
y[]r \
\
"'" ",,.,;
/ 6 ""("-
/O " ::'.,.._ .. " "
.r 0[~ ~
/
ZX
"-.
/
........
0 .'4 5 ..........
0
\
/
,'" /0
\
k
-" \i -
\
-
,,
"..
........
v!
~. /
7"
"'-.
~ = 30 m i ~ , I = 60 m i n t = 90 m i n I "*._**..~ t = 150 "rain -~A__A~_Aw i t h o u f h e a t ~vea~'r~ent
30894o
--El...
~O /
/
/
500
~
"".
"..
-
"...:
0 55 .
Fig. 5.4. A n e x a m p l e of the e v a l u a t i o n o f c o m p o s i t e s t r e n g t h at Vf • 0.50. T h e m a t r i x is A1-6Mg alloy, a n d the heat t r e a t m e n t t e m p e r a t u r e is 350~ (After Mileiko et al. [448].)
Strength and fracture toughness
240
Ch. V, w
1000
\ b 750
50O F
\
t-
250
0
ooor~o ~r = 85o'_c
\
~
50
~
I00
t /
-
5oo*_c
-
150
rain
200
Fig. 5.5. Strength of boron/aluminium composites with AI-6Mg matrix as a function of heat treatment time. (After Mileiko et al. [448].)
The dependencies of the strength upon the heat treatment time are presented in fig. 5.5 for three temperatures. Figure 5.6 reveals a sharp decrease in strength when the temperature reaches a critical value. The appearance of the failure surfaces of composites also changes with heat treatment. In specimens corresponding to the
ii00
~
.
.
.
,
.
.
.
.
,
.
,
,
o
o
'
'
1000 900
b
800
o
700
600 -
o
500
400
300
'
'
'
'
' 400
'
'
'
T / ~
'
'
500
'
' 600
Fig. 5.6. Strength of boron/aluminium composites with AI-6Mg matrix as a function of the heat treatment temperature. The heat treatment time is 30 min. (After Mileiko et al. [448].)
Ch. V, w
Brittle-l~'bre/ductile-matrix." strength
241
Fig. 5.7. Failure surfaces of specimens heat treated at (a) 350~ for 30min, (b) 460~ - 30min, and (c) 560~ 30 min. (After Mileiko et al. [448].) initial part o f the a*/vf curve, w h e n the strength is increasing, fibre pull-out can be observed (fig. 5.7a). After the strength falls, the fibre pull-out c a n n o t be detected (fig. 5.7b). It can be interpreted as being a result o f crack p r o p a g a t i o n t h r o u g h the fibres linked to each other by the brittle zones.
Strength and fracture toughness
242
Ch. V, w
1800 . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . o o
00
0
o
\ b 1400
8
0
o o
o
o o
o
o o
o o
L 1000
0
o
o
6000........
5'o .......
0 o
i 6b ....... i 'o....... z, t /
rain
Fig. 5.8. Strength of boron/aluminium composites with AD-33 alloy matrix as a function of heat treatment time. The heat treatment temperature is 520~ (After Mileiko et al. [448].)
In the case of the matrix with 1% Mg, specimens were quenched after heat treatment at 520~ and then aged at 160-170~ for 10-12 h. The dependence of the composite strength on heat treatment time is given in fig. 5.8. The large scatter of the data obtained is certainly due to the absence of the fibre volume fraction correction. The strength increment corresponding to the initial part of the curve is not as great as in the case of composites with the AI-6Mg matrix. That is certainly because the difference between the ultimate strength of the matrix hardened by boride particles in the brittle zones and that of the matrix in a hardened state resulting from the conventional aging treatment may not be as large as in the case of an originally non-hardened matrix. Comparison of the effect of heat treatment on composites with the matrix containing 6% Mg and on that with 1% Mg shows that within the considered interval of the heat treatment conditions, at least for the former case, heat treatment by itself does not influence the strength of boron fibres so as to cause a decrease in the composite strength. It is also clear that as the magnesium content decreases the temperature of the heat treatment to cause a sharp fall in the composite strength should be increased or heat treatment should be carried out longer. Therefore, the change in composite strength predicted by the model is supported qualitatively by the experiment if slower kinetic of the brittle zones formation is assumed in the case of the matrix with a lower content of the element providing boride formation in the matrix. The dependence of the composite strength on the size of the containing boron zone, obtained by measuring the size of the zone around a fibre where aluminium content decreases, is shown in fig. 5.9. It can be seen that the drop in the strength corresponds, nearly exactly, to half of the average interfibre distance for a composite
Brittle-fibre/ductile-matrix." strength
Ch. V, w i 0 0 0
\
.
.
.
.
Oo
800
,
,
,
243
,
0 o
b
600
400
20015
i,,
I
I
i
I
20
i
i
~1/
1
I
25
i
i
i
i
I
30
i
i
i
i
35
Fig. 5.9. Strength of boron/aluminium composites with A1-6Mg matrix as a function of size of the characteristic zone around a fibre in the matrix. (After Mileiko et al. [448].)
containing 50% boron fibres of a diameter 140 gm. This provides an important support to the model described above. A possible explanation of the experimental data by an influence of the growing interface zone, which would contain, in the case under consideration, pure aluminium borides (see Section 11.6.1), should be excluded. In fact, the thickness, 6, of such a layer should increase with the heat treatment time, t, as 6 cx t 1/2 (Chapter 10). By using the experimental data corresponding to the falling part of the curve a* versus t (fig. 5.8) to represent the relationship a~ = (a*-Om/)m)/l~ f a s a power function of time t, we find the value of the exponent to be -0.825. At the same time, the fibre strength a~ o( 6 -1/2, according to consideration of the interface zone as a possible room for the cracks entering the fibre (see Section 10.3.3), and we should observe a~ oct -1/4. Thus, dependence a~(t) appears much stronger than predicted by the considering of the behaviour of interface cracks. Such a strong dependence can be explained if to assume a jump-like transition from one bundle structure of the composite to another. Now we present experimental evidences of the existence of the maximum point on the dependence of the strength on fibre volume fraction. The corresponding experimental data are shown in Table 5.1. The same type of the behaviour is revealed by boron/aluminium composites (figs. 5.10 and 5.11). The influence of a degree of homogeneity of the fibre packing was demonstrated in a special experiment with boron/aluminium composites [419]. Specimens of the composite were distinctively prepared with non-homogeneous fibre packing (fig. 5.13). Figure 5.12 shows the result of shifting point B in fig. 5.3 to the left simply due to increase in the value of no.
Strength and.fracture toughness
244
Ch. V, w
TABLE 5.1 Non-monotonic dependence of the mean strength of brittle fibre/ductile matrix composites on fibre volume fraction. (v~n~X is the value of vr corresponding to the maximum composite strength) Fibre
Matrix
Temperature, ~
v~?....
Authors and sources
Boron Carbon Carbon Carbon w Carbon Silicon carbide Silicon carbide w Sapphire Sapphire Sapphire
Magnesium alloy Aluminium alloy Aluminium alloy Aluminium alloy Copper Aluminium alloy Aluminium alloy Molybdenum Magnesium alloy Magnesium alloy
RT RT RT RT RT RT RT 1300 RT 200
0.28 0.3 0.3 0.2 0.2 0.2 0.32 0.3 0.35 0.40
Ahmad and Barranco [9] Jackson et al. [274] Jackson et al. [274] Pechersky et al. [43] Kuzmin et al. [348] Yajima et al. [419] Moritomo et al. [43] Mileiko and Kazmin [430] Nunes et al. [498] Nunes et al. [498]
" Random fibre orientation.
2000
.ooooo'u,~ ~b24'-~ 'C, ; s ' g b ~ d ~ Matr~
ooooo
e
1800
"b
1800
2024-T0
9
0 0
0
\ o
80
1400 o
1200
.
0.4
1
6
o
0
o
(~) . . . . . 0.5
0
0
~
i
0.8
,
,
,
0.7
.
1400
\
1200 b
1000 800 600
~oooo ~ ~
0.3
oooo
eoel-p
M~t~tr 6 0 6 t - T 6
0.4
0.5
vt
(,.,, .fabr.tco~e~)"
0.6
0.7
Fig. 5.10. The strength of boron/aluminium composites versus fibre volume fraction. Experimental data after Prewo and Kreider [553].
Ch. V, w 1250
Brittle-fibre/ductile-matrix." s t r e n g t h .
.
.
245
.
5
1000
o
~g~o
o
750
500
\ *b
F i b r e : I k i n d , d = 0.1 ~ n ~ Matrix: 2024-T6 250
1000
-
F i b r e : H k i n d , d = 0.1 m m Maf;'PLr 2 0 2 4 - T 6 o 0
% o
750
0
o 500
250
0.0
J
0.1
0.~2
0.3
vl
i
0.4
0.5
Fig. 5.11. The strength of boron/aluminium composites versus fibre volume fraction. Experimental data after Mileiko et al. [452]. ,0
I
'
'
,
i
oo ,tr
0.8 0 0 0
0.6 o~
O0
0
~
0.4
0.2 I
0.2
i
I
0.3
l
Vf
0.4
Fig. 5.12. Strength of the boron/aluminium composite with a D16-matrix. Dark points correspond to specimens with non-homogeneous fibre packing as shown in fig. 5.13, open points are for normal fibre packing. (Experiments by Sorokin [419]).
246
Strength and fracture toughness
Ch. V, {}5.2
Fig. 5.13. Examples of non-homogeneous fibre packing in boron/aluminium composites. Further evidence is supplied by the results of testing silicon-carbide-whisker/ silver-matrix composites performed by Takahashi [639]. Fibres in these composites were oriented three-dimensionally and randomly packed. The comparison of values of the strength of the composites fabricated on Earth and in Space (where low gravitation provides conditions for more uniform distribution of the whiskers and, actually, better quality of the matrix) is presented in fig. 5.14. One can see that the difference in the behaviour of these two sets of samples corresponds qualitatively to the prediction of the model. Finally, we should make two notes. First, the influence of fibre-packing nonhomogeneity just discussed was also supported by computer simulation results which have been obtained after the above model was developed. This will be shown below, in Section 5.2.3. Secondly, the important question on the evaluation of values of c~ and (ar(l,)) in eq. (5.8) will also discussed later, in Section 5.3.1, when !
,
!
45 a. 4 0
"~ 35 30 25 20 15
o, )5
0.10
v,
Fig. 5.14. The strength of the SiC/Ag composites prepared on Earth (dark points) and in Space (open points). The experimental data after Takahashi [639].
Ch. V, w
Brittle-fibre~ductile-matrix." strength
247
discussing a general question of evaluating fibre strength characteristics that determine composite strength.
5.2.3. Computer simulation The development of a particular composite element (tube under longitudinal load, turbine blade, nozzle blade, etc.) and a corresponding fabrication route should include such stages as 9 choice of a fibre/matrix combination, 9 designing of various fibre configuration to meet the loads, 9 choice of a fabrication scheme, 9 fabrication of composite specimens, 9 testing composite 9 choice of a variant that is a result of a compromise. This non-complete schedule is really time consuming. Hence, one should be looking for shortening some stages, especially those related to physical experiments. This calls for physical or/and engineering experiments to be complemented by a mathematical experiment, that is computer simulation. If we manage to exploit computer simulation instead of most of the bulk of physical experiments by making and testing specimens and composite elements in a computer simulation procedure, we will proceed much quicker. How reliable can the final result be? The answer depends on the choice of the proportions between physical and mathematical parts of the experiment, on the reliability of models of the composite behaviour which have been used in computer algorithms, on the chosen system of reference points for tying up the outputs of physical and computer experiments, etc. This is perhaps a point to be discussed elsewhere. However, before coming back to a particular topic of computer simulation of the behaviour of a composite specimen under tension, we should make a remark about the structure of the computer simulation procedure. This includes * a mechanical model of a process to evaluate stress/strain state in a composite under loading, . a statistical procedure to transform a random set of composites into a numerable set of specimens with defined microstructures, 9 a necessary fracture and/or failure criteria. To illustrate possibilities we will mainly use here results by Kopiev and Ovchinsky [329, 519]. A mechanical model here focuses on the evaluation of stress/ strain state around a fibre break. To simplify the calculation, Kopiev and Ovchinsky adopted a kind of the shear-lag analysis (Section 3.6) taking into account the elasticplastic behaviour of the matrix and considering an axisymmetrical situation. In a simplest model they neglect the dynamic effects caused by a fibre break. 1 ! Dynamiceffects are taking into account by Sakharova and Ovchinsky [579]in the framework of a simple uniaxial model.
Strength andfracture toughness
248
Ch. V, w
iii@iiiii iiiljiiiii@iiiiiiiiiiiiiiii@iiiiiiiiiiiiiii@ [email protected]@iiii.i.i i [email protected] i.i (/'6%j i(~.i i@i.(~i.i .. ,. i.i~i(~i i.i....-i-ii ( ~ iii 9 iilji [email protected]~ii [email protected][email protected]~i.i, i i.>@[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
Strength and fracture toughness
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
Strength and fracture toughness
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
Strength and fracture toughness
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<}f/j~[ Fzh(l)g(O)dl dO.
(5.44)
Hence, the total load on the bridging fibres in zone/~[ is (5.45)
FT -- NcqbFz.
Neglecting the load carried by the ending fibres, the ultimate stress for the composite can be written as ~,
FT
-- ~--~nt- O'mUm.
(5.46)
Substituting eqs. (3.200), (5.43), (5.37), (5.41)-(5.45) into eq. (5.46), yields - af vf
g(O) cos 0 dO
9
x x
1
(I"
/c- O
i~[-;h(l)dl +
g(O) cos 3 0 dO
fo ~176
g(O) dOh(1)dl
'*) )
1- ~
h(l)dl
+ a*Vm.
(5.47)
Equation (5.47) allows to analyze dependencies of the ultimate stress upon both the texture of the structure given by a particular form of function g(O) and the fibre length distribution. Corresponding results can be found in [85]. Another line of the same kind of the treatment is based on the averaging, with respect to the angle, of a macrostructural criterion of failure such as that given by eq. (3.130) (see for example [43]). An attempt to account for dispersion hardening by short fibres and an increase in dislocation density as a result of differential thermal contraction of the components was reported by Zhu et al. [730]. To arrive at quantitative results, the authors simplified the situation by assuming all the fibres to have the same tensile strength and the same length. Under an additional assumption of a random fibres orientation they used an averaging procedure slightly different from that adopted by Fukuda and Chou, which included, in particular, an assumption of a given ultimate strain of the composite, to derive a dependence of the composite strength on fibre volume fraction and effective fibre and matrix characteristics in the form o-*
ll+v2( = 8 1; v
1") 1 - ~-~ 8-~vf+ O-mVm.
(5.48)
Strength and fracture toughness
258
Ch. V, w
Here v is the Poisson's ratio of the composite and ~-~ - a~ + A~'
(5.49)
where Atr' is the residual compressive stress in the fibre and O"m -- O"m + Ao'ml d- Atrm2
(5.50)
where Arrml and Ao'm2 a r e increments to the effective matrix strength, tim, due to dispersion strengthening and dislocation strengthening arose as a result of the thermal strain. It is found AO-ml ~ lamb df '
(5.51)
Ao'm2 ~/-tmbP 1/2
(5.52)
where [tm is the shear modulus, b and p the Burgers vector and dislocation density, respectively. A coefficient to appear in eq. (5.51) is calculated in [730] by constructing a geometrical model composed of elementary cubes containing just one fibre (a part of a fibre). Then the fibre in the cube is replaced by a spherical particle of the equivalent volume which acts as a well known obstacle to the dislocation motion. So the coefficient depends on the fibre volume fraction only. As we see, a complete theory of strength of metal matrix composite with short non-aligned fibres remains to be built up. Therefore, at the present time, technological recommendations formulated below, in Section 5.2.5, on the basis of microstructual models of continuous fibre composites are to be applied to discontinuous fibre composites as well. However, a qualitative nature of the recommendation for this case is to be remembered. One point to be pointed out is related to a 'fabrication - structure - properties' dependencies in case of using powder metallurgy processes to produce composites. During the blending stage of the process, the distribution function of fibre length is changing (Section 12.1.1). This causes changing strength properties of fibres as a result of both shortening the fibre with the corresponding scale effect (Section 2.1) and changing a defect population similar to that observed in the loading process of a composite (see for details Section 5.3). If a theory of fracture of discontinuous fibre composites pretends to be of technological importance such effects are to be also taken into account. A semi-qualitative approach to a solution of the problem is discussed in Section 12.4.1.
5.2.5. Technological impact We will here formulate some technological recommendations mainly based on the mechanical model considered in Sections 5.2. land 5.2.2. In doing this, we will often refer to fig. 5.8, so to make reading quicker we repeat a simplified version of this graph in fig. 5.18.
Ch. V, w
Brittle-fibre/ductile-matrix." strength
259
O-
Fig. 5.18. A simplified version of fig. 5.8. Here only the fibre bundles of an equal size are taking into account.
Looking for a reliable composite, do not exceed a certain fibre volume fraction When the value of vf increases beyond vA (fig. 5.18) the average composite strength can decrease, and the strength scatter increases drastically. The latter is proved by numerous experimental results presented above. We will see similar results below, in Part III, when discussing the behaviour of particular metal matrixcomposites. The value of vA can be easily estimated as shown above. Some ways to control this value shall be evaluated in what follows. Try to achieve fibre distribution as homogeneous as possible Suppose we have two composites identical in all but homogeneity of fibre distribution. The first one is characterized by fibre bundles containing n fibres and the second one contains the bundles with m fibres, m > n (fig. 5.19). When the number of fibres in a bundle increases, point A moves to the left, making the critical fibre fraction lower. Hence, we are to lower fibre content, this lowers some mechanical properties of the composite as for example elastic modulus. Choose a matrix alloy with fracture toughness as high as possible It is likely to shift point A to the right and therefore to make an admissible fibre volume fraction higher by using a tougher matrix as shown in fig. 5.20. In this respect, solid state fabrication methods look preferable because of the generally low fracture toughness of cast alloys. It is important to note that stronger fibres call for employing a tougher matrix. The situation is illustrated by the scheme given in fig. 5.21. Increasing the fibre strength may be represented by turning line OA around point O. Therefore, these lines will intersect curve an, (vf) at a lower value of the fibre volume fraction. To restore a high enough value of vA one needs to use a matrix with a higher value of the fracture toughness (O'n2 in fig. 5.21).
260
Strength and fracture toughness
Ch. V, w
, , , ~
An
0 772
vl Fig. 5.19. Influence of fibre packing homogeneity on the strength of brittle fibre/ductile matrix composites, m > n.
Try to use composite modes to increase the matrix fracture toughness It will be shown below, in Section 5.7, that the reinforcing of a metal with fibres can bring an increase in the fracture t o u g h n e s s of the material. This p h e n o m e n o n can be used to i m p r o v e the metal alloy p e r f o r m a n c e as a matrix. Obviously, the characteristic size of the m a t r i x structure has to be m u c h less than that of the c o m p o s i t e with regard to the m a i n reinforcement. Actually, this m e a n s that the d i a m e t e r of the secondary fibre has to be m u c h less than that of the main fibre.
A2 0
I
c?
Fig. 5.20. Influence of the matrix fracture toughness on the strength of brittle fibre/ductile matrix composites. Curves OA~B~C and OA2B2C correspond to one type of fibre and two matrices characterized by the values of effective surface energy G~l < G~2.
Brittle-fibre~ductile-matrix." strength
Ch. V, w
261
, . *
0 O_n2
Fig. 5.21. The higher the fibre strength, the tougher the matrix should be to preserve reasonably high fibre volume fraction.
An example of such a double composite is boron/steel/aluminium [419], that is the boron/aluminium with boron fibres of diameter 100 lam and a Dural type matrix reinforced additionally (about 5% ) with steel wires of diameter 30 lam. Both fibres are aligned in the same direction. The behaviour of this composite is illustrated in fig. 5.22. A small strength scatter accompanies a high value of the average strength
i
,
i
,
I
,
1.6 (.9
%
i
,
i
,
o% ooo
1.4
,. 9 O00O
1.2 1.0 0.8
%
g 9
oV
~
0.6
0"4.0 ' O'.1 ' 01.2 ' 013 ' 014 ' 015 ' 0.6
Vf
Fig. 5.22. The strength of boron/aluminium composites and boron/steel/aluminium composites versus volume fraction of boron fibres. The structure of the latter (open points) contains boron fibres (100 ~tm) with additional steel fibres (30 ~m) in the same direction. (After Mileiko [419].)
262
Strength andfracture toughness
Ch. V, w
at the fibre volume fraction essentially higher than that for a pure boron/aluminium composite containing the same matrix and boron fibre.
Regulate fabrication parameters to eliminate combining fibres in the bundles During fabrication process, the components of a fibre material can diffuse into the matrix which change the matrix properties in the vicinity of the fibre. An example of such a process will be discussed in Section 11.6.1 to show a possibility of matrix strengthening and embrittlement which yields combining of one fibre to another, at a definite stage of the process, and forming brittle bundles. Hence, geometrically, one obtains a structure with the fibres separated by the matrix. However, physically, some volumes of a cylindrical shape are really brittle bundles because the matrix has become sufficiently brittle. In the case of boron/aluminium composites, the embrittlement of the matrix can be a result of the formation of magnesium boride particles in a magnesium containing aluminium alloy. Regulate .fabrication parameters to achieve sufficiently high fibre~matrix interface strength Increasing the interface strength leads to decreasing the fibre critical length (see eq. (3.198)). This offers larger values of the effective fibre strength, according to eq. (2.10). Hence, the fibre contribution to the composite strength (see eq. (5.8)) increases. Obviously, making the process temperature higher and/or time longer attains this aim. However, the previous instruction can be broken at some point and the composite strength will go down. Hence, an optimum temperature/time regime is to be chosen. Another reason for the composite strength to ,go down can be the growth of the brittle interface layer during the fabrication process. Defects contained in such layers can trigger microcracking and so decrease the strength of a brittle fibre (see Section 10.3.3). This puts an additional limitation to making the temperature/time regime of a fabrication process such as to provide the largest value of the interface strength. I f some consideration requires higher.fibre volume.fraction than the above recommended advice, then regulate fabrication parameters to achieve sufficiently low fibre/matrix interface strength In that case one should rely on the microcrack arrest mechanisms based on the fibre/matrix interface debonding. This mechanism is more relevant to the behaviour of brittle matrix composites, it was discussed in general terms in Section 4.5 and will be discussed in more detail in Section 5.8. Estimating the tensile strength of a future composite, rely on appropriate data on the .fibre strength Even now, some authors compare their results of the tensile tests of composites with prediction of the so called Rule of Mixtures using as the fibre strength values those obtained by testing separate fibres of a definite length which is usually much larger than the critical length. Obviously, these values have often nothing to do with the composite strength. To obtain reliable fibre strength data one needs to perform fibre testing according to either fibre fragmentation procedure or some other procedure giving the fibre
Ch. V, w
Composite strength and fibre properties
263
characteristics which determine the composite strength. It is also obvious that such procedures demand more labour, but the output can be very useful.
Final remarks One should remember that the above instructions aim at the strength of a metal-matrix composite loaded in tension in the fibre direction. Strictly speaking, we may apply them to unidirectional composites only because the basic model describes the behaviour of such materials. However, fibrous composites, being developed for much more specified applications, unlike most of the metal alloys, are still to withstand a variety of the loading conditions. For example, a part of the boron/aluminium tubes in the frames of a space shuttle to be launched by a rocket and then landing as aeroplane, are loaded in tension during taking-off and in compression during landing, and vice versa. Requirements to the composite microstructure loaded by compression (Chapter 8) and by tension can be different. Hence, a compromise is needed when the condition of the minimum weight of a structure under prescribed loading conditions is to be fulfilled.
5.3. Composite strength and fibre properties As we see now, the evaluation of the tensile strength of metal-matrix composites needs appropriate fibre characteristics. The first approach to the description of fibre strength properties outlined above, Section 2.1, obviously, cannot yield characteristics such as, for example, (a~(l,)) used to estimate composite strength, eq. (5.8). Also, fibre properties in a composite can differ from those measured by testing separate fibres because fibre/matrix interaction occurring during composite fabrication effects fibre strength. Thus the problem of determination of fibre properties appears to be closely related to a micromechanical problem of modelling failure processes in composites. Certainly, the most advanced method of correlating the two problems is the socalled fibre fragmentation test. Actually, the idea of observation of sequential fibre breaks under increasing load was first formulated clearly by Kelly and Tyson [306] who did experiments with a metal-matrix composite and used idea to correlate the strength of the composite to that of the fibre and the fibre/matrix interface. Since that work the idea of singlefilament-composite testing has been mainly exploited to evaluate the fibre/matrix strength in fibre-reinforced polymers. A review of corresponding results with criticism of composite models involved as well as algorithms of fibre breaking processes used can be found in [165, 539]. In this section we consider both theories of fibre fragmentation test (FFT) and some experimental data obtained mainly for metal-matrix composites to show the applicability of the theory to evaluation of the fibre in fibre/matrix interface strength.
5.3.1. Fibre fragmentation test." theory We start with a simple approach to the problem, then proceed with an exact theory and a development of the theory to make it more strict.
Strength and fracture toughness
264
Ch. V, w
A simple approach Consider a sequence of the events happening when a single-fibre composite is being loaded in tension. Suppose, the fibre carries a number of defects. When the fibre stress reaches a minimum fibre strength, a~~), which occurs at a random point, say xl, the fibre breaks, and the fibre axial stress drops down to zero at the break (I) point, then the stress recovers up to the regular value of stress, that is af , on the length that can be called the recovery length. If the interface shear stress, ~*, is constant, the fibre stress builds up along the fibre linearly, and the recovery length (fig. 5.23) is exactly the critical fibre length according to Kelly-Tyson model, eq. (3.198). We have 1-
2(A/c)(a'/r*)
(5.53)
- 2~r'
O.('III)
O.(zz)
r--
O-(Z)
g,--~
v ,
!=
Z
=!
Fig. 5.23. Schematic illustration of the fibre fragmentation process in a single-fibre composite specimen. The sequence of stress profiles along the fibre axis is shown, the applied stress increase from top to bottom of the scheme.
Composite strength andfibre properties
Ch. V, w
265
where A and c are the cross-section area and circumference of the fibre, and or' the regular fibre stress. So after the first fibre break:
ll-
(5.54)
~0"~I)
Obviously, defects located on the fibre around the break within the distance + l l / 2 will never be active because within this distance the stress will not increase with the increasing applied load. When the fibre stress reaches the next value of the defect or(II) , a new recovery length arises. The new strength (among the active defects), say vf one may occur at such a point shown in fig. 5.23, that it does not intersect either the first one or the fibre end. A further increase in the fibre stress causes a next fibre break. The situation depicted in fig. 5.23 illustrates a possibility for a new recovery length to intersect an existing one, this may lead to a decrease in the fibre stress on a part of the fibre. Finally, the situation reaches the configuration depicted in the top of the scheme in fig. 5.23, this configuration may not change in such a manner as to increase the fibre stress at any point. In the framework of the simple approach [428] we do not make any assumption of a particular defect or strength distribution along the fibre. We are searching for a dependence of the fibre strength on the fibre length not overspread by the recovery zones, that is what does determine the strength of a composite with sufficiently low fibre volume fraction. Let the whole fibre length before loading be L = L1. The first fibre break at O ' f - O'~I) cut out from the whole fibre length 11 is given by eq. (5.54). The point of the break, x = Xl, may take any position along length L1 with equal probability. Therefore, the mean length cut out from the whole fibre length will be
(ll)--L1
ll
---4-x
~l-k (L1-
/1)
-- ll
1-
1 ~(1
1
With the applied load increasing, the first cut-out length increases according to it(IX), the cut-out eq. (5.53), so when the fibre stress reaches a next critical value, vr length reaches the value
t2[1 1:/LI(1 -
-
t:/4L1)].
So the second fibre break occurs on the mean fibre length
L2 -- L1 -- 12 [1 --
12/L1 - 1/4(12/L1)2].
Similarly L3 -- Zl - 2/3 [1 - 1 3 / L 2 -
1/4(13/L2) 2]
Strength and fracture toughness
266
Ch. V, {}5.3
and Zn - Zl - ( n
- l ) / n [1 -
ln/Zn_l]
(5.55)
if ln/Ln-1 << 1. Note that In depends upon o~n). We will call length Ln as the effective remaining length or just the effective length. In a physical experiment, to obtain necessary information we need to register the load versus elongation eurve using a sufficiently rigid testing machine to provide marks on this curve that correspond to fibre breaks (fig. 5.24). Let us evaluate now a~n) for the nth fibre break. Suppose the stress-strain curve of the matrix is known, that is function a(e), and the Young's modulus of the fibre is El. The average strain on the stress/strain curve following from the curve shown schematically in fig. 5.24 does not coincide with the matrix strain in the specimen cross-section where a fibre breaks because of longitudinal strain concentration in the matrix in the vicinity of existing fibre breaks. So a corresponding iteration procedure is to be used for calculation of the stress state in the composite. If the composite stress at nth fibre break is a (n) and the ith approximation for the fibre stress is O'~n)'i then ~(m n)'i+l - o'~n)'i/Ef,
and we can
(5.56)
obtain O'(m n)'i+l via ~(m n)'i+l. Hence,
d//f ELONGATION Fig. 5.24. Schematic load versus elongation curve resulting from fibre fragmentation test of a single-fibre composite specimen.
Ch. V, w
Composite strength and fibre properties
0"~n)'i+l -- 0"(n)'i+l -- O'(~)'i+l(1 -- Vf)/Vf
267 (5.57)
Now we are to apply, first, an iteration procedure according to eqs. (5.56) and (5.57) to determine the fibre stress values corresponding to fibre breaks. Then we calculate the effective length, Ln, corresponding to the nth fibre break. If we do not know beforehand the ultimate interface stress, r*, we need again to use an iteration procedure. Let us choose an appropriate number of the values of the interface stress, ~*, relate a~n) to the cut-out length In by using eq.(5.53), and follow the procedure given by eq. (5.55). Let the largest value of n in this procedure be N, then we calculate dependence LN(r*). The dependence will have either point L~(~**) = 0, or L~ I ~** = minLN(~*). The value of ~** is assumed to be the ultimate interface stress.
Curtin's theory Curtin [111, 112] staying on the Widom's theory of random sequential addition of hard spheres to a volume [700] with a special case of such a kind of addition of hard rods of a constant strength on a line, developed an exact theory of fibre fragmentation. The problem is formulated in terms of the statistics and distribution function for hard rods of length 6(a), which is the recovery length dependent on the fibre strength cr, placed at fixed random positions along the line. Results of the theory can be applied to many problems. So we shall present the Curtin's theory here. In the Widom case of unique strength fibre (in Weibull distribution/3 = oc, the recovery length, 6, is constant, and the minimum center-to-center separation of hard rods, or equivalently, the fragment length, is 6 with the maximum separation 26), the distribution function, p(x;n,6), for fragment length, which is the fraction of fragments of length between x and dx at recovery length 6 and break density n6 is constructed as
1 p(x; n, 6) - --n-~q(x; n, 6 )
(5.58)
where
q(x; n, 6) -
2 f~ ~ ( ~ ) e x p ( - ( x / 6 - 1)~,(~))d~ ~'2~('7)e x p ( - ( x / 6 - 2)~,(r/))dr/
6<_x<_26, 26<x
(5.59)
in terms of an auxiliary function O(r/) defined implicitly by q--
j~0qJ( - 2 f0 t 1 - eXp(-S) s ds ) dt
(5.60)
As n6 approaches the limit of n 6 - q*, the available space to accommodate new breaks, L*, goes to zero. Thus O'(r/*) ~ oc and, from the second line in eq. (5.59) O(r/) ~ oc, and so from eq. (5.60)
Strength and fracture toughness
268
r/* -- f 0 ~ ( - 2 f0 t 1 - exp(-s) ds ) d t s
0.7476...
Ch. V, w
(5.61)
This is a well known value of the mean normalized fragment length 2/6 ..~ 1.337. It is shown that a good approximation for function ~(r/) for r < l(r/_< 0.47) is ~(~)
1 (1 - q/r/*)'7"
(5.62)
Returning to the original problem when 6 depends on a, we note that at any stress a fragments shorter than 6(a) can exist due to the previous fragmentation process. Imagine, we throw hard rods of length 6(5) on length LT, when stress 5 goes up to a. Suppose, we have obtained AfT -- 1 breaks to the point when the stress reaches a. Let NR be the number of fragments of length x < 6(a) which occupy a total length LR of the fibre. Removing from the consideration these fragments and storing them in a distribution pR (x) we will see the remaining ArT - N R -- 1 breaks distributed along the effective length L T - LR. The fragment length distribution for this remaining portion of the fibre is identical to the fragment distribution for the case of a fibre of a unique strength a, the fibre length being L T - LR, the number of breaks being ArT- RR, and a dimensionless break density is Nv-N~
n6 = ~ , 5 LT -- LR
(5.63)
with eqs. (5.58) and (5.59) working. Let us now evaluate function pR(x) for x < 6. An increase of the fibre stress by a small increment, Aa, and hence an increase of the recovery length by A6 yields removing from the further play fragments smaller than 6 + A6; they are to be added to the fragments which have been already removed from the consideration. The number of those fragments is aNR = (NT -- NR)p(~; , , ~)a~
(5.64)
and their total length is ALR = ANR6
(5.65)
This changes the distribution function pR(x) by ApR (x) --
ANR 0
6 < x < 6 + A6 , all other x .
(5.66)
Substituting eq. (5.64) into eq. (5.66) and replacing small increments with differentials yields dpR(x) = Np(6" r/) d6 6D(X - 6) d~r
'
(5.67)
Ch. V, w
Composite strength and fibre properties
269
where 6o(.) is the Dirac delta function, and N = nL. The total length of the fibre at a + Aa which does not contain the recovery zones is now L = L T - L R - ALR. The number of breaks in this portion of the fibre is N = N T - N R - ANR + Z~A, where ~ A is the number of new breaks occurred between a and a + Aa. A new break can be accommodated only by a fragment of length x > 26(a) and such a break can only occur in the available length x - 2&. Therefore, &NA = ~t(a,L*)Aa
(5.68)
where ~t(a, l) is the number of defects with strength between a and a + Aa in a length l of the fibre, and L* the total length available for new breaks, that is
L* - N
( x - 2&)p(x; rl)dx
(5.69)
Equations (5.58)-(5.60) and (5.69) yield
L*L - NLf )
(x - 2&)p(x; n, &)dx - -~ 2 f0" ~(()d( - ~,(q) l = O(q)
(5.70)
The evaluation of the length L and number of breaks N in the remaining fibre, which determines r / = N&/L, is thus described by the differential equations dL
d&
dcr = -Np(6; rl)& d---~
(5.71)
dN da = -Np(6" ' 17)~d6 + la(a,L*)
(5.72)
Function ~t(a, l) is that of the Weibull type, that is
t)
-
(5.73)
To o-o
Making use of eq. (5.58) and (5.69), the differential equations of the problem, eqs. (5.67), (5.71) and (5.72) are rewritten in the form dpR (X; O') __ L
da
dL d--~ = dN ~ -
-
(~2
db
q(6; n6) ~
&D(X -- &)
L df &q(&; n&)d a L db L ~-1 62 q(&; n&) - ~ + ~p(n6) ~ fl(a/ao)
(5.74) (5.75) (5.76)
Equations obtained can be written in non-dimensional form by introducing variables
Strength and fracture toughness
270
Ch. V, w
S - - O'/O'R,
(5.77)
h = 6/6R,
L=L/6R,
LT -- LT/bR,
Lo = L0/6R.
where O"R - - ~r~ and 6 R = /~ with a0 and 60 are given by eqs. (4.78) and (4.73). Then the differential equations of the problem become dL L d--~ = - s q(6; rl)
(5.78)
dN ds
(5.79)
-
L s2 q(6; q) + L~(q)fls ~-I "~0 ~ (O'R/O'0)P
with the initial conditions L(cr = 0 ) = L v
N(a) = 0
(5.80)
p R ( X ; O" = 0 ) = 0
A further development due to Glushko et al. [192]
Equation (5.68) in the Curtin's consideration means that the strength distribution for a decreasing effective fibre length is assumed to be not influenced by the removing of weak points in the fibre during the preceding fragmentation process. Such an assumption certainly leads to a decrease of exactness of the theory and the difference between the solution obtained and the real situation remains unknown. So we recount here the theory of fibre fragmentation built up without the Curtin's assumption [192]. First, we note that for convenience, some notation of the present section will be different from those accepted in the previous one. Let p be the fibre stress outside the recovery zone, and l -- 89 ~. Let the fibre carry a set of defects statistically distributed along it. Fibre stress p sufficient to cause a defect to transform into the fibre break is to be called the defect strength. We introduce a strength distribution function z(~r) for a set of the defects as well as a distribution function of defect points characterized by mean number, 2, of defects per unit length. When the axial fibre stress is p, all defects of strength ~r < p are located within the recovery lengths and, consequently, may not transform into the fibre breaks. Hence, the rest of the defects, which may transform into the breaks, are located outside the recovery length and have strength greater than p. Therefore, with increasing stress p, the distortion of the strength distribution function Z(a) will take place for a set of defects located outside the recovery lengths. Let the current strength distribution function be denoted Z(cr/p), that is the probability for a defect to have strength cr* < cr, while fibre stress is equal to p. We can write
Composite strength and fibre properties
Ch. V, {}5.3 - z(,,) - x(p) 1 -
z(p)
271 (5.81)
"
Let a single-filament composite embody the fibre of total length L >> l(p). Suppose the number of the current fibre breaks, k(p), is sufficiently large, k(p) >> 1. Obviously, the value of k is physically a discrete one, but to make the analysis simpler we treat k as a continuous function of p. Let the total average length of the recovery zones be A(p). If a distance between neighbour breaks is larger than 2/(p), then the value of A(p) can be easily determined, A(p) = 2k(p)/(p). But due to the stochastic nature of defect distribution this is true for low values of p only, for a general case we have A(P) _< 2kl(P)A(P) <_ L If fibre stress, p, gains increment, dp, then the number of the breaks increases by value dk. This increment is proportional to the density of defects which can be broken, 2, and to the remaining effective fibre length L T - A(p), that is dk = 2(LT -- A(p)){Z(p + dp/p) - Z(P)/P}
(5.82)
The expression in the curly brackets defines the probability for a defect to have strength between p and p + dp. Taking into account eq. (5.81), we can rewrite the last equation as dk = 2(LT -- A ( p ) ) ctp
Z'(P) 1 -
(5.83)
Z(P)
where X(P) is the original strength distribution function. Equation (5.83) accounts for new fibre breaks to arise outside the recovery length only. This equation describes the fibre breakage process provided function A(p) is known. Let the recovery zones be combined in n(k) "islands", (n < k). The average distance between neighbour islands can be written as d - LT -- A(p)
(5.84)
When a new fibre break occurs, one of three events can follow. First, a new island arises, and n(k + 1) = n(k) + 1. That is the case, when the new recovery length does not intersect with a previous one. Second, the new recovery length intersects with a previous zone, then the number of islands remains unchanged, n(k + 1) = n(k) + O. Third, a new recovery length intersects with two previous ones, then the number of islands decreases, n(k + 1) = n(k) - 1. This means
n(k --b 1) - n(k) -- 6n
(5.85)
272
Strength and fracture toughness
Ch. V, w
where n takes one of the values, + l, 0, -1. Let the corresponding probabilities be P+I, P0 and P-l, respectively. Assuming function n(k) to be continuous, eq. (5.85) written in the finite difference form can be approximated by the ordinary differential equation for average values
dn dk
~ ~n = /19+1 --e-1
(5.86)
Here the bar over a symbol means averaging. Therefore, we have written the equation to describe the evolution of the islands configuration when k is increasing. Strictly speaking, an island size increase should be taken into account with increasing value of p. That brings the possibility of joining two islands between two breaks events, but it can be shown that these effects play a secondary role. So we have kdn d---k= k'(p)(P+l - P-1 ).
(5.87)
Probabilities P+1, P0 and P-1 can be easily obtained if the distribution function of the distances between neighbouring islands, O(d), is known. The average distance between islands defined by eq. (5.84) can be written as d -
f0 ~
(5.88)
xd(x)dx.
Using eq. (5.88) and assuming that a new fibre break can arise with equal probability at any point of the fibre outside the recovery length, we obtain
l(p) dP 1 =
[21(p) x(.ot(x)dx +
JO
(21(p) - x)mt(x)dx
J l(p)
r
- - J t(p) 2t(P)(x- l(p))o)'(x)dx + dPo
(p) 2l(p)co'(x)dx
(5.89)
- [2t(P)(x - 2l(p))of(x)dx dP+l at(p) Now we calculate the rate of change in the total recovery length with increasing fibre stress, p. A contribution to the value of the change in the sizes of the islands at a constant number of fibre breaks will be ~A
ep
[k=const= 2n(p)l' (p)
(5.90)
The contribution of a new fibre break can be written as ~A Ip=const-- 31 ~k
(5.91)
Composite strength and fibre properties
Ch. V, w
273
where 6 1 - x at x < l(p) and 61 1 + ( x - / ) - t at x > l(p). Here x is the distance between neighbouring islands. Averaging the value of 61 over all probable values of x, we obtain X
,~-~ -
/o
x~o'(x)~ +
)
2t(p) -
o~'(x)~
(5.92)
Note that dA(p) - -8A(p) - - ~ p d p + -8Adk ~~pdp. Therefore, eqs. (5.90) and (5.91) yield dA(p)
dp
= 2n(p)/'(p)
(5.93)
+ kt(p)~l
Now, setting initial conditions, corresponding to a lower value of defects strength Pmin, for the system of ordinary differential equations as k(Pmin) -
r/(Pmin) - -
1,
A(Pmin) -- 2l(Pmin)
(5.94)
,
we can find a solution to the system given by eqs. (5.83), (5.87), (5.93), and describe the process of fibre breakage. A solution to this system of equations can be easily obtained for the case of low fibre stresses, that is 0
- Pmin << Pmax - Pmin-
It is a set of asymptotic expansions of the expression for the corresponding parameters, , ( p ) - k(p), A(p) - 2/(p)k(p), k(p) - --2LT ln(1 -- Z(P)).
Here Z(Pmin) = 0 is assumed, and Pmax is the maximum value o f p corresponding to A(Pmax)= LT and n(Pmax) = 1. TO proceed with a numerical solution of the system of equations obtained we need to evaluate distribution function, co(x) (see eq. (5.88)). To simplify the evaluation, let us approximate function co'(x) by the Dirac function, co'(x) - 6D(X -- d)
(5.95) m
where average distance between neighbour islands d is defined by eq. (5.84).
274
Strength and fracture toughness
Ch. V, w
Then eqs. (5.86), (5.89), and (5.92) yield 1 ~n~
2l(p)n(p) LT-A(p)
-1
(~ > l(p),
(5.96)
,~ < l(p),
l(p) (2 LT-A(p) n~,/ LT-A(p) n(p)
~(p)t(p).] LT-A(p)J n(p)12(p) l(p) LT-A(p)
d > 21(p), m
l(p) < d < 2l(p),
(5.97)
0 < d < l(p).
Therefore we can write the system of the differential equations of the problem as dn __ dk~n
@
ap
dA -- dk61,
(5.98)
@
dp dk - ~ r @
A(p)~ z'(p) - L~ ] 1 - ~p)"
For the distribution function of the defects strength we shall use a Weibull type function, Z(P) = 1 -- exp(-~
9
Parameters e and/3 are defined via average value p and standard deviation Op"
-(r(l
+ 1/fl)(fi--Pmin)) fl
Dp - 00-pmin)
~/ r(1 + 2/fl)
F2(1 + 1 / f l ) - 1
The system of equations, eqs. (5.98), with initial conditions given by eq. (5.94) is to be solved numerically. An example of such a solution is presented in figs. 5.25 and 5.26. One can see the evolvement of the breaks configuration with the fibre stress increasing. Final remark
It should be noted that an F F T model can be regarded as a failure model of a multifilament composite. Then such a composite is assumed to fail as a result of the accumulation of independent fibre breaks. Obviously, in the case of metal-matrix composites this occurs at sufficiently low fibre volume fractions. Hence, the first term in eq. (5.8) can be calculated by using one of the available F F T models.
Composite strength and fibre properties
Ch. V, w i
i
;
!
I
|
i
!
i
I
!
i
l
!
I
f
I
|
275
i
=2
_
_
1
i
5
0
10 k
i
15
1
20
Fig. 5.25. The average number of islands of the recovery zones versus the number of fibre breaks. Composite parameters are ~*= 40MPa, ~ = 1000MPa, Pmin--500MPa, 2 = 4mm -1, LT = 35mm, r = 0.18 mm. (After Glushko et al. [192].) 1.0
;
I
J
0.8 -
0.6
0.4
0.2
0.0 0.50 p
0.75 Z/ G P a
1.00
Fig. 5.26. The total length of the recovery zones normalized by the original fibre length versus fibre strength. The composite is the same as described in fig. 5.25. (After Glushko et al. [192].)
5.3.2. Fibre fragmentation test." an experiment B o t h c o m p u t e r a n d p h y s i c a l e x p e r i m e n t s h a v e b e e n p e r f o r m e d in t h e f r a m e w o r k of s i n g l e - f i l a m e n t - c o m p o s i t e systems. A m a j o r i t y o f t h e m relates to p o l y m e r - m a t r i x
Strength and fracture toughness
276
Ch. V, {}5.3
composites. For example, numerical results of Curtin's theory were compared with computer simulation results by Henstenburg and Phoenix [243] of such a type, as well as a known direct application of this theory to analyzing physical experiments [114] is also of such a type. In physical experiments field, we point at two series both performed on oxide fibres in a copper matrix.
Experimental procedures The basis for a physical experiment is the load/elongation curve shown in fig. 5.24, which exhibits sequential load drops associated with fibre breaks. The micromechanics of the load drop is simple [258]: an assumption of the stepwise axial fibre stress distribution around a fibre break, with a half of the recovery length being just free of stress, yields the value of the load drop for a specimen is tested in a sufficiently rigid machine as AQ -
(5.99)
Avr~
1+
where A and L are the specimen cross-sectional area and the gauge length, respectively, a~ the fibre strength at the break point, and E' the average secant modulus of the matrix at a point of the event. A simplest behaviour is observed if the fibre has no strength scatter; in this case Kelly-Tyson relationship, eq. (3.198), is valid, that is
lqm- l, lr. In terms of the average values, this relationship can be rewritten as [243] (O'peak) ak
gfm
= -
(5.100)
r
!
w h e r e (O'peak) is the mean_ value of the peak tensile test achieved in every fragment at
the ultimate state and 6 the mean fragment length. A statistical analysis yields
O'peak --
~
(0"6)
(5.101)
where p(fl)=6/h is the non-dimensional mean aspect ratio with h as the characteristic length for which the mean strength is defined. The values of p was determined by Monte Carlo simulation [243]. It occurs that p lies between 1.34 (for fl ---+ ~ ) and 1.96 (for fl = 4), so the constant on the right hand side of eq. (5.101) varies between 0.67 and 0.97.
Composite strength andfibre properties
Ch. V, w
277
Experiments: the first series
In this series performed by Mileiko and Glushko [428], M I G L fibres (Section 2.2.4) based on the eutectic mixture of oxides AlzO3/ZrOz(+Y203) embedded in a copper matrix were tested. All specimens were obtained by diffusion bonding at a temperature of 600~ a pressure of 80 MPa, and a time of 30 min. The effective fibre diameter was 0.38 mm, the length of a specimen and the fibre length 35 ram. The specimen thickness was about 0.5 mm. Tensile tests were carried out using a sufficiently rigid machine to record load drops through a normal load cell bringing its signal to the Y-input of an X-Y recorder. The experiment, which also includes testing pure matrix specimens (fig. 5.27) that had undergone the same treatment as the composite specimens, provides all the necessary data to calculate both the fibre and fibre/matrix interface strength.
Fibre properties according to the simple approach to the analysis of fibre fragmentation test As a result of the procedure outlined above, the dependencies of the fibre strength on the effective fibre length in single-fibre composites are obtained which are shown in fig. 5.28. It should be noted that after some rough defects at large fibre length have worked out, the further breaking process proceeds in such a way that the dependence a~n) (Ln) can be approximated by a power function. It may call for the use of the Weibull type distribution function. Note also that the procedure results in attaining the interface strength. Corresponding values are also given in fig. 5.28.
150
..,
,
..
\ b 100
50 o.oo
i
]
0.02
E
o 04
Fig. 5.27. The stress/strain curve of the copper matrix, averaged after three tests. (After Mileiko and Glushko [428].)
278
Ch. V, w
Strength and fracture toughness
I
I
I
I
I
|
I
,
!
A
x
,
*
9
~
=
31.9 13.0 20.8 66.6
9**** 9, ' , "
2
,
4
,
,
6
,
,
8
+,
0
,
O0
O0
_
.~x4 ,,g . ,oVo ~ ~r x O +
+
~x
" ~ *
~.
l
i 2
+++++
***** i
,
6
/,,v"mm
,
i 8
x
x,
50.2
xxxxx
4
.~
44. I 18.0
r
tP *
I
o
4*
ooooo-r" = 64.3 MPa anna,, 39.0
%
10
0
,~ +
;~,
17.5
o ~
%*
-,%
MPa
, n
_
~ ~
25.0 21.0
,,xxx,,
x~,+ +
~2Xx
.
9
a,,aa,, 9****
u
O
g.*~Oo~
ooooo-r .a,-.,,.., a r',
,
Xx
oo
9~
8
,
Z~
2, o
1000
,
0
A
31.4 |
1
10
i 2
i
! 4
Ln/'rnrrL
Fig. 5.28. Fibre strength versus effective fibre length in single-fibre composites. AI203/ZrO2(+Y203)based fibre/copper matrix composites.
Fibre properties according to the exact theory due to Glushko et al. Three specimens from the set described in fig. 5.28 are chosen to obtain the fibre strength characteristics according to the exact theory of fibre fragmentation. For the same configurations the Cauchy problem was solved for various sets of the input parameters, #, fl, z, and fixed values of 2 = 4 mm -1 . Also, for a particular specimen, the values of Pmin and Pmax were chosen and then fixed (see above figs. 5.25 and 5.26). The least squares procedure was then used to get the best fit of the calculated curves k(p) to the experimental points (fig. 5.29) and to choose on this basis proper 16
|
I
l
I
J
I
/oo//
12
-
r I' 1 I0/
-
"
0 250
~)5 ......... , ,
500
( ooooe
**p*•
'19
"
No 48 (experiment~ No 49 (calculation).] b(o 4 0 ( e z ~ e r i m e n Q _ J
750
1000
(Y / / M P a
1250
Fig. 5.29. The experimental and calculated dependencies of the number of fibre breaks on fibre stress. The fibre strength parameters providing the best fit are shown in Table 5.2.
Composite strength and fibre properties
Ch. V, w
279
values of these parameters characterizing a given fibre. The parameters are shown in Table 5.2. Experiments: the second series In the second series performed by Houpert et al. [258], A1203 fibres were embedded in a copper matrix by diffusion bonding at two temperatures (950 and 1050~ for 90 min and a pressure of 30 MPa. The interpretation of the tests is given by the authors in terms of eqs. (5.99)(5.101). Using eq. (5.99) with a constant value of E' yields f l - 6 . 6 7 and a 0 - 3440 MPa for 10 = 40 mm (10 is chosen equal to the gauge length of the FFT specimen). It occurs that in the case under consideration, in situ fibre strength characteristics are the same as those for free fibres. Then scaling fibre strength to the mean fragment length 6, -
.0
permits using eqs. (5.100) and (5.101) to obtain the results presented in Table 5.3. One can clearly see that (i) increasing the processing temperature and, hence, improving the interface strength properties, yields an essential increase in the effective fibre strength; (ii) further increase in the interface strength is provided by coating the fibre with a thin layer of niobium metal. It should be also noted that in [258], a method of calculation of the value of E is suggested via measuring the slope of the stress/strain curve on the reloading stage following the load drop. TABLE 5.2 Strength characteristics of A1203/ZrO2 (+Y203)-based fibres Specimen NO
fl
~ MPa
z* MPa
Pmin MPa
Pmax MPa
38 48 49
4.48 4.20 4.55
1061 1001 775
42.8 38.1 40.1
141 184 173
1266 1195 1039
TABLE 5.3 The results of the F F T of A1203/Cu composites. (After Houpert et al. [258].) Fibre:
Diffusion bonding temperature, ~ Average number of fibre breaks Mean strength of the fibre on the length, MPa Interfacial shear strength, MPa
A1203
A1203
A1203 coated with Nb
950
1050
1050
8 4454 50
12.4 4730 80
23.8 5200 170
280
Strength and fracture toughness
Ch. V, w
5.4. Brittle-fibre/ductile-matrix: Fracture toughness The macrocrack in a composite containing brittle fibres can cause multiply cracking in a fibre system. This spreads fracture process zone and can enhance an apparent fracture toughness of the composite. In the previous chapter (Section 4.3), we considered the problem in rather general terms. In the present section we intend to evaluate fracture toughness of brittle-fibre/metal-matrix composites for which the crack propagation through a zone of fibre cracking (see fig. 5.30) is a c o m m o n way and, generally, a wanted one. Other ways of the interaction of the crack with fibre system, as crack bridging (Section 4.4) or fibre pull-out (Section 4.2.1), can contribute into the total energy dissipation at the crack tip not so much. Hence, we shall consider here mainly the first mechanism of the energy dissipation. We start with a discussion of experimental data and then proceed with their interpretation and technological recommendations. However, even before presenting experimental data obtained in testing m e t a l - m a t r i x composites we need to discuss briefly experimental methods of measuring fracture toughness of fibrous composites.
Fig. 5.30. The cracking in the outer layer of a boron/aluminium specimen (vf --- 0.3). The crack tip is at the right side of the picture. The specimen of a thickness 1.2mm was unloaded at K--59 MPa. m I/2 After Mileiko et al. [453].
Ch. V, w
Brittle-fibre~ductile-matrix." fracture toughness
281
5.4.1. Experimental methods Unlike testing metal alloys, that for fibrous composites with any kind of matrix has not been so meaningfully documented. As we see in the previous chapter, the fracture of composites includes a number of processes that normally delocalizes the process zone at the crack tip, applicability of the linear fracture mechanics concept seems to be a priori questionable. Experiments, performed mainly on fibrereinforced plastics, are known which make the questions being reasonable but really make an experimentator to be careful in testing and interpreting testing results. In testing unidirectional glass-fibre/and graphite-fibre/epoxy-matrix composites, Sih [608] determined critical values of the strain energy release rate, G, by using the compliance method, and critical stress intensity factor, K, by measuring critical values of the applied load, and then compared couples of the values obtained by using eq. (4.33). At usual values of fibre volume fraction the comparison appeared to be sufficiently good. On the other hand, at vf -- 0.1-0.2, eq. (4.33) does not seem to work, a possible reason being non-homogeneity of the system. Analyzing experimental data on the tensile strength, a,, and fracture toughness, KIc, obtained for a variety of fibre-reinforced plastics of various structure (unidirectional and laminates as well as random), Harris et al [231] found a linear regression between a, and Kic that occurred to be invariant with regard to composition or structure. Each type of materials behaves as though it contained defects of the order of 1 mm in size. This makes the authors to conclude that the very concept of linear elastic fracture mechanics has no validity for composites. We should remember that fracture criteria like Kic, Gc are local ones by their nature and a degree of the success in the application of such criteria depends on correctness of the procedure. Therefore, as in the case of metals, the original fracture mechanics concept and its modifications are applied to approximate fracture behaviour of composites. When choosing specimen sizes, characteristic load, method of making notch, etc. one is to rely on rather sporadic publications. Specimens used in fracture toughness testing are shown in fig. 5.31. For convenience we give the formulas for calculation critical stress intensity factors [337]. For tensile specimens (fig. 5.3 la-d): g~
~
O:r
- t---~ Y ( l / b )
where t is the specimen thickness and ya _
(l/b)1/2(1.99 _ 0.41(l/b) + 18.7(l/b) 2 - 38.5(l/b) 3 + 53.8(l/b)4),
yb _ (l/b)l/2 (1.98 + 0.72(l/b) - 8.48(l/b) 2 + 27.36(l/b)3), yc _ (lib)l~2 (1.77 + 0.454(1/b) - 2.04(l/b) 2 + 21.6(l/b)3),
(5.102)
Strength andfracture toughness
282
Ch. V, w
(a)
1
!
I ~Q
[ Z
(d)
....
(~)
Fig. 5.31. Specimen configurations for fracture toughness. (a) Single edge notch (SEN) specimen. 1 - - ( 0 . 2 - . - 0 . 6 ) . b. When testing metal alloys, it is recommended to load the specimen via pins of a diameter d -- 0.65b, the corresponding holes being located on the central axis of the specimen at a distance of 1.5b from the notch. (b) Double edge notch (DEN) specimen, l = (0.15-..0.25) 9bb. For testing metal alloys, it is recommended to have gauge length of the specimen equal to 3b. (c) Central notch (CN) specimen, l = (0.3... 0.5) 9bb. When testing metal alloys, it is recommended to have gauge length of the specimen equal to 3b. (d) Compact specimen. / = (0.45...0.55)b. For testing metal alloys, it is recommended to have the width of the specimen equal to 1.25b and the height equal to 1.2b. (e) Single edge notch specimen for bending test. I = (0.45...0.55).bb. When testing metal alloys, it is recommended to have L - 4b. (After the compilation by Kovchik and Morozov [337].)
yd = (l/b)1/2(29.6_ 185(l/b)+ 655(l/b) 2 - lO17(l/b) 3 +639(l/b)4). F o r b e n d i n g s p e c i m e n (fig. 5.31e)"
x* = O__L*~ r(l/b) tv~b
(5 103)
Ch. V, w
Brittle-fibre/ductile-matrix." fracture toughness
ooooo v I - 0 . 3 ***** v I = O. I
SEN-specimen SEN-specimen Bending Compact specimen
. . . . . v I = 0.3 ve = 0 . 2 5
ooooo
80
,
|
i
-
1
I
,
|
i
o o
70
283
i
i
9
0 0
0
6O
o
o
~40
o
3O 0
1
0
i
I
,
1
2
i
t
t /
i
I
i
4 ~t~zz
i
i
i
.,
I
6
Fig. 5.32. The apparent fracture toughness of boron/aluminium composite versus specimen thickness. Specimen configuration according to fig. 5.31. Matrix is 2024-T6 alloy, fibre diameter is 0.1 mm. Experimental data after Mileiko et al. [453].
where Y~ - 3 ( l / b ) 1/2 (1.93 - 3 . 0 7 ( 1 / b ) 1 / 2 + 1 4 . 5 ( 1 / b ) 3 / 2 - 2 5 . 1 ( 1 / b ) 5 / 2 + 2 5 . 8 ( l / b ) 7 / 2 ) .
Here superscript at Y relates the value to a particular specimen configuration shown in fig. 5.31. Note that for four-point-bending specimen L = ( L 1 - L 2 ) / 2 where L] and L2 are the distances between the corresponding points and Q* the total load. Consider now the influence of characteristic parameters and peculiarities of the testing procedure on the values assumed to be critical stress intensity factors. S p e c i m e n t h i c k n e s s When measuring fracture toughness of elastic-plastic metal alloys, it is important to use thick specimens to ensure plane-strain conditions over major part of the crack propagation zone. The stress/strain state of the matrix of a unidirectional fibre composite is three-dimensional and obviously does not depend on the location of the composite layer in the specimen except for the outer layers. (The difference in the behaviour of external and internal layers in a composite specimen is qualitatively illustrated in fig. 5.33 showing fibre cracking at the vicinity of the macrocrack path.) So one cannot expect a dependence of the apparent fracture toughness values on specimen thickness for specimens with, say, more than
284
Strength andfracture toughness
Ch. V, w
Fig. 5.33. (a) The first layer in a boron/aluminium-composite specimen after crack propagation. (b) The fourth layer in the same specimen. After Mileiko et al. [453]. ten layers in total. The experimental data shown in fig. 5.32 3 are in accord with that conclusion. Note that at small thickness (~4 layers) of SEN-specimen the scatter of the data is too large, then the scatter decreases and the fracture toughness values are slightly decreasing with the thickness increasing. Hancock and Swanson reported [228] the same dependence of K* on specimen thickness, t, for boron-fibre/6061aluminium-alloy-matrix composite: K* does not change with t at t > 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.
Strength and fracture toughness
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
Strength and fracture toughness
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.
Strength and fracture toughness
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.
Strength and fracture toughness
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
Strength and fracture toughness
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.
Brittle-fibre~brittle-matrix
V , {}5.8
299
45
~TTTTT
I<,,d, ll
~TTTT I
IIIII!1 llillii, kl!!iiiii IIII!11 ! !.! i.l . . . .
__
Iii!i[II Lllll ll III!1!! III!iiii .......
lll.li 1 1
2
3
[!lll[[ .......
I!lll!ll 4
IJJJ,TTTT,,,! ~'-
lli llii _
I!lll!l I 5
Fig. 5.42. A schematic picture of the sequence of events going on during the failure process of a ceramicmatrix composite. length around a fibre break. The second case applies to composites where the matrix cracks are widely spaced relative to the recovery length in the fibres. Following C u r t i n [ l l 3 ] , consider the first case, that is lm < lf, where If----l,/2 =ra'/2T, Fibre stress a' in the non-disturbed state, should satisfy inequalities O"
-
-
0"** Em t)f
G
_< ~' _ < Vf
(5.12o)
where cr is the average stress in the composite that has passed the onset of matrix cracking. Obviously, a fibre cannot break within a distance If from the previous break. Outside of the +lf zone around the break, an additional break is possible. So an element of fibre length may be called 'intact' if within any interval If of this element there are no breaks. In the Curtin's model, three vital assumptions are made. First, the load dropped in any plane by fibres broken within +lf of that plane is shared equally by all intact fibres in that plane. Second, the variations in axial fibre stress caused by the matrix cracks are neglected, by effectively setting the matrix stress to zero (the variations are really small as shown by eq. (5.120)). Third, the composite contains a sufficiently large number of fibres to neglect variations in the axial fibre stress in each intact region between the neighbouring breaks. These assumptions make the situation completely analogous to a process of fibre fragmentation under the loading of a single-fibre composite. A nature of the problem was considered in detail in Section 5.3. Here we just obtain the distribution of fragment lengths created in each fibre (or equivalently, in a single fibre containing all the fibres in the composite).
300
Strength and fracture toughness
Ch. V, {}5.8
Let p(x) be the fibre fragment distribution function (see Section 5.3), that is the fraction of fragments of a length between x and dx. To obtain the pullout length distribution p(lp) via p(x), we should find the statistics of the locations of the fibre breaks relative to the matrix crack. Because of the first assumption, the matrix cracking and fibre breakage are independent events; therefore, the probability of intersecting a fragment of length x, pi (x), is proportional to length x and top(x), that is x pi(x) - -(-Qp(x).
(5.121)
Because a shorter part of the fragment is pulled out, the distribution of pullout length px (lp) is
2/x
px(lp) --
0
at 0 < L < x/2, at L >_x/2.
(5.122)
Hence, p(lp) --
J0oopi(x)px(lp)dx=-~2
lp p(x)dx.
(5.123)
All the moments of the distribution function p(lp) can be now calculated; however, Curtin determines the mean pullout length (lp) in a more instructive way. For a fragment of size x, the average pullout length is (5.124)
(lpx} -- fo ~ lpPx(lp)dlp.
Averaging this result over all fragment sizes x yields the mean pullout length (lp) --
fo
1 (x
(/px)Pi (x)dx - 4 (x)
(5.125)
where (x"} denotes the nth moment of the distribution function. Ifp(x) is known then (xn) are also known, Curtin shows that eq. (5.125) yields 1 22(fl)60 (/P} -- 4 2,(/3)
(5.126)
where the characteristic length 6~ is given by eq. (4.73). Now averaging work to pull out a piece of a single fragment of length x over all possible values of x we obtain the average pullout work per fibre. 2 ~rg 2. A(x) - rtr~f c ~ l~px(Ip)dlp --~X -
-
(5.127)
Brittle-fibre~brittle-matrix
Ch. V, w
301
The work per unit area of the entire composite is A = vf f0 ~ ~zrr 2 vf "C{X3) Vf 23 (/~) 0060 ~cr---5 - - ~ x pi(x)dx - 12 r (x) = 1--2(fl----~ 21
(5 128)
where the characteristic fibre strength o ~ is given by eq. (4.78). Therefore, we obtain a very simple and predictable result: the average pullout length is proportional to the characteristic length, and the work of pullout is proportional to the characteristic fibre stress times the characteristic length. This should be truly independent of particular hypotheses assumed for the calculations made above. Changing the hypotheses can change numerical factors only. Also changing the statistical procedures can yield a change in numerical factor. Note the difference in the results obtained by Curtin and those by Sutcu outlined in Section 4.2.1 which arose as a result of the two slightly different statistical procedures applied. Returning to the evaluation of the composite strength, let us smooth here a moderate saw-tooth type variation in the fibre axial stress caused by the matrix cracks as this variation hold within the interval defined by eq. (5.120) are small. During loading, the average stress per fibre is a/vf. This load is shared between the fibres with no breaks within + If of the matrix crack and those fibres that are broken within the zone shown in fig. 5.42(4-5). Note that 0!
If = r - 2r
(5.129)
and the equation of equilibrium is obvious: _
Vf
=
-
+--
2~
(5.130)
r
Here s is the stress carried by an unbroken fibre, (/p (s)) the mean pullout length, and Vb a portion of broken fibres that is, actually, the probability P(s) that at least one break occurs in a given fibre within +/f(s) of the crack plane. The first term in the equation is the basic relationship of the Daniels loose bundle theory (Section 5.1). Because we neglect, for the case under consideration, the variations of fibre stress along its length, eq. (5.129) with o' = s may be substituted into eq. (5.130) that yields --=s vf
1-
1
If
Vb(S).
(5.131)
The equation obtained can be solved numerically to obtain a maximum of o which is supposed to be the strength of the composite. An approximate solution can be found analytically after making two assumptions. First, it is assumed that at the critical state, the probability of finding two breaks on the same fibre within + If of the matrix crack is zero. It follows that the average pullout length (/p(S)) = lf/2, independent of s, and eq. (5.131) reduces to
Strength and fracture toughness
302
-- = S Vf
1 --
2
Vb(lf)
Ch. V, w
(5.132)
"
The second assumption is to approximate vb(lf) _ by the fraction m of breaks occurring in a gauge length 2lf. Then vb(lf)--N(2lf, s) where N is given by eq. (2.17). Hence, eq. (5.132) does further reduce to
0 (= s 1 -t)f
' (~00)/~+1)
2
(5.133)
"
The maximum of a(s) is now easily found, that is
I cr* = ~Ovf j~ + 2
(5134/
/~ +-----5"
Equation (5.130) which yields eq. (5.134) can be further simplified assuming [65, 66], that a fibre broken within the entire sample length L does not carry load. Then the second term on the right hand side of eq. (5.130) disappears and the composite strength is
0" - o~
fl-l/e exp(-1/fl)
(5.135)
which is a corrected bundle strength given by eq. (5.3). The stress/strain curve of a composite shown in fig. 4.14, in its part after matrix cracking, is determined by the stress/strain behaviour of intact fibres since the matrix stress is small. Thus the average composite strain is the average strain of the intact fibres at the matrix cracking plane, e = s/Ef. The stress/strain curve of the composite at o > 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
Brittle-fibre~brittle-matrix
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
Strength and fracture toughness
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
Brittle-fibre~brittle-matrix
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.
This Page Intentionally Left Blank
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
Creep and creep rupture
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
Short fibre composites
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
Creep and creep rupture
/ /
"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
Short fibre composites
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
Creep and creep rupture
Here -- I]/O'm(~/qm) 1/m
Z1 < Z2 <
< Z6
The random values X,. are such that 0 ~ Y i -"[ ~i/Zk [--1 (Zi--Zk)/Zk I ~ 1
at i < k ,
)~ - 0
at i - k ,
0 _< Xi --[ ~i/(1 -- Zk) [--I (zi -- Zk)/(1 -- Zk) I _< 1
at i > k .
Since the distribution of value [ ~i [ is that of equal probability, values X~ must also have the distribution of equal probability over the interval (0, 1). Therefore, the central limit theorem will permit us to make a very rough approximation of the distribution of sum ~A~ by the normal distribution. The expectation E(a') and variance p(a') can be found through the corresponding values of the distribution of X]. We have E ( X ] ) - 1/2,
#(ZX])
#(Aq) = 1/12,
- 5/12,
E(a'(Zk)) = (17/12)r
P(1ZAq) -- Zk),
- 5/432,
la(a'(Zk)) -- (5/432)(2z2(1 -- Zk)2
Now we can obtain an approximate expression for the part of the ~(cr) curve depicted by the dotted line in fig. 6.4 in the vicinity of ~*. Neglecting a contribution of the chains which include broken fibres to the total load we have the stress which produces the strain rate ~ as
f0'f" o'f
~7 -- Vf
o'tf (~ t (Zk)) dtr' dz,
(6.24)
oo
Here f ( x ) is the density of normal distribution with the expectation and standard deviation determined above. The result of integrating in eq. (6.24) yields - ~ofl
(6.25)
where eo is given by eq. (6.11), and a complicated expression for fl can be found in [4151.
Short fibre composites
C h . VI, w
315
Kelly-Street's model and its developments Ferris [167] performed a strict analysis of plane motion of linear viscous liquid reinforced with a regular system of the rigid plate with zero transverse dimension. Although the analysis shows that the model just considered describes the situation adequately, limitations of the model are obvious. In particular, the model being of shear-lag type may only be applied to composites with sufficiently high fibre volume fractions because tensile stresses in the matrix are neglected. Equation (6.18) was obtained in a somewhat artificial way. Kelly and Street [305] found, actually, the stress distribution in the fibre by using the shear-lag analysis and then also adding the average tensile stress in the matrix, #', corresponding to a prescribed value of the composite strain rate, ~, to the average stress in the fibre, ~'. They obtained 6" t =
~)O'm
(6.26) m
where d is the fibre diameter and
m
-ljm
Finally O" =
'm+l'm O'tt~f -Jr- O"m t~m =
O"m
l)f + Vm
) 9
(6.27)
m
Obviously, eq. (6.27) being solved for the creep rate degenerates to the behaviour of the unreinforced matrix at vf = 0. Goto and McLean [205, 206] introduced in the Kelly-Street's model an interface layer with creep properties different from those of both the matrix and the fibre. This layer is expected to occur as a result of interaction of the matrix dislocations with the fibre. The elastic fibre is an obstacle to dislocation motion, so approaching a fibre, the dislocation bows leaving a loop around the fibre. If the fibre/matrix interface is totally incoherent, this occurs to be a sink for dislocation loops. Therefore, the dislocation density in the matrix does not expect to grow, but it weakens the interface providing a possibility for boundary slipping. The width of the process zone is of atomic dimension, thus if the zone is to influence composite behaviour this will not depend on the fibre diameter. If the interface is fully coherent and Ef > 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
Creep and creep rupture
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
Short fibre composites
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
Creep and creep rupture
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
Short fibre composites
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
Creep and creep rupture
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 and ~, - (1 - 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
Short fibre composites
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
Creep and creep rupture
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
Creep and creep rupture
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
Creep and creep rupture
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
Continuous fibre composites
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) << 1. The value of the ultimate strain is
(6.68)
328
Ch. VI, w
Creep and creep rupture
200
vf=O.19 vf=O.10 o
vf=O.14
100
vf=0.08
vf=0.19 vf=O.08
vf=O.12 0
|
1
|
i
|
|
i
9
|
'10
.
.
t./h
.
.
.
.
.
iO0
Fig. 6.9. The original experimental data on stress rupture of composites with A1203 - ZrO2 fibre and nickel superalloy matrix at a temperature of l l00~ The points are labelled with values of the fibre volume fraction. After Mileiko and Glushko [428].
e, =
27, 1 - t~m-l)/2 p
(6.69)
m-1
where ,/, is the ultimate shear strain at the interface. Hence,
2B(m- l)+2m r/m
0"o
/~7(/~+1) "
(6.70)
Finally, we can write the rupture time in the form
t, -
+
(6.71)
where A and B are the constants which are determined by eqs. (6.64) and (6.70). To illustrate the appliance of the model to the analysis of the creep rupture b e h a v i o u r of a composite, we give the results of testing a c o m p o s i t e with the fibre based on a eutectic in the A 1 2 0 3 - Z r O 2 - Y203 system and nickel superalloy matrix G S - 6K 1 [428]. Creep rupture tests were p e r f o r m e d in bending at a t e m p e r a t u r e of 1100~ The specimens contained different fibre volume fraction, so a traditional representation of the test results as shown in fig. 6.9 does not permit to
1Russian trade name
Continuousfibre composites
Ch. VI, w
329
draw any conclusions. Therefore, we have to account for the fibre volume fraction when analyzing the results. If we assume that the main part of the creep life is the fibre breaking process then the creep-rupture time will be given in eq. (6.64). Because we do not know exact values of m beforehand, so we should assume appropriate values of m within a reasonable interval. Generally, m ~ 1 with the temperature increasing [184, 559]. Some data compiled in [95] show that for the particular matrix alloy under consideration the value of m should be about 2 at 1100~ However, assuming m < 4 yields to negative value of F(vf) given by eq. (6.66), and the relevance of the assumption that t, ~ t! 1), looks doubtful. The results obtained for assumed values of m equal to 4, 5 and 6 are shown in fig. 6.10 together with the values of m giving the best fit of the experimental data to the power approximation according to eq. (6.64). One can see that the reasonable consistency between the assumed and resulted values of m takes place at too large value of m to support the original hypothesis on the fibre breaking process being the longest process within the creep life of a specimen. Now let us assume that the main part of the creep life of a specimen is creep of the specimen with fibres broken to length approximately equal to the critical length, p*. We can rewrite eq. (6.70) as t! 2) --
R(ffo/Vf)-m(~ +1)
(6.72)
where the constant, R, does not depend on the fibre volume fraction. The experimented data presented according to eq. (6.72) are shown in fig. 6.11. If we take into account the values of/~ for the fibres used which are obtained in an independent series of the experiments at room temperature (see Section 2.1), that is //,~ 2.5, we obtain m ~ 1.5. This estimation for the value of m is quite reasonable, so we may conclude that the second hypothesis is supported by the experimental data for the composites under consideration within a specific interval of fibre volume fractions. Finally we note that experimental study of fibre breaking kinetic during creep is really necessary to make proper corrections to the simple model. Such studies are now appearing [ 151].
6.2.3. Off-axis loading Off-axis loading, in addition to a clear case of the unidirectionally reinforced composites, is expected to happen in a cell of the composite with the randomly oriented reinforcement. Miles and McLean [399, 463] observed anisotropy of creep in an experiment with (Co, Cr/CrvC3) in-situ composite and interpreted the data obtained in a rather simple fashion, just adopting Kelly's description of off-axis properties (Section 7.4.1). Considering creep of a unidirectionally reinforced composite loaded in tension at an angle q9 to the fibre direction, they assumed the composite strain in the load direction to be the sum of three independent components. The first is that due to
Creep and creep rupture
330
Ch. VI, w
~ = 4
\ 13 0
10 o
0 o i
i
i
l
s
1
i
i
i
i
u
I
i
s
10
B
2
,
,
100
,
,
, , f
9"n=5
\ b o o
0
~
10
o
"
o
o
o 0 l
I
,
,
I
,
,
,,I
I
,
,
i
,
I
10
,
,
,l
,
, , ,
9
100
t/h
,
,
,
,
,
rr~=6
\ b o o
10
0
""
o
0 o i II
i
i
i 8
i
i
l
i
I
10
i
,
9
9
i
I
t/h
a
1
i
1
100
Fig. 6.10. Stress rupture data given in fig. 6.9 presented according to eq. (6.64) assuming the value of m are equal to 4, 5 and 6. The straight lines, corresponding to the power approximation, are labelled with the value of m which provide the best fit to the experimental points. After Mileiko and Glushko [427].
Ch. VI, w
331
Concluding r e m a r k s
,
,
~
v.f
ooooo
T
-
vf -
Do~GD
i
,
v
0.1
0.08
i
!
-
0.2
0
(:3
1000 -
[]
1
4
2
1
,
|
i
4
!
I
6
t
6
l
l
10
|
2
t
~
t/h '
,
J
~
J
ii
"100
Fig. 6.11. Stress rupture data given in fig. 6.9 presented according to eq. 6.72. The approximation shown does not take into account two experimental points (vf --0.08). After Mileiko and Glushko [427].
creep of the composite in the fibre direction together with corresponding transverse contraction. The second component is arisen as a result of the matrix shear under the shear stress component acting in the matrix. The third is caused by the matrix tensile creep. The final result, i.e. the dependence of creep rate of the composite on the angle q~ provided the applied stress is constant, is -- ~o
(
,
)
COS2(n+l) (./9 _ 2 Cos2n 99 sin 2 q9
+ ego
"7-
)
(sin qocos (./9)m+ 1+ sin 2(m+1) qo
where ~0 and ~90 are the creep rates for q~ = 0 and 90 ~ respectively, m and n are the exponents in the power laws for the matrix and composite, respectively. Note that a purely phenomenological approach to this problem is also known [478].
6.3. C o n c l u d i n g r e m a r k s
We see that the creep behaviour of various metal matrix composites has been theoretically studied nearly thoroughly. One can obtain the creep and creep-rupture properties via those of the components and known composite structure, with the accuracy that satisfies present practical requirements. However, the theories developed are mainly of an approximate nature, so numerical analyses of more rigorous models will provide a means to sort out the theories. The experimental works in this field do not seem to have revealed all the features of creep and creeprupture processes. Also some effort is needed to connect the results of microscopical
332
Creep and creep rupture
Ch. VI, w
theories with the mechanical ones. In particular, including a characteristic size of the work hardened zone into mechanical calculations seems to be useful. Also, similarly to the effect of influence zones on the failure behaviour of composites discussed above, in Section 5.2.2, an effect of the reinforcement on the matrix microstructure can be important. Creep behaviour of composites with random reinforcement by short fibres is to be analyzed to complete the mechanical theory. Perhaps this should be done along the lines of the theory of plastic deformation of polycrystals. This is important because a better understanding of the high temperature behaviour of ceramic matrix composites with random reinforcement geometry is necessary. Despite the last decade having brought about a number of the experimental data on creep of ceramic based composites (see, for example [161]) as well as some approaches to analyzing high temperature creep of short-fibre reinforced ceramics (for example, [702]), this field remains to be systematized and developed.
Chapter VII FATIGUE AND BALLISTIC IMPACT
Fatigue of metal and ceramic based composites has been studied mainly in experimental works. Micromechanical models of a predictive nature are rather deficient. A similar situation in the field of composites aimed at armour applications. Moreover, there are some experimental data on fatigue and ballistic penetration for the same simple composite structures which occur to have common features. So we combine the discussion of these two properties of composites in one chapter. 7.1. Fatigue of a metal laminate
Fatigue cracking of a simple laminate containing homogeneous metal lamina assembled by diffusion bonding provides a good illustration of the possibilities to arrest cracks by weak interfaces discussed in Section 4.5. A model experiment was carried out [16, 418] by using specimens composed of 10 layers of the A1-6% Mg alloy, connected to each other by diffusion bonding in vacuum. Temperature and pressure of the process were 525~ and 30 MPa. The interlamina strength was varied by changing the process time from 10 to 120 min. Fatigue tests were conducted by vibrating a bending specimen with natural frequency, which was between 400 and 500 Hz, at a constant amplitude of the free end. During the test, the natural frequency of a specimen decreased as a result of stable cracking, and after the frequency decrease reached 5% of the initial value the test was stopped assuming the specimen gained an essential damage. The corresponding points are black in fig. 7.1. If after 106 cycles the specimen did not reach the presupposed value of the frequency decrease, the test was also stopped and the specimen was labelled as non-failed. The related points are white in fig. 7.1. Fatigue strength of the laminated metal composite occurs to be essentially larger than that of the corresponding homogeneous material. In a homogeneous specimen, a crack initiating at a most dangerous defect, that is a defect that generates a largest amplitude of the stress intensity coefficient, AK, promptly goes ahead of all other cracks due to non-linear dependence of the crack rate on AK [634]. This crack is practically alone and its rate determines the fatigue life of the specimen (fig. 7.2a). The crack approaching an interface in a laminated specimen can cause the localized delamination and this means usually the crack arrest. Some examples of such a situation can be seen in fig. 7.2b, c. In a case of the strong local interface bond, the 333
Fatigue and ball&tic impact
334
1.4
,
,
i
I
i
i
l
i
'
'
I
I
i
t
i
I
'
i
i
I
Ch. VII, w '
i
,
b D
( o
b
~P
1.3
b
9
9
9
9
S
o
/
)
9
0
/ o
1.2
()
9
9
9 9 9
9 9
I.I
0
l
0
J
~
]
20
,
J
I
i
40
I
I
I
t /
J
60
I
I
rain
l
[
80
I
i
I
[
100
I
I
I
120
Fig. 7.1. The fatigue strength of the laminated aluminium composite normalized by that of a homogeneous metal versus diffusion bonding time; the fatigue base is 106 cycles. Homogeneous specimens have been heattreated at 525~ for 10 min, a0 = 80 MPa. (After Anishshenkov and Mileiko [16].)
crack penetrates into the neighbouring layer. Such examples can also be seen in fig. 7.2b, c. We may definitely assume a non-homogeneous structure of the interface if we take into account Shioiri's experiments described in [419]: during an essential portion of diffusion bonding time long wave roughness of the surfaces are smoothing, then during even longer time, short wave roughness are disappearing. Therefore, the crack can meet various parts of the interface. For short diffusion bonding time, the delamination may not be localized and the global delamination can lead to fatigue life of a laminated metal shorter than that of a homogeneous metal. The experimental data presented call for computer simulation to make the situation more clear. This has been done in [152] and the procedure is as follows. It is assumed that the surface of a specimen carries a number of defectsmicrocracks (fig. 7.3). Let ~ be the distance of a point of the surface profile from the average line of the profile, that is a random value with the normal distribution, the dispersion being 2. The dispersion is determined by the arithmetic mean R~ of the profile hills and valleys, i. e. 2 - (~R~) 2. The number of defects is assumed to be such that the average distance between them is 2R~. A realization of random value represents a particular set of initial lengths of the microcracks. The interface is modelled by an array of the ideal bonding parts and with no bond in between. The portion of the ideal bond is s. The size of a single portion of the
Ch. VII, w
Fatigue o[ a metal laminate
335
Fig. 7.2. (a) The fatigue crack in a homogeneous specimen. (b, c) The multiply cracking of laminated specimens under fatigue conditions. The fatigue strength of the materials is presented in fig. 7.1. (After Anishshenkov and Mileiko [16].) interface is a r a n d o m value, r, distributed h o m o g e n e o u s l y on the interval [-(1 - s)d, sd] with density p ( r ) = 1/d. If r > 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].)
Fatigue and ballistic impact
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
Fatigue and ballistic impact
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
Fatigue and ballistic impact
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
Fatigue and ballistic impact
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.
Fatigue and ballistic impact
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
Fatigue and ballistic impact
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
Fatigue of metal matrix composites
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].)
Fatigue and ballistic impact
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].
Fatigue of metal matrix composites
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
Fatigue and ballistic impact
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"
Fatigue of metal matrix composites
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
Fatigue of metal matrix composites
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
Fatigue and ballistic impact
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
Fatigue of metal matrix composites
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].
Fatigue of metal matrix composites
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 _<
j=~l ( l o g (Nj(n)(o'o)) - l o g ( N i ( n ) ( o ' o ) ) )
where N(n) is the mean value of the cycle number in M computer simulation procedures at a given value of a. Table 7.5 contains all the data characterized fatigue of boron/aluminium composite with the foiled matrix (Table 7.3). The values of p are evaluated by comparing computer simulation data with experimental values of the ultimate cycle numbers, the values of s are estimated by comparing experimental naturalfrequency/cycle-number dependencies and simulated ones (an example of the latter dependencies is shown in fig. 7.22). A comparison of the experimental and simulated SN curves is presented in fig. 7.24. Figure 7.25 shows characteristic patterns of fatigue fracture obtained in computer experiments. Two points should be emphasized. First, the accuracy of the approximate evaluation of the values of n a n d / / b y using eq. (7.20) appears to be very high (see Table 7.5). This means that the damage accumulation is the main process in the fatigue of metal-matrix composites.
TABLE 7.5 Characteristics of B/Al(F) composites (see Table 7.3) which were used in the computer experiments and those obtained in these experiments (see text for the notations).
Vf
/~f
2m
My
no
n1
n(opt)
fll
fl(opt)
s
p
0.09 0.30 0.53
1 0.5 0.5
0.5 0.5 1
4 7 9
4.7 13.7 8.8
4.70 13.50 8.81
4.5 13.6 8.6
1.75. 10-3 9.75. 10-5 2.87. 10-6
12. 10-4 4. 10-5 8. 10-7
0.1 0.5 0.8
0.3 0.7 0.8
Mx -- 125, an = 981 MPa The value of no is obtained by using eq. (7.20), set (n 1, ill) is obtained assuming s = 0, p = 0.
Fatigue and ballistic impact
362
40"~
xu
8
,.
,,
[]
,,
|.
"El
[]
%, =
,
|
|
,
|
Ch. VII, w
|I
cii
I
0.30
121
\ oO
vs =
0.09
oO Oo
O~ o0
N
Fig. 7.24. Experimental and calculated fatigue curves for boron/aluminium composites with a foiled matrix. The calculation was carried out using the optimal sets of the material parameters. Open points stand for the physical experiment and solid points are for calculated results. (After Mileiko and Suleimanov [457].
Element
~Element (~)
accumulated
ultimate
damage
f r a c t u r e d by the m i c r o c r a c k j u m p s = O
D ~ I I)QXD
(b)
s = 0.5
x~ • 1 6 2
x x > ~ x xlI
,OOO@@~
l U U U I~ l U U U l~ IUPJ PJ 9
m
,,, ,,,,,,,
igln glgl i~Iololololol
Fig. 7.25. Two patterns of the fatigue fracture obtained by the computer simulation. (After Mileiko and Suleimanov [457].) S e c o n d , r a t h e r u n e x p e c t e d l y , the values o f p r o b a b i l i t i e s p a n d s a p p e a r to be m u t u a l l y i n d e p e n d e n t . T h e v a l u e o f s d o e s definitely d e p e n d on the fibre v o l u m e fraction.
Ch. VII, w
Fatigue."further experiments
363
For low values of the fibre volume fraction (up to about 10%), s ~ 0-0.1. This means that the damage does not penetrate the specimen severely (fig. 7.25a). At large fibre volume fraction (about 50%), s ~ 0.6-0.8, and the fracture pattern (fig. 7.25b) differs drastically from the previous one. This difference is also obvious in the dependencies of the natural frequency on the cycle number (fig. 7.14 and fig. 7.22). We say that composites with low values of vf are fracturing in the manner that could be called as cyclically stable. On the other hand, composites with high values of vf are fracturing spontaneously at low levels of the damage accumulation. A physical reason for this is certainly the existence of two types of the interfaces in composites, those being of the fibre/matrix and matrix/matrix types. The properties of these interfaces and the conditions for cracks to occur and to be arrested at the vicinities of the interfaces may be different, this yields a two-parameter description of the propagation of two types of the cracks of different characteristic lengths.
7.4. Fatigue: Further experiments To analyze experimental data, we need a means for their classification. For the classical stress-life approach, a Talreja's scheme [641] provides such a means. Considering unidirectional fibre-reinforced plastics and staying with two extreme damage mechanisms, fibre breakage and matrix cracking, he plotted the fatigue-life diagram as a strain/cycle-number dependence containing three areas (fig. 7.26). The first area is a band parallel to the N-axis centered around the composite fracture strain, e*, this corresponds to fibre breakage. The second area is that below strain em in which no cracks or only non-propagating cracks may occur in the matrix material, that is the fatigue limit of the matrix. These two areas are connected by a sloping band corresponding to matrix cracking and interfacial shear failure. In the first area, the fatigue-life dependence is weak. Actually, during cycle loading, the number of broken fibres is increasing, while the distribution of the broken-fibre sites remains random and the process continues until a cross-section, having a stress sufficiently high to break the remaining fibres in it, appears. Hence, the main reason for the very dependence is the fact that the probability of finding a cross-section with enough broken fibres to cause failure increases with increasing number of cycles. In the sloping band, the matrix cracking is cycle-dependent and its rate is straindependent. This is the reason for a strong fatigue-life dependence. In fact, a case corresponding to cm > c, has also been considered. In this case again, all the events take place around the value of ~ = ~,. It is useful to rely, in some cases, on the Talreja's scheme analyzing l~ehaviour of metal-matrix and ceramic-matrix composites. So let us generalize this scheme, the characteristic value of e* remaining and introducing the value of ~min to characterize the fatigue limit of a component which is not cracking under fatigue loading of the composite. The generalized Talreja's scheme is presented in fig. 7.26.
Fatigue and ballistic impact
364
~
t
Ch. VII, w
cracking,
~mi~
iV Fig. 7.26. Generalized Talreja's fatigue-life diagram for unidirectional composites under loading in the fibre direction. The first component is that with a lower value of the ultimate strain. In the case of metalmatrix composites, this is normally the fibre, in the case of ceramic-matrix composites, this is the matrix.
When we considered fatigue of laminated metal plates in Section 7.1 we used, in fact, a defect-tolerance design approach to fatigue, which is based on the assumption that the fatigue life is the time or the number of cycles to propagate a dominant flaw or microcrack of an assumed or measured initial size to a critical dimension [634]. Usually the dependence of crack propagation rate on the stress intensity factor amplitude AK looks as shown schematically in fig. 7.27. In the log-log plot of this dependence, three distinct regimes of crack propagation can be identified. In the slow-growth rate regime, in which the average crack increment per cycle is of an order of magnitude of crystal lattice parameter, the dependence is non-linear in the log-log plot. An essential feature is the existence of the fatigue threshold AK0. In the mid-growth rate regime (Paris regime), linear variation of with log(AK) is observed, which corresponds to the Paris law, eq. (7.17), with m = 4 for metal alloys. We will see that for composites the value of m is much larger. In the highgrowth rate regime, crack size is such that
dc/dN
log(dc/dN)
Kmax "-
AK 1 -R
~ K,
where, as usual, R = O'min/O'max, K , is the critical stress intensity factor for "instantaneous" fracture. In some cases, cracks of the sizes smaller than those corresponding to the fatigue threshold, AK0, propagate with rates that are an anomalous high as schematically shown in fig. 7.27. Another anomaly of the behaviour of such cracks called usually
Fatigue:further experiments
Ch. VII, w
lO -z
Slow-growth rate
Mid-~rowth rate
S m a l l cracks
\ 10 -~
~
High-~rowth rate
'I
x~10 -3
-,
t K,
I I
\\
1 0 -5
365
\
1 0 -6 10
-v
l0
-s
! l~o
I
I
~og(~)
Fig. 7.27. A schematic representation of the different regimes of fatigue crack propagation. The solid lines correspond to long cracks, dashed lines describe the behaviour of two possible kinds of short cracks. (Compiled after Suresh [634].)
small cracks [634] is that their rate decreases with stress intensity factor amplitude increasing up to nearly the threshold for long cracks. 7.4.1. Metal matrix composites
We shall start with continuous fibre composites, in which fatigue mechanisms have been clarified, and then proceed with discontinuous fibre composites in which fatigue crack behaviour has not been studied in detail. However, in view of growing importance of the latter materials a brief discussion of their fatigue behaviour will be presented. Unidirectional
composites
The SN curves for boron/aluminium and steel/aluminium composites (fig. 7.15) cannot be compared with the Talreja's scheme because of lack of the data for low cycle numbers. Also we should not expect this scheme to be satisfactory with regard to the boron/aluminium type composites of low fibre volume fractions. On the other hand, the results obtained by Vauterin and Bunsell [672] in testing boron/aluminium composites with large values of vf and various alloys as the matrix over a sufficiently large interval of N values, seem to follow this scheme. Actually, the plateau over low cycle numbers for the composites tested as well as an apparent dependence of the threshold on the matrix fatigue limit, correspond to the scheme. The authors' microstructural observations reveal that at the beginning of the fatigue loading, fibre breakage occurs, this incites the intensive matrix cracking corresponding to change from the plateau to the sloping band.
366
Fatigue and ballistic impact
Ch. VII, w
Leis and Peters' data obtained on boron/aluminium (vf = 42%) and siliconcarbide-fibre/aluminium-matrix (vf = 60%) composites [359] fit also to the Talreja's scheme: there is a weak dependence of the fatigue strength on the cycle number, N, at low values of N and the ratio of the fatigue limit on the base of 5 9106 cycles to the ultimate tensile strength is about 0.72 for the former material and about 0.81 for the latter one. The second fact may be explained by the quicker stress transfer from the breaking fibres to the matrix in the case of boron/aluminium (with lower fibre volume fraction) and, correspondingly, by the quicker shift from the horizontal band to the sloping one in the fatigue-life diagram. In a systematical study of the fatigue crack propagation in a B/B4C-fibre/ Ti-6A1-4V-matrix composite (vf = 0.55), Chan and Davidson [72] measured crack rate in SEN specimens which were tested in an as-received condition and after two thermal treatments leading to a decrease in the fibre/matrix interface strength. Fatigue crack in the as-received composite propagates through the matrix and fibres remaining generally the initial configuration of mode I type. On the other hand, in thermally exposed materials, interfaces are observed to debond while the main crack is still far from the fibre (about 100 lam). In the latter case, the crack is disintegrated into a set of the cracks of two modes, those being mode I and II. This enhances the crack resistance but at the same time the scatter of experimental data becomes very large (fig. 7.28).
10 6
,
.
.
.
.
!
900~
(D
9
4.5 h
,
500~
168 h
1 0 "7
!1
1 0 -8
< Matrix
l O .9
'
9
'
'
'
'
' lO
'
'
AK / MPs. m u2
Fig. 7.28. Fatigue crack rate for B4C-fibre/titanium-matrix composites. The matrix data are for a laminate material used in fabrication of the composite. (Experimental data after Chan and Davidson [72]).
Fatigue."further experiments
Ch. VII, w
367
In a model developed by the authors, they used a simplified plane geometry of the composite to consider the behaviour of the main crack approaching two fibres embedded into the matrix. (To obtain the geometry of the model, withdraw all the fibre, except two ones at the front of the macrocrack, from the model depicted in fig. 4.10). Each fibre is divided into a number of longitudinal segments. The strength of the segments and interface strength vary according to the Weibull distribution. The criteria for fibre fracture, interface tensile debonding and interface shear debonding are written in terms of the stresses which are calculated assuming the composite being an anisotropic continuum. The co.nfiguration of a crack system determines a value of the stress intensity factor at the tip of each crack. Amplitudes of these values predetermine the crack rates according to twocomponent equation
dNda_
A1
1/2/~
+ A2
~
(7.22)
where A1, A2, nl, and n2 are constants evaluated from experimental crack growth data of the matrix material. The coefficient stands to correct the nominal AK to estimate the stress intensity factor in the matrix. Calculations performed in [72], assuming a particular set of the material constants including values of the interface strength, yield a surprisingly good agreement with the experimental data (fig. 7.28). Qualitative conclusions drawn by the authors from their experiments and computational analysis are important. First, low interfacial strength and strong, high modulus fibres lead to interface debonding, crack deflection at the fibres, and extended crack propagation along the fibre/matrix interface, with the consequence of increasing the fatigue threshold and reducing crack propagation rate. Secondly, the improved crack resistance of the thermally exposed B/B4C-fibre/titanium matrix composites originates from a reduction in the interfacial strength without a reduction in fibre strength. The experiments also show [72] that the stress intensity amplitude in a composite for a crack to grow transversely to the fibres is much larger than that in the matrix to cause the same crack propagation rate if the rate is relatively low. As the crack rate increases the difference is tending to zero. It should be noted that the crack rate depends on the stress intensity amplitude very strongly (see fig. 7.28), the exponent in the Paris law (7.17) measured for unidirectionally reinforced composites is much larger compared with typical values for metals. For example, fatigue testing of AlzO3(FP)/magnesium composites in [378] yields m - 16-18. So the log-log plot of the dependence of crack rate on either stress intensity amplitude AK or maximum stress Kmax in the cycle looks rather like a vertical band of experimental data. Perhaps, it will be useful to introduce AK,, a characteristic stress intensity amplitude, such that if AK < AK, then the crack may be assumed not to propagate, and if AK _> AK, the crack propagates in a brittle manner. For the composites under consideration [378], Kmax ~ 89 The authors noted a striking observation, i.e. for the same values of
(Era~E)1/2
368
Fatigue and ballistic impact
Ch. VII, w
Kmax, the crack propagation rates varied from one composite plate to another over several orders of magnitude. The variation of the values of Kmaxfor the same values of crack rate was also noticeable. This fact makes the introduction of the characteristic value of AK* o r Kmax even more practicable. When a load is applied in the direction normal to the fibre direction, the fatigue crack grows either through the matrix or along the fibre/matrix interface. Such a transition takes place in both AlzO3(FP)-fibre/magnesium matrix [378] and SiCfibre/aluminium (6061) matrix [674] composites. Perhaps, this may be explained [378] by enlargement of size r of the fracture process zone when AK increases, since rcx (AK/o'm) 2. In the case of AlzO3(FP)/Mg composites, the value of r varies from 11 to 70 lam as AK increases during the test from 4 to 10 MPa m 1/2, with a mean centre-to-centre fibre spacing being 60 gm. The fatigue crack growth under such conditions can be faster than in the unreinforced matrix [674]. A characteristic macroscopical feature of the fatigue behaviour of unidirectional metal-matrix composites is a high ratio of the fatigue strength (on the base, say, of 106 cycles) to the tensile strength (RFS) as compared to this ratio for metal alloys. The foregoing discussion has revealed some reasons for this. Figure 7.29 presents some experimental data for a variety of the composites that show a tendency for the RFS value to go up with the fibre volume fraction increasing over interval (0.3, 0.6). This trend may be certainly caused just by a slow growth of the tensile strength or even decrease in its value over the interval of vr (see Section 5.2).
0.9
~
w
ooooo oDooo
aaaaa
0.8
I
~
w
~
I
Gerold Leis & Peters Nzenes et al
0.7 tJo/tJ*
A O
0.6
A
O
O
0.5
.4
i
0.3
|
|
1
0.4
l
|
vt
J
i
0.5
i
a
i
0.6
Fig. 7.29. Ratio of the fatigue limit to tensile strength of metal-matrix composites versus fibre volume fraction. (Experimental data for B/A1 composites by Leis and Peters [359]; for AIzO3(FP)/Mgcomposites by Nunes et al. [498]; for B/A1, SiC/AI, AIzO3/AI, and AlzO3/Mg are compiled by Gerold [201].)
Ch. VII, w
Fatigue."further experiments
369
Laminates
Fatigue of metal-matrix-composite laminates was reviewed by Johnson [286]. It occurs that the fatigue behaviour of the laminates inherits the mechanisms of fatigue failure of unidirectional composites. However, the presence of off-axis plies means their inevitable cracking under sufficiently low applied stresses that is very similar to the well documented case of fibre reinforced polymers (see, for example, [182]). In the case of matrix dominated damage, in laminates containing 0 ~ plies, the damage accumulation, expressed as a dependence of the effective modulus of the laminate, normalized by its initial value, on the cycle number, appears to reach a "saturation damage state". The higher the applied stress the lower is the effective modulus at the saturation level. If the laminate contains plies at 45 ~ crack in the matrix in these plies are growing parallel to the fibre direction. These cracks do not appear to penetrate into 0 ~ plies. When a laminate does not contain 0 ~ plies, there is no fatigue damage provided the applied stress induces the matrix stress/strain state below the shakedown limit, otherwise matrix cracking takes place. Johnson [286] suggested a guideline for a designer using metal-matrix laminates under fatigue loading by plotting a dimensionless map to formalize the above considerations. Discontinuous fibres
Fatigue behaviour of metal-matrix composites reinforced with randomly oriented short fibres, possessing the characteristic features of behaviour of unidirectionally reinforced composites, reveals some special features. According to Komai et al. [322], two kinds of fatigue crack initiation sites in SiCw/AI composites can be observed, i.e. surface sites and internal ones. In the latter case, the fatigue failure process includes: (i) debonding at whisker clusters and its subsequent development; (ii) initiation of a fatigue crack at a whisker debond when the size of the debond exceeds a critical size, that is, the stress intensity exceeds a threshold value, AKth; (iii) fatigue crack propagation and final unstable failure. Fatigue cracking of the SiCw/Al(7075) composite at low values of the stress intensity factor propagates with a zig-zag path, avoiding whisker clusters, whereas at higher stress intensity levels, cracks grow rather straight with whisker pull-out and whisker breakage. 7.4.2. Ceramic matrix composites
Silicon-carbide-fibre/aluminosilicate-glass-matrix composites are perhaps the only composite system that has been systematically studied under fatigue condition [382, 549, 552, 728]. Matrix cracking causes non-linearity of the stress/strain behaviour of such composites (see Section 4.4.1) starting from point (a**, e**) in fig. 4.14. If the maximum stress in a cycle, ama• does not exceed a**, then there is n o damage accumulation under fatigue loading, at least up to 105-106 cycles [549, 728]. The Young's modulus of the material either remains nearly constant or undergoes a small decrease during initial 104 cycles as shown schematically in fig. 7.30. The
Fatigue and baH&tic impact
370
Ch. VII, w
--" O'm~ < O'.,
__e &
O'm~ >O'o,
0.8
10 ~
I 10 a
I 10 6
N
Fig. 7.30. A schematic representation of the fatigue behaviour of SiC/aluminosilicate-glass composite.
situation is within zone 2 of the Talreja's diagram (fig. 7.26) that means that the specimen does not fail until 106 cycles. In this case, the residual strength of a specimen is equal to the initial one. If O'max > 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
Fatigue and ballistic impact
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.
Compressive strength
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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i
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,
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el
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i
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i
i
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15
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2.0
b
o% % %
1.0
1.0
%
go 0.0
I
,
,
,
i
0
,
q9
i
i
I
20
|
|
i
oo i
40
i
i
,
,
i
60
Q
0.0
,0
80
0.00
100
0.01
0.02
0.03
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
100
u
!
!
i
,
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i
u
,
600
,
i ....
i . . . .
u ....
el
75 el
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o~ 9
o
200
I
o
25
i
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i
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400
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0.01
|
,
,
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I
n
n
0:02
|
,
I
,
,
l
0.03
,
0.04
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].)
Compressive strength
378 I000
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Ch. VIII, w
800
.
~
.
i 800
\
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800
600 400 400 200
200 0
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600
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.
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=
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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%.
Compressive strength
380
i
l
u
n
l
'
u
u
u
I
n
i
u
Ch. VIII, w
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o
,,6 &
\ "o
e
0
0
B
El
0
O0
oooo~v!
= = -
66&&.. v! v!
ooooo
nonnnvl I
I
I
0 B o
0.4!
0.41 0.27 0.09
I
I
0
I
I
5
I
i
I
,
i
n
i
10
t/h
15
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].)
3.0
i
i
i
i
I
1
i
u
i
o
0
9
~2.0 0
1.0
as
ooooo
received
***** mal;ri~
0.0 0.00
I
J
I
I
0.25 vl
t
I
aged i
I
0.50
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
Compressive strength
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~ << 1, h / R o << 1.
The energy associated with the bending of a strip is U2 =
Ez
x nFL.
(8.12)
The energy of newly formed surfaces is U3 = n~hL
(8.13)
where ~ is the effective surface energy for the splitting of the material along the fibre direction. The criterion for transition from the state of homogeneous compression to the splitting and strips buckling may be written as
h Fig. 8.9. The cross-section of a strip.
Tubes
Ch. VIII, {}8.2 U0 = U1 + U2 + U3.
383 (8.14)
Substituting eqs. (8.10)-(8.13) into eq. (8.14) yields
O', -- O"E ~- CEz~(0~R0).
(8.15)
The Euler stress is
rc2EzI O'E- FL 2
(8.16)
where I is the moment of inertia of the strip cross-section. In view of the above assumptions, h/Ro << 1 and ~ << 1, we may express the moment of inertia as
I = R3oho~
~4 + g
.
(8.17)
Assuming h/Ro << ~2, we neglect the second member in the brackets. Substituting eq. (8.16), with the value of I given by eq. (8.17), into eq. (8.15) and differentiating the result with respect to c~, we find the value of c~corresponding to minimum critical stress, a,, that is
(2025"~1/9(, ~ )l/9(t) 4/9 EzRo
~Zo = \64rc4 j
~
.
(8.18)
Corresponding value of the critical stress is
(892~1/9( o-** = 1.125 \ 45 /
'~ ) 4 / 9 ( L ) - 2 / 9 EzRo -~o Ez.
(8.19)
It should be noted that eq. (8.18) gives a very weak dependence of the strip number, n, on the material parameters, that is c~0 o((7/EzRo) 1/9, and a rather weak dependence of n on the tube aspect ratio, ~0 e((L/Ro) 4/9. The dependence of the critical strength on the material parameters given by eq. (8.19) is rather weak and that on the tube aspect ratio is very weak, ~r** o((L/Ro) -2/9. A limited number of experimental data obtained in testing tubes made of a glass fibre reinforced plastic seem to support this prediction. Note also that, as was shown in [147], the dependence given by eq. (8.19) is valid if only the circumferential strain, e22, does not reach the critical value, s that is
O" -
-
v12
___
where V12 is the corresponding Poisson's ratio. This corresponds to
Compressive strength
384
Ch. VIII, w
Z _ 2.25( ~ )2(Y12~9/2. Ro
ks
EzRo
8.2.2. Computer simulation model We consider now a more detailed version of the energy balance in the failure process of a tube with longitudinal reinforcement [335]. The model to be discussed is based on experimental observation of the behaviour of boron-aluminium tubes. So we start with the discussion of experimental results and then, before presenting a computer simulation model, we consider an auxiliary problem of the buckling of a strip in a composite plate. Experimental data for boron/aluminium tubes
Testing boron-aluminium tubes of various diameter/length values [426] shows dissimilarities in the behaviour of the simple model considered in the previous section and real metal-matrix composite tubes. First, the dependence of the ultimate stress on the length/radius ratio of a tube is much stronger than that predicted by the simple model. Figure 8.10 illustrates such a dependence. The approximation of the data by a power function leads to r cx (L/R) -078 9 This contradicts the model prediction with the exponent equal to 29 " Second, the growth of the strips in long tubes may be limited as shown in fig. 8.11. This can contribute to the dependence discussed above. 2000
1500
x. b
1000
5OO
0
0
20
L/R
40
60
Fig. 8.10. The ultimate compressive stress versus length/radius ratio for boron/aluminium tubes. The length/diameter values for the tubes are 1200/40, 1000/65, 1500/75 and 300/30 (mm). The tubes have aluminium screw fittings at the ends for applying the axial load, shown in fig. 11.29 (Experimental data after Mileiko et al. [426].)
Ch. VIII, w
Tubes
385
Fig. 8.11. A long boron-aluminium tube after compression testing. These experimental facts (some others shall be presented later in this section) call for a more precise model that would have a technological impact.
Buckling of a strip The problem to be analyzed as an auxiliary one has been considered in several publications varying from very rough approximations to strict approaches. Let a strip in a plate of thickness h be formed by two parallel cracks of length l, the distance between the cracks being H (fig. 8.12). At infinity, the plate is loaded by the compressive stress, - a . Assuming the ends of the strip are clamped rigidly, we may write the Euler stress as
OE =
4 rc2E I SI 2
(8.20)
Here I = H h 3 / 1 2 and S = hH. Consider behaviour of the strip at a > aE assuming the difference, A, of the displacements of the strip ends being determined by the average plate deformation, that is A = __g/. E I
r/ . / ,j
// ,j ,~
f
Fig. 8.12. A delaminating strip.
386
Compressive strength
Ch. VIII, w
Then the Euler's analysis of the supercritical behaviour of an elastic rod [335, 560] yields
{
E~S
P(A) -
64rc2EI ( 4 / + A0 - A) 2
A < Ao; A > A0
(8.21)
where
4x2I Ao-
Sl
"
Because we shall apply the relationship given by eq. (8.21) for A << l, a good approximation will be
= j"
P(0-)
PE;
_< 0- > 0-E
(8.22)
Let now the applied stress, P = aS, to increase. This can make the cracks, creating a supercritical strip, to grow. When the applied load reaches the critical value, PE, the elastic energy of the strip is U0(l) = 89 PEA0- The increment of the elastic energy of the system after the buckling starts will be
1
A U ( I , 0-) = -~0-SA - Uo(l, O-E) -- PE(I) (A - A0(/))
(8.23)
where A = (0-/E)I. Using the Griffith energy balance we obtain the stability condition for the strip in the form 6,4 < 6AU + 4h76l.
(8.24)
Here 6A is the work done by external forces to increase the strip length by 61, 7 is the effective surface energy of the material. At fixed boundary conditions, which means that the value of 0- does not change, we have 6A - 2 6 A U . Hence, eq. (8.24) may be written as OAU
_< 4h7.
(8.25)
Substituting eq. (8.23) into eq. (8.25) yields
0-28 4rc210- 24rc4El 2 2E
~
l2
SI n
< 4h~,.
(8.26)
-
At a - aE the left hand side of eq. (8.26) is zero, so the stability condition is satisfied. Solving eq. (8.26) gives the critical stress" 0-. ( l ) - 0-E(I) (2~/1 q
27E h
E(ll S
1'~
J
(8.27)
Tubes
Ch. VIII, w
387
1.50
~1.25
b 1.00
S 0.75
i
0
I
2
i
I
4
i
Vto
I
6
i
I
8
i
10
Fig. 8.13. The dependence of the critical delamination stress, a*, on the strip length normalized by the characteristic length. (After Kovalenko et al. [335].)
When stress a reaches the value of a* the strip of length l may grow. 1/4 The dependence a*(l) is illustrated in fig. 8 913. The value l0 - ~zI1/2 (2E) 'K
/
involved in the presentation is a characteristic length. If the initial length li is such that li _< 10, then the strip will propagate infinitely when the stress reaches the critical value, a*. At l0 < li < 10v/-3, the strip stops when its length reaches a definite value. At li _> 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
Compressive strength
388
l >_ rc
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
Compressive strength
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
Tubes
,
,
i
,
,
-
:
!
I
i
i
i
,
,
:
9
i
,
250 * $
t i
~200 \
"~ ~5o 9
J
-
100
5O
i
i
7
10
I
i
,
I
13 N
i
i
16
19
Fig. 8.15. The ultimate load of a tube versus the number of strips realized in a particular computer experiment.
300
I
'
I
I
I
-'
I
'
I
'
250 ~200 \
"~ 15o I00
50
10
12
14
N
16
18
20
Fig. 8.16. The ultimate load (the average and standard deviation) of a tube versus average strip number. (/) = 7 5 m m , fl1=24, (z)= 500mm, flz = 16, fl~0=0.1, 7 = 3, • = 0 . 5 J/m 2, L = 1000mm. (After Kovalenko et al. [335].) T o o b t a i n a d e p e n d e n c e o f t h e t u b e s t r e n g t h o n its l e n g t h , w e c o n s i d e r a t u b e as a c h a i n o f e l e m e n t s o f a c h a r a c t e r i s t i c l e n g t h 2. So t h e t u b e is c o m p o s e d o f n e l e m e n t s , L = n2. E a c h e l e m e n t c a r r i e s its o w n s y s t e m o f t h e q u a s i - c r a c k s , all e l e m e n t s a r e s t a t i s t i c a l l y h o m o g e n e o u s . T h e f a i l u r e p r o c e s s in e a c h e l e m e n t d o e s n o t d e p e n d o n
Compressive strength
392
Ch. VIII, w
failure in other elements. The tube is assumed to fail if at least one element fails. Obviously, the value of 2 is to be chosen by comparing a calculated dependence on the ultimate load on the tube length with the experimental one. Figures 8.17 to 8.20 reveal the dependencies of the ultimate load on the characteristic parameters of the material microstructure, the latter are usually strongly effected by technological parameters.
8.3. Shells We shall consider the buckling of a cylindrical shell under external hydrostatic pressure. The behaviour of metal-matrix-composite shells differs from that intrinsic for shells made of polymer matrix composites as the stress/strain behaviour of metal matrix composites differs essentially from that of polymer matrix composites. The elastic anisotropy of metal matrix composites is much less than that inherent to fibre-reinforced plastics (see Section 3.2.5). Therefore, the usual problems aroused when simplified shell theories based on either the Love-Kirchhoff or Timoshenko hypotheses are applied [218], are rather of little importance in this case. On the other hand, the non-linearity of the stress/strain curve of a metal matrix composite, which can be essential at small deformations (see Section 3.3), can be an important feature of the material behaviour. Because of the lack of well-established buckling theories for anisotropic elasticplastic shells, in what follows, we deviate from a rather traditional way of describing
300
. . . .
i
250
. . . .
i
. . . .
w
. . . .
\ \ k \
2O0
\ \
\
"~ 15o 100
0
|
0
,
,
|
i
250
,
i
i
i
1
500
|
/; / r a m
i
|
i
i
750
I
|
i
|
I000
Fig. 8.17. The ultimate load (the average and standard deviation) of a composite tube versus tube length. N = 13, (l) = 7 5 m m , fll = 2, (z) = 100mm, flz = 2, 2 = 2 0 0 m m , fl~0 - - 0 . 1 , • = 0.5, 7 = 3 J / m 2. (After K o v a l e n k o et al. [335].)
Ch. VIII, {}8.3
393
Shells
250 x . . . . . . . . . . . . . . . . . . . \ \ \ \ \
200
\15o
100
0
,
i
,
I
50
i
,
I
i
60
i
70
,
,
I
,
,
,
I
80
,
i
,
90
/ m m
I00
Fig. 8.18. The ultimate load (the average and standard deviation) of a composite tube versus the average length of the quasi-crack. N = 13, 0.1, 7 = 3 J/m 2, ~c = 0.5, L - 1000mm. (After Kovalenko et al. [335].) 300
,
,
200
,
,
t
.
.
.
.
i
.
.
.
.
t
,
,
,"
,
.- 1 i "
50 0
0
5
10
15
20
Fig. 8.19. The ultimate load (the average and standard deviation) of a composite tube versus the effective surface energy. N = 13,
Moreover,
not be accepted
traditional
ways based on a detailed review of shell
h e r e s i n c e it w o u l d
b o o k s u b j e c t . H e n c e , first, w e s h a l l g i v e e x p e r i m e n t a l elastic/plastic
shells made
of an unreinforced
l e a d far a w a y observations
aluminium
from
the present
of the buckling
of
alloy, then experimental
Compressive strength
394
'
200
I
v
I
/ 150
/
/ t
w
I
'
Ch. VIII, w
/
1
t i
/////
\ /
//
J
~
//// / / / / / -
I00
/
/ ~0 0.0
n
I
0.2
/
/
I
I
l
0.4
I
0.6
I
I
0.8
J
1.0
Fig. 8.20. The ultimate load (the average and standard deviation) of a composite tube versus the p a r a m e t e r of continuity. N = 13, (l) = 7 5 m m , fll = 24, (z) = 5 0 0 m m , flz = 16, fl~ - 0 . 1 , ? = 3 J / m 2, L = 1000ram. (After K o v a l e n k o et al. [335].)
data on boron-aluminium shells will be presented followed by a buckling analysis of the cylindrical shell. We shall mainly concern with a descriptive analysis based on an expansion of simple Papkovich's formula derived originally to give the critical pressure of an elastic isotropic shell, to the case of an anisotropic elastic/plastic shell. This provides a means to classify experimental data and to evaluate the effectiveness of both various fabrication parameters and reinforcement schemes. Finally, a rigorous theory of the buckling of an anisotropic elastic/plastic shell will be outlined.
8.3.1. Experimental observations." Metal shells We start with the results of testing of carefully made homogeneous cylindrical shells to get a basis for analyzing the behaviour of composite shells. In the experimental series [323], all the specimens were made from thick-walled tubes of an aluminium alloy of the 2024-T6 type by turning over the external diameter. The internal diameter of the shells was about 43 mm. The length and wall thickness values were such that 0.04 < R/L < 0.5, 0.025 < h/R < 0.14. The first aim of the experiments was to verity the Papkovich's formula for the critical pressure of an elastic shell under hydrostatic loading [525]: p , -- k E ( R / L ) . I 1 6 ( h / R ) 258
(8.35)
where R, L, and h are geometrical factors, k is a constant which depends on the end conditions. The formula is obviously simple and, if an extension of such a simple
Ch. VIII, {}8.3
395
Shells
dependence into the elastic-plastic region occurs to be possible, it may be useful in the presenting and classification of experimental data. Values of the critical pressure for shells buckled in the elastic region are used to correct the exponent value, 2.58, in the Papkovich's formula. This is necessary because it occurs that if one uses the original Papkovich's formula then the values of kE fitting the experimental data reveal too large scatter. The analysis of these data yields m = 2.266, k = 0.850. Writing a relationship similar to Papkovich's formula, that is p , -- k E e ( R / L ) n ( h / R ) m,
(8.36)
we try to fit the experimental data for the elastic-plastic region to it by taking n = 1.16, m = 2.266, k = 0.850 and choosing an appropriate value of the effective modulus which is supposed to be either tangent or secant one, or a combination of them. To understand which modulus is applicable, the stress/strain curve of the material was also determined in the fashion described in Section 8.1.2. Plane specimens to be tested in compression to obtain the tangent modulus and then to restore the stress/ strain curve, were cut off from the tubes which were used to make the shell specimens. The cutting of the plane specimens was performed along chords of the tube cross-section, to make the directions of main stresses in a plane specimen and in a shell to be the same with respect to the material texture. The dependence of the tangent modulus determined according to eq. (8.9) on the stress is presented in fig. 8.21.
0.15
'
I
'
'
'
I
'
'
'
0
7
.\
0.10
I
r~ 0.05
00 0.00
I
0
I
i
1 O0
z.
i
i
i
t
200 o"
I
/
t
I
300
,
,
i
400
MPa
Fig. 8.21. The tangent modulus of the 2024 aluminium alloy versus stress. (After Kondakov and Mileiko [323].)
Compressivestrength
396 8o
~
~
I
'
o
w
,
I
,
o
i
\',
,
i
I
~
'
: ."
d'
0
'
Ch. VIII, w
6O
\ 40
~
20
0
,
0
~
;
!
100
~
;
J
I
z
200
i
\\k\
I
I
300
~ / MPa
i
400
Fig. 8.22. The effective modulus of the shell material versus circumferential stress at the critical pressure. The solid and dashed lines are the tangent and secant moduli of the material, respectively. (Experimental data after Kondakov and Mileiko [323].) By plotting the experimental dependence of the critical pressure versus parameter
R/L in log-log scale it was found that eq. (8.36) may by used if only 0.10 < R/L < 0.33. Now we show values of the effective modulus, Ee, determined from the experimental data by using eq. (8.36) and compare them with the tangent and secant moduli of the material (fig. 8.22). One can see that to describe the experimental data reasonably well it is necessary to take a combination of the Young's and secant moduli as the effective one. However, we shall use the secant modulus further on.
8.3.2. Experimental observations." Composite shells Boron/aluminium shells of four reinforcement schemes (fig. 8.23) made by hot pressing in gaseous isostat (Section 11.3.2) have been tested under external hydrostatic pressure [438, 440]. The matrices and fabrication parameters used are shown in Table 8.1 together with the prescribed numbers to be used further on. The external diameter of the shells was 122 mm. The length of a specimen, L, varies from 50 to 200 mm. Values of the critical pressure obtained in testing the shells with circumferential homogeneous reinforcement H in fig. 8.23 are given in Table 8.2, those for two-dimensional reinforcement (schemes CP and 3-F in fig. 8.23) as well as for scheme 3 - 0 are in Table 8.3. During the loading of boron-aluminium shells, the strains in the central plane of a shell have been measured by strain gauges glued on the external surface of the shell at 8 points homogeneously located along the circle. The results of these measurements (an example is presented in fig. 8.24) show clearly the growth of
Ch. VIII, w
3-0
H
I l
000000006 000000000 000000000 000000000 ooooooooo 000000000
,~
I
000000000 000000000 000000000
000000000 000000000 000000000 000000000
CP
I
397
Shells
ooooooooo
0 00000000
000000000 0 00000000 000000000
R
/7~~
3-F
ooooooooo Tooooooooo
0 00000000
ooooooooo
ooooooooo 0 00000000
j
,~ 000000000 0 00000000/1 000000000
~
Fig. 8.23. Schemes of the reinforcement of the boron-aluminium-shell walls.
TABLE 8.1 Chemical composition of the matrix materials and fabrication parameters, pressure-temperature-time, 0 (After used for making boron-aluminium cylindrical shells, and the effective matrix yield stress, am. Mileiko and Khvostunkov [438].) Number
Matrix alloy
q MPa
T ~
t min
a 0m MPa
1 2 3 4
A1- Mg-Zn A 1 - M g - Zn A1- Mg-Zn A 1 - 6 % Mg
550 550 460 550
520 460 500 500
20 120 30 30
189 239 191 -
initial imperfections with the pressure increasing. When the pressure is approaching the critical value, the wave amplitudes become very large, this may cause the unloading in some volumes of the material just before buckling. A relatively slow growth of the initial imperfections up to quite large values with pressure increasing means that rather rough imperfections do not influence critical pressure vigorously. Actually, three shell models having the differences between a maximum and minimum diameter equal to 2, 4 and 6 mm, were tested [440] and the hypothesis formulated was proved.
8.3.3. Empirical modification of Papkovich formula Following [422] we assume the effective modulus for the homogeneous reinforcement scheme as
Compressive strength
398
Ch. VIII, w
T A B L E 8.2 Boron-aluminium cylindrical shells with unidirectional homogeneous reinforcement of the wall in the circumferential direction. The matrix is A I - Z n - M g - a l l o y for all specimens except for those marked by two stars. (After Mileiko et al. [438, 440].) Specimen number
Fabrication regime
R/L
979 976.1 976.2 978 975.1 975.2 964 963.1 963.2 950 1002 1003 1004 1005 953 1244, 1493, 1494, 114 1890 937** 938** 1492,, 173,,
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4
0.3362 1.1518 0.5650 0.3326 1.1738 0.5701 0.3380 1.2547 0.5791 0.3383 0.3314 0.3313 0.3283 0.3276 0.4309 0.3184 0.3236 0.3253 0.3224 0.3275 0.3380 0.3380 0.3270 0.3270
h/R
vf
p, MPa
0.0384 0.0369 0.0369 0.0376 0.0379 0.0379 0.0276 0.0257 0.0257 0.0376 0.0451 0.0458 0.0530 0.0460 0.0518 0.0384 0.0379 0.0384 0.0453 0.0470 0.0460 0.0451 0.0364 0.0361
0.394 0.410 0.410 0.402 0.400 0.400 0.328 0.350 0.350 0.345 0.277 0.266 0.287 0.330 0.393 0.460 0.370 0.360 0.290 0.340 0.296 0.296 0.380 0.380
10.98 24.72 16.18 11.18 21.77 14.12 5.29 10.79 5.59 10.00 12.85 14.22 15.79 14.81 21.58 11.77 13.43 12.55 12.65 14.12 17.45 12.75 10.30 9.70
9) Fibre diameter is 140gm. 9,) Matrix is A I - 6 % Mg-alloy.
Ee-
(E;)2/3(E[) 1/3
(8.37)
e and E~ are the secant moduli of the composite in the circumferential and where E~o longitudinal directions, respectively. Because the fibre volume fraction changes in the experiments, we have to calculate the stress/strain dependencies for corresponding values of yr. An approximate nature of eqs. (8.36) and (8.37) calls for approximate determination of the elastic and secant moduli involved in the corresponding calculations. Hence, we use simple averaging for the values of moduli: E~o - - E f v f +
and
Emvm
(8.38)
Shells
Ch. VIII, w
399
T A B L E 8.3 Boron-aluminium cylindrical shells with two-directional reinforcement, the matrix is A 1 - Z n - M g - a l l o y . Subscripts q~ and L relate corresponding values to circumferential and longitudinal directions, respectively. (After Mileiko et al. [440].) Specimen number
Fabrication regime
924 955 954 1014 1015 1012 1008 1009 1011 1007 1013.1 1013.2 1742
Structure of the wall
1 1 1 1 2 2 2 2 2 2 2 2 1
400
Scheme fig. 8.23
N~o
NL
vf
h/R
R/L
CP CP 3-F 3-F 3-F 3-F 3-0 3-0 3-0 3-0 3-F 3-F 3-F
10 15 14 13 12 10 4 4 4 8 10 10 10
5 4 4 6 6 8 8 8 4
0.35 0.40 0.38 0.43 0.43 0.44 0.15 0.15 0.16 0.31 0.45 0.45 0.34
0.0449 0.0470 0.0473 0.0461 0.0436 0.0430 0.0544 0.0554 0.0566 0.0521 0.0410 0.0410 0.0359
0.331 0.341 0.304 0.331 0.330 0.332 0.328 0.330 0.328 0.391 1.172 0.623 0.333
,
300
Geometry
,
p, MPa
p~, MPa
13.73 19.62 16.78 18.84 15.89 16.78 18.25 19.42 19.72 21.38 32.37 23.54 9.71
14.3 17.2 15.8 17.0 15.3 15.1 14.3 14.3 15.8 19.7 29.1 20.2 9.2
,
9.3 MPa 15.5 MPa 15. 7 MPa 17.85 MPa
/'~ [ \
200
X_//
~_ '/
I00
,
.. ,,
,
" A p ~ Ap
? 0
!
0
I
~o
I
2~
Fig. 8.24. The circumferential strains in the central plane of a boron-aluminium shell with circumferential and axial reinforcement, R = 60mm, h/R = 0.046, L/R -- 3, c = 0.032, vf = 0.43, p* = 18.8 MPa. (After Mileiko et al. [440].)
400
Compressive strength
(8.39)
EL = EfFv
where
f) I / (Vm-Vfm2/l' Vf -['-~m-mVm 1--
fv--
Ch. VIII, w
l+~e_~
vf Ef
,
(8.40)
as well as for the stress/strain curve in the plastic region: a -- Efevf +
0 m amy
(8.41)
where vf, Vm, and O"om are the Poisson's ratios for the fibre and matrix and the matrix yield stress, respectively 9 So the secant modulus in the circumferencial direction is e
~
Eq~
a
B
Efvf
n
e
9
(8.42)
1 - ~Vm O"
Note that in the case under consideration we may take the value of EL as the elastic modulus. Therefore, eq. (8.36)can be rewritten as p, - - k f f t ~ / 3
(1
amt~m ~)-2/3 F 1/3 ( h ) m (R) n P,
(8.43)
which is a non-linear algebraic equation for the critical pressure of a shell made of a composite containing elastic fibres unidirectionally and homogeneously distributed in a perfect elastoplastic matrix according to scheme H in fig. 8.23. The equation is valid provided the solution yields the stress which exceeds the elastic limit of the composite. Otherwise eq. (8.36) should be used directly. 0 which determines the composite secant To connect the effective matrix stress, a m, modulus according to eq. (8.42), to the fabrication parameters, we shall use the presentation of experimental dependencies of the matrix strength, a m, on fabrication parameters, the pressure, q, temperature, T, and time, t, by introducing (Section 11.1.1) the equivalent time according to eq. (11.21) - t
q
exp
-
where n, a,, and T, are constants, and the dependence of the matrix strength, am, on the equivalent time, z, given by eq. (11.22). 0 to obtain N o w we need to accept values of the effective matrix yield stress, am, corresponding stress/strain curves of the composites. Having the dependence of the 9 on the equivalent time, we can assume 6 m 0 _ ~a m where tensile matrix strength, am,
Ch. VIII, w
Shells
401
o o~
I
x - ~ . / xaFig. 8.25. The approximating surface corresponding to eq. (8.36) with k = 0.07296, m--1.938, and n = 0.58. The lower point of a vertical line indicates the experimental value for a shell with the A 1 - Z n - M g alloy matrix. (After Mileiko and Khvostunkov [438].)
is to be chosen from consideration of the stress/strain curve of the matrix. However, can be seen as a free parameter which can be chosen arbitrarily within a reasonable framework, and the final result is expected to fit the experimental data if an appropriate choice of the constant in the modified Papkovich formula, eq. (8.36), is made. Assuming ~ = 2/3 for the A 1 - Z n - M g alloy matrix (the corresponding values of 0 are shown in Table 8.1) we calculate the effective modulus, Ee, by using Gm eqs. (8.37), (11.22) and (11.21), for each shell in Table 8.2. Then a procedure of fitting the experimental values of p, normalized by the corresponding values of Ee obtained by using eq. (8.37) to the dependence given by eq. (8.36) yields the following values of the constants: k = 0.07296, m = 1.938, and n = 0.58. Figure 8.25 is an attempt to visualize the corresponding surface, p,/Ee - F(h/R,R/L), together with the experimental data.
8.3.4. Classification of experimental data The approximation evolved allows to systematize the experimental data and draw some quantitative conclusions. Because in a series of physical experiments it is really
Compressive strength
402
Ch. VIII, w
impossible to keep the fibre volume fraction and geometrical parameters of the shells strictly constant, small variations of these values are to be either neglected, and that is a source of the inevitable scatter, or taken into account by a correction procedure. The approximation given by eq. (8.36) provides a means for the correction procedure. Let us acquire a dependence of the critical pressure on the fibre volume fraction making use of eq. (8.36) to bring all the data to some sets of values of (L/R)o and (h/R)o. First, we obtain corrected values of the critical pressure,/~,, applying to the experimental values of p,, the following transformation: p. -- p, ( ( R / Z ) o / ( R / Z ) ) n ( ( h / R ) o / ( h / R ) ) m.
Then calculating a new value of the stress in the shell wall, #~o = _
P* -
P,/(h/R)o,
we get
Ee
P * t~'~e
where Ee is the calculated value of the effective modulus corresponding to the corrected value of the wall stress, ~o. The dependencies calculated according to eq. (8.36) and experimental data corrected according to the procedure just outlined are presented in fig. 8.26. Note that the higher geometrical rigidity of the shell (this means increasing h/R and/or decreasing L/R) the stronger is the dependence of the critical pressure on the fibre volume fraction. Consider now the behaviour of a shells reinforced according to schemes CP, 3-F, and 3-0 (Table 8.3). From the viewpoint of the usage of an expensive reinforcement, the effectiveness of a reinforcement scheme is given by ratio of the critical pressure of a shell of a particular reinforcement scheme, p,, to that of a shell with the same geometrical parameters and the same fibre volume fraction but of reinforcement scheme H, p,h. The values of p,h are calculated and shown in Table 8.3. The first conclusion is nearly obvious. Homogeneous two-dimensional reinforcement (scheme CP) does not yield any fibre saving. The second conclusion is obvious in qualitative terms: it is worthwhile to dispose fibres in the periphery of the wall (scheme 3-0). The experimental data presented in fig. 8.27a give the quantitative measure of the effectiveness of fibre reinforcement according to scheme 3-0. These data are to be considered together with the dependence of the critical pressure on relative height of the unreinforced part of the wall, h~ presented in fig. 8.27b. These two dependencies allow to choose a compromise when designing a shell. Note that increasing the value of h~ up to about 0.5 and keeping fibre volume fraction in the reinforced zones constant yield just a small decrease in the critical pressure. The third conclusion concerns shells with partial reinforcement in the longitudinal direction (scheme 3-F). The experimental data presented in fig. 8.28a show the quantitative measure of the advantage of such reinforcement scheme as comparing
Ch. V I I I , w
Shells
9 9 9 9 9F a b v q , co~tAon wegi, r r t e 2 ooooo F a b r i , cc~tAon w e g i , ~ e s 1 & * * * * * F a b r ' i , ca, t A o n v'egi,'rr~e 4
25
,
,
,
,
I
,
,
,
L/R=3,
20
*
15
9 ~*
1 0
. . . .
I
0.25
i
i
J
L
0.30
3o
,
,
,
,
,
!
,
,
,
,
I
,
',
,
,
1
,
,
,
,
h/R=O.05
I
,
,
,
0.35 ,
|
;
|
I
,
,
,
0.40
,
,
I
|
;
,
I
!
i
I
0.45 ;
,
,
,
403
3
J I
I
0.50 ,
,
,
h / R = O. 0 3 7 5
25
.08 1 ~ / ~ *~/~
._ t'[i~
20
"
o
15
o ~ 10
5
0.3
..___----r-
I
'
l
'
l
l
l
l
'
l
l
l
l
l
0.4
i
l
l
l
l
0.5
Vf Fig. 8.26. The critical pressure of boron/aluminium shell versus fibre volume fraction according to eq. (8.36). The shell wall is reinforced in the circumferential direction. The experimental points are brought to the values of (L/R)o and (h/R)o shown by the correction procedure outlined in the text. For the AI-Zn-Mg alloy matrix, the corresponding values o f (70 m a r e s h o w n in T a b l e 8.1, for t h e A 1 - 6 % M g a l l o y m a t r i x , o-0m = 100 M P a was assumed.
to the homogeneous fibre distribution. However, if one compares these data with those shown in fig. 8.27a one can see that the former scheme looks more attractive from the viewpoint of fibre saving. In terms of the absolute values, the substitution of the longitudinal reinforcement instead of circumferential one in a central part of the wall, yields just a small increase in the critical pressure (fig. 8.28b).
8.3.5. Calculation of the critical pressure of composite shells." an approximate scheme Let us compare values of the critical pressure of composite shells with material parameters representative for metal matrix composites calculated assuming elastic behaviour of the material (fig. 8.29) and elastoplastic behaviour (fig. 8.30). We see
Compressive strength
404 1.4
,
,
.
,
,
,
,
,
,
,
,
,
,
,
,
,',
,
Ch. VIII, w
,
(~)
0
1.3
,t.
1.2
1.1
1.0
25
'
'
'
'
i
|
. . .
~ d
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
i
.
.
.
J
!
.
.
.
|
!
w
f
.
i
.
.
.
(b)
.
2O
\
-~15 lO 0.00
,
r
.
.
0.25
i
i
,
,
,
0.50
h~
,
i
.
.
0.75
!
.
.
1.00
Fig. 8.27. The critical pressure of a boron/aluminium shell reinforced according to scheme 3-0
(h/R ~ 0 . 0 5 , L/R ~ 3) versus relative thickness of unreinforced part of the shell wall. (a) That normalized by the critical pressure of a shell with homogeneous distribution of the same quantity of the fibres. (b) Absolute values of the critical pressure. The theoretical line is due to Dymkov (see the next section).
that the accounting for plastic deformation of the matrix yields to a decrease in the critical pressure by no more than 20% . The higher Young's modulus of the fibre and yield stress of the matrix, as well as the lower geometrical rigidity, the smaller is the decrease. This observation can be a base for an approximate calculation of the critical pressure of metal-matrix-composite shell. First we calculate the critical pressure of a shell assuming elastic behaviour of the material. This can be done by using any rigorous or approximate theory (see for example, [49, 218]). Then the correction deduced from the ratio of a solution of eq. (8.43) to a value o f f given by eq. (8.36) is to be used to reduce the critical pressure to account for the plastic deformation of the matrix. Because the correction is sufficiently small, the expected error should not be too large. Finally, it should be noted that the ratio of the values of exponents m and n actually obtained after fitting the experimental data for boron/aluminium shells into eq. (8.36) makes it more reasonable to use exponents 3/4 and 1/4 in eq. (8.37) instead of 2/3 and 1/3 which have been used. This will slightly change the final results.
Shells
Ch. VIII, w 1.20
.
405
,
0
1.15
,~. 1 . 1 0
o
o
o
1.05
1.00
20
~
(b)
8
0
15
0
0
\
-~10 i
oo
,
|
i
,
0.25
h~
i
|
|
l
0.50
Fig. 8.28. The critical pressure of a boron/aluminium shell reinforced according to scheme 3-F (the values of h/R and L/R) are in Table 8.3 versus relative thickness of the wall part reinforced in the longitudinal direction. (a) That normalized by the critical pressure of a shell with homogeneous distribution of the same quantity of the fibres. (b) Absolute values of the critical pressure.
Also we should note that for sufficiently large values of geometrical rigidity (values of h/R and R/L increasing), a failure mode similar to that happening when the shear band occurs in a rod under compression (Section 8.1.1) can be expected. An experimental observation of stable growth of initial imperfection with the pressure increasing (fig. 8.24) corroborates such an expectation. However, this failure mode can be actually important only for the behaviour of rather thick-walled shells with ribs.
8.3.6. Buckling analysis A complete analysis of the stability of shells is a subject of the shell theory and this is far away from the subject of this book. Hence, we shall describe just a scheme following Refs [146, 148]. The problem is solved assuming the Love-Kirchhoff hypotheses. Until the bifurcation point is reached, the membrane state of the shell wall is assumed. An influence of unloading zones which may arise at buckling is neglected. The vectorial form of the representation of the stresses and strains is accepted and the bifurcation criterion is written as
Compressive strength
406
250
I
\
200
150
\
\
\
I
/ / ~ = 0" 0 5 =0.10
\
\
X
\
\
\ 9
Ch. VIII, w
\
100 I
50
0
I
0.5
l.O
1.5
I
2.0 L/R
I
2.5
3.0
3.5
Fig. 8.29. Critical pressure of composite shells versus relative shell length. The dependencies were calculated by using eq. (8.36) with k = 0.07296, m = 1.938, n = 0.58, Em = 70 GPa, vf = 0.26, Vm = 0.30. Ef = 400 and 200 G P a for the top and bottom curves in each pair, respectively.
fl/2~j[o'h(6eTzaz+6e_Ta~ fl fo2~6llTAlJdxRdq)--O
(8.44)
Here l and R are the length and radius of the middle surface, respectively, u is the vector of additional displacements of the middle surface, Cz and az are the strain and stress vectors at distance z from the middle surface, a ~ is the vector of pre-buckling stresses, p is the applied pressure. Presumably the pressure, p, is applied to the middle surface of the shell and the work on the additional displacements of the pressure applied to the plane end surfaces of the shell is zero. Taking into account geometrical considerations and the stress/strain relationships given in Section 3.3, eq. (8.44) yields the following equation in the matrix form (K + R + A)u - 0
(8.45)
where K-
L*CL,
L
I
R - F*SF,
and
0
~ R~p
R~0 ~ a
~
0
0
1
0
-~V
e
0
0
0
~~
- ~
0
I T
28 R ~x
--2R-~-~x
,
Shells
Ch. VIII, w
1.00
"
I
/
I
~0
;g, I
I
=
h/n=o. 10
. .
. +
0.95
tI- .~ - f
t~
or,,, ~
E 0.90
_-----(-a;~:
F ,
I
I
1.0
0.5
400 GPa -
I
1.5
0.95
.
I
2.0
L/R
~
I-
~
0
i
0
t~"
~
_
(b) g~ = 2 o o , 1.o
3.5
~.-"
~ ~
-
0.5
3.0
//
,
0.80
I
2.5
h/R-O.O5 h / R - O . ~0 0a
.=. 0.90
-
-a,
~////
b
I
hi(.R O.05
/~"
"
407
J z.5
I
z.0 Z/R
t
z.5
cPa i 30
3.5
Fig. 8.30. Critical pressure p, of a composite shell normalized by that of the same shell, p ,el, calculated assuming the elastic behaviour of the material, versus relative shell length. Values of p, are given by solution of eq. (8.43) with the same parameters and material constants as in fig.. 8.29. (a) Ef = 400, (b) Ef -- 200 GPa.
408
Compressive strength
F
0 0
0 ~1
C-
Cll C12 1 cT2 c22 '
Ch. VIII, w
~ Raq,
S
C ll -- fo h Bzdz,
A =
0 0 1 0 ~ a__ a
~x R~cp
C12 -- fo h Bzzdz, ax Ra,p 1
C ll - f0 h Bzz 2dz,
9
R
Here, matrix S is that of pre-buckling stress resultants, A is the operator determining Ap via u, that is Ap = An, matrices L* and F* are obtained transposing matrices L and F and changing a sign at their members containing differential operators of odd powers. The material properties are given by matrix B introduced by eq. (3.121). The bifurcation load for a shell with simply supported ends is given by a smallest root of the following equation
det(Kmn + Rmn + Amn)
(8.46)
which is obtained from eq. (8.45) by introducing a particular buckling mode, m and n being the wave parameters. The calculation procedure outlined can obviously be adjusted to any reinforcement scheme of the shell wall. In fact this was done for the reinforcement schemes shown in fig. 8.23 [146, 148]. To illustrate the reliability of the calculation results we have shown them (fig. 8.27) in comparison with the experimental data obtained by testing boron-aluminium shells. As an example of the theory prediction we show the dependence of the critical pressure of boron-aluminium shells on the relative thickness of the middle part of the wall reinforced in the axial direction (fig. 8.31).
8.4. Technological impact We have considered three typical cases of applications of metal matrix composites in structural elements subjected to compressive loads. We see now that the failure mode depends strongly on both composite microstructure and a shape and
Technological impact
Ch. VIII, w
25
i
:/
i
i
i
i
i
!
i
,
I
i
i
i
"!
i
i
I
,
409
i
~20 \
*~15
I0 0.00
I
I
i
i
,
0.25
i
I
0.50
J
I
0.75
,
J
1.00
C
Fig. 8.31. Calculated dependence of the critical pressure of the boron-aluminium shell on the relative thickness, c - ho/h, of the layer around the middle surface reinforced in the axial direction. L/R = 3, h/R = 0.05, vf = 0.4. (After Dymkov [146].) geometrical parameters of the element. Therefore, a technological recommendation has to be addressed to a particular case. 8.4.1. Plane specimens Perhaps, the plane unidirectionally reinforced specimen under compression is a special case due to relative simplicity of making and testing it and, which is more important, such specimens are characterized by highest values of the compressive strength achieved. Hence, looking at the conclusions from the model and experimental observations described above which are important from the point of view of requirements to the structure of a composite as well to the fabrication technology, we will certainly obtain recommendations as to how to make metal matrix composites of highest compressive strength. 9 The model predicts a maximum strength for a composite with v f - 1/3. Although experimental dependencies show no maximum, they do not also show a significant increment in the strength with the fibre volume fraction increasing above 1/3. Still, this difference has no practical importance. If a rod is designed to have a limit load determined by the shear mode of buckling, then there is no reason to have vf > 1/3; if the bending mode is to be expected, then increasing of fibre volume fraction is useful. 9 The model instructs when and how important the fibre packing can be (see Fig. 8.32). The value of ao/h describes, i n g e n e r a i terms, a quality of a fabrication method. 9 The choice of a matrix and a fabrication method to provide a high enough 0 is also important. Equation (8.8) shows that if the shear mode of value of o-m buckling takes place, changing this value is more important than changing Er. See also Fig. 8.32 for the influence of the matrix yield stress on the change of the buckling mode.
Compressive strength
410
Ch. VIII, w
1.00
/// / / / ///
[//""'i'" /[/'/,'.t'
/// ,','/
0.75
!
,
!
0.50
I
I
i
0
l
J
25
'
*
'
i
I
50
,
,
,
,
l
75
,
i
i
i
I00
Fig. 8.32. Influence of the effective initial imperfection of a composite rod with vf = 1/3 on the compressive strength. Solid lines are for Ef/60 -- 103 and the values of ao/h = 0.05, 0.1,0.15, 0.2 (from top to bottom). Dashed lines are for Ef/60 -- 104 and the same values of ao/h. A drop of the limiting stress at compression takes place at ao/h ~ 20 for the stronger matrix, and at much larger values of ao/h for the weaker matrix. As the drop is observed, the strength value is essentially influenced by the initial imperfection value. (After Mileiko [422].)
9 Obviously, the shear strength of the interface should be as large as possible because any deviation from the ideal value, which is equal to the shear yield 0 stress of the matrix, is equivalent to a decrease in the effective value of a m. 8.4.2. Tubes
If we need to enhance the tube performance under compressive loading we should arrest the propagation of cracks that form strips in the tube. We can do it in a number of ways, but perhaps the easiest and most efficient way is to realize a kind of the bamboo structure or that of the stalk of a grass. An example of the realization of such a structure is shown in fig. 8.33. A photograph of such tubes is also presented in fig. 2.17. Two steel rings on a tube of the length of 1000 mm and diameter of 65 mm yield an essential reduction in the strength scatter and increase the minimum ultimate stress to about 1000 MPa. The comparison of this value with those shown in fig. 8.10 makes the beneficial effect of the " b a m b o o " structure clear. Note that shrink fit of the rings should be used to provide the necessary performance.
Technologicalimpact
Ch. VIII, w
I" 350 ..........
1000
50
.......__i__t._. t..............!
~Steel
F- 10-~ /
411
] ............
Boron//aluminium
Fig. 8.33. A boron/aluminium tube with steel rings providing the "bamboo" structure of the tube.
8.4.3. Shells
We formulate main technological conclusions that follow from the experimental and theoretical studies of cylindrical shells under hydrostatic pressure. These conclusions are valid in qualitative terms for various shell structures. 9 If one is looking for a highest value of the critical pressure, it is necessary to lay out a part of the fibres in the longitudinal direction; in the case of boron/ aluminium shells this portion is about 1/5. 9 If one is looking for a most effective use of a limited quantity of high modulus fibres, it is necessary to lay out the fibres on the periphery of the shell wall in the circumferential direction. Expected values of the critical pressure can be evaluated following the approximate procedure outlined. 9 To make shorter the time of optimization of shell structure and fabrication process, an approximate calculation of the critical pressure based on introducing a plastic correction to the pressure calculated assuming the elastic behaviour can be used.
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Part I11 TECHNOLOGICAL PROCESSES AND MATERIALS
The scientific requirements for a composite structure to provide a material with particular properties, as well as general guidelines to organize the technology were discussed in the two preceding parts. Here we shall give a brief summary of the main technological schemes, putting emphasis not on engineering details (which can vary to a large extent), but on techniques to satisfy the requirements mentioned above. We shall also put emphasis on the processes either invented or developed at the author's laboratory, examples being the internal crystallization method, "blotting" paper technology, boron/aluminium technology, etc. Because of the lack of space, we have to avoid a description of some useful technological process, examples being unidirectional solidification of eutectic melts, a process that has been successfully employed to produce both metal- and ceramic-matrix composite. To discuss this method, as well as some others, in an informative manner, one should go into rather special details. We shall also describe structure and properties of some metal-matrix and ceramicmatrix composites linking the description to the most appropriate fabrication process.
This Page Intentionally Left Blank
Chapter IX INTERFACES AND WETTING
The interfaces in a composite provide the very existence of a non-homogeneous structure. The interface behaviour effects the mechanical behaviour of metal-matrix composites to a large extent. In brittle-matrix composites based on ceramics and intermetallics, the interface behaviour determines crack resistance and so predetermines the potential of a practical use of these types of materials. Hence, it is necessary to control the interface in order to 9 optimize the interface performance with respect to crack resistance of a composite, transverse properties of a composite, etc.; 9 make possible to produce a composite by using a liquid infiltration procedure due to improving the wetting in the system; 9 preserve the fibre from degradation resulting from chemical interaction in the system during either composite fabrication or service at sufficiently high temperatures; 9 preserve the fibre from environmental influences such as oxidation; 9 reduce the effects of coefficients of thermal expansion mismatch in the fibre/matrix system. Physics, chemistry and mechanics of the surface of solids have been rapidly developing during the last decades. This appears to be because, firstly, an importance of the problem for a number of technological fields including that of composite materials has been appreciated, and secondly, many powerful experimental techniques (like a variety of methods for local analysis, surface spectroscopy of various kinds [365], atomic force microscopy [160], and so on) as well as numerical methods have become available. Wetting of a solid surface with a liquid determines a possibility to form strong and controllable interface bond. On the other hand, analyzing wetting is a useful tool to describe fundamentals of interface properties. So we combine these two topics in one chapter although the problem of interface properties is equally relevant to other chapters of this Part. An examination of wetting conditions is also important because of the necessity to evaluate a possibility of using liquid infiltration methods in composite technology.
415
416
Interfaces and wetting
Ch. IX, w
Fig. 9.1. A model of the fibre pull-out in the case of a rough interface. Note that the facets of the sawtooth profile, oriented at angle - ~ , lose contact; on the other hand, the opposite facets are experiencing extra normal pressure. The model is due to Parthasarathy et al. [527].
9.1. Structures of the interface
The interface structure in a composite is a result of a particular fabrication route used to produce the composite. The fibre/matrix interface can be composed of a number of layers of products of chemical reactions between materials of the fibre, fibre coating, and matrix. The structure changes when a composite is exposed to a high temperature, this is a subject of the next chapter. Purely mechanical bonds which provide shear load transfer can arise as a result of friction along the interface because of normal compressive stress or/and penetration of the matrix into microcavities on the fibre surface. Atomic force microscope images of the surfaces of various fibres [309] show that the average wavelength of the surface roughness is about 100, 1000, and 5000 nm for SiC of Nicalon type, A1203 of FP type, and CVD-SiC fibres, respectively. The amplitude of the roughness is nearly equal to the wavelength. In the case of frictional (purely mechanical) interface, its properties can be evaluated on the basis of a mechanical model. If the interface is smooth then for the t axial stress in the fibre, a x, we have the shear lag equation" do-Ix 2/tO'r = ~ dx rf
(9.1)
where/t is the coefficient of friction and O"r the radial stress at the interface. The case of the interface between the fibre and matrix with equal values of the Poisson's ratio, v, characterized by roughness as shown in fig. 9.1, was considered by Parthasarathy et al. [527]. In this case, eq. (9.1) transforms into
Structures of the interface
Ch. IX, {}9.1
417
/
Fig. 9.2. Schematic representation of (a) purely mechanical interface bonding and (b) atomic coherent bonding.
datx 2fiO'r = ~ dx rf
(9.2)
where /t 4 -
/~ -
1 -/~
tana tan
and the radial stress, ar, differs from that for the smooth surface. Namely, o
(9.3)
rYr -- r7r -[- ~ I u ( x ) I + baAor(X)
is the radial residual stress ahead of the debond, cklu(x)l the contribution due to the roughness misfit, and blAO-r(X) the change in the radial stress due to a change in the axial stress. The sliding displacement, u(x) is
where o 0 r
ljo x a '(x)ax. u(x) - -Gb
(9.4)
Here the prime relates the value to the fibre. It was also found that -
(gEm~ \
rf ,/
tan
where B
m
El(1 - Of) (l+v)Ef+(1-v)E
'
E - E f v f q- E m Vm
and A a r -- Eb
du(x) dx
(9.5)
Interfaces and wetting
418
Ch. IX, w
where Eb--
EfEm (1 -
vf) A
E(1 + v)
,
A =
(1 --[-v)Ef + ( 1 - v)E Ef + ( 1 - 2v)E
also bl
Ev (1 + v)Ef -t- (1 - v ) E "
Combining eqs. (9.2), (9.3), and (9.5) yields an ordinary differential equation of the second order with respect to u(x) which is solved in [527] under boundary conditions clearly shown in fig. 9.1. The solution is presented as a dependence of axial fibre stress aa, normalized by a characteristic stress, on a normalized sliding displacement, ft. Dependencies #a(fi) for pullout and pushout conditions are different, both are strongly influenced by the roughness parameter x-
1
2BEm~ tan bl-~Eb,] ~
.
(9.6)
On the other extreme, atomic bonding can be seen. This is either a large or small angle grain boundary of the coherent atomic structure. So in this case the well developed physical techniques of analyzing grain boundaries is to be applied to analyze the structure of the interface. These two cases (see fig. 9.2) are obviously extreme ones. The first can be observed nearly always and can dominate the behaviour of ceramic/ceramic systems. The second is rather an exception in structural composites, being observed in unidirectionally solidified eutectics 1. Between the two cases any possible bonding mechanism can work, including Van der Waals bonds, covalent bonding, noncoherent crystal boundary, chemical bonding, and so on. Various types of bonding are coexisting in a composite structure. The structure of the interface is influenced to a large extent by possible mutual diffusion of fibre and matrix elements. This process leads to solid solution regions in the vicinity of the interface, and to formation of a third phase layer, or a number of layers of new phases. Hence, at first, diffusion bond arises, but further on bonding between the layers and the original components as well as between the layers themselves can be of various types. The kinetics of the process and corresponding structures will be considered in the next chapter. An example of a possible variety of physical and chemical bonding is observed in carbon/carbon composites [338]. If low-modulus graphite fibre is used as a 1 Actually with the advent of multifunctional or smart composites which are to carry load and perform other function (electromagnetic, piezo-electric, etc.), such bonds can be expected to be a case in this exiting field. Therefore, investigations aimed at the understanding of metal-oxide/metal bond occurring on the lattice level (see [575]) will be very relevant to structural composites with metal and ceramic matrices.
Ch. IX, w
Interface layers
419
reinforcement via a pitch-precursor route, then similar crystal scales of the fibre surface and the matrix facilitate a covalent bonding between the active sites of the exposed prismatic edges of the two components. On the other hand, bonding between the matrix and high-modulus fibre can be primarily of Van der Waals type since the fibre has a high degree of preferred orientation and allows much less possibilities to establish the covalent bond. Fibre/matrix bonding of a chemical nature in C / C composites with a matrix converted from a phenolic resin, can be established at a pre-carbonization stage as a result of the carbon fibre surface carboxylic group reacting with a resin carboxylic group. Some works are known which attempt to connect fabrication parameters to the structure of the interface and parameters of this structure to the interface strength. For example, the interface energy has been calculated [724] assuming a usual potential function of the pair interaction between atoms of two sorts within the diffusion zone at the vicinity of the interface plane. The structure of this zone is determined by temperature, pressure, and time of the process of interface formation. A clear physical result has been obtained, namely that the bond strength goes up monotonically with the parameters mentioned above, but it appears to be impossible to connect this result to the choice of an optimal set of fabrication parameters. Work in this direction is going on (see [149, 366, 575]); perhaps, results applicable to composite technology can be expected in future. The interface strength is certainly determined by defects of the interface structure. The defects can be macroscopic, as in the case of mechanical bonding. On the other hand the defects can be introduced into a coherent atomic structure on atomic level. Calculations of the real strength of the interface can be done only by taking into account the interface defects. This situation is quite similar to the evaluation of the real strength of a homogeneous body which is impossible without consideration of cracks, dislocations, and other defects. This makes most of the theoretical evaluations of the interface properties not reliable from the engineering point of view. Hence, direct methods to assess the interface characteristics in mechanical tests (Section 9.4.2) are very important in composite technology.
9.2. Interface layers The interface layer in a composite either occurs as a result of the fibre/matrix interaction during fabrication and/or service or is introduced on purpose to facilitate the fabrication process or to prevent the interaction mentioned. An interface layer is normally introduced into the composite structure by fibre coating. When we have chosen a chemically suitable material for the coating, we can meet either the fibre/coating or coating/fibre thermal expansion coefficients misfit, or drastic degradation of fibre strength due to easy penetration of the crack generated in the coating layer into the fibre (see below, Section 11.3.3), or some other problems arisen as the whole system becomes more complicated. Sometimes this yields to a necessity to produce duplex coating to solve the problems. Moreover, there are known discussions of multilayered coatings to balance a variety of the
420
Interfaces and wetting
Ch. IX, w
requirements including those generated by each new layer [534]. A particular coating procedure is usually chosen for a particular fibre/matrix couple. It should be also noted that if the interface layer in a brittle-matrix composite is introduced to provide conditions for the interface debonding at the matrix crack tip, it can be constructed either to supply necessary properties to the layer itself or to the matrix/ layer interface, or to the layer/fibre interface. There are known a large number of the coating methods. We describe here some of them.
9.2.1. Chemical and electrochemical methods The story about carbon fibre coating started quite a long time ago when such fibres were metallized by an electrolytic process. The fibres have been coated with nickel [273, 689], copper [273], tantalum [581], and other metals. However, there are some evidences [670] that electroplating does not provide a strong enough bond between fibre and matrix. Some authors noted that it looked preferable to use chemical coating methods, which provide a more uniform surface film. For example, there was suggested [331] the deposition of nickel and copper on the fibre surface from solutions of appropriate salts. In this method, the fibre surface has to be activated before the metal deposition. The activation essentially provides the surface with catalytical properties by surface oxidation and subsequent treatment in nitric acid, chlorous tin and chlorous palladium. Metal coatings of carbon fibres allow them to be wetted by metals with a lower melting point, but they appear to be unstable in metal melts, i.e. fibre degradation, leading to a drop of composite properties, is still observed. Hence, one has to look for more thermodynamically stable substances for fibre coating. As is known (see below, Table 9.1) many molten metals of practical interest wet some carbides, nitrides and oxides. So mixtures of such substances are widely used for carbon fibres coating. A method of coating carbon fibres with carbides has been suggested and studied in [562]. The fibres are pulled through a molten metal bath which is chemically inert to carbon (examples being copper or tin) and contains elements to react with carbon to form a carbide. The thickness of a titanium carbide layer, for example, depends on the titanium content in the melt, temperature and time of the reaction. At a subsequent stage of infiltration of a fibre bundle by an aluminium alloy, aluminium carbide is formed at the titanium carbide-aluminium interface. The rate of this process can be decreased by doping aluminium with such elements as silicon, magnesium and gallium. Himbeault et al. applied this method for coating graphite fibres with chromium carbide via copper-chromium and tin-chromium melts [249] and titanium carbide via tin-titanium melt [251]. It occurs that a tin transfer medium produces a smoother and more uniform coating. The thinnest chromium carbide coating in the tinchromium bath is about 0.14 ~tm, the thinnest titanium carbide coating is about 0.05 jam. This is important because the coating is a layer with a high density of
Ch. IX, w
Interface layers
421
defects and the strength of coated fibres occurs to follow the Griffith equation, eq. (1.40) with 7 ~ 2J/m2-2 Coating carbon fibres with refractory metal carbides can also be done by using a liquid metal as a transfer medium. In particular, it has been shown [250] that even tungsten despite a very limited solubility in copper, can be used as a carbide former at the fibre surface. A liquid-metal-bath procedure can be also used in a different way [195, 196]. A bundle of fibres, which can be carbon or polycrystalline sapphire, is pulled first through a potassium bath. For carbon fibres, the temperature of the bath should be about 550~ and the time around 10 minutes. Then if a metal to be used as a matrix (for example lead) wets potassium, the coated fibres are pulled through the molten matrix, which wets the coating and replaces it at the fibre surface. When a matrix (for example aluminium) does not wet potassium, an intermediate bath is necessary. It can be a tin for example, as tin wets potassium and then aluminium wets tin. To enhance the oxidation resistance of carbon fibre, an experiment is performed [600] in which silicon carbide coating is obtained by carbothermic reduction of silica enveloping the fibre at the first stage of the process. The formation of the protective layer is given by the overall reaction: SiO2 ( s ) + 3C (s) = SiC (s) + 2CO (g) which goes at a temperature of about 1000~ Coating of both carbon and silicon carbide fibres with K2ZrF6 is known to provide wetting of the fibres by aluminium melt due to dissolution of alumina layer, which envelopes molten metal, in the reaction products (particularly K3A1F6) between liquid aluminium and KzZrF6 [569]. The coating procedure is simple. Fibres are first treated in an aqueous solution of KzZrF6 near its boiling point and then they are heated in vacuum up to temperature above ~ 600~ At the first stage the precipitation of KzZrF6 microcrystals on the fibre surface is observed, at the second stage a fluoride layer becomes continuous certainly because of a peritectic reaction which goes at 590~ In the experiments with CVD silicon carbide fibres, it was discovered [586] that the fluoride layer does not reduce the fibre strength essentially. However, after the fibres contacted aluminium, aluminium carbide particles have arisen at the interface. Moreover, an increasing rate of the carbide formation can be viewed certainly due to the cleaning effect of the fluoride species on the fibre/matrix interface. The notch effect produced by the carbide particles on the fibre leads to an essential loss of the fibre strength. To decrease such an effect, the authors recommend to optimize the fluoride layer thickness as well as casting parameters and to use SiC fibres coated
2 In fact, Himbeault et al. [251] found the value of 7 mentioned for coatings thinner than 0.15 ~tm.On the other hand, for thicker coatings, the value of 7 approaches 1 J/m 2. This is interpreted as a result of the existence of a sheath on the outer fibre surface consisting of highly ordered crystallites with basal planes oriented parallel to the fibre surface.
422
Interfaces and wetting
Ch. IX, w
with a layer of pyrocarbon which, being a soft material, serves to decrease the stress concentration factor. Oxide coating can be deposited on carbon fibres by passing them through a bath containing a corresponding organometallic compound solution followed by either pyrolysis or hydrolysis of the organometallic compound [298]. For example, alkoxide tetraethoxy silane can be hydrolyzed as Si(OC2Hs)4+2H20 = SiO2 + 4C2HsOH or pyrolized as Si(OC2H5)4 = SiO2 + 4C2HsOH + 2C2H4.
Coating of whiskers can be performed by using a combination of powder metallurgy and pyrolysis procedures. Matsui et al. [391] coated SiC whiskers with A1203 by decomposition of aluminium stearate at 900~ during 2 h in air as well as with ZrO2 by decomposing zirconyl nitrate under the same conditions. Oxide precursors were attached to the whiskers by mixing equal amounts of them and precursor in ethanol.
9.2.2. CVD-methods Similar to technological schemes used to produce fibres (boron, silicon c a r b i d e Section 2.2), various CVD-methods have been used for fibre coating. Normally, carbide, boride, and nitride coatings employ conventional atmospheric pressure CVD processing. Examples of two reactions involved in these processes are: TIC13 + BCI3 + H2 --~ TiB2 + HC1, BC13 + NH3 --, BN + HC1. Both reactions need the substrate (a fibre) to be heated up to l l00~ or higher. Therefore, practical realization of the reactions, which requires elaborate reactor design and sophisticated control of the reagent gas flow rates, occurs to add significantly to the cost of composite fabrication technology. Hence, the choice of a reactor type becomes an important issue [350]. A cold wall reactor, similar to that used in fibre production by CVD, in which the fibre is heated by passing a current through it, has an advantage of reducing reagent consumption. At the same time, it is obviously applicable to conducting fibres only and then it needs mercury seals which have a number of disadvantages including problems in realization of the chamber for a tow of the fibres. So a hot wall reactor is often used. Carbide coatings of carbon fibres do not appear to be stable enough in metal matrices. So there has been tested [530] coating fibres with a mixture of silicon carbide and silicon oxide by using a CVD-method; silicon tetrachloride is supplied to the fibre surface in a flow of hydrogen gas and the reduction of silicon tetrachloride takes place in the presence of an oxygen-containing gas. The thickness of the coating is in the range of about 10 nm to 1 Jam. CVD of a thin layer of boron
Ch. IX, w
Interface layers
423
nitride on the carbon fibres in a tow produces amorphous structure as X-ray analysis shows [318]. CVD-coating of carbon fibre by a mixture of titanium and boron has also been used [405]. In this case, a mixture of tetrachlorides of titanium and boron is reduced by zinc vapour. The thickness of the coating is about 20 nm. Both stoichiometric TiB2 and TiC coatings of the thickness of about 1 to 5 gm on CVD-produced silicon carbide-fibre inhibit the SiC/Ti-alloy interfacial reactions at the conditions of fabrication of the composite by diffusion bonding (1100~ 1 h)[87]. However, the TiC-coating reacts with silicon carbide as a result of diffusion of silicon from the fibre into the coating. A promising process can be organized if one finds (or is designed) a precursor to contain the main coating material, a "compositional" core of the precursor, surrounded by "extraneous substituents" which bring desirable physical properties (the most important property is volatility) to the molecule [170]. An example of such a precursor for deposition of A1N is a volatile solid comprised of cyclohexane-like [(CH3)2A1NH2]3 molecules which have a central "core" consisting of [CH313 rings held together by strong A 1 - N bonds and is appended by CH3 and H "substituents" through relatively weak bonds. This complex decomposes to aluminium nitride and methane on heating above 500~ Stability of the byproduct contributes to forming a stoichiometric A1N coating at a low temperature with minimum contamination from the "extraneous substituents" [170]. It is worth to note that CVD method allows to deposit high temperature superconductive layer of YBazCu3Ov-x oxide on a ceramic fibre [350]. It opens a possibility to use strong fibres in high-Tc superconductive wires.
9.2.3. MOCVD-methods A further reduction of the deposition temperature can be achieved by using metalorganic chemical vapour deposition (MOCVD). The method has been developed mainly because of the needs of the electronic industry. It is reviewed recently by Fisher et al. [170]. It occurs that during the last decade a number of metal organic compounds have been discovered that are volatile and properly decompose at relatively low temperatures to form oxide, nitride, carbide, boride, and mixed deposits. Oxide coatings can be obtained by using so-called alkoxides and//-diketonates which require deposition temperature from 250 to 800~ For example, yttria coating on silicon carbide fibre was produced by decomposing the /%deconate complex in the presence of water vapour in a hot wall CVD reactor [170] . The coating was found to survive sintering with the Ni3A1 matrix at a temperature exceeding 1300~ and subsequent annealing for extended periods at 1000~ Boron nitride, silicon nitride, silicon carbide, boron carbide, titanium carbide films can also be deposited from appropriate compounds.
9.2.4. Sol-gel techniques Sol-gel techniques look more promising for thin layers deposition because it is cheaper than CVD-methods. However, a straightforward usage of these techniques
424
Interfaces and wetting
Ch. IX, w
does not certainly lead to a success: when attempting to coat a tow of the fibres, that allows to decrease the cost, the coating occurs to be non-uniform throughout the tow, bridging between the fibres and tracking occur due to capillary effect [227]. To overcome these problems, two processes proved to be successful. The first one [237] is based on passing a tow of the fibres through a vertical reservoir with sol on the bottom and an immiscible liquid on top (top-liquid). When the tow enters the reservoir it contacts first the sol, which penetrates between the fibres. Then the tow goes to top-liquid which should displace the sol on the fibre, but if the sol wets fibre/top-liquid interface, a thin layer of the sol remains. When the tow comes out the top-liquid, a thin layer of two-phase liquid will be present on the fibre provided top-liquid wets sol-air interface. In particular, the conditions formulated are fulfilled for alumina fibre, sol precursors for a number of oxides, and hexane as the top-liquid. To go on with the process, the tow is moved into a furnace for heat treatment to evaporate the liquids and to form a coating. The final thickness of the sol layer may be from 0.3 to 5 ~tm, coating thickness from 0.005 to 0.1 lam is applied per pass. The second process [227] is based on growing the coating on the tow while the tow is in a solution. It prevents the tow from fibre bridging and tracking.
9.2.5. Sputter deposition Kieschke et al. [311] developed a process of magnetron sputtering to deposit a duplex Y/Y203 layer on the unprotected silicon carbide fibre. Both batch and continuous versions of the process were presented. In the latter version, the fibre is passing between four magnetron sources. The distance between yttrium target and the fibre is such that under the magnetron power used, that is between 100 and 400 W per unit, the fibre temperature is about 400-500~ The whole device is located in an ultra high vacuum chamber with a base pressure of less than 10 -9 torr. The sputtering gas is argon. The duplex coating is formed by reactive sputter deposition of yttria onto an yttrium underlayer and by partial oxidation of a sputtered yttrium layer. The latter is performed either by heating the coated fibre in oxygen or by in situ heat treatment after composite fabrication in the presence of oxygen dissolved in a titanium matrix. Choy [86] reported using such a technique in modifying CVD-produced SiC fibre by depositing bilayer C / T i C coatings. The first layer is carbon (0.2~tm thick) deposited by using C target, the second is TiC (about 1 Jam thick) by using C and Ti targets. The coating appeared to be stable with SiC and exhibited no interfacial reaction or diffusion up to 950~
9.3. Surface energy and wetting The present understanding of wetting, as well as of liquid-solid interaction (which determines wetting), can be found in a number of books and review papers (see, for example [169, 332, 475, 480, 576]). Theoretical treatments of solid/liquid, solid/ vapour, and liquid/vapour interactions involved in the wetting phenomenon are
Ch. IX, w
425
Surface energy and wetting
known (a recent review is in [576]) but they are still of little engineering importance because of drastic simplifications of a real situation. 9.3.1. Wetting conditions
The main relationships between characteristic parameters to determine wetting are written as follows" - - O,
(9.7)
7LS -- 7SV + 7LV COS 0 - - 0.
(9.8)
--~
7LS
-~- p g x
Here 7 is the value of the interface energy related to the particular interfaces by indexes L (liquid), S (solid) and V (vapour), R1 and R2 the main radii of the curvature at a point of the LV interface, x the vertical coordinate of the point, r the liquid density, and 0 the angle of contact. The second expression is called the Young's equation. Equation (9.8) is illustrated by the equilibrium of the forces of surface tension shown in fig. 9.3a. If the angle of contact 0 > ~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
////////////
~',s
%
'~
9
'
-T
(c)
J
Fig. 9.3. (a) An illustration of the Young's equation; (b) Liquid-solid contact when wetting is absent, 0 > ~z/2; (c) Liquid-solid contact at wetting, 0 < 7z/2.
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
X
Source
u
9
T
-C
Cu O dcgree
Sourcc
750
K 90
1000 1OW 1100 1150 1200
145 60 I 19 33
28
[480] [321] [480] [4X0] [321]
700 l I50
I lX 20
14x01 [4X0]
1285
120
14891
(g)
14891 13211 [489]
~
[7Zg
[4XO]
24 32
[4XO] [480]
1 100 1200
135 126
[.I801 [4XO]
HfC
1 3x0
23
[4XO]
1200
I32
[4XOl
I I50
45
13211
60
16671 0',
N~
TiN
900
TiRz
O(H1
ZrBl
900
106
[580]
Z
ZrOz
XOO 900 1000
100 I20 90
[JYO] [490] [190]
!I
[SXO] 1650
'~,~ --...I ~
125
(4801
1500 1500
125 130
[480] [480]
1100 1027 IlOO 1157
174 1 15 I I5 1 15
14801 [272] [6h7] [272]
1130 l SO0
155 148
[480j 14801
1480
1500 l OSO
K1
N
-i~
1500
[4XO]
t~
N
I'~
[4XO]
Tit,
Si7Nd
',.~
1 SO
k.__J Lm..,
I I00
SiOl
1500
~
13641 [48O] [480]
~mJ
140 90 48
Lm.~ L._..J
1000 11 SO 1250
40 25 25
[480] [SXO] [SXO]
1 120 1 130
142 143
14801 (5801
42 78 78
14891 [5XO] [SXO]
1 100 1300 1400
123 135
36
[480] 15801 [480]
I150
115
[4YO]
~
A1201
L
132 I ] ~
0
-.,
I000
m
B
k_..~ L__J L_.~
1380 1450
~m,
Zr(.'
L__~ L__J , - J
I08
--
l I00
0
[4XO] [480] 14801
,-.,
23 17
.--
[4XOj [4RO] (3211 [524] [524] [4XCL]
/
137 136 134 126 126 U
...-
13x0 1450 1500
, m . . -
960 1\00 1 150 1150 1300 15UU
--
[480] [480] 15241
m
X8 87 43
_,
1460 1500 1480
Tic'
Lm~
cs~ 9
Ni 0
..-,
YO 75 90
| 0",
T "C
dcgrcc
800 1000 I020
SIC SCS-h
B4('
Source
0 degrcc
0 C
0
-1
0
0"
.A I
T "C
-1
Mctals =. Fibrc n~atcrial
o
0
9
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
Surface energy and wetting
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
Interfaces and wetting
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
Surface energy and wetting
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
Interfaces and wetting
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
Interfaces and wetting
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
Interface properties
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)
Interface properties
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 . ~ .
Interface properties
.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.
This Page Intentionally Left Blank
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].
Diffusion throughfibre~matrix interface
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) --
Kinetics in the case o f chemical reactions
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
Kinetics in the case o f chemical reactions
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
Kinetics in the case o f chemical reactions
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
Kinetics in the case o f chemical reactions
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
Diffusion throughfibre~matrix interface
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
Effects of component interaction on composite properties
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.
Diffusion throughfibre~matrix interface
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
Effects of component interaction on composite properties
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
Diffusion throughfibre~matrix interface
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
Effects of component interaction on composite properties
(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)
Diffusion throughfibre~matrix interface
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.
Diffusion throughfibre~matrix interface
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
Diffusion throughfibre~matrix interface
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
Diffusion throughfibre~matrix interface
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)
Diffusion throughfibre~matrix interface
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
Diffusion throughfibre~matrix interface
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)
Processing parameters
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.
Processing parameters
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
Processing parameters
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].
Processing parameters
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
Processing parameters
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'<\'L<,~\4X,~?\41 I1 V,,'X6 . . . . . ~,L.r////A II
.
!i
I
II-
'
fi" "v)/22 "''rd- ....
-~A-F,7,4
.......
/~
\\
...........
~,/"Jv-/KR\\%"~\~\~\\\;\\~-\\~
;///A \ ~ ' / . 4 / K / / / / A / / / / / / / / / d F////////~/./.4//././,/4~ \ | I
/
./..
.....
////A
....
...
'
-"
"///'~Fitting
Fig. 11.15. The assemblage for producing boron/aluminium tubes. After Sarkissyan et al. [582].
This method has many advantages as well as disadvantages. A scheme suggested by Hearn et al. [240] permits decreasing the volume of gas and producing composite tubes without using an isostat. A similar equipment (fig. 11.15) was used by Sarkissyan et al. [582]. The equipment installed in an ordinary furnace allows to produce tubes with the outer diameter of 20 to 75 mm and the length up to 1.5 m. The time/temperature profile is controlled by monitoring the temperature at a reference point of the outer surface of the die. Keeping input energy of the furnace as well as the boundary conditions (that is especially important for the cooling stage) constant, makes the equivalent time given by eq. (11.24) be a function of the maximum temperature To (see fig. 11.16). Therefore, it is not necessary to measure temperatures at various points of the blank. A scheme of the densification with the internal diameter being constant (fig. 11.14b) can be effectively used for making tubes of relatively small diameter, up to 20-30 mm.
~r
_ To
o
\ o
Fig. 11.16. Schematic of the temperature change on the die and composite blank. After Sarkissyan et al. [582].
498
Ch. XI, w
Hot pressing
11.3.3. Shells
Fabrication of metal-matrix-composite shells is also possible by using a 'soft' tool. Again, this can be done either in a gaseous isostat or in a die designed to reduce the volume of the gas. Choosing a scheme of those shown in fig. 11.14, a and b, we should take precautions to prevent both excessive fibre breaks or matrix cracking during densification of the blank when the external diameter is constant or a possible fibre kinking [441] (internal diameter is constant), the latter problem being a more difficult problem to conquer. So the scheme of fig. 11.14a looks preferable. Mileiko and Khvostunkov [438] described fabrication of boron/aluminium shells. The blank, after plasma spraying, is encapsulated into a container shown schematically in fig. 11.17. Winding the fibres and spraying the matrix are performed onto the shell-mandrel made of stainless steel of 18-8 type. The wall thickness of the shell-mandrel is about 1.5 mm. The blank after spraying is inserted, with a minimal gap, into a segmented die two halves of which are kept together by the end rings. The whole set is then inserted into the external shell which encapsulates the blank hermetically by welding. Then the container with a blank is subjected to hot pressing in a gas isostat. The extraction of the boron/aluminium shell from the container is performed by turning both the shell-mandrel and sealing welds and then cutting the external shell in the longitudinal direction. The quality of a shell obtained depends, in particular, on fabrication parameters. They determine a configuration of the fibre-break system, in addition to properties of the matrix and fibre/matrix interface. When a set of parameters is chosen appropriately, then the fibre breaks occurred during consolidation of the blank are distributed homogeneously, so they do not influence the shell performance. Otherwise, in a sufficiently thick shell, the fibre breaks are concentrated along one plane as can be seen in fig. 11.18 where specimen 1006 exhibits white spots illuminating the breaks.
Segmented die N
~
Internal steel shell
~
rl#llll#llllllllWllllllllllllZ#llllil.llllilllll/llll111111lli111~
/
steel shell
Fig. 11.17. The container with a blank. After Mileiko and Khvostunkov [438].
Ch. Xl, w
Techniques
499
Fig. 11.18. Boron/aluminium shells with external diameter 122 mm and wall thickness between 3 and 4 mm. The white spots on the shell surface illuminate the collective fibre breaks. After Mileiko and Khvostunkov [438].
In the case of densification of the blank with the external diameter being constant, the fibres in the internal layer experience strain e ~ (h/r)p where h, r, and p are the wall thickness, internal radius and porosity of the blank, respectively. Hence, for usual values of the porosity, p = 0.2, and ultimate fibre strain, e* = 0.007, we have a m a x i m u m relative thickness of the shell wall h/r ~ 0.035 to be densified without a danger of fibre breaks. Actually, because of the presence of rough defects in fibres and gaps in the setup shown in fig. 11.17, the m a x i m u m wall thickness appears to be even smaller. Consequently, one should be looking for such a densification route as to exclude a possibility of the collective fibre breaks and to provide a structure with r a n d o m fibre breakage. Obviously, to stimulate r a n d o m fibre breakage the main densification stage is to be performed when the fibre/matrix interface strength is relatively low. Experiments with various pressure/time and temperature/time routes (fig. 11.19) were done on shells with h/r = 0.07 to choose a consolidation regime. The results show that in order to exclude both collective fibre breaks along some lines (fig. 11.18, Specimen 1006) and longitudinal cracks resulted from the coalescence of the collective breaks (fig. 11.18, Specimen 995) to achieve r a n d o m fibre breakage as shown in fig. 11.18 (Specimen 1008), it is necessary to accept the following regime: p0 ~ ps ~ 55 MPa, Ts ~ 460-500~ The fabrication route described has been used to produce boron-aluminium shells of various configuration, some examples of the shells are also shown in fig. 2.17. A simplification of the equipment by drastic decreasing the gas volume can be done as shown in fig. 11.20. In this case, a gas-tight bag welded of the sheets of stainless steel is located between the blank and the segments with a conical internal surface pushing outward by a conical puncher loaded by a hydraulic press.
500
Ch. XI, w
Hot pressing
71, p
2ZTC
P~
--
~perature
,.
Fig. 11.19. A scheme of possible technological regimes for the fabrication of composite shells in gaseous isostat. See the text for details.
11.4. Rolling and drawing Rolling and drawing can be used to fabricate composites, mainly processing a composite when the matrix is in a semi-solid state. This case is a part of so called compocasting technology, so it will be considered in the next chapter (Section 13.2.5). Still, occasionally these processes have been used for processing solidstate-matrix blank. Also, rolling and drawing can improve the microstructure and so enhance properties of metal matrix composites. A possibility of producing metal-metal composites by rolling was established in 60s. Perhaps the only essential feature of the process is a necessity to conduct it in vacuum or with a protective atmosphere if the foil is used as a matrix precursor. It can be achieved either using a vacuum-tight container or a vacuum rolling machine. In this way the molybdenum-nickel specimens were obtained by Kopecky et al. [328]. To use rolling for making brittle fibre composites one needs to choose processing parameters more carefully. Gusev et al. [215] were able to get a boron-aluminium composite by rolling plasma sprayed monotapes, the matrix being an A1-Zn-Mg alloy. A composite of good quality was obtained after 5 to 6 passes with total reduction of about 50% , at a temperature of 400-500~ Doble and Toth [132] have reported to obtain a boron-aluminium composite by rolling a package of the matrix foils and the filament mat, kept together by a polystyrene binder. The package is placed in between two stainless steel cover plates
RollhTgand drawh~g
Ch. XI, w
Argon gas
501
Conical s e g m e n t s P r e s s u r i z e d bag / B lank
I
////////,' AA
i i ' l
! "//////// ente d die
Fig. 11.20. An equipment for consolidation of composite shells and tubes. After Mileiko et al. [462]
of about 8 mm thickness. The assembly is preheated either in an argon atmosphere or in air. (In the latter case, it has to be wrapped with a protective foil.) The most important parameters of the process are temperature and rolling pressure. When temperature increases, the pressure necessary to form a bond goes down and at high values of the pressure, fibres tend to fracture. An optimal temperature for the 6061alloy is 565~ which is slightly higher than that for static hot pressing. Rolling speed does not seem to influence the composite strength. The same method has also been used for producing precursor tapes. The largest sizes of the tape obtained by the authors are 1200 x 15 mm. Drawing in an evacuated container, as a method of composite fabrication (fig. 11.21), can be found in some publications. For obtaining graphite-aluminium composites by using precursor wires [189], the temperature of the process is about 500~ the speed about 5 cm/s and the material of the container is inconel. To obtain a composite with a sufficiently high strength, the value of the reduction should be between 5 and 15 % . It has been noted that a minor modification to the fabrication scheme allows tubes with various profiles to be made.
502
Ch. XI, w
Hot pressing
Die o
HeATeR
o
CONTAINER
~_.>..., ;...,;
~,
"' ~
~x~~ ooo ~
/o
o
//
o
COMPOSITe WIRES
Fig. 11.21. Drawing in an evacuated container.
11.5. Explosive welding Explosive welding can be considered as a kind of hot-pressing. The phenomenon, discovered by Lavrentiev at the end of the W W II [122], is an efficient way to bond together two similar or dissimilar materials. In 70s, the explosive welding was intensively used in attempts to develop an economically effective fabrication route for metal matrix composites. Then the activity started to diminish. There is perhaps a hope to reanimate the interest in using the method because it seems to be an easy way to utilize the explosive after the cold war has ended. In the process of explosive welding, the impact of two plates is accompanied, as a rule, by a stationary periodic process leading to a characteristic modulating waved interface (see fig. 11.22). All the important events take place in the vicinity of point
Fig. 11.22. Wave formed at the interface obtained by explosive welding. After Mileiko [419].
Ch. XI, w
Boron/aluminium composites
503
eXpLOsive ~
PLATe
GROUND PLATe
Fig. 11.23. A scheme of explosive welding of two plates.
A, which runs behind the detonation front (see fig. 11.23). Certainly, the conditions for forming the cumulative jet occur periodically here. Processes that occur around point A in the presence of fibres between the plates located either in the direction of the welding front or along normal to the front, are analyzed in [715]. In the first attempts to produce metal matrix composites by explosive welding (see, for example, [108, 281]), the direction of detonation front propagation was chosen to coincide with the fibre direction. In this case some new features were observed and they were associated with the flow of the matrix layer around a more rigid fibre [615]. Many experimental data show that the choice of technological parameters in this case is not a difficult task. The situation is easily reached when the mechanical behaviour of a composite obtained by explosive welding corresponds to that of a composite with an ideal interface bond. Moreover, the very short time of the process excludes the formation of brittle interface layers as well as annealing the reinforcing wires. Another advantage of explosive welding is the possibility of making sheets and plates of large sizes with a high productivity rate. When making composite tubes and shells with non-axial fibre directions it is not clear a priori what will be the result of the interaction of wave-forming processes at the matrix-matrix interface with a set of the reinforcing fibres lying, for example, along the impact front. A corresponding experiment is described by Mileiko et al. [441,443]. An assembly before detonation is shown in fig. 11.24, and an example of a structure can be seen in fig. 11.25. An interesting situation arises when the wave formation is completely excluded, which is a case, for example, when the reinforcing element is a wire mesh [50]. In this case, the bonding mechanism appears to differ from that described above. In going through the mesh, the matrix surfaces are cleaned and then the physical contact arises and the bond forms.
11.6. Boron/aluminium composites It is difficult to distinguish between the material and structural element when dealing with composites. Still, we shall divide the following discussion of the two topics mainly from the point of view of convenience. Although boron/aluminium composites have been constantly mentioned in illustrations of various aspects of the
Hot pressing
504
9
9
9
,
,
,
,
.
.
"
.
.
,
,
Ch. XI, w
.
r
e
c
-
.
.
. ,
,
.
.
.
~
.
9 -
:
9
.
. .
-
.
.
.
Fig. 11.24. An assembly to fabricate a composite tube with circumferencial reinforcement 9 l - mandrel; 2 and 4 - matrix layers: 3 reinforcement layer; 5 - protective shell; 6 - explosive; 7 - detonator; 8 - cone. After Mileiko and K o n d a k o v [441].
Fig. 11.25. The steel-aluminium composite obtained by explosive welding (Kasperovich and K o n d a k o v ) .
behaviour of MMCs, we are coming back to these composites because, first, it is a material thoroughly studied and its behaviour models the behaviour of various metal matrix composites rather well.
Boron/aluminium composites
Ch. XI, w 1.6
505
11.6.1. Materials Formation of the microstructure of boron/aluminium composites is determined by the processes discussed in detail above, those being plastic yielding and interface bonding as well as diffusion through the interface. The latter process is determined by a low solubility of boron in aluminium and a larger affinity of boron for magnesium which is a common alloying element in aluminium. It was shown in Section 10.2.2 that a heat treatment of a boron/aluminium composite with a matrix containing magnesium can yield formation of magnesium borides in the matrix rather than aluminium borides at the interface. The situation is similar to well studied phenomenon of the internal oxidation of alloys [116]. In that case the diffusion of oxygen from the surface of a solid into the volume leads to oxide precipitation. For the element to react with oxygen it must have an affinity for oxygen higher than that of the main element of the solid solution. When boron penetrates an alloy, such as an aluminium alloy containing magnesium, the sequence of events may be almost the same, only internal boriding occurs instead of internal oxidation [448]. The condition for the process to proceed are clear [116], namely cSDB >> CMgDMg, where c s is the boron concentration on a boron-source/alloy interface, CMg is the magnesium concentration in the solid solution, DB and DMg a r e the diffusion coefficients of boron and magnesium in the alloy, respectively. For the simple case where c s << CMg and the absolute value of the free energy of formation of boride Mg~B~ is large, as in the case of boron/aluminium, the concentration of boron and magnesium in the alloy should be as shown in fig. 11.26.
CMg C
s CB
x:
x
Fig. 11.26. Schematic representation of the dependence of boron and magnesium concentrations in the aluminium matrix on the distance from the interface. After Mileiko et al. [448].
506
Hot pressing
Ch. XI, w
Following the analogy with internal oxidation, the internal boriding front may be assumed to be quite sharp, which means that the thickness of the front layer is small in comparison with xf, the concentration of free magnesium in a solid solution at x < xf may be neglected, and the boride content within the internal boriding volume may be assumed to be independent of x. The larger the value o f ] - G ] the smaller the particle size. The particles are becoming finer as the rate of movement of the boriding front increases. Again, following the same analogy, we can write the equation for the rate of the boriding front as dxf
o~r S 1
~ ~ o d---t-= DB -fl- CMg xf
(11.25)
Obviously, in the axi-symmetric case eq. (11.25) transforms to drf a cs 1 d--t- = DB fl CMg rf l n ( r f / R )
(11.26)
where R is the radius of a cylindrical source of boron (a fibre) which is constant in the case under consideration. Just before linking zones, eq. (11.26) may not describe the situation well enough because of a decrease in the magnesium content at r > rf. The kinetics of the internal boriding just described can yield the formation of dispersion hardened zones around the fibre in a boron reinforced aluminium/ magnesium alloy. The influence of the zones on the failure behaviour and the strength of boron/aluminium composites was analyzed in Section 5.2.2. Here we continue the discussion introducing the dimensionless effective heat-treatment time as - (t/to)exp(-Q/RT)
(11.27)
where to is an arbitrary constant and Q is the activation energy for an appropriate physical process which may be the diffusion of either magnesium or boron in aluminium. With the obvious difference of the diffusion constants for boron and magnesium in mind we take Q = 38.5 kcal/mol as the activation energy for magnesium diffusion in aluminium [314] and obtain the strength versus effective time plot fig. 11.27 for various combinations of temperature and time used in [448,456] to produce boron/aluminium composite specimens. Obviously for changing temperature an integral representation of the effective time similar to that given by eq. (11.24) should be used, or in the case of piececonstant temperature: z = Z(ti/to)exp(-Q/RTi).
(11.28)
i
Figure 11.27 justifies the idea of the effective time. A shift of the critical point to smaller values of the effective time for set 2 of experimental data can be explained by the presence of a larger fibre volume fraction for this set.
Boron/aluminium composites
Ch. XI, w
1400
,
,
,
,,,,i
I
,
,
,
,,,uu
I
,
9
0
,
,
,u,,v
I
u
,
507
u
~3 1200 1000 *
O
800
600
0
0 0
400
~ooooo Set;
! Set 2
-oooo, 200
'
1 0 _1,e
'
I ~ .... I
1 0 -11
~
'
' ~ .... I
|
1 0 -1o~
I ' ~ .... I
1 0 -9
l~.
I
I
Fig. 11.27. Strength of boron/aluminium composites with A1-6Mg matrix as a function of effective heattreatment time according to eqs. (11.27) and (11.28) with to -- 10-l~ min. Set 1 of the experimental data is from Ref. [456], set 2 is from Ref. [448]. The average fibre volume fraction for set 1 is about 0.5, for set 2 is about 0.07 higher. After Mileiko et al. [448].
11.6.2. Structural elements Boron/aluminium composites, as mentioned in Section 2.3.1, have been used in aerospace structures. Therefore, fabrication methods to produce structural elements exist, some of them, namely those for making tubes and shells were described above. The behaviour of shells under hydrostatic pressure was discussed in Section 8.3.2 and that of tubes under axial c o m p r e s s i o n - in Section 8.2. Here we continue the discussion of the mechanical behaviour of b o r o n / a l u m i n i u m tubes with longitudinal reinforcement.
Tube strength versus fabrication regime Introducing the effective time for composite consolidation during hot-pressing according to eqs. (11.23) and (11.24) and following the assumption made above, Section 11.3.2, about dependence of the effective time on a m a x i m u m die temperature, Tmax, reached in the consolidation process (fig. 11.16), we present the tube strength as a function of Tmax. Figure 11.28 shows, first, that such a representation is a reasonable one, and second, similar to the behaviour of the strength of plane specimens (fig. 11.27), the tensile and compressive strength of tubes reaches a m a x i m u m at some value of Tmax, or equivalently, of the effective time.
Hot pressing
508
o
1800
Ch. XI, w
Tension 9 Compression 9 Compression, fai/ure at the fitting end 9
I
"
I
"
I
"
1600
\14oo b
1200 1000 800 0
600 =
480
I
490
=
l
500
I
510
T/~
l
520
Fig. 11.28. Strength of boron/aluminium tubes with AI-6Mg alloy as a matrix versus the maximum temperature of the outer surface of the die. The tube diameter is 60 mm, the length is ~ 400 mm, the wall thickness is ,-~ 1.5 mm. After Sarkissyan et al. [582].
Fitting ends Unlike the case of tubes as well as other structural elements made of fibre reinforced polymers where the problem of load transfer to the element is usually solved by quite a heavy design of the fitting ends (the problem is similar to that of gripping tube specimen for its testing which has been discussed in [630]), the load can be transferred to a boron/aluminium tube through sufficiently light fitting ends [582]. An example of an effective fitting end design is shown in fig. 11.29. Despite some guidance for choosing characteristic sizes and angles for such a fitting end design can be found as a result of consideration of the stress state and failure behaviour of overlapping joints (see Section 4.7.3), a large a m o u n t of the experimental work is still necessary to optimize the design.
11.7. Silicon-carbide/titanium composites We consider here both composites with titanium alloy and titanium aluminides matrices.
Silicon-carbide/titanium composites
Ch. XI, w
509
I
1.5 I
::!1 Fig. 11.29. An example of the fitting end design for the boron/aluminium tube. The design provides an appropriate load transfer if some precautions are beheld. After Sarkissyan et al. [582].
11.7.1. Titanium alloy matrix
A composite with Sigma BP SiC-fibre coated with a thin TiB2 layer and (~ +/~) titanium matrix (Ti-6A1-4V) reveals a fibre break accumulation failure mechanism [693]. A degradation of the fibre strength during the composite processing lowers the low-strength part of the fibre strength distribution which has no effect on the composite strength. To decrease processing temperature and so decrease the thickness of the reaction zone, it was suggested a special titanium alloy, Ti-4.5A1-3V-2Fe-2Mo, of "high formability" at sufficiently low temperatures [179] to be used as a matrix. It occurred that the consolidation temperature of 750~ for 2 h under a pressure of 150 MPa yields a maximum value of the room temperature composite strength. The composite retained its strength up to 500~ 11.7.2. Titanium-aluminide matrix
Titanium aluminide matrices are less tough than titanium alloy matrices. An example of tensile behaviour of such a composite is given by Brindley et al. [59] who produced and tested Sigma BP SiC-fibre/Ti-24A1-6Al-matrix composite. The fibre is coated by a diffusion barrier composed of TiB2. The matrix consists of ~2(Ti3A1) and/~ phases the latter being introduced to enhance the ductility of the intermetallic alloy. It is known (see Section 10.2.2) that the reaction zone in such a composite is surrounded by a matrix zone depleted of//-phase. Then, the alloy ductility is strongly influenced by oxygen content. The authors revealed just a limited fibre breakage in the composites that had occurred in the vicinity of the failure surface and only immediately before the failure, although cracks within the reaction zone and the/~-depleted zone were revealed at early stages of loading. This
Hot pressing
510
Ch. XI, w
microstructural observations mean that the fibre volume fraction in the composite lies within interval v) - vfB in fig. 5.2 that actually corresponds to a large scatter of the composite strength (fig. 11.30). Because fracture toughness of the matrix is relatively low, some contribution to the microcrack arrest comes from the interface. In fact, the authors measured the interface stress at debonding, r,, and that during sliding restrained by the friction, r**, and found ~, ,,~ 105 MPa and ~** ~ 60 MPa at room temperature. The composite behaviour at elevated temperatures is the same, although experimental data here are even more scarce (fig. 11.31). Still, it is clear that SCS6 SiC/(Ti3A1 + Nb) composite has superior specific strength and creep-rupture strength as compared with the corresponding properties of an IN-100 superalloy (see figs. 11.32 and 11.33). However, the superiority inheres with regard only to the properties along the fibre direction, the transverse properties are too low. To enhance fracture toughness at room temperature of titanium-aluminide-matrix composites, it is necessary to introduce a weak interface (Section 5.8) which can decrease creep and creep rupture resistance of the composites (Sections 6.1.2 and 6.2.2). Hence, it is necessary to optimize the interface properties to balance the two characteristics mentioned. Weber et al. [694] showed that single-crystallinealumina/TiAl-matrix composites with graphite interlayer providing a weak interface can be an example of such a balance. Their creep resistance at about 1000~ as well as effective surface energy at room temperatures look promising.
1500
,
,
~
o 0
r,..b
o
, ,
G
o o o o
1000 -It
0
b
8 ~
500
0 0.0
o o o o o C~= ooooo *****
I
0.1
600 1200 1600
1000 1300 1900
-
I
0.2
ppm ppm ppm
I
0.3
0.4
Fig. 11.30. SiC-fibre/Ti-24Al-11Nb-matrix composite: strength versus fibre volume fraction. The matrix contains various amounts of oxygen. Experimental data after Brindley et al. [59].
Ch. XI, w
Silicon-carbide~titanium composites
1500
I
I
!
511
I
r~ o
1000
8
b
500
425~ 650~ 815~
ooooo ooooo ooooo I
0
I
0.I
0.o
I
0.2
I
0.3
0.4
0.5
"Of Fig. 11.31. SiC-fibre/Ti-24Al-11Nb-matrix composite: elevated temperature strength versus fibre volume fraction. Experimental data after Brindley et al. [59].
~ . 0
,
I
,
I
'
I
|
..4 "~-. 1.5 b Q. "~'- 1,0 b
_/
TisA I + Nb
0.5
S'~C/(Ti~4 I+Nb) -
0.0
i
0
I
200
i
I
transverse
400
I
T /
I
600
~
I
800
Fig. 11.32. Specific strength of Ti3A1 + Nb matrix and SiC/(Ti3A1 + Nb) composite normalized by that for IN-100 superalloy versus testing temperature. Experimental data after Larsen et al. taken from [345].
512
Ch. XI, w
Hot pressing
2.5
I
|
|
i
|
I
|
|
I
l
I
|
i
|
,
I
!
i
i
|
,--~ 2.0 b ~
~
s'c
_
b
It~___~ ~*~'~
i
1.0 T'i,sA l +Nb
0.5 Ss
0.0
l
21
TisA l,+Nb) |
|
i
I
22
i
i
i
i
trans~erse_._..._..__.~ I
i
J
22
i
i
I
2a
i
i
i
i
23
LM
Fig. 11.33. Specific stress of Ti3AI + Nb matrix and SiC/(Ti3AI + Nb) composite normalized by that for IN-100 superalloy versus Larsson-Miller parameter, LM = [20 + log(t/to)]T where t is rupture time in hrs, to = 1 h, and T is testing temperature. Experimental data after Larsen et al. taken from [345].
11.8. Glass- and glass-ceramic matrix composites Two driving forces for the development of glass and glass-ceramic matrix composites are known, these being (i) a versatility of the matrices due to a variety of their physical properties and (ii) processing temperatures lower than that for ceramic matrix composites. Such composites are candidates for elevated temperature structural applications, especially when some physical properties are necessary, missile radomes being an example. Hence, this type of composites has been intensively studied starting certainly with early work by Phillips and co-workers (see Section 2.3.2) and that by Prewo and his co-workers [30, 550]. As a matrix, borosilicate and aluminosilicate glasses are utilized. Glasses can be processed at relatively low temperatures, this minimizes unwanted fibre/matrix interactions, but lowers an upper service temperature by 600-700~ Glass-ceramics are normally found in the following systems: Li20-A1203-MgO-SiO2 - lithium aluminosilicate (LAS), CaO-AI203-SiO2 - c a l c i u m aluminosilicate (CAS), MgO-AI203-SiO2 - magnesium aluminosilicate (MAS), BaO-MgO-AI203-SiO2 - barium/magnesium aluminosilicate (BMAS), BaO-AI203-SiO2 - barium aluminosilicate. The compositions can include some oxide additions as nucleating and fluxing agents. Processing of glass-ceramic matrix composites including crystallization, needs higher temperature than glass-matrix composites. Service temperatures for the LAS system containing /~-spodumene crystalline phase are between 1000 and 1200~
Glass- and glass-ceramic matrix composites
Ch. XI, w
513
those for MAS based on the cordierite composition -1200~ [551] . The highest service temperature is certainly characteristic for barium aluminosilicate matrix based on hexacelsian, although there seems to be no reports on a real usage of such glass-ceramics as a matrix. Moreover, a doubt is expressed in a very possibility of such a use due to a phase transformation of hexacelsian with a large volume effect [721]. Varying heat-treatment regime yields variations in crystal/glass composition of a glass-ceramics; among properties which vary accordingly, is thermal expansion coefficient, e. This makes glass-ceramic matrices be very convenient due to a possibility to adjust values of e for the fibre and matrix and to lower the residual stresses in the composite. Normally, hot pressing of slurry infiltrated monotapes (Section 11.1.2) are used to produce the composite, although some other techniques such as hot matrix transfer into woven preform and hot injection molding of chopped fibres compounds in preforms can be also used to produce elements of complicated shapes [551]. If hot pressing and heat treatment parameters are chosen adequately, the fracture toughness and strength of a composite can be sufficiently high. Some typical values of mechanical properties of such composites are presented in Table 11.3. The failure of a brittle-fibre/brittle-matrix composite (see Section 5.8) is a process involved matrix cracking, fibre pullout, interface debonding, etc. An interaction of those processes determines strength and fracture toughness of the composites and is also revealed in the shape of stress/strain curve. Note that niobium oxide present in some LAS matrices reacts with silicon carbide of the fibre that yields a formation of a thin C / N b C layer at the interface [551] which occurs to satisfy requirements to the interface fracture toughness to deviate the matrix crack [159]. Relatively high fracture toughness of ceramic-fibre/glass-ceramic-matrix composites yields a sufficiently small strength scatter (large value of the Weibull parameter /~). At the same time, a variety of fracture mechanisms involved in failure process of the composites makes them very sensitive to both testing method (loading pattern) and specimen sizes. Jansson and Leckie [279] analyzed the results of testing of a unidirectional SiC/LAS composite in tension, four-point bending and three-point
TABLE 11.3 Mechanical properties of some unidirectional glass- and glass-ceramic composites Fibre
Matrix
vf
Graphite SiC Tyranno SiC Nicalon SiC Nicalon SiC Nicalon
BSG MAS CAS LAS LAS
0.44 0.40 0.37 0.35
Tp~
810
Tc~
1100
1200
Young's Tensile modulus strength GPa MPa 170 80 124
1400
1 Borosilicate glass. Tp- temperature of hot pressing, Tc -crystallization temperature.
Bending Work of Source strength fracture MPa kJ/m 2
785 334
1000 793 475 750
30 50
[4831 [403] [2321 [601] [551]
514
Hot pressing
Ch. XI, w11.8
bending of specimens of rectangular and triangular cross-sections. They observed a few patterns of the crack behaviour as well as the shapes of load/displacement curves. The most important finding is perhaps that the comparison of the strength values obtained from various tests yield an effective value of/3 equal to 21 which corresponds to a very small scatter of the strength values. Composites in a SiC/LAS system retain their room temperature strength up to temperatures of about l l00~ [551]. Work of fracture of SiC/MAS composite changes with the testing temperature reaching a maximum, 45 kJ/m 2, at 500~ Because of high anisotropy of unidirectional composites, cross-plied laminates are to be considered for many possible applications. Pryce and Smith [555] studied microcracking in longitudinal lamina of (0x/90y)s SiC-Nicalon/CAS laminates. They found that the progressive cracking of these lamina is reasonably similar to that observed in unidirectional composites and described approximately by the ACK model (Section 4.4.1). Although, the interaction of the neighbouring plies leads to some differences from what the model predicts. A simple shear-lag model [296] of a lamina subjected to microcracking, incorporated into a stress analysis of a laminate predicts the stress/strain behaviour of SiC-Nicalon/CAS laminates. The strength behaviour of angle-ply carbon-fibre-reinforced borosilicate glass presented in fig. 11.34 can be certainly described by Tsai-Hill failure criterion, eq. (3.132). At least, the authors made such a statement.
lO0O (
800(y tl,
-
b 600400 200
%
0 I
,0
45 ~ /
,
,
grad
9O
Fig. 11.34. Nardone and Prewo's experimental data on strength of carbon-fibre/borosilicate-glass-matrix + 0 angle-plies [483].
Ch. XI, w
Graphite-aluminium composites
515
11.9. Graphite-aluminium c o m p o s i t e s
Such composites are normally produced by liquid-phase technologies, so they shall be described in detail in an appropriate place (Section 13.3.1). Hot pressing is just sometimes used to obtain such composites. For example, composites of such a kind can be produced by vacuum hot pressing of prepregs obtained by ion-plating technology disclosed by Ohsaki et al. [513]. The prepregs have been pressed at a temperature of about 540~ and pressure of about 60 MPa for 1 h. The fibres used were of low m o d u l u s - high strength type, the matrix was pure aluminium. The mean composite tensile strength occurs to be low, that is about 800 MPa, the strength scatter at a volume fraction just above 40% seems to be very large which is obviously not unexpected. To optimize fabrication parameters of hot-pressing of graphite/aluminium prepregs, Masson et al. [389] varied time/temperature conditions when performing the consolidation process in vacuum under a pressure of 25 MPa. The results of tensile testing of specimens obtained is presented in fig. 11.35. The failure surface of a composite obtained at 600~ min shows the crack propagation in one plane, that for a composite obtained at 580~ min reveals the failure process based on the fibre breaks accumulation. 700
I
I
I
I
I
i
I
I
'
I
ooooo T = 5 8 0 " C 9,,,, T = 6 0 0 " C
\
600
*b 500
400 -
30fl "t
0
,
,
I
25
,
,
t
I
/
40
,
,
I
55
,
70
Fig. 11.35. Tensile strength of graphite/aluminium composites obtained by hot-pressing of prepregs at two temperatures versus consolidation time. The fibre volume fraction changes irregularly from 0.39 + 0.10 to 0.43 + 0.06. Experimental data are after Masson et al. [389].
This Page Intentionally Left Blank
Chapter XII POWDER METALLURGY METHODS
Powder metallurgy methods are suitable for making composites with short fibres, as well as those with continuous fibres. It should be noted that powder metallurgy methods permit a number of variations to yield new possibilities of making fibrous structures. Some of them are rather traditional for powder metallurgy, some have specially been invented for making composites. In the present chapter, we shall also give a brief description of methods based on pyrolysis of either liquid or gaseous precursor. They actually have much in common with powder metallurgy methods considering the philosophy of the making and optimizing the microstructure of composites.
12.1. Basic scheme
Usually a fabrication route (fig. 12.1) includes at least three steps: (i) preparation of a mixture of the matrix material powder and reinforcing fibres (blending); (ii) shaping either a semi-product or structural element to be made; (iii) sintering. The two latter steps can be swapped.
12.1.1. Blending When making short-fibre composites, blending procedures developed in powder metallurgy are normally used. Mechanical properties of short-fibre/brittle-matrix composites are essentially dependent on the homogeneity of the composite macrostructure. In addition to the requirement of obtaining sufficiently homogeneous mixtures, we should also be looking for a process which introduces an acceptable degree of fibre damage. The two requirements contradict each other, this means that if we are analyzing a dependence of a mechanical property, say the composite strength, on a blending parameter, say blending time, we expect to see a maximum on the dependence. The mixing can be either dry or wet. In wet blending processes, homogeneity of the mixture can be enhanced by some means. An example is the surface-induced process developed in [277] which is based on careful consideration of thermodynamic condition for achieving stable colloidal suspension of the matrix and fibre. The consideration yields the necessary pH-value for the fibre and matrix suspension 517
518
Powder metallurgy methods
Ch. XII, w
Fibres
Powder
~\~',,,~',,,\\\\\\' 4,\
/ / / / / / / / / / / / /
Blending H~ Cold p r e s s i n g
j,
~
2
3 Hot pressing
Sintering
+
l
Heat treatment
Shaping
(Rolling)
Shaping
(Extrusion)
Fig. 12.1. Schematic draw of basic powder metallurgy fabrication routes for short-fibre Composites.
to achieve the state of minimum floccation. Actually, a mixture of alumina matrix, silicon-carbide whisker and partially stabilized zirconia was obtained under carefully controlled blending conditions and the influence of homogeneity of fibre spatial distribution was demonstrated. This is shown in Table 12.1. The fibre length distribution is changing during blending. This influences fracture behaviour of the composites. A predictive model of the fibre breaking process remains to be built up, so at present, only measurements of the fibre length supply data for the evaluation of a fibre length distribution function. To perform the measurements [439], a quantity of the fibre/powder mixture is taken out of a batch and after drying, ethyl alcohol is added to that part of the mixture to make easier the
Ch. XII, w
Basic scheme
519
dispergation which is done on a vibrating plate. After evaporation of the liquid, a sheet of the mixture is observed by using optical microscope at a moderate magnification, between • and • (an example is shown in fig. 12.2). The microphotographs are served for the measurements of the length distribution function. Some distribution function obtained are shown in fig. 12.3. The experimental data show, first, that the higher the fibre volume fraction, the quicker the fibre attrition proceeds. Secondly, the velocity of the fibre breakage is decreasing with the mean fibre length decreasing. This is certainly due to eliminating weakest points in the fibres.
T A B L E 12. l
Fracture toughness of SiC-whisker/(Al203-ZrO2)-matrix composites (vf--0.2) obtained by using various blending methods [277]. Fabrication route
K { / M P a 9m 1,/2
Surface-induced coating; sintering 1700~ MPa-40 min Conventional mixing; sintering 1700~ MPa-40 min Surface-induced coating; sintering 1600~ MPa-20 rain
12.5• 1.1 9.7 • 0.9 6.3•
Fig. 12.2. An example of the micrograph used for measuring fibre length distribution function in a mixture of graphite fibres and boron carbide powder. After Mileiko et al. [439].
520
Ch. Xll,
Powder metallurgy methods
1.Of
~~
='~
o.81- ~_~,i,"
o01/g' " F ~'~
8x
..
%'-"
'
v,:o.~
Mixingtime
IIw
o I, v x
r~
oo
=
"
9[ ~ l = l = [~" ,A',=,=. 0"2 I ~ l l
0.0 ~
'
04
I
1.0
*'
~,o'~ g ' ' '
"I-"
i 0.8
t ]
/ 0.4
Mixing time
06 I / , . 0 8
lo
o * "
J
/ 1
1
"
Vf = 0.2
0.6
10 min 20rain 40 min 120 min
o,,,o r.,n
02
="
w
30 min 120 min 180 min
0"2 0.0
oo
02
o14
o16 I / o 1 8
~o
Fig. 12.3. Experimentaldistribution functions of the fibre length in mixtures of graphite fibres and boron carbide powder. After Mileiko et al. [439] 9
Because the mechanism of shortening fibres during blending process is unknown, a choice of an approximation for the distribution function may be rather arbitrary. The Weibull type function, that is
F(x)- l - e x p
-
x
(12.1)
with the mean length 1-- x()I-'(l +
l/m)
(12.2)
and the standard deviation
D-x(){r(l + 2/m)-
1-'2(1 +
l/m)} '/2.
(12.3)
can be adopted. It should be noted that a possibility of a change in the fibre-length-distribution function after sintering is not to be excluded, however such a change can hardly be essential.
12.1.2. Shapin~ Shaping can be done by a variety of techniques. During the extrusion, fibre aligning as well as breaking takes place. Hot extrusion of a loose blend swaps two processes, shaping and sintering. This can be done with the blend poured into a can which is closed thereafter [203, 612]. A special study of the influence of processing parameters on the fibre length in extruded short-carbon-fibre/aluminium-matrix composites performed in [612] have supported the conclusion derived in early works, that is, an increase in the billet temperature results in matrix shear stresses decreasing and, therefore, in fibre tensile stresses decreasing. This yields a less fibre damage.
Ch. XII, w
Variations of the basic scheme
521
A relatively easy shaping can be achieved by using reaction-bonding technique for producing silicon carbide based composites (see Section 12.2.1). In the ceramic technology, a mixture of silicon powder with an appropriate substance is undergone to either injection molding, or pressing, or slip-casting. The green body is then converted into silicon nitride by heat treatment in a nitrogen atmosphere at 1200-1400~ The volume change of the body is very small because the expansion, as a result of the conversion, is accompanied by the pore formation in the powder compact. Before sintering, short fibres can be oriented along a chosen direction. This can be done by using either extrusion or doctor blade method [391].
12.1.3. Sintering The general theory of sintering (Section 10.5) yields recommendations which can be used when choosing fabrication parameters. In practice, sintering can be performed either without pressure, after forming a "green" body as a result of densification under pressure at a low temperature, or with pressure. A choice of the sintering regime in addition to usual considerations for sintering ceramics should be also based on consideration of the fibre/matrix interface. The regime has to provide necessary interface properties. A preliminary thermochemical analysis is useful for the determination of bounds for the fabrication parameter. For example, such an analysis for processing of SiCfibre/Si3N4-matrix composites has been performed by Misra [466]. His analysis shows that unless there is excess carbon in SiC reinforcement, thermodynamic stability of the fibre/matrix interface does not limit sintering temperature. To promote sintering, the matrix is usually doped with appropriate agents. For alumina matrix such agents can be magnesia or rare earth oxides. Doping matrix with an appropriate agent can also be necessary to prevent strong chemical interaction between the fibre and matrix yielding a too strong fibre/matrix interface. For example, sintering SiC matrix and SiC fibre with the outer layer enriched with carbon leads to a sufficiently weak interface provided the matrix contains an excess of carbon [468].
12.2. Variations of the basic scheme
12.2.1. Reaction-bonding A reaction bonding procedure used to obtain a ceramic material, in particular silicon nitride, can also be applied for obtaining a ceramic matrix. The method is known to decrease processing temperature and matrix shrinkage as compared to traditional sintering ways; on the other hand it yields a rather high matrix porosity [731]. In particular, silicon-nitride-matrix composites were produced by this method. To obtain a continuous-silicon-carbide(SCS-6)-fibre/silicon-nitridematrix composite [153], both SiC fibre mat and pure silicon cloth containing a small
522
Powder metallurgy methods
Ch. XII, w
concentration of MgO to stimulate further sintering were first prepared with polymer fugitive binders. Then alternate layers of the mats and cloths are stacked and pressed in vacuum at 1000~ under pressure of about 70 MPa for 1 h. Finally, the SiC/Si preform is heat-treated in high-purity nitrogen at 1200~ for 40 h to obtain the silicon nitride matrix. Samples obtained have a rather high porosity, so that they normally need densification by HIP'ing. It has been reported [516] that using microwave furnace for nitridation of silicon allows to lower reaction bonding temperature and to preserve Nicalon fibre from high temperature degradation. Nitridizing silicon in a mixture of silicon carbide whisker and silicon powder can yield, at the same time, conversion of a part of the whisker to siliCon nitride also [57]. This leads to a drastic degradation of the composite microstructure. The whisker can also disappear as a result of its interaction with a liquid phase arisen in a system containing sintering additives [598]. Such factors calls for a fine adjustment of fabrication regimes for a complicated system to that satisfying requirements to a final composite microstructure. Reaction-bonding as a method to obtain the matrix is not restricted to well known silicon nitride. Fabrication of other matrices has also been reported, nickel aluminide reinforced with either silicon carbide whiskers [706] or polycrystalline alumina fibres [13] are just two examples. In that case, a mixture of the matrix ingredients yields the intermetallic matrix after heat treatment under pressure. Another example is a sapphire-fibre/alumina-ziconia-matrix composite, in which case the matrix is obtained by oxidation of an AI/AIzO3/ZrO2 mixture containing a large fraction of ductile aluminium particles [698]. In the oxide/oxide composite, an attempt to develop a suitable interface was also performed by using the fibres coated with titanium. The TiO2/AlzTiO5 interface arisen seems to deflect the matrix crack although a problem of an interface/fibre interaction should be solved to prevent fibre degradation. It is important to note that the reaction-bonding process applied to obtaining composites needs to be optimized to produce the fibre/matrix interface microstructure which corresponds to a sufficiently low strength of it. Introducing SiC-SCS-6 fibres into silicon nitride matrix by using this way can yield a sufficiently weak interface, with shear strength from about 6 to 18 MPa [731] provided special precautions have been taken. In particular, it was shown that such factors as mean silicon powder size, additives and impurities, changes in the nitridation atmosphere controlled the microstructure and properties of the interface. It was also found that the shear stress of the interface does strongly depend on the carbon coating thickness decreasing from about 25 MPa for thickness of 30 nm to about 4 MPa for thickness of 1.25 ~tm [354]. There has also been reported a possibility to produce a composite by impregnating the preform with a slurry containing reagents to form the matrix. In particular, impregnating a stuck of the fabric plies made of silicon carbide of SiCNicalon fibres with a slurry containing equal parts of fine alumina powder and monoaluminium phosphate and then consolidating the mixture at very low temperature, below 200~ yields a composite with the aluminium phosphate
Ch. XII, w
Pyrolysis of matrix precursor
523
matrix [36]. The low temperature processing cycle excludes completely the degradation of the fibre. Hence, bending strength of the two-dimensional reinforced composites occurs to be rather high up to a temperature of 850~ Weakening the fibre/matrix interface by carbon coating the fibre yields an increase in the crack resistance. Titanium zirconate matrix within the fibre tows (SiC-Nicalon) was created [47, 283] by a procedure containing the following steps: (i) preparing the fibrous preform by enveloping the fibre tows into a wire cage, (ii) infiltrating the preform with a solution of zirconium n-propoxide and titanium isopropoxide in absolute ethanol (a mixture to yield ZrO2 :TiO2 = 1 : 1), (iii) drying the preform for 4 h at 110~ to hydrolyze the alkoxides and to leave a mixture of Zr and Ti hydroxides, (iv) calcinating the pre-matrix in flowing air at 540~ (v) hot-pressing the composite at 1270~ for 30 min in a CO atmosphere at a pressure of 17.25 MPa. The infiltrationcalcination steps are repeated for a number of times to reduce porosity of the prematrix. Mechanical testing of two batches of the SiC/TiZrO4 composites revealed [283] that they differed drastically in the bending strength. The first batch was characterized by a mean strength of about 1000 MPa, and the second one by about 650 MPa. The only visible reason for such a difference was found to be a difference in fibre packing homogeneity. Hence, a large effect of the fibre packing on the tensile strength of metal matrix composites, which is now fully understood (Section 5.2), seems to be observed in ceramic matrix composites as well. It should be pointed out that the importance of packing homogeneity cited was verified by a special quantitative metallography technique. 12.2.2. Co-extrusion
Although the method of producing a fibrous structure by extrusion of a mixture of different powders was suggested long ago [563, 564], certainly it has not been really used to develop new composite materials. However, it can merge in future as it looks rather promising in making relatively non-expensive materials.
12.3. Pyrolysis of matrix precursor Supplying a matrix precursor in either liquid or gaseous form into premade fibrous carcass provides an excellent means to combine the fibre and matrix in such a way as to preserve an initial fibrous carcass configuration and its exterior. On the other hand, the process of transforming the precursor into a solid matrix can take too long a time that makes a corresponding fabrication technology too expensive for a majority of applications. Still, applications like aerospace structures require such technologies. The first stage in both fabrication routes is manufacturing a fibrous carcass. Usually a three-dimensional structure is required in such a case, hence so called multidirectional substrates are made. Carbon fibre substrates is an example of
524
Powder metallurgy methods
Ch. XII, w
elaborated structures and manufacturing processes. McAllister and Lachman [395] presented general schemes of the processes which are in use up to now. They include weaving dry yarns, piercing of fabrics, combining rigid premade rods (from yarns), and filament winding. 12.3.1. Chemical vapour impregnation CVD process described briefly in Section 2.3.3 with regard to fibre fabrication has also been applied to deposit refractory compounds such as carbides, nitrides, and oxides into a fibrous carcass to form ceramic matrix composites. In that case the method is called chemical vapour impregnation (CVI). Although a number of the matrices are known to be deposited into fibrous carcass, CVI-process to obtain C / C and SiC/SiC composites have only been well documented [395, 484, 625]. The corresponding chemical reactions are mainly CH4 ---+ C + 2H2,
CH3SiC13 --+ SiC + 3HC1. In the latter case, hydrogen as a carrying gas is used to deliver CH3SiC13 to a reaction zone. The main problem to be solved to achieve a sufficiently low porosity of the matrix is that of controlling the deposition kinetic in such a way as to ensure relatively uniform deposit thickness. Otherwise, "bottlenecks" arise and make the very possibility of getting a solid composite material problematic. In addition to the chemical reactions, mentioned above for two particular matrices, transport of reagents to the substrate determines the deposition kinetics. It has been shown (see [484]) that since with temperature and total pressure decreasing, the rate of chemical reaction is obviously decreasing rather drastically, and at the same time, the mean free path of gaseous molecules is increasing, the uniformity of the deposit thickness is increasing. This is illustrated schematically in fig. 12.4. The process described is going on under isothermal and isobaric conditions. Because it is normally performed under such conditions that yield an extremely low deposition rate, it takes a very long time which enhances the product cost. To speed up the process, a thermal gradient technique is used [395, 625]. It requires more sophisticated equipment: a simple hot-wall reactor suitable for an isothermal process (see also Section 10.2.2) should be changed to a cold-wall reactor to allow to control temperature fields in the substrate. The same goal, that is shortening the process time, can be reached by introducing a controllable pressure gradient in the process zone. There is another way to preserve the advantages of the isothermal process, the main being a possibility to use simple equipment of large sizes to treat rather large preforms or a number of preforms simultaneously, and at the same time, to reduce processing time. This is called pulse CVI [142]. In a cycle of such a process, the fibrous preform or substrate is rapidly filled with the feed gas by forced convection,
Pyrolysis of matrix precursor
Ch. XII, w
525
9
x ~-~
•
iJJ JJi
iii'llli,i/iiilliiiiiiiJJii
-Substrate
--~
IIIIII IIIIIIIIXIII IIIII ill I!1 II ! I11 I I I I I I I II IXI I III I II I II111 I 11 II I I l | l l l l l l l l J ' l l l l l l IIII1[1111111 III1111111 I P l I I I I I I I I I I IIIIII11 ! I I I l l l f l l l l l l l l l l l l l l l l IIIIIIIIII IIIIIIIIIll IIIII I1111 I1111 =11/111111 II11111 IIIIII I I1 II11111
j
/
-j
t
h
Ill Ul HI HI HI il II Hi
I'tlllll I IM i I I I I ]~llll I I~[lll I I I\1 I I ] [l~kll I lli~il I I f l I~i
~ ........ ~ . ~ : .
[11 l i I III III II I III II
[I f I 11 ]1 I I I il i i
~,I .... , ~ . ,
I I I I I I a I
Fig. 12.4. Schematic of dependencies of the deposit thickness on infiltration depth; p2 > P~, T2 > T].
then the gas is trapped in the preform to achieve sufficiently uniform matrix deposition, and finally the residual gas products are evacuated. Despite there being no known reports of a successful usage of the process in industry as yet, it seems to be a good base for an industrial development. Chemical vapour infiltration is convenient in making ceramic matrix composites since it allows to include in the fabrication process a stage of depositing an appropriate interface material. In particular, to make the fibre/matrix interface in SiC/SiC composites weaker, a layer of either pyrocarbon or boron nitride is usually deposited [485]. Moreover, there can be organized the interface of a multilayered structure, for example that containing a number of pyrocarbon layers of some tens of nanometers stacked with silicon carbide layers to improve oxidation performance of the composite as well as to optimize mechanical behaviour of the interface [485].
12.3.2. Liquid impregnation Liquid impregnation techniques were first applied to produce carbon/carbon composites. Advantages and drawbacks of the process were analyzed at the early stage of the technology. Also, ways to overcome problems arisen were traced. McAllister and Lachman [395] described corresponding routines in details. An appropriate selection of the carbon matrix precursor is important since its characteristics influence both the fabrication cycle and its parameters. Most important characteristics are (i) viscosity, (ii) coke yield, (iii) coke microstructure,
526
Powder metallurgy methods
Ch. XII, w
(iv) coke crystal structure. Two kinds of the carbon matrix precursors are normally used, those being thermosetting resins and pitches. Viscosity values of both types of the possible precursors are sufficiently low, just coal-tar pitch requires higher temperatures to infiltrate (softening point is around 100~ Coke yield for pitches is larger, especially when carbonization is performed under pressure of about 10 MPa. The yield can be as high as 90% as compared to maximum values of about 70% for phenolic resins. Coke from pitches is of graphitic structure with a high density, of about 2 g/cm 3, that from phenolic resin has a glassy structure, does not graphitize up to 3000~ and its density is very low, less than 1.5 g/cm 3. Pressure is an important fabrication parameter since it influences the coke yield as well as promotes the impregnation and keeps the impregnant inside the preform during pitch pyrolysis. It should also be noted that to decrease porosity of the matrix, it is necessary to repeat impregnation-carbonization-cycle a few times. Impregnation of the rigid preform with an appropriate liquid substance and subsequent pyrolysis is now used to produce various non-oxide ceramic matrices. Oligomeric compounds as precursors for many carbides and nitrides, which match requirements mentioned (viscosity, ceramic yield, etc.), are known [731]. Chemistry of oligomeric compounds has been sufficiently developed to instruct how to control properties of the precursors. High volume shrinkage can be suppressed by adding either passive or active fillers to the liquid [731]. A precursor to make the SiC matrix can be the same polymer as that used to produce SiC fibres, that is polycarbosilane (see Section 2.2.2). To reduce shrinkage and porosity of the matrix, adding titanium and molybdenum powder is expected to give a positive result; adding silicium aids in the formation of crystalline SiC [253]. It should be noted that a simple model to estimate a possible reduction in shrinkage as a result of reaction of active filler with decomposition products I was developed [207].
12.4. Short-fibre composites Composites reinforced with randomly oriented short fibres are normally produced by using a basic fabrication route of powder metallurgy scheme although a variety of other methods are also known to be used, especially to fabricate metal matrix composites. We concentrate on discussion of the microstructure and properties of such materials in this section. Reliable methods of evaluation of the strength of such composites that would account for fabrication parameters and microstructural characteristics of such composites have not been developed yet. That is why we consider approximate approaches here but not in Chapter 5. A brief discussion of the problem is necessary, otherwise experimental data compiled below can seem rather confusing. 1Actually, in that case, the procedure becomes very similar to reaction-bonding (Section 12.2.1)
Ch. XII, w
Short-fibre composites
527
12.4.1. Strength It was found in systematical studies of both metal and ceramic matrix composites reinforced with short carbon fibre that dependencies of the composite strength on fibre volume fraction are non-monotonic [425, 583]. An example of such a dependence is presented in fig. 5.17. An occurrence of a maximum on the a*(vf) dependence normally reveals a competition of two or more processes determining the failure. However, for the case under consideration, it is important to point out that with the fibre volume fraction changing a significant microstructural parameter also changes. Note that when making a short-fibre composites with various fibre volume fractions by mixing powder and fibres, one looks often for a constant degree of mixture homogeneity independent of yr. This yields a dependence of the blending time on the fibre volume fraction which, in turn, makes the mean fibre length to be dependent on vf as demonstrated in Section 12.1.1. Therefore, plotting the experimental results in the usual way, that is a strength/fibre-volume-fraction curve, is not completely representative. At least one parameter, either a mean fibre length or blending time, has to be added to the representation. Hence, the model described in Section 5.2.4 can hardly be applied to analyze technology-properties chain for composites produced by powder metallurgy methods which include the blending stage. Let us consider a powder technology fabrication route to produce a short-fibre composite in more details. The stage of structure formation includes two phases: blending the fibre and matrix powder and sintering the mixture. The blending phase predetermines such structural parameters as 9 fibre length distribution and, therefore, strength characteristics of the fibre, provided the fibre microstructure does not change at the sintering phase; 9 a degree of homogeneity of the structure on the sub-fibrous level. Figure 12.3 illustrates changes in the fibre length distribution function in a particular case of carbon fibres and boron carbide powder under blending in a ball mill under constant conditions (rotation speed, load of the mixture in a vessel, processing time). An appropriate statistical procedure to describe changes in the mixture homogeneity versus blending parameters has not been applied yet to the case under consideration. Corresponding works aim mainly at the problem of mechanical alloying (see for example [268]). The sintering phase predetermines 9 microstructure and properties of the matrix; 9 structure and properties of the fibre/matrix interface. Under some sintering conditions, fibre properties can also be changing, however, as a rule, such changes are unwanted and normally they are to be avoided by an appropriate choice of the regimes. Obviously, the structural parameters mentioned determine properties of representative elements of different scales (levels). There can be seen the following
528
Powder metallurgy methods
Ch. XII, w
hierarchy of the levels controlled by both the scale and composition of a representative element: 9 a single fibre with surrounding matrix, both components properties and the fibre aspect ratio being known; 9 a set of the fibre in the matrix containing fibres of characteristic reinforcement directions, the parameters mentioned above also being known; 9 a set of elements of either first or second type, a configuration of the set being a result of a statistical procedure; 9 a macrospecimen to be tested and so related to a phenomenological model studied previously. When studying the failure behaviour of composites with short fibres, perhaps the first structural level has been mainly analyzed (see Chapter 5). Actually, we are going to use these results to build up a mesoscopic model as a basis for computer simulation. First, we note that the critical fibre length determines the contribution of a fibre of a length l into the composite strength given by eq. (3.200), that is
\-----]
where a~ (l*) is the effective fibre strength on a critical fibre length, l*, s is the average fibre stress on the length within from its end normalized by a maximum fibre stress, a = Then, normally when analyzing composite strength in the case of a composite with either short fibres or those breaking under the load, the Weibull distribution for the fibre strength is used, although the procedure is clearly justified only under a condition of statistical homogeneity of the object (Section 2.1.1). This yields a scale effect, as
1"/2
l/l*.
a;(')
_
o.;(2)
(l(,)/l(2))-'/#
(12.5)
Finally we shall use a linear dependence of the Young's modulus of a composite on the fibre volume fraction, that is E = Em + 2AEvf
(12.6)
where 2 is a constant depending on the geometry of the reinforcement, AE is a contribution of the fibre system to the Young's modulus of the composite. The model we are now to formulate consists of two blocks. The first one describes formation of the composite structure and the second one describes failure of the structure.
Short-fibre composites
Ch. XII, w
529
Formation of the structure
In the absence of a reliable model of blending fibre/powder mixtures, we accept the following plane algorithm of the process. Let the fibre volume fraction determined by a load of the fibre before mixing be v~. We may imagine the ballmilling process as a sequence of 1, 2, ..., n steps. The first step of mixing (n = 1) produces a representative structural element containing N1 cells. The fibre volume fraction in the cells obeys the homogeneous distribution in the interval (0, 2v~ The second step duplicates the number of cells, the third one triplicates the number, and so on, so that
N=Nln
(12.7)
and the bounds for fibre volume fraction v~ in the ith cell become more narrow, namely (vf(1 - ~),vf(1 + 1)). The distribution function remains homogeneous. A random value of the cell strength is taken as 0"~ -- O"m -k- Ao'(n)v~ i)
(12.8)
where
Aaon~/2P(1- nl/2/n~
Aa(n)-
(12.9)
Here eqs. (12.4) and (12.5) are taken into account and it is assumed that l - L/n 1/2, l*--L/n ~ Aao, and n o are constants to characterize a particular material and process, L is a characteristic size of the specimen. Neglecting possible changes in the Young's modulus of a cell due to a decrease in the fibre length, we use eq. (12.6) for the cell modulus with v~ being a random value,
Ei -- Em + 2AEv~i)
(12.1 O)
Fracturing structure
Assuming homogeneous straining a plane set of the cells yields the stress distribution in a cell layer as
~
eEi, 0
if if
eEi < O'i, ~gi ~ o-i.
(12.11)
Here e is the homogeneous strain in the specimen. Let us neglect possible interaction of mesocracks occurred at sites of the cells that failed (ai -- 0). Then the ultimate strength of the model is equal to a maximum value of the average stress in a layer. Hence, the strength is written as - max(
For N1 - 1.
n0
i)
(12.12)
530
Ch. XII, w
Powder metallurgy methods
Computer simulation The algorithm of the computer simulation is now clear. We produce a set of the raw specimens ( n - 1) with expected fibre volume fraction v~ by using a random value generator to obtain a homogeneous distribution over the interval (0, 2v~ The raw specimens undergo n steps of mixing. Obviously, mixing time t and value n are related to each other, a particular form of the relationship being certainly dependent on blending parameters. In a particular experimental series, a set of the specimens with a constant fibre volume fraction v~ contains four samples. Some results of the calculations are presented in fig. 12.5. The following observation should be noted. The strength scatter of the raw specimens is enormously large. With blending time increasing the scatter decreases reaching nearly zero at a time corresponding n = 32. At short blending times, the strength scatter is still so large that it is impossible to see a trend in the change of composite strength with increasing fibre volume fraction. At moderate blending times (n = 4 - 8 ) , the composite strength is definitely increasing with the fibre volume fraction increasing. There is a blending time (that is somewhat between n = 16 and n = 32 in the case under consideration) at which the strength growth changes to a strength decrease. Since the plane model was used in the calculations, a correction to the final results can be made by choosing a lowest value of the strength from a set of the four
n=J
~ ~
oO
E1.1 1.0 0.9
o
~ oo
o
o
o~_ ~o ~ ~o o: ~o o~ ~ o oo~oo ~) o
8o o
0.8
o
o9 o
1.2
o o
o o
o
o
1.0
1.2
o~o~O
OoOoo
o
o
o
o
Oo
~o
~o o oo
^o o o~o~oo oooo %oO%#~ %oOo'~o oO o
o
~o
~
o
o
~
o
o
o
o
o
o
o.g
0.1
0.2
0.3
.
.
o
0.4
.
0.5
0.6
0.7
.
0.1
0.0
1.1
n=4 9
9 E
o
o
o
0.7 0.6 0.0
n=2
1.1
o o
o~
o
o
0.2 ,
o
0.3 ,
0.4 ,
n=16 9 n=32
n=8
0.5 ,
~
% 1.1
o
1.0
oo
o o
0
o~
9
o
~176 Q
1.0
o.o
o.1
02
0.3
v,
04
0.5
09
0.0
o11
0'2
0'3
v,
0.4
Fig. 12.5. Calculated dependencies of the normalized composite strength on fibre volume fraction. The structural parameters accepted are characteristic for carbon-fibre/carbide-matrix composites.
Short-fibre composites
Ch. XII, w12.4
J
J O
J
531
J
J
G
Fig. 12.6. Schematic of a dependence of the composite strength on fibre volume fraction for composites obtained by applying different blending times at vf < v~ and vf > 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
Powder metallurgy methods
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.
Short-fibre composites
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].
Powder metallurgy methods
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
Short-fibre composites
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~
Powder metallurgy methods
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
Powder metallurgy methods
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
Powder metallurgy methods
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
Continuous fibre composites
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
Powder metallurgy methods
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].
Continuous fibre composites
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
Powder metallurgy methods
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.
Infiltration mechanics
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
Infiltration mechanics
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 <_ i <_n)
and n is the number of the interspaces of a radius of r in a unit preform. Since the area fraction of the same dimension of the interspaces is equal to its volume fraction, that is r v(r), the value of flux across unit cross-section, q(r), through the interspaces with dimension r is
q ( r ) - U(r)qg(r). The total flux across a unit cross-section of the preform will be
Q(x) -
fg R-',,
q(r)dr- g(l -
(5 dp ) dx
(13.44)
mi
where B-
3D(1 - vf)R22 40/~
and
D-
"D r d r - fN COS N(r)~i di
(0 <_ i < N)
(13.45)
Techniques
Ch. XlIl, w
561
P
g. %
,z"
Fig. 13.7. Schematic of the dependence of external infiltration pressure on the ram displacement. After Long et al. [373].
and N is the total number of the interspaces in a unit preform 9 The system of differential equations, eqs. (13.40) and (13.43), together with the condition dQ dS = ~ dx dx
(13.46)
allows to exclude variable ~ and obtain function p(x). The authors [373] analyzed, in a qualitative manner, a possible solution for the case of a constant velocity, Vram, of the ram applying the pressure to the melt, so that G = //ram. The result of their analysis is shown in fig. 13.7. It is important to note that after the initial stage of infiltration related to the capillary effects, the stable infiltration follows which is characterized by zero back pressure since air can go out of the preform through the open porosity. However, when the melt reaches the far end of the die, the air-venting ducts occur to be sealed and the back pressure, Pback, builds up with the infiltration proceeding. It can be calculated [373] and take into account when analyzing the infiltration kinetics. This pressure effects the stress in the preform and, hence, stability of its shape.
13.2. Techniques In choosing a liquid infiltration method, one has bear in mind the following points 9 A uniform fibre packing 9 The importance of this was discussed in Section 5.2.5. 9 A choice of a matrix with satisfactory mechanical properties. This is to be pointed out since casting alloys do not usually possess high toughness and ductility.
562
Liquid infiltration
Ch. XIII, w
9 Providing the means to control the time of contact between the fibre and the molten matrix. Results of considerations of capillary effects, infiltration kinetics, thermal effects and matrix crystallization (Section 13.1) provide a guidance in choosing fabrication parameters. On the other hand, fibre/matrix interaction also discussed above, Chapter 10, sets limitations on the fabrication parameters. So up to the present time, most studies of liquid-phase fabrication routes have been based on intensive experimental work. To prevent fibre dissolution during infiltration, the following ways are known: 9 Decreasing solidus temperature of the matrix by heavy alloying it as was done to produce C/AI and C/Ti composites, see Sections 13.3. land 13.5, respectively. 9 Introducing an intermediate layer of a material with low melting temperature, an example will be given in the next chapter, Section 14.3. 9 Decreasing a time of the fibre/liquid-matrix contact by applying a special way of the rapid heating, see Section 13.2.6.
13.2.1. Pressureless infiltration This is an easiest way to produce composites, however there are not so many useful fibre/matrix couples with sufficiently good wetting. Still, Kazmin et al. [299] and Hillig [248] have demonstrated a possibility to make ceramic-matrix composites relying on wetting only. The former authors produced molybdenum-wire/oxidematrix composites, the latter one made SiC-whisker/powder-mixture reinforced a number of matrices such as CaF2, SrSiO3 and similar ceramics. In the first case, the infiltration was performed in vacuum, in the latter one, this was done in an appropriate gas (carbon oxide, argon, or a mixture of the two).
13.2.2. Vacuum infiltration A scheme of the vacuum infiltration of a bundle of fibres wetted by a molten matrix was suggested at the beginning of composite technology [71, 121]. In this case, a driving force is the sum of capillary and atmospheric pressures which occurs to be sufficient to lift a liquid column to a necessary height if fibre volume fraction is large enough. A practical realization of the method can be done as shown in fig. 13.8. A mold made of a material stable in the molten matrix is filled with the fibre and placed into the crucible containing the matrix melt. The upper end is connected to the vacuum system to provide a pressure difference caused by the atmospheric pressure. Kun et al. [346] preheated a bundle of CVD SiC fibres placed in a steel tube in a small furnace while keeping a vacuum environment in the tube sealed at one end by an aluminium stopper. Then the tube was immersed in molten aluminium located in another furnace. After melting the stopper, aluminium infiltrated the fibre bundle for sufficiently short time.
Techniques
Ch. Xlll, w
0 0 0 0 0 0 0 0 0
563
9 0
Fibre
0 0
Heater Matrix
0
melt
Crucible
0
Io 7//////////////// Fig. 13.8. Schematic of vacuum infiltration process. Actually, an inert gas pressure is preferable. Hence, the process can be p e r f o r m e d in a v a c u u m c h a m b e r (see for e x a m p l e [194]). After e v a c u a t i n g air a n d m e l t i n g m a t r i x material, the c h a m b e r is filled by a r g o n gas at a pressure j u s t a b o v e the a t m o s p h e r i c (fig. 13.9).
(a)
(b)
Fig. 13.9. Schematic diagram of the infiltration of the fibre bundle with the alloy melt. The scheme was used by Glushko et al. [194] to produce oxide-based-fibre/NisAl-matrix composites, a) The initial state of the crucible with the fibre and solid matrix after pumping out the furnace, b) Applying the argon gas pressure.
Liquid infiltration
564
Ch. XlII, w
13.2.3. Pressure infiltration The early paper by Morris [473] contains a scheme of infiltration under pressure by pressing a container with aluminium as matrix material and unidirectionally oriented carbon fibres. The pressure is to be applied after melting of the matrix, a schematic of the method is depicted in fig. 13.1. Using inert gas pressure to propel the matrix melt into the fibre preform is especially convenient when dealing with matrices of high melting points such as nickel or intermetallics. In the case of a sufficiently high pressure, an autoclave shown schematically in fig. 13.10 is to be used. A particular design of the apparatus depends on particular fabrication parameters (temperature/time/pressure). Nourbakhsh et al. [493] described in detail such an apparatus to process composites with matrix of a melting point as high as 1 4 5 0 - 1750~ They used it to produce both Al203-FP-fibre/NiAl-matrix and AlzO3-FP-fibre/Ni3Al-matrix composites. Khvostunkov and Mileiko [310] used a similar apparatus to fabricate composites with Ni_~Al-alloy matrix reinforced with oxide fibres produced by using the internal crystallization method. Figure 13.11 illustrates a distribution of the fibres of an unusual cross-sectional shape (see fig. 14.1d) in the matrix. Jarfors et al. [280] presented schematically an apparatus for processing aluminium matrix composites. In an autoclave described by Masur et al. [390], there are two independent heaters, the upper one preheats fibre preform placed in a tubular quartz mold, the lower one melts the matrix in a crucible with the bottom end of the mold placed in it. Since that
\
\\
9 \ , "\ "\
\
-
/ // ///:/
~ : ....... M o l d //iJ"i
//~
Heater
/// / /
~ //
Matri.r
melt
/~/ C~"ucible , ~ / ~ --///.,
Cas
7'he'+'~r~ocouplc
irtle [ I
Fig. 13.10. Schematic of an autoclave used for pressure infiltration of a fibre preform.
Ch. XIII, w
Techniques
565
Fig. 13.11. A cross-section of the A1203/Ni3Al-composite obtained by a pressure infiltration method. After Khvostunkov and Mileiko [310].
apparatus was used for studying the infiltration kinetics, it had two interesting features. First, to perform the infiltration at a constant pressure, the autoclave was pressurized from a holding tank, from which nitrogen gas was released via orifice ball valves of a sufficiently large diameter. Secondly, to sense a position of liquid metal front, a CVD SiC filament inserted into the preform and the molten metal served as a variable resistor. The driving force to make the melt to infiltrate the fibre preform can also be body forces, for example of centrifugal or Lorenz type. It should be mentioned that for the latter case, Andrews and Mortensen [15] calculated infiltration velocity and infiltration distance by solving numerically Maxwell equation for electromagnetic field together with eq. (13.5) to which the inertial term was added. Calculation results occur to correspond to experimental ones obtained in a specially designed apparatus with the energy source being a capacitor bank which discharges through a primary coil causing finally a pressure in the premelted matrix as a result of the interaction of magnetic field and eddy currents with a frequency between 2 and 3 kHz.
13.2.4. Squeeze casting Squeeze casting (fig. 13.12) is actually pressure casting, just the pressure in the molten matrix is determined by the displacement velocity of the ram and the
566
Liquid infiltration
~
Ch. XIII, w 3.2
2~eh
/ -D-i e
/
0 .
-
II]I iiii L~i
-~--J\
\
Heater
Magrix
melt
Fibre
preform
"\0 ///M Fig. 13.12. A schematic of squeeze casting process.
filtration velocity in the fibre preform. On the other hand, this is a variant of the well known process in foundry technology employing pressure to enhance casting quality [102]. Obvious advantages of this method as compared with the pressure infiltration method, are usage of a simple equipment, high speed operation and possibility to produce near-net-shape elements. This points are especially seen well when dealing with relatively low-melting point matrices. Bader et al. [31] were certainly the first to use and study systematically the method and composites obtained in such a way. In experiments with Saffil 6-alumina-fibre/ aluminium-alloy matrix composites, they registered the ram displacement, temperature at the upper and bottom surfaces of the preform, and pressure in the melt versus time and found the conditions for good infiltration without essential distortion of the preform. It is important to prepare sufficiently rigid preform which is usually reached by adding an organic binder to the fibres. Normally, the organic binder contains small quantities of silica which remain after the binder is burnt off during preheating the preform. As we have seen (Section 5.2.5), homogeneity of fibre packing effects essentially the maximum strength of the composite. So preform should be sufficiently homogeneous. To achieve it, Ju et al. [290] used ultrasonic dispersion of short carbon fibres in water containing 10% of sodium silicate. The preform should be preheated to make conditions of the infiltration more favorable. To make fibre distribution in the preform more homogeneous and, at the same time, to control fibre volume fraction more accurately, some authors use the so called hybridization [80], that means adding particles or whiskers to the fibrous preform. This is done, for example, by impregnating fibres into aqueous suspension of particles (or whiskers). A polymer is used as a binding agent and an organicmetallic
Ch. XIII, w13.2
Techniques
567
compound serves as a dispersing agent. Mechanical tests show that an optimal content of the particulate exists that certainly depends on the fibre volume fraction. Another important parameter is the infiltration pressure. Enhancing the pressure (or the ram velocity) yields decreasing the infiltration time and, therefore, limits unwanted fibre/matrix interactions. On the other hand, this requires more rigidity and strength of the preform and, perhaps, will call for using the pressing equipment of higher cost. Usually, a pressure up to 100 MPa is used. A design of the mold/ram configuration can ease the requirements mentioned. In particular, a suggestion to envelop the preform with a porous ceramic filter [317] provides (i) a way for the melt to infiltrate the preform from its sides as well from the top, allowing preform deformation to be hydrostatic, (ii) trapping air containing in the preform in the lower part of the filter. Hot-rolling of a blank made by squeeze-casting improves microstructure and properties of the composites certainly because of void healing [290]. Hot-extrusion after squeeze-casting of randomly-oriented-short-fibre preform provides fibre alignment [368, 578, 713]. For a particular combination of the temperature and strain rate, there exists a maximum reduction value which can be observed without the cracking of the material during hot-extrusion [713]. For SiC-whisker/6061aluminium-alloy-matrix composite (vf = 0.2), the reduction value of about 45% seems to be critical at 400~ and about 55% at 600~
13.2.5. Compocasting Compocasting, or rheocasting, suggested by Mehrabian et al. [402] in early 70s, is used to produce metal-matrix composites, mainly short-fibre/aluminium-matrix composites. The process is now known to have some variations but the main steps are as follows [190, 292, 364]: 9 flow-casting of a semi-liquid alloy at a temperature just above the solidus; 9 vigorous mixing of the alloy and adding fibres into the liquid/solid mixture; 9 rolling, die-casting, extrusion, or some other procedure to shape a composite. Obviously, a matrix characterized by a large difference between solidus and liquidus temperatures makes easier to process the composites, especially at the final step. A mixture is contained in a crucible with an opening at the bottom used to pour out the mixture for further processing. Vigorous mixing at the second step is necessary to keep the solid/liquid mixture fluid to prevent the formation and growth of primary phase dendrites, that is a kind thixotropic behaviour of the slurry. The fibres are added to the matrix when it contains about 50% of the solid phase. Going on with stirring at the stage of adding fibres prevents floccation of the fibres and promotes wetting as a result of disruption of the contaminated layer on the fibre. If the fibre volume fraction is sufficiently low, the mixture can be cast directly, otherwise it should be heated again. The microstructure of a composite obtained in such a way is characterized by a globular shape of the primary phase and a homogeneous distribution of fibres within the eutectic areas [190], so that fibres are surrounded by the matrix of a composition different from the average one. Possibly, this is a reason for enhanced wettability in the fibre/matrix system.
Liquid infiltration
568
MIXING CHAMBER 0 "[, ~ o
Ch. XlII, w
HEATER
ijj ,
O
"'/I. (-~,.40 HEATER . . . ~
lr
FIBRE/SOLID/LIQUID _ /SOLIDIFICATION FRONT
WA TEN COOLED ~ SOLID C___OOPPE____ROLLER R_ ~ C O M P O S I T E PLA TE
Fig. 13.13. Rolling of a mixture of short fibre and metal matrix containing initially solid and liquid phases. After Kang et al. [292].
Figure 13.13 presents a schematic example of rolling as a final step of the fabrication route as produced by Kang et al. [292] who fabricated a Saffil fibrereinforced aluminium matrix heavily alloyed so that the difference between solidus (477~ and liquidus (635~ temperatures occurred to be large. This allowed to pour a mixture of the fibre and matrix heated up to a temperature within an interval of the existence of a liquid phase in the matrix, in the gap between rollers and to keep a solidification front at a constant position in the processing zone. There exists a combination of the pressure on the slab, roller speed, and initial temperature of the mixture that provides a composite microstructure which looks like a usual one. Compocasting combined advantages and shortcomings of the fabrication routes using liquid and solid phases. Simplicity of the process and facilities, as well as easy shaping have come from liquid state technologies, a possibility to clean fibre surface are from the powder metallurgy. On the other hand, the process inherits the problems arising due to fibre breaking during processing.
13.2.6. Rapid infiltration Shortening time of the exposure of a fibre to a molten matrix of high chemical reactivity toward the fibre limits unwanted chemical interactions at the interface and allows sometimes to produce composites by using a more convenient liquid phase
Ch. Xlll, w
Techniques
569
routes instead of solid state processes such as diffusion bonding or powder metallurgy. The time can be made shorter by either using a eutectic alloy which can wet the fibre sufficiently well or enhancing the infiltration temperature to a sufficiently high level to make the matrix alloy to wet the fibre much better than at temperatures normally used for the infiltration. The former way shall be discussed in the next chapter, Section 14.3, the latter one has been suggested by Warrier et al. [369, 686, 687]. With regard to aluminium matrix reinforced with either graphite or silicon carbide fibres, they have found that enhancing the infiltration temperature above 1000~ makes wetting and flow characteristics of a casting aluminium alloy favourable for pressureless infiltration. Performing such a procedure called for the development of a controllable method of fast heating and cooling. Warrier et al. use the infrared heating for that purpose which occurs to allow to keep the fibre in contact with molten matrix for times ranging from 5 to 30 s. Experiments with graphite fibres in aluminium matrix have shown [687] that the rate of the reaction zone thickness growth at 1100~ is about 3. 10 -6 c m / s . The dissolution rate of silicon carbide (Nicalon) fibre in aluminium occurs to be 4.7 9 10 -6 c m / s . This means that when the exposure time is controllable, which is the case, the macrostructure of the composites appears also to be controllable. In the case of titanium matrix reinforced with thin fibres such as graphite fibre it is obviously wanted to use a liquid infiltration process, so the problem of the fibre/ matrix interaction arises as one of major importance. A systematic study of the problem with regard to silicon carbide fibres and titanium alloys matrix with sufficiently low solidus temperatures was carried out by Warrier and Lin [686]. They were dealing with SCS-0 (unprotected), SCS-2 and SCS-5 silicon carbide fibres and titanium-nickel alloys with some ternary additions containing 40, 60, 70, and 80 wt% of titanium (marked as Ti40 - Ti80 further on). The fibre loads were rapidly infiltrated with a matrix at temperatures 1200~ (all matrix alloys) and 980~ (Ti40 - Ti70 alloys). The time of the infiltration was 30 s. The authors have found that 9 The higher the titanium concentration in the matrix alloy, the more extensive are the fibre/matrix reactions. 9 Unprotected silicon carbide fibres react extensively with the matrix. In the case of Ti80 alloy the fibre is drastically decreasing in the diameter after the processing at 1200~ The carbon-rich layer in SCS-2 and SCS-6 fibres is able to prevent the reaction of SiC with the matrix certainly due to the formation of a continuous TiC layer. A titanium matrix containing 15 wt % of nickel and some ternary additions was also reinforced by graphite fibres at 1350~ and 10 s [685].
13.2.7. Preparation of a composite wire Pulling a fibre bundle through a bath of a molten metal (see fig. 13.14) can be used to produce a composite precursor cable with a large e n o u g h fibre volume fraction. The time of contact between the fibre and the molten matrix, as well as the coating
570
Liquid infiltration
Ch. XIII, w
FtBRe ReeLs
SEPARATOR CRUCtBLe METAL
MeLT
COMPOSITe wire SHAPER Fig. 13.14. A scheme of producing a composite cable by pulling a fibre bundle through a matrix melt.
necessary to form a diffusion barrier or to promote wetting, can be estimated prior to the experiments. Expected mechanical properties can also be estimated and analyzed. We shall mention a technical scheme of producing boron-aluminium cables with a matrix of commercially pure aluminium [593] as an example. The rate of pulling is about 1 m/s. The composite precursor emerges after infiltration with a melt which begins to crystallize at a temperature about 90~ below the melting point. Crystallization starts at the fibre surface and gives a matrix with properties better than after normal casting. A similar scheme was used [189] to produce a cable containing seven filaments of silicon carbide in an aluminium matrix (the 6061-alloy). The fibres are in contact with the melt for less than 1 s, and the fibre volume fraction can be varied by changing the melt temperature, pulling rate and the distance between fibres. The third example is the production of the maraging-aluminium tapes [176]. A plane array of the steel cables is pulled through the melt, the pulling rate being equal to about 1.5 m/s and after the bath the tape is slightly compressed. The width of the tape is about 30 mm, and the fibre volume fraction is up to 60 per cent. Certainly the most effective use of liquid infiltration to form composite cables is in the case of carbon fibres. The fabrication scheme includes two main stages [189, 405]. At first, fibres are coated by a mixture of titanium and boron to promote wetting. The procedure is based on the reduction of titanium tetrachloride and boron tetrachloride by zinc vapour and yields thin (10-20 nm) and uniform coating of each fibre in a bundle. Then the fibre bundle is pulled through the melt of an aluminium alloy and a compact composite wire is obtained. The strength of the
Ch. XIII, w13.3
Aluminium-matrix composites
571
cables at v f - - 0 . 3 5 - 0 . 4 0 is about 1.0 to 1.3 GPa. Photomicrographs of the crosssection reveal rather non-homogeneous fibre packing. In such a way, there can be also obtained Nicalon-fibre/aluminium-matrix wires [164].
13.3. Aluminium-matrix composites
13.3.1. Graphite/aluminium composites Graphite-aluminium is a promising metal-matrix composite which is composed of relatively inexpensive components and expected to reveal sufficiently good mechanical properties. So if a reasonable fabrication technology of such composites existed and material properties were sufficiently stable and reliable then the outlook of the whole field of metal-matrix composites would be quite different from what we see now. The understanding of this has been always subsisted since the very beginning of an intensive work on metal-matrix composites. Perhaps the most effort in this field has been actually put into graphite fibre/aluminium matrix-composites. Small diameter of the fibre calls either for vapour or liquid phase technology route to be executed. Involving a vapour phase stage into the technology gives a rise to a cost of the material because of a relatively low production rate of such processes. Hence, a liquid phase technology is mainly used. Then, aluminium is a strong carbide former, therefore it is difficult to prevent formation of A14C3 on the fibre/ matrix interface. Aluminium carbide occurred is hygroscopic, so a composite can change its structure drastically when it contains the carbide on the interface and is exposed to normal environments. To avoid such changes, a fibre coating is used as a diffusion barrier with a corresponding increase in the material cost. At the same time, there are some evidences that an optimum carbide content, from viewpoint of the composite strength, exists (see fig. 10.8 above and fig. 13.17 below). Perhaps, a way to either overcome or to make more controllable problems related to carbide formation is to lower infiltration temperature by using heavily alloyed m a t r i c e s industrial aluminium alloys [675].
Fabrication
Techniques Liquid phase fabrication route is a most convenient base for a technology of graphite fibre/aluminium-matrix composites, at least for continuous fibre composites. Liquid phase technology is used either to produce prepreg wires and layers containing continuous fibres which are then undergo hot pressing to form a composite element or to make a composite body of a structural element directly. In the latter case, various techniques can be used, pressure casting seems to be most popular. Choice of technological parameters Obviously, temperature and time of exposing a fibre to molten matrix effect strongly the composite strength through the fibre and
Liquid infiltration
572 '
'
'1
"
I
9
I
'
Ch. XIII, w
'
I
800 o
C/A/ 9 B/B4C/AI
x
SiC/AI
*b 600
400
-
200 640
,
l
660
,
i
680
,
i
700
,
T~ ~
i
720
Fig. 13.15. Bending strength of three composites obtained by pressure infiltration method. The fibre volume fraction and exposure timc arc tixcd for each compositc. After Zabolotsky [725].
fibre/matrix interface strength. An example of such a strong dependence is shown in fig. 13.15. Zabolotsky [725] has suggested an algorithm for an error and trial procedure which replaces an analysis similar, for example, to that described in Section 5.2.5. He measured strength, o-~, on a fixed length, of a fibre extracted from a composite, and shear strength, rr*m, of a composite. Then he noticed that the composite strength can be presented as a function of the parameter M -- (a~ / at*.,,)2 ('rt~n/I"m ) where af*,o is an initial value of the fibre strength (before processing) and Tm is the matrix strength, fig. 13.16. Despite the experimental data presented in fig. 13.16 do not reveal maxima on the dependencies, Zabolotsky's assumption of the existence of a value M0 of the parameter that corresponds to a maximum composite strength seems to be justified, at least on a qualitative level. Hence, parameter M determined experimentally plays about the same role as equivalent time, eq. ( 11.21). Introducing M allows to analyze experimental data as a function of a single argument. Zabolotsky's experimental data have also clearly shown that coating the fibre and alloying matrix with elements reacting with either the fibre or coating produces an essential effect on the choice of optimum fabrication parameters. This can be illustrated by dependencies of the strength of graphite-fibre/aluminium-matrix composites on the content of aluminium carbide on the interface, see fig. 13.17 which is the same plot as shown in fig. 10.8.
Ch. XIII, w
573
A luminium-matrix composites
900
I
'
i
'
I
'
0
/y
o
/o
% BIB 4
600
-
"
/s
300
2
I
,
I
,
0.4
0.2
I
06
M
0.8
Fig. 13.16. Bending strength of the composites described in fig. 13.15 as a function of parameter M (O'~/O'~-0)2('r~-m/'Cm).After Zabolotsky [725]. =
1000
'
I
o
i
Uncoated fibrelAI 9 Uncoated fibre/M-7%Si
% 800
a
Coated fibre/M-7%Si
~
600
o
D
400 9
200 0.00
i
I
I
0
0.04
0.02 VAI4C3
Fig. 13.17. Strength of graphite-fibre/aluminium-matrix composites versus the volume fraction of aluminium carbide. After Zabolotsky [725].
We should note that approaches to optimization of the fabrication parameters similar to those just described have certainly a limited importance since they do not take into account strength/fibre-volume-fraction dependence discussed in detail in
Liquid infiltration
574
900
i
'
I
,
I
I
'
I
i,,
Ch. XIII, w
I
800 1o
700
600 500
4000
i
I
50
i
100
I
150
i
200
I
250
"~*fm / MPa
i
300
Fig. 13.18. The composite strength versus interface shear strength for graphite/aluminium composites. The matrix is AI-7Si-0.3Mg-0.3Fe alloy, various fibres and fibre surface treatment, vr=0.35. Experimental data after Diwanji and Hall [130].
Chapter 5. This means that a change in the composite macrostructure (changing vf) can cause a shift in the optimum set of the fabrication parameters. An illustration to such an expectation can be found in experimental data provided by Diwanji and Hall [130] (fig. 13.18). Certainly, an increase in the interface strength yields an increase in the fibre effective strength ((c~(l*)) in eq. (5.8)) due to a decrease in the fibre critical length, l*. To keep the fibre volume fraction within the limits of the favourable fracture mechanism (vf < vA in fig. 5.21), one should somehow enhance the matrix fracture toughness if the initial value of vf is sufficiently large, which is the case in [130] (vf = 0.35). Since the matrix in all these experiments was the same, the results obtained is not unexpected. The temperature/time parameters of pressing of prepregs are obviously limited by carbide formation on the interface. Hence, an optimum set of the parameters are to be found in experiments.
Mechanical properties Longitudinal Young's modulus of graphite/aluminium composites can be high since the corresponding modulus of the fibres can be very high. The same is true with regard to the specific values of the modulus. On the other hand, both transverse and shear moduli are low because of high anisotropy of graphite fibres. The following matrix of elastic constants Cij (in GPa)
Ch. XIII, w
575
A luminium-matrix composites
24.0
ll.2 23.5
12.6 12.6 271.1
0 0 0 17.7
0 0 0 0 17.9
0 0 0 0 0 6.29
"
measured by ultrasonic pulse method [356] being compared with analogous measurements for boron/aluminium (figs. 3.8, 3.9), alumina/aluminium (Table 3.1) and other composites with rather isotropic fibres, shows clearly a strong anisotropy of graphite/aluminium composites. As mentioned in Chapter 5 (Table 5.1) the dependence of the strength of graphite/aluminium composites on fibre volume fraction is typical for brittle-fibre/ ductile-matrix composites, a nearly linear dependence with relatively small strength scatter is observed up to vf ,,~ 0.3 - 0.4 and then the average strength stops to go up any more, the scatter becomes very large. Additional evidences of such a behaviour are shown in figs. 13.19 to 13.21. High temperature strength of the composites depends on the time/temperature conditions following the carbide formation (Section 10.2.2). Experimental observations of such a behaviour can be found in numerous publications, see i.e. [274, 513, 529]. Creep-rupture behaviour is also typical for such kind of materials, see fig. 13.22. As shown in Section 6.2.2, creep-rupture of composites with fibre volume
•)0
I
I
Graphitised
"1
i
oo
(a)
~400
i
i
"
i
500
(b)
0
0
%
0
o
0
0
0
0
0
0
400
o
0
0
0
0
300
o~
0
0 o
0O 0
n I
0 0
0 0 0
o
20~t.1
' 0'.2
'
0:3
'
0'.4
' O'"5 ~'
g
300 o
0.6 0.1
.
9 C.~rt:x~se(:/, type I Carbonised, type II
. 012
.
. . o'.a
o'.4
v,
015
0.6
Fig. 13.19. Tensile strength of composites with aluminium matrix reinforced with (a) high modulus graphite fibre and (b) high strength carbon fibres of two various types, versus fibre volume fraction. Composites are made by hot-pressing of precursors at 550~ MPa-1 h. Experimental data after Jackson et al. [274].
Liquid infiltration
576 800
~b
'
i
,
i
'
Ch. XIII, w
i
7OO 600
t
500
400 300
01
0.0
'
o12
013
Fig. 13.20. Average values and scatter of the tensile strength of graphite/aluminium composites obtained by hot-pressing of composite wires. Matrix is the AI-13Si alloy. Experimental data after Pepper and Penty [529]. fractions vr > v~, (v A is a critical value o f vr shown in fig. 5.18) is characterized by weak-link model failure and, c o r r e s p o n d i n g l y , by large scatter.
13.3.2. Alumina/aluminium composites A l u m i n a - f i b r e / a l u m i n i u m - a l l o y - m a t r i x c o m p o s i t e s c a n n o t offer as high specific rigidity values of in the r e i n f o r c e m e n t direction as those characteristic for graphite-
~300 % 200
100
~i0
'
012 ..... v,
014
'
0.6
Fig. 13.21. Tensile strength of composites with 6061-aluminium matrix and planar random graphite fibre obtained by squeeze casting. Experimental data Pechersky et ai. Cited after Baxter [43].
Ch. XIII, w
577
Aluminium-matrix composites
300
'
'
'
'
I
'
'
'
'
I
~
'
'
'
I
'
'
'
'
I
'
'
'
'
t~
~200 b
100
0.01
.
,
,
.
I
0.1
|
,
,
|
I
1
,
,
|
,
I
,
10 t / h
J
,
,
I
100
,
.
,
.
1000
Fig. 13.22. Creep-rupture properties at T--400~ of graphite/aluminium composites described in fig. 13.19a. Experimental data after Jackson et al. [274].
fibre composites. Also they are based on fibres which certainly will never be produced in such large quantities and in so many variants as carbon or graphite fibres. However, the elastic anisotropy of alumina-based composites is expected to be essentially less than that for graphite-based composites. Then, no problems caused by a special interface behaviour arise. Corrosion properties of the composites look much more attractive. All this determined a great interest in such composites in the last two decades. F P - A I 2 0 3 fibre
Composites with such fibres are known to have been obtained by b o t h vacuum infiltration method [71] and squeeze-casting [317]. In the former case, a matrix alloyed with lithium was used to promote wetting. This led also to a decrease in the fibre strength certainly due to formation of a brittle layer on the fibre/ matrix interface. So an optimum lithium content in an alloy seems to be between 2 and 3 . Unidirectionally reinforced composites obtained by both methods are characterized by a linear strength/fibre-volume-fraction dependence up to vr ~ 0.6 (fig. 13.23) which means that combination of fibre, matrix, fibre/matrix interface properties, as well as composite geometry shifts critical point vA in fig. 5.18 to sufficiently high values of yr. Composites retain the room temperature strength up to temperatures of about 300~ the fatigue behaviour (fig. 13.24) is of a normal type for metal-matrix composites.
Liquid infiltration
578
--600
Ch. XIII, w
0 A
b
400
200
z~ AI-2.2%Limatnx (Champion et al.) o AI-4%Mgmatrix (Hu et al.) 9 AI matrix (Hu et al.)
~
o12
'
o14 'Vf o16
'
Fig. 13.23. Tensile strength of FP-Al203-fibre/aluminium-matrix composites versus fibre volume fraction. Experimental data by Champion et al. [71] and Hu et al. [263].
Observation of fracture processes in composites with discontinues fibres randomly oriented in plane has shown [317] that microcracks in fibres lying side-by-side locate in one plane. On the other hand, the matrix (AI-4% Mg alloy) is capable of arresting the microcracks of a size of at least two fibre diameters. This leads to an intensive fibre breakage during loading which changes essentially the distribution of the fibre aspect ratio, p: the average value p changes from about 25 to a b o u t 8.
~ o ~ AI-2.2%Li matnx, 20~ - - e - - Mg alloy matrix, 20~
........ _~. ~. ~ a,o~.~,x,. 2.6~c ........
0.8 %
~
0.6 ~
/,.
E0.4
AI and Mg alloys at 25~ ,,~ i
0.2 0.0 102
Mg alloys at 260~ ,
.
,
,..,,I
103
.
,
, ,,,,,I
104
,
,
,
,.,,,I
,
,
,
,,,,.I
105N 106
"---'t ,
,
,
,,,,,I
107
Fig. 13.24. Fatigue strength of composites with FP-AI203 fibre normalized by the ultimate stress. Stress ratio R = 0.1. Experimental data by Champion et al. [71].
Ch. XIII, w
579
Magnesium matrix composites
6 - A ! 2 0 3 fibre
Nearly the same sequence of events with fibre microcracking was observed in testing composites with a planar array of discontinues 6-A1203 fibres [93]. The strength/fibrevolume-fraction dependence (fig. 13.25) is characterized by a possible maximum at very low values of fibre volume fractions for room temperature testing data. With increasing testing temperature, maximum of the strength shifts to larger fibre volume fractions exactly as observed for other metal-matrix composites with short fibres (see for example, fig. 5.17). Such a behaviour perfectly corresponds to fracture behaviour of the composites revealed by measuring fracture energy in instrumented impact tests [178]: introducing 25 vol% of 6-alumina into A1-4Zn-2Mg and AI-12Si alloys yields a decrease in fracture energy from 184 to 30 and from 35 to 23 kJm -2, respectively.
13.4. Magnesium matrix composites Magnesium-matrix composites look attractive due to a low density of the matrix comparable with that of polymers. Infiltration pressure during squeeze casting effects essentially matrix microstructure and properties [83]: the grain size of the Mg8.9A1-0.7Zn-0.2Mn matrix alloy decreases from 14 to 10 m/t as the pressure increases from 35 MPa to 75 MPa. At the same time, the fibre volume fraction increases from 0.22 to 0.26 due to compression of the preform. This leads to an increase in bending strength of the composites from about 360 to about 460 MPa as compared with corresponding increase in the matrix strength from about 280 to 400 MPa. 350
,
,
,
,
.
,
.-"
,
0
D
300
0
~'250
% 2oo 150 100
o
25~
Baderet aL
9 250~ o
50 0
0.00
25~
Bader et aL Ackermann et aL
9 250~ ,
I
0.05
~
I
0.10
Ackermann et al.
9 350~ i
......
I
0.15
Ackermann et aL ,
v,
I
0.2O
,
0.25
Fig. 13.25. Strength of composites with randomly oriented 6-A1203 fibres and aluminium alloy matrices versus fibre volume fraction. Experimental data by Bader et al. [31] (AI-9Si-3Cu alloy matrix) and Ackermann et al. [4]. (AI- 12Si- 1Cu- 1Ni alloy matrix).
580
Liquid infiltration
600
.
.
.
.
.
500
Ch. Xlll, w
1.0 0
0
9
~'1.1.
o"
7
~400
0.8
300
200
:
100 oL-
Strength
0.6
I? 0
'
0.0
o:2
'
o14
'
Vf
o. o."
Fig. 13.26. Tensile strength, a*, and fatigue strength, av, normalized by a* (R=0, N = 106 cycles) of AlzO3-FP-fibres/ZE41A-magnesium-alloy matrix composites. Experimental data after Nunes et al. [498].
13.5. Titanium-matrix composites T i t a n i u m - m a t r i x c o m p o s i t e are mainly p r o d u c e d by hot-pressing or p o w d e r metallurgy m e t h o d s (Sections 11.7.1 and 12.4.2) because o f a high melting point of titanium and high chemical reactivity o f the matrix in contact with such possible reinforcement as c a r b o n or silicon carbide (Section 10.2.2). Still, obtaining carbonfibre/titanium-matrix c o m p o s i t e via liquid phase route looks very attractive. There are k n o w n at least two a t t e m p t s of such a kind. 5O0
.
.
.
.
I
.
.
.
.
I
.
.
.
.
0
7"400
300
200 o
T i - 35Cu 9
T i - 25Cu
100 . . . . . . . . . . . . . . 0.0 0.1 0.2
Vf
0.3
Fig. 13.27. Tensile strength versus fibre volume fraction for graphite-fibre/titanium-copper-matrix composites. Experimental data after Toloui [657].
Ch. X I I I, w13.6
In termetallic matrix composites
581
In the first one, Toloui [657] infiltrated carbon fibres with titanium-copper alloy. However, to reduce the melting temperature of the matrix and to decrease the thickness of the C/Ti interface zone composed of titanium carbide, the copper content has to be too high to consider the product as a real titanium-matrix composite. Strength characteristics of the composites with matrices containing 25 and 35 w% of copper (with the liquidus temperatures of about 1280 and l l00~ respectively) shown in fig. 13.27 reveal a usual kind of the behaviour with a maximum strength at a rather low fibre volume fraction. In the second attempt, carbon fibres were introduced in the titanium matrix through an interphase of a eutectic in the Ti/TisSi3-system (see Section 14.3 below).
13.6. Intermetallic matrix composites Titanium and nickel based intermetallic compounds, because of the reasons discussed above, Section 1.3.1, are expected to be widely used as matrices for fibrous composites. Such composites produced by using liquid infiltration techniques have been described in a number of the publications, however the composite microstructure has mainly been a subject of the discussion (see Sections 10.2.2 and 10.4). Here we shall describe strength characteristics of a composite with M I G L fibre based on AlzO3/ZrO2 eutectic (see Section 2.2.4) and matrix of an alloy based on Ni3A1 [ 194]. Composite specimens were obtained by using vacuum infiltration process, an atmospheric pressure being applied by argon gas (fig. 13.9). A microstructure of the composites is illustrated in fig. 13.28. A typical composite structure of the AlzO3/ZrO2 + Y203 fibres should be noted. Actually, not all the features of both the fibre/matrix interaction and crystallization of the matrix are clear. Still, it is interesting to note that X-ray microanalysis shows a decrease in A1 and Cr content in the matrix within about 2 lam from the interface. This is just one of the possible features mentioned that yields a matrix microstructure in the composite different from that of the pure matrix. The phenomenon seems to depend on the fibre volume fraction. This can explain an unusual dependence of the strength on fibre volume fraction at room temperature, presented in fig. 13.29a. By extrapolating the experimental dependence r~*(vf), obtained testing composites with vf > 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
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o
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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
~-~
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9
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..~
,-,
.~"
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-<.
~...,.~
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>~-~
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~ ~"~
1 . i ~ .14 5. Typicit1 hts~ogri~l)l\ 171' it11gI~4 h c t w t : ~Ihc ~ ~ ( . - ~ I X I S 11f 1 1 ) ~ ' ~ i ~ p p h ~tihrch r c i ~ n dihc 1ihl.c axis. Di;tl~ictcrormc~lyhdcnuniwirc IS 50 pm. p~~llilig ritk I Y 7 ~lllil:'lni~i. Fihrc vr)lumc fraction: (;I-d)0.17. (c,l) 0.3.5. (g.h) 0.46. .AILlcrMilciko ;lnd K;t/min [4.1.\].
;..,.
~--"
=~-. ~'=~
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--.
~:~
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9
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:~" ~" ~ ,--. ~
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Variatinns of fihre volume Fraction in spccinlclis f-21bricatc by the intcrnal by variations of both ch~~racteristic sizes crystallization method can he :~ccol~lpanicJ and proportions ol'the fibre cross-sectional shiipe. This nlilst he taken into account i~nalyzingtho fijilure beh:~viouro f the eomposi~es. According to eq. (5.8). k i t low values ol'tihre volu~nef'raction. 1~1.. the st]-ength of a composite is
~.
9
~" ~
~rO
~ X
...,..
~ ,---
~
v
o"~
where {cr;(l*))is the 111can fibrc strength on length 1'. I' is the average distance betweon fibre breaks. is the matrix strength, ct is a constant. r =. I .
!
-2
~2
....:
4"--
Fihrc1.r cottl c ortrpo.o~c.cohtctrrtc,tl hl- I( ' M
i
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C'h. XIV, 314.2
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,
-
C~
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C~
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r.O
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tin
C~
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,
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C~ C~
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(P
Fig 14 5 c h
v
=
~"
9
If fibre of the cross-section shape shown in fig. 14.ld is loaded lo I'i~ilurc by shear stress 7' distributed on its surf~~ce, then the cqi~ilibriun~ of the Libre Icuds to
+
,~.,e = uo ~ o o
u .~
r~ "'"
u
~:~ ~'~. ~
o~ =
~... ~= . ~~ ~ r~
C;
~..u ~ ' ~
~ "' "-i
o o .~ =~ ~ . ~ ~, .~
I
~"~
~ ~
9= ~ : ~
r~
~''~ ~
~,~,~
tvhcrc p = 2t/d x,J ' = 4 t / d - rt. On thc other hand, [here exists a scale dependence of the fibrc strength which can bc taken to be of' the Weibull type, but it is not clear a priori what should be the dimension of the scale. A variation of the fibre cross-section with the variation of fibrc volume fraction provides us with a means of investigaling what dimension might be the appropriate scale for fibre strength. whethcr i t should be tho length or the surface or the volume. We can check all three hypotheses on the basis of experimental data. Let us write down the scale dependence as
594
Ch. XIV, w
Internal crystallization
(14.2) where /~ -
1,2,3, and k~---l* , k ~ - l * p d ,
k *3 -
~ l * f d 2 correspond to particular
(
hypothesizes. The same is true of the constants cr0. k - A ~ . It is clear that a particular value of/~ depends upon/t. Comparing eqs. (14.2) and (14.2) and excluding the value of l* which is not measured in direct mechanical tests, we obtain (a~(l*)) - B~21~,/('+~)
(14.3)
where B~ -- A ~ / ( l + ~ ) ( z * ) ~/(!+~) ,
B3 -- A~(+ /l ~)(4z*()+ /l ~)3 21 = p / f d , ,~2 - - 1 / f d 2 ,
23 = p / f Z d 3
9
Therefore the initial equation for the composite strength can be rewritten as S* - C~,fl'~/(' +~)
(14.4)
where C~-~B~,, S * - O"m * q - - ( O ' * - - O ' *m ) / t ) f. S o approximating the experimental dependence S*(2) by a power function gives a value of [/for each hypothesis. On the other extreme of composite failure mechanisms, is the failure of a specimen immediately after a first fibre break at a weakest point. It involves quite a large total fibre length, of the order of metres. To evaluate/3 for this case, let us assume the "volume" hypothesis to be true. If the total fibre length in a specimen with a fixed number of layers and a constant length of its homogeneous part is L = nb/t, then eq. (14.2)can be rewritten as a~(L) - C2 '/~
(14.5)
Here C is a constant, 2 = f d 2 / t , 2 is introduced through 23. 14.2.2. A l 2 0 3 / M o
composites
A1203/Mo composite family is a convenient model to study structures obtained by internal crystallization. The description of the fibre and composite structures for these composites was given above to illustrate the fabrication method. Now we present some mechanical properties of the composites and fibres. Figure 14.6 shows experimental data for the tensile strength at 1300 ~ approximated by power functions according to eq. (14.4). The values o f / / appear to be
Fibres and composites obtained by ICM
Ch. XIV, w
595
3.0
2.8
B
o
1/
9
0
2.6 o,A,
2.4 , ,
oct
2.2
,
i
ooooo ***** I
2
i
~ -,~ Z /z = 3 l
i
og(X/Xo)
4
Fig. 14.6. Tensile strength of A1203/Mo composites 1300~ presented according to eq. (14.4) to determine of the scale dependence of the strength of a sapphire fibre in a molybdenum matrix. Logarithmic plot, So = 1 MPa. After Mileiko and Kazmin [433].
equal to 0.350 for p = 1, 1.82 for p = 2, and 3.24 for p = 3. Obviously the value of fl corresponding to the linear scale in eq. (14.2) is non-admissible, but there are no strict grounds to choose a proper value of fl between those for p = 2 and p = 3. Still the value for the " v o l u m e " hypothesis (fl = 3.24) looks more realistic. This conclusion is confirmed by the observation of fracture surfaces of fibres. An example of a typical surface shown in fig. 14.7 illustrates a usual defect which initiates the fracture. Such voids are distributed within the volume of a fibre. A possible reason for their occurrence is a shortage of the melt near the crystallization front because of the volume effect at crystallization. This leads to a d e m a n d to decrease the pulling rate, u, at composite fabrication. The experiments (fig. 14.8) point to a value u = 7 m m / m i n as a reasonable upper limit. This also satisfies a d e m a n d to have a small enough size of the polycrystalline region in a specimen. To analyze experimental data corresponding to composites with high fibre volume fractions by using eq. (14.5) one needs to account for residual stresses. They are estimated as
a~r - E A ~ A r / ( 1 + vf/Vm) where E = EF = Em (a case when the moduli of the fibre and matrix are nearly the same), AT = 1400~ for the reasons discussed in Section 5.5. The values of composite strength at 1300~ (presented in Table II of [433]) are shown in fig. 14.9 in accordance with eq. (14.5), they lead to fl = 3.45. The difference between this value and the value of fl obtained for short-lengths basis of the same fibres (fl = 3.24) is, obviously, negligible.
596
Internal crystallization
Ch. XIV, w
Fig. 14.7. Fracture surface of a sapphire fibre. The pore, certainly initiating tile failure, can be seen.
The experimental data for temperatures other than 1300~ are insufficient to obtain statistical characteristics of the fibre strength. So the data given in fig. 14.10, can be used to evaluate a first approximation to the temperature dependence of the fibre strength, as a value of ~(a~(l*)) obtained by direct application of eq. (5.8) to
Ch. XIV, w
Fibres and composites obtained by ICM
597
300 0
~
0
200
80
\ b
[]
100
o
!~ -o~
n
[]
ooooo v i = 0 . 3 3 , dw=O.05 m m vf=0.33, d~=0.2 m m vf=0. 46, d.w=0.2 m m
oonoo
~ ~
0
i
I
0
I
i
I00
i
200
I
i
I
300
u /
i
400
mm/min
500
Fig. 14.8. The dependence of the strength of sapphire/molybdenum composites (for two fibre volume fractions, vf, and two wire diameters, dw) on crystallization rate of fibres. After Mileiko and Kazmin [434]. 600
!
!
!
i
i
i
!
i
i
|
!
i
,
,
I
i
i
i
0
500
.
400
300 200 i o I00
0.0
'
0.2
i
i
0.4
i
0.6
Fig. 14.9. Determination of the scale dependence of the strength of a sapphire fibre for fibre length of the order of metres at 1300~ After Mileiko and Kazmin [433].
the experimental data. Such a p r o c e d u r e yields the result presented in fig. 14.11 t o g e t h e r with the t e m p e r a t u r e dependencies for sapphire fibres p r o d u c e d by the E F G - m e t h o d (see Section 2.2). It should be noted that different batches of fibres o b t a i n e d and tested using b o t h different fabrication schemes and technological p a r a m e t e r s have nearly equal characteristics in a t e m p e r a t u r e interval to be considered of m o s t interest f r o m the point of view of high t e m p e r a t u r e materials.
598
Ch. XIV, w
In ternal crystallization
600
I
9
I
'
I
~
'
I
'
o_o__o__oo 0.33
0 0 0 0 0 '01 vt
9. .O.g Q. O
=
=
'~.)f
0
0.17
"--
_
~400 \ b
200
0
m
I
0
i
500
I
i
1000
T / ~
.
l
i
2000
1500
Fig. 14.10. Temperature dependence of the tensile strength of sapphire-fibres/molybdenum-matrix composites. Diameter of molybdenum wire is 50 Jam, crystallization rate is 7 mm/min. After Mileiko and Kazmin [434]. 1500
~1000 \ "b
500 ooooo
Mileiko Hurley
~, A
Shahi~tian
ooooo
0
i
0
~
I
500
&
i
Ka
"
I
I000
T / ~
i
l
1500
i
2000
Fig. 14.11. Temperature dependence of the tensile strength of sapphire fibres obtained by both the internal crystallization method and EFG-method. Experimental data are after Mileiko and Kazmin [434], Hurley [267], and Shahinian [594]. W h e n considering high t e m p e r a t u r e properties of materials, it is obviously m o s t i m p o r t a n t to evaluate the creep properties. Therefore, the results of creep tests of A l z O 3 / M o c o m p o s i t e s at 1300 a n d 1700~ will now be discussed. The creep
Fibres and composites obtained by ICM
Ch. XIV, w
599
behaviour of the composite under consideration at these two temperatures differs qualitatively, and that is very convenient to illustrate different cases of creep of composites. At temperatures up to 1300~ creep of sapphire fibres can be neglected, at least to a first approximation. Therefore, we shall use here the model of Section 6.2.2 to interpret the experimental data. Typical creep curves of the composites with fibre effective diameter 200 ~tm are shown in fig. 14.12. The creep characteristics of the matrix (the matrix being a specimen prepared to melt the infiltration and undergo thermal treatment at 2100~ have been determined in a special experiment. In the power creep law for the matrix, expressed in the usual way, eq. (6.1), we can assume m - 6, 6 m - - 16 MPa, qm -- 10-4 h - I Fibre stress a'(t), normalized by the value of fibre stress ao/vf at t - 0, increases according to eq. (6.59). Calculated curves (~r'/ao)vf for AIzO3/Mo composites (fig. 14.13) reveal a very quick stress redistribution. It means that the time of the first fibre break in a composite should vary in a very broad interval even if fibre strength scatter is not too large. The interval should become broader with increasing fibre volume fraction. Qualitatively these conclusions correspond to the shape of experimental creep curves shown in fig. 14.12 as well as to creep-rupture data given in fig. 14.14. When a specimen creeps under the stress equal to 70 MPa (fig. 14.12), the average fibre stress as a result of the pure stress redistribution without taking into account fibre creep will be not more than 140 MPa. Creep-rupture time for this condition varies from 0.5 to 60 h [433]. This is a confirmation of the qualitative conclusion about a large scatter of rupture time. On the other hand, such values of the fibre 0.006
'
vf=0"400"
0.004
'
I
I
'
I
'
40 .3
cO
0.002
0.000
i
o
3
1 I
I
0.004 cO
vf=O.
0.002 O. 0 0 0
0
,
I 20
,
..
t/h
.I 40
Fig. 14.12. Creep curves of AI203/Mo composites at 1300~ Kazmin [433].
60
tr = 70 MPa. After Mileiko and
600
Internal c r y s t a l l i z a t i o n
~
Ch. XIV, w
.00
"b
/
090
/ / /
0,85
/
'ul = 0.33. ~o = 90 MPot. vI
.....
vI
vf oo,ooov,,, 0 . 8 0
10-'
.
.
.
1
.
.
.
0 33, 0146, 0.46, .46, '
Oo = 1 1 0 ~. = 60 o'0 = 90 o'0 = 1 2 0
........
10./ h
'
10 ~
'
MPa. MPoc. MPa
MPot :
103
Fig. 14.13. Fibre stress versus time for AI203/Mo composites at 1300~
stress are well below the values of fibre strength (fig. 14.5). This suggests that a stress redistribution between the fibres due to their various crystallographic orientation is essential. The crystallographic orientation of a part of the fibres is such that the basal plane of the sapphire crystal lies at an angle of about 45 ~ to the fibre axis (see fig. 14.5). Such fibres creep easily, the stresses in these fibres decrease, and as a result, well-oriented fibres take an extra load. When the fibre diameter decreases to 50 lam then obviously the average strength of the fibre increases, leaving a general view of composite failure process without essential changes. Nevertheless, a shift of fibre volume fractions is to be noted for weakly interactive fibre breaks towards lower values (this corresponds to an increase in the slope of the OA-line in fig. 5.2). The other important point is quite a large increase in creep-rupture stresses (fig. 14.14b). However, the creep strength remains not high enough, the scatter is too large, and an increase in fibre volume fraction does not lead to an essential increase in creep strength. To enhance the creep strength of composites of the type under consideration, it is necessary either to achieve a homogeneous fibre orientation which would be preferable from the viewpoint of creep resistance, or to retard creep of fibres in some way. The first route can complicate the fabrication process. So the second one looks more worthwhile. This can be performed using eutectic mixtures of oxides as the fibre materials. It shall be described in detail below, in this chapter. At the temperature of 1700~ sapphire crystals creep intensively [76, 200]. So the creep of composites is described by the model presented in Section 6.2.1. Typical creep curves of the composites at 1700~ are presented in fig. 14.15. The matrix creeps at this temperature according to power law given by eq. (6.1) with m -- 3.61, o m - 4.29 MPa, qm - 10 -4 h -I.
Ch. XlV, w
Fibres and composites obtained by ICM 1
120
i
I
i
i
i
i ii
10
I
i
i
i
o
i
i
i
ii
601
100
I
I
i
!
i
i
i i i
o
8O
40
(~) o o o o o v I = 0.33 ODaODVI = 0.40 .oooov s = 0.46 160
~
-
140
I00
-
80
i
o
b 120
i 0
(b) i
1 0 -2
i
i iiiiii
I
10 -1
I
I iiiiii
I
1
i
~o
i iiiii
I
10
I
I
I illlll
0
I
10 8
i
I i
lO s
Fig. 14.14. Creep strength of A1203/Mo composites at 1300~ and 50 ~tm (b). After Mileiko and Kazmin [433].
Molybdenum wire diameter is 200 ~tm (a)
The dependencies of the creep rates of the matrix, composites, and fibres on the applied stress are shown in fig. 14.16. The fibre stress during creep is as follows from eq. (6.53) ~, _ (~ - ~ , , ( ~ ) ) ~ , / ~ f
+ ~"(~)
where a" is the matrix stress. Assuming all the fibres have the same crystallographic orientation, one obtains a dependence of the creep rate of the fibre material on stress (fig. 14.16) which is too strong (n ~ 40 in creep law ~ = qn(a/an)"). N o t e that the points corresponding to highest values o f the creep rate are not taken into account. Such a value does not correlate with the data for the values of n previously obtained for sapphire crystals [200]. It appears that for the slip along the basal plane
Internal crystallization
602 I
|
|
I
I
,
Ch. XIV, w
'
6
4
2
0
0
5
10
15
20
t/h Fig. 14.15. Creep curves of sapphire-molybdenum composites at 1700~ Kazmin [433]. '
'
'
'
'
'
' ' ;
/
ooooo v t = 0 . 2 4 o o o o o v t = 0.:33 9~ o
10
-~
+ + + + +
vt
~x~
vt = o.4s
=
0.40
/
'
* 0 '
/
* x
'
'
'
9 .
/
9**** m a ~ 9" . . .
~bre
1 0 -3
I
1 0 -"
'
a = 30 MPa. After Mileiko and
1
. . . . . . . .
5 '
10
+
*.
'<"
"e
9
9
. . . . . . . . . /
MPa
100
Fig. 14.16. Stress dependencies for creep rate of the matrix and sapphire-molybdenum composites. Calculated values of the fibre stress at the stationary stage of creep of the composites are also plotted. Test temperature is 1700~ After Mileiko and Kazmin [433].
at 1550~ n ~ 4. For a single crystal, when its c-axis deviates from a tension (compression) direction by 8 ~ ~ within temperature interval 1200-1750~ n = 19 + 4. At the same time, experiments show [76] that the yield stress does strongly depend on temperature and crystallographic orientation. Therefore, for
Fibres and composites obtained by ICM
Ch. XIV, w
603
different orientations, the values of n and an should be very different. The values of n ~ 40 for sapphire fibres can be claimed to be too large for any orientation. Such values can be explained by a s s u m i n g fibre orientations in tested specimens have varied within a specimen as well as f r o m specimen to specimen. P e r h a p s it m a y also explain the large scatter in creep rates of c o m p o s i t e s with a fixed fibre v o l u m e fraction.
14.2.3. Complex-oxide-fibres/molybdenum matrix composites A variety of inherent physical properties of complex oxides of rare earth metals (piezoelectricity, ferroelectricity, superconductivity are just some examples [546]) m a k e s fibres a n d composites with these c o m p o u n d s be interesting f r o m the viewpoint of possible applications. We shall give here some d a t a on the strength of fibres obtained f r o m the oxides shown in Table 14.1 [434]. The original experimental d a t a are given in figs. 14.17 and 14.18. It should be noted that because of particular c o m b i n a t i o n s of the fibre strength and the m a t r i x TABLE 14.1 The melting points (in ~
of some oxides. After Portnoi and Timofeeva [546].
M =
La
Sm
Y
MA103 M3AIsO12 M2Ti207 MTaO4
1650 1930
2100 -
1875 1930 2000 -
600
!
~ ~
|
|
o oooo.' ro',, ; =6b ~ + + + TieY20~,
|
!
|
|
~
|
/~m, ~ = T m m / m i ~
d=200
.•400
\
b
[]
.
200
r'l " 9- - . . . . _ .
0
0
J
I
I
I
400
i
,
o _.
i-i
,
T/~
I
800
I
I
II slZl
I
I
I
1200
Fig. 14.17. The temperature dependence of the strength of composites with a molybdenum matrix and fibres shown on the figure field. Fibre volume fraction is 33%. After Mileiko and Kazmin [434].
Internal crystallization
604
6 0 0
,
,
,
i
,
,
,
w
,
,
,
I
,
,
l
,
A tSrr~Os LaTaO,
ooooo . . . . .
~
,
Ch. XIV, w
x x x x x A tYOs AA,,AA T~2Y~07
400
\
*b 200
0
600
o~ ,
,
,
I
800
I
I
,
I
J
1000 r / ~
f
,
I
1200
~
J
I
i
1400
Fig. 14.18. The temperature dependence (at high temperatures) of the strength of composites with a molybdenum matrix and fibres shown on the figure field. Fibre volume fraction is 33%. Diameter of molybdenum wire in composites with Ti2Y2Ov fibres is 50 lam, in all other c a s e s - 65 ~tm, crystallization rate 4 - 7 mm/min. After Mileiko and Kazmin [434].
yield strength, as well as the corresponding values of thermal expansion coefficients, the fibres of YAIO3 and SmAIO3 have been breaking during the cooling of specimens in the fabrication process, and the average length of fibre pieces is close to the critical fibre length at room temperature. Therefore, when being tested at high temperatures these composites already have fibres with their lengths less than critical. This remark is important when considering the data in figs. 14.17 to 14.19. These fibres are not loaded up to their ultimate stresses so the corresponding data in fig. 14.19 do not represent inherent fibre strength characteristics. It should also be noted that a single crystalline structure of the fibres under consideration was not constantly attained. For example, in case of SmA103 fibres, the shape and distribution of the reflections in the Laue pattern obtained on an isolated fibre correspond to a large crystalline structure. Results of the microprobe X-ray analysis of the fibres reveal their single-phase, chemically homogeneous structure. The latter appears to be important: a special experiment with nonstoichiometric Y3A15O12 fibres have shown [434] that small deviations from stoichiometry of the complex oxide can yield a drastic decrease in the fibre strength. 14.2.4. AlzO3-AlsY30]2-eutectic-~'bre/molyhdenum matrix composites
As shown above, the creep strength of composites with single-crystalline, continuous sapphire fibres obtained by ICM cannot be large enough at high temperatures, first, because of the existence of operative slip systems at relatively low temperatures (for example, the A1203 crystal starts to creep at temperatures not higher than
Ch. XIV, w
605
Fibres and composites obtained by ICM l
u
u
I
00000
800
7
x x x x x A IYOs
-
o o o o o A IS~'l~,Os " 9 9 9 9 9
600
LaTa04 Al20s
\ *b,.,400
200
0
700
oX 0
n
n
i
I
900
i
i
9
i
I
l
11 O0
T /
~
n
n
I
1300
i
i
i
1500
Fig. 14.19. The temperature dependence (at high temperatures) of the strength of fibres shown on the figure field. After Mileiko and Kazmin [434].
1200~ This causes redistribution of the fibre stresses due to non-homogeneous crystallographic fibre orientation in a specimen, which goes on quite quickly. A convenient and simple method of preventing a high fibre creep rate can be based on the use of eutectic mixture melts for the crystallization of fibres. The crystallization of the melt of such a mixture can form a structure with an ordered system of obstacles to dislocation motion at high temperature. At the same time, the volume of a fibre is divided into smaller volumes and the size of potential microcracks is therefore limited. Furthermore, the melting point of a eutectic is lower than those of the ingredients and that can be useful in real processing. If we consider a possible structure of the crystallized AlzO3-A15Y30~2 eutectic, it can be seen as two continuous carcasses bonded to each other continuously. It has been already shown that one of them, namely sapphire, can be the single crystal. The boundary between the two crystals, one being hexagonal with a = 4.759 A and c - 12.991 A (A1203) and the other being cubic with a - 12.005 ,~i [5], cannot be coherent. So plastic deformation in such a structure is expected to be suppressed to a large degree by the incoherent boundary, and the creep rate should therefore be lower by comparison with that for corresponding single crystals under similar conditions. This yields an expectation of having the creep strength of a composite with such composite fibres higher than that of a composite containing separate single oxide fibres. The creep properties of a eutectic fibre are expected to depend on a characteristic structure size which, in turn, depends on the crystallization rate. Such motives have led to the systematical study of AlzO3-A15Y3Olz-eutectic-fibre/ molybdenum matrix composites in comparison to AlzO3-fibre/molybdenum matrix composites [434]. The matrix materials for all of the specimens is again commer-
606
Internal clTstallization
Ch. XIV, w
cially pure molybdenum. The melt of fibre material has been made from a mixture of chemically pure oxides of both aluminum and yttrium. The composition corresponds to the mass ratio A1203 : Y203 equal to 1.96:1, that is to produce the eutectic in the system [532] and the volume fraction of sapphire in the fibre is 44%. The melting point of the eutectic is 1760~ so the melt temperature before infiltration should be about 1800~ The expected structure of the fibres described above is actually observed. Figure 14.20 shows the appearance of transverse sections of fibres. The longitudinal ones are of the same pattern, no preferred orientation of any carcass can be seen, at least within the interval of the crystallization rates used. Neither the fibrous nor plate structures inherent in unidirectionally solidified eutectics have been observed in this case, almost certainly because a stable flat crystallization surface in a separate channel within the matrix does not exist. When the pulling rate increases the characteristic size of structure decreases, and when it reaches 7.5 mm/min this size becomes 1-5 gm, which is the limit for the optical resolution. A further increase of the rate results in a structure which is optically transparent, as illustrated by a series of photographs of separate fibres extracted from composites (fig. 14.21). The structure of the sapphire constituent of a fibre changes from mono-crystalline to polycrystalline in about the same fashion as in the case of pure sapphire fibres. A block structure can be detected in the sapphire crystals in fibres crystallized at a pulling rate between 60 and 240 mm/min (figs. 14.21c and d). Increasing the crystallization rate of a eutectic fibre thus leads possibly to an increase in fibre strength because of a decrease in the characteristic size of the
Fig. 14.20. Transverse section of an A1203-Y3A15O12 eutectic fibre obtained at a crystallization rate of 1 mrn/min. (Magnification x 1000.) After Mileiko and Kazmin [434].
Ch. XIV, w
Fibres and composites obtained by ICM
607
Fig. 14.21. A1203-Y3A15012 eutectic fibres obtained at crystallization rates (a) 1 mm/min, (b) 30 mm/ min), (c) 60 mm/min, (d) 240 mm/min. Photographs (b) and (c) were taken in transmitted light, (a) is obtained in reflected light. After Mileiko and Kazmin [434].
structure but, on the other hand, it can decrease the strength because of the appearance of the block structure of sapphire. Therefore, there should be an optimal pulling rate corresponding to a m a x i m u m value of the fibre strength. Figure 14.22 shows the high-temperature composite strength as a function of the crystallization rate. An optimal pulling rate during crystallization lies apparently between 10 and 100 mm/min. N o w let us use eq. (14.4) to evaluate the Weibull parameter, /~. The necessary plots are presented in fig. 14.23, and the values of/~ obtained are as follows: 0.928 for kt = 1, 2.83 for /~ = 2, and 4.71 for g = 3. As in the case of sapphire fibres, assuming either the length or surface defect distribution (/~ = 1 a n d / t = 2) do not seem to bring a realistic result. So it remains to accept the volume defect distribution
Internal crystallization
608
I
ooooo ooooo
I I I !
0.05 = 0.2.0
d
d
u
ram, ram,
= I
,
u i ,
l
u
n
"of = v F' = I
0.42 0.45 u
o
o
250
u l I
Ch. X I V , w
-
o
\ 9b 2 0 0
150
100
....
,
i
,
1
,
........
, ,,,,i
u
I
/l'Omm,/mA
,
n 1 O0
Fig. 14.22. T h e d e p e n d e n c e o f the s t r e n g t h o f c o m p o s i t e s with e u t e c t i c fibres on c r y s t a l l i z a t i o n rate. A f t e r M i l e i k o a n d K a z m i n [434].
2.8
! ~,,~2.6
,/o/
0
9
2.4 ~
i
*
o o o o o
~
ooooo/x, ***** /~
2.2
, 0
,
,
i 2
,
,
-- ! = =
2 3 ,
og(X/Xo)
4
Fig. 14.23. D e t e r m i n a t i o n o f scale d e p e n d e n c e o f the s t r e n g t h o f A I 2 0 3 - Y 3 A I s O i 2 e u t e c t i c fibres o b t a i n e d at c r y s t a l l i z a t i o n rates 30 m m / m i n in m o l y b d e n u m m a t r i x at 1300~ . T h e p r o c e d u r e is a c c o r d i n g to eq. (14.4). L o g a r i t h m i c plot, So = ! M P a , 2 = I m m -! for IL = I, 2 = I m m -2 f o r / t = 2, 2 = 1 m m -3 for /x = 3. A f t e r M i l e i k o a n d K a z m i n [434].
Ch. XIV, w
Fibres and composites obtained by ICM
609
hypothesis. The difference between corresponding values of /~ for sapphire (see above),/~ = 3.24, and the eutectic,/3 = 4.71, should be emphasized. It relates to the ratio of the variation coefficients of the strength of the fibres at a critical length equal to about 1.5. The failure surfaces of eutectic fibres (fig. 14.24) differ essentially from those of single-crystalline sapphire fibres (fig. 14.7). Their appearance changes from purely brittle (sapphire) to quasi-tough (composite eutectic). The dependence of the creep strength of the composites with A1203-A15Y3012eutectic fibres on the crystallization rate is presented in fig. 14.25. For an applied stress equal to 180 MPa, quite a sharp maximum is observed. Together with the corresponding strength data (fig. 14.22) it defines an optimum pulling rate equal to 30 mm/min. Creep strength data are displayed in fig. 14.26. The main conclusion which can be drawn if one compares these data with those for short-term strength is that up to 1300~ the creep strength, ~ , is quite high in comparison with the short-term strength, ~ . In fact, at 1000~ the value of the ratio cr~/a~ is equal to about 0.9 for rupture time 100 h, while at 1300~ the value is equal to about 0.7. It can be explained if only one assumes a sufficiently high creep resistance of the eutectic fibres. Certainly it is much higher than that of single-crystalline homogeneous fibres. It should be noted that a deviation of the fibre composition from the eutectic point leads to a fibre structure with typical dendrites. Consequently, the composite strength falls and its creep strength is drastically reduced.
Fig. 14.24. Failure surface of A1203-Y3A15012 eutectic fibres obtained at crystallization rate 30 mm/min. The testing temperature is 1300~ • 2500. After Mileiko and Kazmin [434].
610
Internal crystall&ation ''''!
'
'
' '
''''u
,
--
,
Ch. XIV, w
, , , , , l
=-
10 ~
0
9
10
\1 1 0 -t 10
-2
1 0
-s
0
--4
|
, ,,,I
,
eoooo
o
=
180
ooooo
o" =
f50
|
|
|
|||i
l
,
,
MPa MPtt
,
,
,|al
i
1 Fig. 14.25. The dependence of rupture time of composites with eutectic fibres on crystallization rate for two applied stresses. After Mileiko and Kazmin [434].
3 0 0
........
w
........
,
........
u
. . . . .
I:$ 0 n
0
\ b
n
250
200
I*1-
-
@
. . . . . . . .
1 5 0
0.1
I
1
. . . . . . . .
n
I0
. . . . . . . .
I
100
Fig. 14.26. Creep strength of composites with eutectic fibres (vi- = 0.42) obtained at a crystallization rate 30 mm/min. The testing temperatures are 1000~ 1150~ and 1300~ (from top bottom). After Mileiko and Kazmin [434].
Finally it is interesting to note that the gas corrosion resistance of A1203/Mo composites at high temperatures does not differ from that of pure molybdenum. Unlike this, AI203-Y3AIsO12/Mo composites reveal some resistance to gas
Ch. XIV, w
Fibres and composites obtained by ICM
611
corrosion, certainly because of yttrium molybdate formation on the surface exposed to oxygen containing atmosphere [432].
14.2.5. Nickel-matrix composites Despite the gas corrosion resistance of molybdenum matrix composites with particular kinds of the fibre can be superior to that for molybdenum alloys, it remains far below the values necessary for a material to be used in an oxygen containing atmosphere for a sufficiently long time. So we should be looking for a possibility to fabricate oxide-fibre/nickel-matrix composites by using internal crystallization method. Doing so, we need to explore either ternary complex oxides or ternary oxide eutectics as candidates for the fibre material. As a first example, we consider here 2A1203- MgO. 3CaO-fibre/nickel-matrix composites [220]. The phase analysis of the AlzO3-MgO-CaO system, was carefully studied by Majumdar [380] who discovered the stable oxide mentioned. Phase diagram of the system is drawn schematically in fig. 14.27. The stable compound, 2A1203. MgO. 3CaO, exists within the ABC-field. It should be noted that the melting point of the compound under consideration changes between 1343 and 1348~ even if the composition remains within the ABC-field. Low melting
AlcOa (a)
CaO
M90
E ~"
B
A
(o)
Fig. 14.27. (a) Schematic drawing of Majumdar ternary phase diagram [380] of A1203-MgO-CaO system with (b) a look at ABC-field.
612
Internal crystallization
Ch. XIV, w14.2
temperature (~ 1350~ provides a chance to use nickel as a matrix material in the internal crystallization method. The melt of the fibre material is obtained by mixing up powders of the simple oxides in an appropriate ratio. Before mixing, the CaO powder is heated up to a temperature between 1100 and 1200~ in an alumina crucible to eliminate water. Melting and all the procedures with the melted oxides are performed in argon gas atmosphere under pressure of about 0.1 MPa. We should stress technical problems arisen because of a small difference between the melting points of the fibre and matrix materials. Because it is a rather tricky thing to keep the infiltration temperature higher than 1380~ and in most experiments the infiltration was carried out at the temperature mentioned, one has to do it overcoming large enough viscosity of the 2A1203 9MgO 93CaO melt. Hence, a special experiment has been performed to obtain a dependence of the strength of composites on the infiltration time. The oxide composition in this experiment was 51.5% A1203, 6.5% MgO, and 42% CaO, the crystallization rate is 8 mm/min. The results are presented in fig. 14.28. There is a significant growth of the strength when the infiltration time increases up to 4.5 h, then it goes down. The observation of cross-sections of the specimens and their failure surfaces reveals a decrease in a number of unfilled channels with increasing the infiltration time. A decrease in the strength at a large infiltration time can be caused by evaporation of the matrix during the infiltration. It has to be noted that a zone of the chemical interaction at the fibre/matrix interface, estimated by measuring both nickel and magnesium content with the X-ray microanalysis, increases with the infiltration time reaching a size of 5-6 lam for 5 h. The dependencies of the strength of the composites on the crystallization rate are shown in fig. 14.29. It is interesting to note the presence of a maximum on each curve. The X-ray analysis shows that the crystallization rate u0 = 8 mm/min corresponds to a single crystalline structure of the fibre and the crystallographic
0
,
,
,
,
,
,
,
,
8o
,
,
,
I
I
I
0
\ 0
*b 70
0
0
60 50
I
0
I
I
I
120
I
t /
I
I
I
240
m~n
360
Fig. 14.28. Strength of 2A1203. MgO. 3CaO/Ni composites versus crystallization rate. Fibre volume fraction is 30%. After Gvozdeva et al. [220].
Ch. XIV, w
100
!
Fibres and composites ohtained hy ICM i
~
u
u
!
|
|
u
n
!
|
v
,
n
i
613
v
|
9
\ b
5O
0
800~ 1200~
ooooo ooooo I
0
I
I
I
I
5
i
I
I
I
I
10
/ mm/m~.n
i
.
15
Fig. 14.29. Strength of 2A1203. M g O - 3 C a O / N i composites versus crystallization rate. Fibre volume fraction is 30%. After Gvozdeva et al. [220].
orientation of the fibres in a specimen seems to be homogeneous, unlike the case of A1203/Mo composites. At higher crystallization rates the fibres appear to get polycrystalline structure with rather large crystallites. Similar Laue-patterns, typical of large grains polycrystals, are observed for specimen obtained at crystallization rates lower than u0. Perhaps this is because of non-homogeneous crystallographic orientation of the fibres in a specimen. Such changes in the fibre structure can certainly be correlated to the strength/crystallization-rate dependencies. The initial fibre compositions in the experiments with 2A1203. M g O . 3CaO/Ni composites were changing in a rather wide interval. The strength data obtained are presented fig. 14.30. The centre of the ABC-field corresponds to a minimum of the strength at 1200~ The direction of the largest strength gradient corresponds to increase in MgO content and decrease in CaO content at nearly constant A1203 content. The maximum is a very pronounced one. A most likely reason for the shift of the stress maximum to the direction of increasing MgO content is evaporation of the MgO component from the melt. At the same time, it should be noted that plastic deformation and fracture of single crystals of the type under consideration have not been studied yet, and the drop in the fibre strength at the vicinity of the centre of the homogeneity area on the phase diagram may be perhaps explained by some features of the deformation and fracture processes of the crystal lattice. It is important to point out nearly the same location of both maximums and minimum on the a* versus the initial fibre composition surface at various temperatures within the whole interval, from 800 to 1200~ The same results were obtained in testing composites with these fibres and molybdenum matrix [220].
Internal crystall&ation
614
Ch. XIV, w
):
b
~"~ ~
''~
~
x~
~ o -~.
C Fig. 14.30. Strength of 2A1203 9MgO. 3CaO/Ni composites versus deviation of the initial fibre-material composition from the central point of the ABC-field in fig. 14.27a. Test temperatures are 800~ (a), 1000~ (b), 1200~ (c). After Gvozdeva et al. [220].
Ch. XIV, w
Fibres and composites obtained by ICM
615
Approximate estimates of the strength of the 2A1203 9MgO 93CaO fibre system in the nickel matrix at 1200~ yield value between 250 and 300 MPa. Therefore, to obtain the creep rupture strength of a composite to be equal to about 150 M P a it is necessary to use a matrix with the creep rupture strength of 5 0 - 70 MPa. We see now the main drawback in using a complex oxide as the fibre material: viscosity of the melt in the vicinity of the melting point is rather high and this makes the fabrication route time consuming. This is unwanted, in particular because of the danger of unfavourable chemical interactions. A possible choice of the fibre material is a system with the triple eutectic. TiOz-MgO-CaO is such a system with the eutectic at a point corresponding to the elements' mass ratio 78.6:5.65:15.75, and the eutectic temperature at ~ 1350~ However, the TiO2 entity sets a problem of its stoichiometry, that need to be solved because of its effect on the fibre strength. The necessary experiment was performed by Rudnev [574] who produced a series of rutile/molybdenum specimens that were drawn one after another, from one melt placed in the vacuum chamber. The results elucidated an essential influence of oxygen evaporation from rutile on the composite strength. Similarly, a row of the specimens TiOz-MgO-CaO-fibre/Mo-matrix (vf = 0.46) produced from one melt (u = 1.75 mm/min) showed the following chain of strength values: 180 - 150 - 144 - 140 MPa. Therefore, the problem of infiltration of the nickel matrix with a rutile containing melt looks controversial: it is necessary to keep the melt under the oxygen containing atmosphere to preserve the stoichiometry; on the other hand, there is a danger of the oxidizing of the matrix. Thus we have to search for an optimum air pressure that possibly exists. Rudnev [574] used a furnace specially designed to perform the infiltration and crystallization under controlled air pressure, between 0 and the normal atmospheric pressure. His experiments showed that the optimum conditions of the infiltration did not yield strength values higher than those for 2A1203 9M g O . 3 C a O / N i composites.
14.2.6. Composites with silicide-based fibres We consider here a special case of using the internal crystallization method to produce composites with matrix material M and fibre material M + N the latter being a eutectic. In this case, wetting of the matrix by the fibre melt is certainly provided, the only point is to prevent the fibre material from overheating that would cause dissolution of the matrix. We illustrate the possibilities by an example of making silicide-based fibres/metal-matrix composites. Suppose we aim at silicide of metal M as a base for the fibre material. Then choose metal M as a matrix material and a eutectic in M-MxSiy system as a material to be melted and then infiltrated into channels pre-made in the matrix. To restrict dissolution of the matrix material in the melt, the infiltration process is to be performed at a temperature just above the eutectic point. After crystallization of the melt, the fibres of a composition close to the eutectic are formed in the channels [445].
Internal crystallization
616
Ch. XIV, {}14.2
Fig. 14.31. Failure surface of a specimen of the Ni~Si-Ni/Ni composite. After Mileiko and Rudnev [445]. 700
,
,
,
,
,
,
,
~600
,
O
\ 41'
,
b 500
400
300 0.0
i
I
0.1
I
I
i
0.2
I
0.3
I
I
0.4
0.5
vs Fig. 14.32. Room temperature strength ot" Ni.~Si-Ni/Ni composites versus fibre volume fraction. After Mileiko and Rudnev [445].
Ni3Si-Ni/Ni
composites
T h e m e l t c o m p o s i t i o n w h i c h c o r r e s p o n d s to the e u t e c t i c p o i n t in the N i - Ni3Si s y s t e m is 88.5 w % Ni a n d 11.5 w % Si, the e u t e c t i c t e m p e r a t u r e is ~ 1155~ [230]. T h e m e l t t e m p e r a t u r e at the i n f i l t r a t i o n s h o u l d be as close to the e u t e c t i c p o i n t as p o s s i b l e to p r e v e n t o u t s p r e a d d i s s o l u t i o n o f the nickel m a t r i x . T h e c r y s t a l l i z a t i o n
Ch. XIV, w 150
\
Fibres and composites obtained by ICM
I
,
I
,
617
]
100
b ooooo
50
oooon
800~ I000~
(
F
o o.o
I
0.1
I
"Oj,
I
0.2
I
0.3
Fig. 14.33. High temperature strength of Ni3Si-Ni/Ni composites versus fibre volume fraction. After Mileiko and Rudnev [445].
rate influences the composite strength, the value of 125 m m / m i n corresponds to a maximum room temperature strength. The composite macrostructure is illustrated in fig. 14.31. The dependence of room temperature strength of the composites on volume fraction of the Ni3Si/Ni-fibre is presented in fig. 14.32. A usual non-monotonic dependence is observed with a maximum value of the strength at vr ~ 0.3. High temperature strength of the composites is illustrated in fig. 14.33. The temperature dependence of strength of the composite, matrix, and Ni3Si/Ni fibres are shown in fig. 14.34. The fibre strength was obtained using a simple procedure based on eq. (5.2). The values of fibre strength obtained look quite promising up to a temperature of 800~ Figure 14.35 illustrates creep behaviour of the composite.
TisSi3-Ti/Ti composites Both commercially pure titanium and/~-alloy VT-151 were used as matrices for the titanium matrix composites. Titanium carcass was produced in a slightly different way from that used for the nickel carcass. Instead of an array of the wires, a titanium foil with regular slots formed by E D M is placed between continuous titanium layers. The regime of diffusion bonding is as follows: temperature 800~ average pressure 2 MPa, time 60 min.
t Russian trade mark
Internal crystallization
618
1200
!
1000
~ \ b
..... nicker (published)
o o o o o m a C:r'~ o n o o o % , - 0.30 xxxxxv~
800
600
Ch. XIV, w
0.49 (talc)
:
* * * * * f~.bre
B
400 200 L
o: o
D |
I
200
l
I
400 T/~
|
I
600
|
I
800
|
1000
Fig. 14.34. Temperature dependence of the strength of Ni3Si-Ni/Ni composites, nickel matrix and Ni3Si-Ni fibres. After Mileiko and Rudnev [445], the data on nickel are after Portnoi et al. [545].
Composition of the eutectic mixture is 86.3 at % Ti and 13.7 at % Si, the eutectic temperature is 1330~ [230]. The infiltration temperature is between 1330 and 1360~ The infiltration time is about 10 min. Changing the crystallization rate between 3 and 30 mm/min does not seem to influence the room temperature strength of the composite that remains on the level of 1000 - 1200 MPa [445]. The strength/fibre-volume-fraction dependence of the composite (fig. 14.36) is of a usual type for a brittle-fibre/ductile-matrix composite. The temperature dependencies of the composite and fibre strength are shown in fig. 14.37. Note that the strength of the TisSi3-Ti eutectic tested in the fibrous form is much higher than that for the same eutectic tested by Crossman and Yue [109] who have dealt with the bulk form and used much lower crystallization rate. This is a normal scale effect. 14.2.7.
Composites
with ceramic matrix
Ceramic-matrix composites are normally produced by using powder metallurgy methods (Chapter 12). The method of internal crystallization is an alternative process which perhaps may be considered as real one only in future. At present, there is not enough experience to estimate the potentialities. Nevertheless, a brief outline of the technology is appropriate since a possible variety of couples of the constituents in such composites is enormous and properties to be expected are very promising. To prepare the matrix, one can use a number of fabrication routes, some of them are based on burning out precursor fibres, either metal or organic, embedded into the matrix. For example, a bundle of molybdenum wires can easily be infiltrated with an Oxide melt. Then heating the blank specimen to the temperature of about
Fibres and composites obtained by ICM
Ch. XIV, w
1.5
.
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150
.
200
Fig. 14.35. Typical creep curves of Ni3Si-Ni/Ni composites (vf = 0.3) at 700~ Rudnev [445].
After Mileiko and
1500~ yields oxidizing molybdenum and clearing away the molybdenum oxide vapour. The result is shown in fig. 14.38. It should be noted that exercising such a procedure with some combinations of the matrix and fibre material yields the occurrence of a non-volatile oxide, like yttrium or magnesium molybdate. The infiltration should be performed with precautions, that is because if the wetting occurs, mutual dissolution of the components is expected. So in a series of the experiments by Mileiko et al. [435], the idea of using a eutectic, containing an element the matrix is composed of (see the previous section), is employed. Table 14.2 shows some results obtained. To illustrate both the possibilities and difficulties, let us consider the fabrication, structure and properties of a composite with the A1203 matrix and AlzO3-AlzTiO5 eutectic as the initial composition for the fibre material.
620
Ch. XlV, w
Internal crystallization 150
.
.
.
,
.
.
.
.
,
.
.
.
.
.
b
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i
1
i
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I
1
i
i
0.1
i
vl
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i
,
i
i
0.2
0.3
Fig. 14.36. Strength of the TisSi3-Ti/Ti composites (the matrix is VT-15 alloy) at 800~ The specimens were obtained at crystallization rate 7.35 mm/min. After Mileiko and Rudnev [445]. 500
~ o
,
,_
I
o o o b o M'at'r~ 9" , v ! = 0 . 1 8
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'
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\
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100
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500
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-....Q 0
600
,
i
700
,
i
800
,
l
900
T/*C
,
0
1000
Fig. 14.37. Temperature dependence of strength of the titanium matrix used for making composite specimens, TisSi3-Ti/Ti composites, and tibrc. After Milciko and Rudncv [445].
First, we s h o u l d note that a l u m i n i u m titanate, A 1 2 0 3 - T i O 2 , has nearly zero thermal expansion coefficient ( 1 . 9 . 1 0 6 g r a d -1), low Young's modulus ( ~ 5 5 G P a ) [ 1 5 6 ] , a n d this m a k e s it attractive for the use in composites. O n the o t h e r h a n d , the oxide c o n t a i n i n g t i t a n i u m sets t r o u b l e s with its stoichiometry. Also
Ch. XIV, w
621
Fibres and composites obtained by ICM
Fig. 14.38. The alumina specimen with the channels obtained by burning out molybdenum wires. After Mileiko et al. [435].
in the t e m p e r a t u r e interval between 700 a n d 1300~ a l u m i n i u m titanate is k n o w n to d e c o m p o s e into a l u m i n a a n d rutile. The changes in the s t o i c h i o m e t r y degree a n d the phase transitions complicates the technology, the latter b e c o m e s flexible and, on the o t h e r hand, difficult for the control. The flexibility m e a n s t h a t one can p r o d u c e a n u m b e r of m i c r o s t r u c t u r e s dealing just with a couple of the c o m p o n e n t s one of t h e m being a l u m i n i u m titanate. After infiltration of the A1203 m a t r i x with the A1203-A12TiO5 eutectic melt, the process being p e r f o r m e d in v a c u u m c h a m b e r at t e m p e r a t u r e ~ 1920~ aluminium
TABLE 14.2 Composites obtained by the internal crystallization method Fibre
Matrix
vr
a, MPa
K, MPa.m 1/2
A1203 -+-ZrO2 A1203 q- ZrO2 A1203 + ZrO2 + Y3A15012 A1203 + ZrO2 + mullite
Y203 + A1203 A1203 A1203 + ZrO2 A1203 + ZrO2
0.40 0.40 0.40 0.40
10-300 160 150 220
0.9 5-6 3--4
a, is the bending strength at room temperature. K, is the fracture toughness.
Internal crystallization
622
Ch. XIV, w
TABLE 14.3 Elastic constants of the oxide/oxide composite (see text for details). The fibres are aligned along the xl-axis Specimen number
1
2 3
CIi GPa
C22
C33
GPa
GPa
73 86 325
195 184 406
159 184 479
titanate in the fibres occurs to be non-stoichiometric. The only phase revealed doubtlessly by the X-ray analysis of the powder, is A1203. The bending strength of such composites is very low, that is ~10 MPa. Certainly, a heat treatment in air is required to restore the necessary oxygen content. The heat treatment at 1300~ for 5 h changes neither microstructure nor the strength. That at 1500~ changes microstructure drastically but does not change strength, despite that the matrix suffers a severe microcracking [435]. The elastic moduli, obtained via ultrasonic waves velocities, are very low in the fibre direction, and reasonably high in the transverse direction (lines 1 and 2 in Table 14.3). If after this high-temperature treatment, the specimens undergo a heat treatment in air at a lower temperature, between 1100 and 1200~ the microstructure does not appear to change, the X-ray analysis reveals the rutile phase definitely, the elastic moduli change (Table 14.3), the strength is dependent on the heat-treatment regimes as shown in fig. 14.39. Therefore, having started with the alumina-aluminium titanate eutectic as fibre material we obtain finally a composite with the alumina matrix, which is heavily cracked (certainly because of the volume effect at the phase transformation), and the fibre composed mainly of alumina/rutile mixture. The mechanical properties of the composites are not good, but a hope for obtaining really useful oxide/oxide structures still remains.
14.3. ICM with pre-made fibres The method of internal crystallization can be made more versatile. A possible modification of the method is based on the adding of pre-made fibres into the channels to be filled with the melt of a basic fibre material. We show the potentialities that arise, by considering a particular example of a carbon fibre/ titanium matrix composite produced via liquid phase fabrication route [446]. A direct use of a casting technology to obtain such composites is impossible because of high chemical reactivity of titanium at high temperatures. On the other hand, attempts to use a liquid phase fabrication route to achieve the aim mentioned are justified by a possible prospect of such composites if they are sufficiently cheap. At least one such attempt is known [657] and it is based on the infiltration of carbon
Ch. XIV, w
ICM with pre-made fibres ,
120
,
,
,
,
,
623
,
~
80
o o
40
0
II00
*
-
1200 ~C
1300
IO0 b
o
0
0
2
4
t/h
6
8
10
Fig. 14.39. Dependence of the strength of the oxide/oxide composite on heat-treatment regime of the low temperature stage of heating. (a) Strength versus temperature, the time is 10 h. Open points stand for regime of the high temperature stage 1500~ h, dark points are for 1400~ (b) Strength versus time of the low temperature stage, temperature is 1200~ Regime of the first stage is 1500~ h. After Mileiko et al. [435].
fibres with titanium-copper alloy, but to reduce the melting temperature of the matrix and to decrease the thickness of the C/Ti interface zone composed of titanium carbide, the copper content has to be too high. Hence, using the scheme of the method of internal crystallization, we take a titanium alloy as a main matrix and a titanium based eutectic, Ti-TisSi3, as a liquid medium contacted with the fibre. The melting temperature of the eutectic is sufficiently low [532], the Ti/Ti-TisSi3-composite can be obtained by using ICM (see Section 14.2.6), so it is expected that carbon fibres can be introduced into the titanium matrix by using the eutectic as an interphase. It should be noted that the unidirectionally solidified eutectic under consideration is a rather brittle material [109]).
624
Internal crystallization
Ch. XIV, w
In the experiments [446], a composite structure of a plane configuration was produced. An assemblage of titanium foils and a unidirectional carbon fibre fabric is prepared as a first step of the fabrication procedure. The whole pile is kept together by winding over either tungsten or molybdenum wire. Then the assemblage is pushing into the Ti-TisSi3 eutectic melt. The infiltration of a plate of about 100 m m length is accomplished in 1 - 2 s. Finally the plate is withdrawn from the hot zone of the furnace with a rate of 125 mm/min. The carbon fibre fabric used here is LU-2 (Russian trade mark [327]) composed of unidirectional fibres with Young's modulus of 230 GPa. The fibre diameter is 8 lam. Commercially pure titanium as well as //-alloy containing 7 M o - l l C r (marked thereof by Russian trade name VT-15) are used for ductile layers. Further on these layers will be called a matrix. A general view of the composite structure is presented in fig. 14.40. Despite a relatively low temperature and short time of the process, titanium carbide is formed at the interface. The titanium content along a line crossing a fibre is changing without observable jumps within the interval of about 1 ~tm as shown in fig. 14.41. Taking into account a wide possible range of the T i / C ratio for the existence of titanium carbide [554], the interface zone presumably containing titanium carbide is assumed to arise as a result of Ti-C interaction. Because the infiltration is performed at a temperature above the eutectic point, titanium goes out of the eutectic mixture to form the carbide at the fibre/eutectic interface, and that is compensated by titanium from the titanium layer. So the thickness of the latter decreases.
Fig. 14.40. The macrostructure of a Ti/Ti-TisSi3/TiC/C composite. The titanium layer is slightly darker than the eutectic layer. After Mileiko et al. [446].
Ch. XIV, w
625
ICM with pre-madefibres
Fig. 14.41. The distribution of Ti along a line crossing a fibre. After Mileiko et al. [446]. To illustrate viable mechanical efficiency of the composites of such a structure, we present longitudinal elastic modulus, strength and fracture toughness at r o o m temperature, the temperature dependence of the strength up to 900~ and creep rupture at 800~ The longitudinal Y o u n g ' s modulus, Ee, was determined by measuring the n a t u r a l frequency of a one-end-clamped specimen. Table 14.4 presents the measured values of Ec which are used below to estimate the carbide layer thickness at the carbonfibre/eutectic interface. Obviously, the longitudinal m o d u l u s of the f o u r - c o m p o n e n t composite under consideration is E -- ETi VTi -1--EE VE + El( 1 - VTi -- VE) Here ETi and EE a r e the Y o u n g ' s moduli of titanium and Ti-TisSi3 eutectic, respectively, Vxi and VE are the corresponding volume fractlo,~ gror convenience, the Y o u n g ' s m o d u l u s of the c a r b o n fibre with the carbide layer is denoted as Ef and
TABLE 14.4 Ti/Ti-TisSi3/TiC/C composites: the composition, density 7, Young's modulus Ee, and calculated thickness of TiC layer Ti 0.65 0.60 0.48
Ti-TisSi3 0.10 0.10 0.12
7 g/cm 3
Ee GPa
c5 ~tm
3.72 3.55 3.22
190 200 210
2.74 2.40 1.68
Internal crystallization
626
Ch. XIV, w
Ef - ETiC/)TiC --[-Ecf(1 - / ) T I C ) where ETi C and Ecf are the corresponding moduli, /)TIC is the volume fraction of the carbide ring around the C / T i C fibre. The carbon fibre diameter decreases with the carbide layer growing because the densities of carbon fibre and titanium carbide, 7el and 7TIC, are different. So assuming carbon goes into carbide only from the fibre, we get the carbon core diameter, d, as a solution of the quadratic equation ~d 2 q- 26d + ((~2 _ ~d(~) -- 0
where 2 = 7cf/TTiC, do is the initial fibre diameter. Taking ETi = 110 GPa, EE -- 160 GPa [109], ETic -- 460 GPa, 7TiC = 4.92 g/cm 3 [554], Ecf = 230GPa, 7cf-- 1.7 g/cm 3 [327], we obtain the values of the titanium carbide layer thickness, 6, to fit the measured values of Ec shown in Table 14.4. The data presented in Table 15.4 yield value of the Young's modulus of the brittle layer, Ti-TisSi3/TiC/C, between 285 and 310 GPa. Tensile strength at room temperature is shown in fig. 14.42. To have a basis for the comparison, some Ti-TisSi3/TiC/C specimens without titanium layers were also tested. Some titanium specimens, without brittle layers, after having undergone heat treatment corresponding to that in the fabrication process of the composites were tested as well. The usual maximum of the strength (see Section 5.2) at about 40% of volume fraction of the brittle phase is observed. This means that the usual failure mechanism of composites with low values of the volume fraction of the brittle layers, that is stable cracking of these layers, is observed in this case. Note also that 600
b
400
200
/ |
0.0
tat~re Nfanilxm m a ~ * ** * 9 WF- ! 5 alloy matr~r
i
00000 ,
1
I
0.2
|
|
|
I
,
|
0.4
|
I
0.6
|
|
|
I
0.8
,
|
,
1.0
V~rr Fig. 14.42. Ti/Ti-TisSi3/TiC/C composite tensile strength at room temperature versus volume fraction of brittle layer. After Mileiko et al. [446].
Ch. XIV, w
ICM with pre-madefibres
627
the strength values for Vbr = 1 are given without any adjusting to the real scale of the corresponding composite specimens. A very rough evaluation of the effective strength, a~,r of pieces of a layer between the cracks is based on the proportionality of the effective strength to the slope of the strength/volume fraction dependence at small volume fraction values (see eq. (5.8)). The data presented in fig. 15.42 yield O'er ~,~ 1.05 GPa. It is one order of magnitude higher than the strength of large specimens presented in fig. 14.42. Note that the values for the strength of carbon fibre used here are between 2 and 2.5 GPa [327]. The dependence of the fracture toughness upon the volume fraction of the brittle layer, Vbr, is given in fig. 14.43. Again, it is important to note that at 40% volume fraction of the brittle phase a maximum for the K*/Vbr dependence can be seen. High temperature strength is illustrated by experimental data presented in figs. 14.44 and 14.45. It should be noted that even at temperatures as high as 600~ the failure surfaces of specimens (fig. 14.46) reveal the brittle appearance of the failure of the Ti-TisSi3/TiC/C layers. The fibre pull-out takes place in zones of poor infiltration only. The preliminary creep rupture data obtained in the tests carried out on specimens of the microstructure corresponding to low tensile strength (the matrix layer is mainly pure titanium, I~r = 0.35) are presented in fig 14.47. Note that the creep rates under the conditions of the experiment occurred to be too small to be measured by an optical device used.
4O
I
3O
820 \ "~1o ooooo unuuu i
0.2
i
i
Pxtre titarti~m m a t r ~ FT- t5 alloy raatr~ I
0.4
i
i
i
!
0.6 Vbr
i
i
,
I
0.8
,
,
|
1.0
Fig. 14.43. Ti/Ti-TisSi3/TiC/C composite fracture toughness versus volume fraction of brittle layer. SEN specimens for fracture toughness of usual configuration with the width between 10 and 13 mm, thickness of 1.2 to 1.5 mm and the notch length about a half of the width, were. After Mileiko et al. [446].
628
hlternal crystallization 300
,
,
,
u
,
,
,
u
o
0
,
,
,
,
,
I
i
900~
ooooo
\
,
6000C 800"C
ooooo ooooo
o
,
Ch. XIV, w
200 b 0
0
12] 0
0
100
I
I
0.2
I
[
I
I
I
0.4
l
i
i
0.6 ~Jbr
i
I
I
1.0
0.8
Fig. 14.44. High temperature strength of Ti/Ti-TisSi3/TiC/C composites with pure titanium matrix versus volume fraction of brittle layer. After Milciko ct al. [446].
600
,
,
,
00000
ooooo
,
,
,
,
,
,
Pure ~itanium ~a~ VT- i 5 a l l o y m a t r i x
\ 400 b
200
00
I
I
200
i
I
I
400 T
/
I
600
l
I
800
i
1000
06"
Fig. 14.45. Temperature dependence of the Ti/Ti-TisSi.~/TiC/C composite strcngth. The volume fraction of the brittlc layer is 0.35. After Miiciko ct al. [446]. To estimate a possible standing of the C / T i c o m p o s i t e s of a type u n d e r c o n s i d e r a t i o n , we should start with placing the area occupied by the strength/ fracture t o u g h n e s s points on the c o r r e s p o n d i n g plane with the d a t a for various t i t a n i u m alloys (fig. 1.15). One can see that the r o o m t e m p e r a t u r e fracture
Ch. XIV, w
ICM with pre-made.[ibres
629
Fig. 14.46. A typical failure surface of the Ti/Ti-TisSi3/TiC/C composite with Vb,-= 0.35 tested at 600~ at various magnifications. The scale is given by the carbon fibre diameter. After Mileiko et al. [446]. properties of the composite is lower than that for titanium alloys, and they are rather close to the aluminum properties. However, it should be noted that the density of the composites is also close to that of aluminum. Therefore, a real advantage of such composites should be expected in the specific stiffness and high temperature properties. The longitudinal Young's modulus is very high (see Table 14.4), it reaches 2.5 times the specific value for metals. The strength of the composites commences to overcome that for titanium alloys at about 7 0 0 - 800~ Certainly this threshold can be made lower if to increase the content of the brittle layers. Titanium alloys are subject to intensive creep at temperatures 600~ and higher. A composite containing creeping matrix reinforced with strong brittle fibres have the advantage being characterized by a recumbent creep rupture curve (see Section 6.2.2). It makes composites to be superior in comparison to metal alloys at longer loading time.
Internal crystallization
630
200
|
[]
,
,
| ,I,|
I
w
|
,
, ,||,I
I
[]
|
,
Ch. XIV, w
, |,,,I
\ b 150
100
0
I
0.1
2
I
l
l ' ,,,I
5
1
2
'
t,/h
'~
....
I
10
~
[]
'
'5'''
,I
100
Fig. 14.47. Creep strength of Ti/Ti-TisSi3/TiC/C composites with Vbr = 0.35 at 600~ in vacuum. Open points are for composites with pure titanium matrix, dark point is for VT-15 titanium alloy matrix. After Mileiko et al. [446].
14.4. "Blotting paper" technology The successful applications of the ICM-based technology to fabricate composites with various fibres set a challenge to apply such a type of the technology to produce composites with a fibre material that does not melt congruently. Actually, this problem became especially important with the discovery of high temperature superconductive oxides [45] which normally decompose when they are heated. So "blotting paper" technology was developed as a modification of the ICM [18, 19] primarily to meet the need for organizing a liquid phase route of making oxide superconductor materials. Liquid phase technologies can provide an oxide superconductor with a texture which is certainly necessary to reach high values of the critical current (see [284]). In a main variant of the technology, a superconductive layer is being obtained under the isothermal exposure of a layer of a particular composition at a temperature within an interval where the latter exists as a mixture of the solid superconductive phase and a liquid. During this exposure the liquid infiltrates a porous inert layer transforming it into a composite matrix of high enough strength. The main technological scheme can be modified along various ways.
14.4.1. Fundamentals of the method Consider the phase diagram of a binary system, AB, where A and B can be either simple substances or arbitrary compounds, as in the Y203-BaO-CuO system shown
"Blotting paper" technology
Ch. XlV, w
T** 211+L T t
T"
91
x"
123+L
T,
l I I I l I I I l I
211
631
123
Xl
I I I I I I I I I I
Xe
Fig. 14.48. A schematic view of Y203-BaO-CuO phase diagram.
in fig. 14.48. Consider a compound, say AxBy (123 in fig. 14.48), growing from the melt; it decomposes into solid .4 (211 in figure) and a liquid as heating takes place to a temperature Too. A substance of composition X1 is transformed into a solid .4xBy and liquid X~ when it is heated up to a temperature T' > 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
Internal crystallization
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
Internal crystallization
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
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I
I
I
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:
:
:
I
635
'
? 400
..... .
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.
.
.
.
. . . . .
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
o
o
o
.9 ," I
20
, 9" "
J
IKx~xxxxx
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
Internal crystallization
636
120 o O ~ 1 7 6 1 7 6 1o7o6o1o7o6o o o o o o o o o
100
eo
80920 ~ 920 ~ 920 ~
.....
\
60
..... .....
3 4 5
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40-
o
aaaaaaaa aaaaaaaaoaaaaaaa
o
20-
o
o
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~
o
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
o
-
[
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.
This Page Intentionally Left Blank
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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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9
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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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9
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 ,...
O~
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9
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N
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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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,..., -., ~
t~
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4~
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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
~
o g ~ ' ~ N N -.~
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] .~
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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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SUBJECT INDEX
acoustic emission, 181 adiabatic shear band, 339 aeroelasticity, 12 aeroplane, 6 engine, 11 wing, 13-14 AI-Li alloy, 287 alloys, 26 alumina/aluminium composites, 576 -6-A1203 fibre, 579 - Al203(Saffil)-fibre, 254 - fatigue, 577 - FP-AI203 fibre, 577 - fracture energy, 579 - fracture toughness, 259, 280, 283, 287 - squeeze-casting, 577 - strength, 254, 572, 575, 577, 579, 581 - vacuum infiltration, 562, 577, 581 aluminium, 4 aluminium alloys, 29 - c r e e p parameters, 484-485 aluminium based composites, 531,553, 569, 571 aluminium matrix, 564 for SiC-whisker/6061-aluminiumalloy-matrix, 567 high-temperature properties, 531 517, 531 - strength, 527, 532, 538 aluminium carbide, 420 aluminium nitride, 423 aluminium titanate, 622 analysis - shear-lag, 124, 134, 194, 307, 514 anisotropic material, 77, 79, 81, 83,84 anti-plane problem, 148 armour, 339 armour plates, 71 Aveston-Cooper-Kelly theory, 177, 289, 298, 586
bending, 15 beryllium, 18 blending, 258, 305, 517, 527, 533 length distribution, 518 wet, 517 bonding, 334, 416, 473, 480, 491 atomic, 418 chemical, 418, 420 covalent, 418 diffusion, 418 - physical, 418 borides, 238, 445 boron/aluminium composite, 65, 102, 225, 231, 238, 250, 261,280, 445, 457, 488, 503, 505, 507 - bearing strength, 225 compressive strength, 494 -critical compressive stress, 377 - d a m a g e , 333, 347, 355, 369 density, 17 - fatigue, 333, 335, 346, 349, 351-353, 355, 357, 359, 361,365, 366, 368-369 - fibre breakage, 170 - fracture surface, 359 heat treatment, 505 - microstructure, 505 plate, 221 Poisson ratio, 49 rolling, 500 shells, 498 strength, 37, 39, 41, 43, 45, 51, 66, 72, 494 stress/strain curves, 120 - structural elements of, 507 - technological parameters, 347 transversal tensile strength, 66 transversal Young's modulus, 66 tubes of, 496 - Young's modulus, 66, 81, 96, 99, 103-107 buckling, 376, 381,385, 392, 405 analysis, 405 Burgers vector, 258 -
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ball mill, 533 ballistic impact, 337, 339, 341,343, 345 ballistic limit, 339 beam, 15 - stiffness 15, 20-21, 23, 25
C/TiC/Ti, 537 capillary tube, 425 carbides, 420
683
684
Subject index
carbon fibre reinforced polymers, 34 - c o k e yield, 525 - CVI, 459 - C V I - p r o c e s s , 524, 544 liquid impregnation, 525 resistance, 546 carbon-fibre plastics, 4 C a u c h y - R i e m a n n relationships, 149 ceramics, 3, 31, 34, 303 chemical vapour impregnation, 524 coating of fibres, 12, 194, 416, 419-420 - A1203 on whiskers, 422 - AIN, 423 BN on mullose fibre, 464 C/TiB2 on silicon carbide fibre. 469 - C / T i C on SiC fibre, 424, 469 on carbon fibre, 420 - c a r b o n on SiC fibre. 451 chemical, 418, 420 carbide on graphite fibre, 420 - c o p p e r on carbon fibre, 420 - C V D - m e t h o d s , 422 - duplex, 424 electroplating, 420 - K2ZrF6 on carbon and SiC fibre. 421 liquid-metal-bath proccdurc, 421 - metal on sapphirc fibrc, 420--421 - M O C V D - m e t h o d s , 423 multilaycred, 419 nickel on carbon fibrc, 420 - SiC/SiO2 on carbon fibrc, 423 sol-gel techniques, 423 - sputter deposition, 424 tantalum on carbon fibrc, 420 TiB2 on SCS-6 SiC fibrc, 469 TiB2 on SiC fibre, 423 - TiC on SiC fibrc, 423 - Y/Y203 on SiC fibrcs, 424 yttria, 423 - ZrO2 on whiskcrs. 422 co-extrusion, 523 cohesive zone, 157 compliancc, 78 compliancc method, 281 compositc. 63.65, 67, 69, 71, 73 2A1203-MgO-3CaO-fibrc/nickci matrix, 611 - A1203-ZrO2-Y203/nickcl-supcralioy, 328 - Al203(FP)/magnesium, 367 Al203(Saffil)-fibre/magncsium-matrix, 448 - AI203-AlsY30 j2-eutectic-fibrc/molybdcnum matrix, 604 Al203-FP-fibre/Ni3Al-matrix, 564 Al203-FP-fibre/NiAl-matrix, 564 -
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Al203-FP-fibres/ZE41 A-magnesium-alloy, 580 AIeO3/80Ni-20Cr, 466 AI203/Al, 468 AleO3/Cu, 279 AI203/Mg, 193 AI203/Mo. 594, 637 AI203/Ni, 466 - AI203/Ni-Cr-Fe. 466 AI203/NiAI, 466 -- AI203/W, 466 -- A1203/ZrO2( + Y203)/copper-matrix, 277 - Al203/ZrO2 + Y203 fibre/nickel aluminide alloy matrix, 584 alumina-based-fibre/(0r + 7)-titaniumaluminide, 452 alumina-based-fibre/Ti-50% Al-matrix, 467 - alumina-fibre/aluminium-ailoy-matrix, 576 aluminium matrix, 445, 467 B-B4C/Ti, 193, 466 B-SiC/Ti, 466 - B/B4C-fibre/Ti3AI + Ni-matrix, 452 - B/Ni, 466 B/Ti, 352, 466 biological, 63 boron-fibre/magnesium-alloy matrix. 244 - boron/aluminium, 65.70, 102, 347, 466, 503 boron/steel/aluminium, 261,346 - brittlc-fibre, 455 brittlc-fibre/brittle-matrix, 297 brittlc-fibrc/ductilc-matrix. 235, 237, 239, 241, 243, 245, 247, 249, 251,253, 255, 257, 259, 261, 280 brittle-matrix, 169, 177 466 - C/B4C, 537 C/Cu, 244, 466 C/Ni, 466 C/Ni-Cr, 466 C/Si3N4, 454 C/Ti, 532, 581,623 carbon fibre rcinforced glass, 181 carbon-fibre/magnesium-matrix, 448 --carbon-fibre/soda-lime glass matrix, 289 -- carbon/carbon, 72-73, 98, 297, 418, 458, 524 -- Co, Cr/Cr7C3, 329 complex-oxide-fibres/molybdenum-matrix, 603 continuous fibre, 298 ---cross-ply, 370 - d i r e c t i o n a l l y solidified, 30 discontinuous fibres, 164. 253, 303 d uctile-fibrc/brittle-matrix, 289 - ductile-fibre/ductile-matrix, 291,295 ductile-matrix, 194
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C
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Suh]ect index effective characteristics, 182 behaviour of, 78, 85, 87, 89, 91,93, 95, 97, 99, 101, 103, 105, 107, 109 behaviour of, 77 - fibre reinforced cements, 181 - glass fibre reinforced plastic, 383 graphite-fibre/epoxy-matrix, 281 - graphite-fibre/titanium-matrix, 255 - graphite/aluminium, 67, 353, 466, 571 resistant, 69 intermetallic based composite, 451 iron aluminide matrix, 453 magnesium-matrix, 448, 579 - metal-fibre/metal-matrix, 295, 455 - Mo-fibre/Al203 + Z2Oz-matrix, 291 - Mo/AI203 + ZrO2, 584, 585 - Mo/AI203, 584, 585 Mo/Ni, 466 - molybdenum-wire/oxide-matrix, 562 - molybdenum/titanium, 325 - N i - N i 3 AI/Cr3C2 317 Ni3Si-Ni/Ni, 616 Nicalon SiC-fibre/C-fibre-coating/LASmatrix, 457 Nicalon-SiC-fibre/ZrTiO4-matrix, 458 nickel aluminide matrix, 452 non-elastic behaviour, 111, 113, 115, 117, 119, 121 - o x i d e matrix/molybdenum fibre, 585 - oxide-based-fibre/Ni3Al-matrix, 563 oxide-fibre/nickel-matrix, 611 - oxide/oxide, 543 - phosphor-bronze-wire/lead-matrix, 319 Poisson's ratio of, 82, 99, 107 - polymer matrix, 63 - porosity of, 107 - sapphire-fibre/alumina-zirconia-matrix, 522 sapphire-fibre/magnesium-alloy-matrix, 244 - sapphire-fibre/molybdenum-matrix, 244 - sapphire/nickel, 470 - SCS-6 SiC-fibre/Ti3Al + Nb-matrix, 487 - SCS-6 SiC/TiA1, 452 - SCS-6-fibre/Ti-15V-3Cr-3Al-3Sn-matrix, 457 - shear modulus of, 107 - short-fibre, 517, 531 short-fibre/aluminium-matrix, 67 short-fibre/brittle-matrix, 517 - SiC-fibre/aluminium-alloy-matrix, 479 SiC-fibre/magnesium-matrix, 448 SiC-fibre/Si3Nn-matrix, 521 - SiC-fibre/Ti3Al-matrix, 479 - SiC-fibre/TiAl-matrix, 479 SiC-fibres/(BaO-SiO2-Al203 + Si3N4)matrices, 454 -
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685
SiC-whisker/aluminium-alloy-matrix, 319 SiC/AI, 448, 466 SiC/mullite, 458 - SiC/Ni3AI, 452 SiC/SiC, 72, 524, 544 SiC/Ti, 466 - SiC/Ti3AI + Nb, 452, 491 - SiC/TiZrO4, 523 - SiCw/AI, 317, 369, 491 - SiCw/AI203, 371 silicide-based fibres, 615 silicon-carbide-fibre/aluminosilicate-glassmatrix, 369 - silicon-carbide-whisker/silver-matrix, 246 - steel/aluminium, 347, 376 - steel/silver, 352 - Ti2Y207/Mo, 604 TisSi3-Ti/Ti, 617 - TiC-fibre/TiB2-coating/Ti-matrix, 467 - Ti/Ti-TisSi3/TiC/C, 624, 630 - tungsten-fibre/copper-matrix, 287, 294, 455 tungsten-fibre/nickel-alloy-matrix, 443, 466 unidirectional, 88 - W/Ag, 319 composite cylinder assemblage model, 89 composite precursors, 475, 477, 479 compressive strength, 373-374, 376, 378, 380, 384, 386, 388, 390, 392, 394, 396, 398, 400, 404, 406, 408, 410 computer simulation, 384 computer simulation, 247 concrete, 3-4 C o o k - G o r d o n microcracks, 147 C o o k - G o r d o n ' s model, 189 copper, 277, 420 corrosion, 12 crack, 147 branching, 32, 34, 304 - bridging, 159, 182, 199, 255, 280, 286, 304, 32, 34, 304 - i n homogeneous solids, 148-149, 151, 153, 157, 159, 161, 163 - interface, 243 longitudinal shear, 200, 204 - opening, 132, 182, 285 - penny-shaped, 154, 186, 188 - periodic array, 187 pinning, 32-33 - propagation, 182, 209, 280 tip, 182, 197, 213, 220, 280, 286 creep, 12, 29, 600 creep curve, 320, 326 - tertiary stage, 320, 323 -
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382, 402,
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545
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155,
686
Subject index
creep law, 473, 483 creep parameters, 485 creep rupture, 29, 601,609, 630 critical energy release rate, 236, 295 critical fibre length, 133 critical stress intensity factor, 24, 204, 281,336, 364 computer simulation, 174 cutting tools, 71 CVD-method, 52 CVI method, 72 Czochralsky-Stepanov's-EFG method, 56 -
damping, 337 D'Arcy law, 548, 557 debond energy, 169 debonding, 458, 545 delamination, 143, 191, 211,333, 339 densification, 473, 480-481,496, 499 design philosophy, 14 die-casting, 567 diffusion, 443 diffusion barriers, 468-469 in intermetallic-matrix composites, 470 in nickel-matrix composites, 470 in titanium-matrix composites, 469 multilayered, 468 diffusion bonding, 277, 279, 333, 588 kinetics, 441-443, 445, 447, 449, 451,453 dislocation, 25, 195, 254, 257, 294, 467 dispersion hardening, 257 drive shaft, 13 dually non-homogeneous solid, 204-205, 207, 209, 211,213,215 ductile-fibre/ductile-matrix composites, 352 Dugdale's model 157 Dundurs' parameter, 194 -
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Earth, 8, 246 effective fibre length, 270 effective fibre strength, 455, 528 effective matrix characteristics, 465 effective modulus, 347 effective properties, 85 effective shear modulus, 211 effective surface energy, 169, 173, 180, 213, 295, 382, 386, 455 effective Young's modulus, 353 effectiveness of materials, 3, 5, 7, 9, I1, 13, 15, 17, 19 eigenstrain, 86, 321 - Eshelby's method, 321 elastic anisotropy, 392 ~ elastic interface, 123
elastic symmetry, 79, 88 isotropy, 81 - orthotropy, 79 transversal isotropy, 80, 89 elastic-plastic strain/stress curve, 248 energy balance, 23, 180, 184, 215, 382, 386 energy dissipation, 164-165, 167, 169, 214, 280, 286, 339, 455, 471 energy release rate, 153, 156-157, 193, 236, 286 - fracture toughness, 259, 280, 283, 287 engineering constants, 82, 89, 104 equivalent time, 400, 489, 494, 497 Eshelby's method, 86 Euler stress, 375, 382 eutectics, 68, 418 directional solidification, 68, 418 explosive welding, 502 extrusion, 567 -
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failure criteria, 122, 217, 253 failure surface, 241 fatigue, 12 fatigue crack, 335 - c o m p u t e r simulation, 334, 353 life diagram, 364 mechanisms, 346, 365 - threshold, 371 ultimate cycle numbers, 361 fibre alignment, 567 fibre breakage, 353, 369 fibre bundle, 233, 569, 590 - loose, 234 - strength of, 233-279, 286-287, 293 fibre cracking, 169, 171, 173, 175, 280, 288 fibre damage, 491 fibre fragmentation test, 263, 275 fibre healing, 464 fibre microcracking, 458 fibre packing, 123, 245 - homogeneity, 260 - non-homogeneity, 236, 245 -- non-uniform, 143 - random, 98, 246, 255 -- regular structure, 100 -- three-dimensional, 98, 141 two-dimensional. 95 uniform, 561 fibre, 37, 39, 41, 43, 45 - AI203-Y3AlsOI2 eutectic, 606 - AI203 of Fp type, 416 --alumina, 60 -- aspect ratio of, 64, 72, 86, 133, 276, 308, 311 - boron, 52-53 - boron carbide, 52 -
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Subject index - c a r b o n , 42, 46, 71,420, 422 - coating, 416, 419-420 creep resistant, 62 fibre length, 264 - CVD-SiC, 416 -degradation, 465, 522 elastic constants, 49 graphite, 460 - ineffective length of, 235, 255 - Kevlar type, 339 oxide, 56 Poisson ratio of, 49 polycrystalline ceramic, 60 - polymer precursor, 50 - produced by CVD-methods, 52 - recovery length of, 264, 270, 298 - sapphire, 42, 57, 421,590 - SCS-6 SiC, 453 SiC of nicalon type, 416 silicon carbide, 42, 50, 52, 55, 60, 62, 421, 423-424 ternary oxide eutectics, 611 - TiO2- MgO-CaO, 615 tungsten, 287, 460 Young's modulus, 48, 51 zirconia-toughened alumina, 453, 460 fibre reinforced concrete, 72 fibre strength, 37-45, 51, 72, 233, 235, 237, 239, 241,243, 245, 247, 249, 251,253, 255, 257, 259, 261-263, 265, 267, 269, 271,273, 275, 277, 279, 286-287, 293, 460, 463, 491,593 bi-modal distribution, 42 - scale dependence, 39 fibre/matrix interface, 123, 165, 177, 194, 286, 293, 298, 348, 441,445, 447, 454, 476 - debonding, 262 - ideal, 293 - layer, 294 non-ideal, 294 shear stress on, 165 - strength, 233, 235, 237, 239, 24,1,243, 245, 247, 249, 251,253, 255, 257, 259, 261-263, 265, 267, 269, 271,273, 275, 277, 279, 286-287, 293, 455 transfer, 123, 125, 127, 129, 131, 133 structure, 445 - weak, 455 fibre-reinforced cement, 297 fibre-reinforced plastics, 217, 281 Fick's second law, 441 flaw population, 42 fracture, 12 fracture criteria, 156, 180, 199, 281 fracture process zone, 220, 280
fracture toughness, 28 measuring, 280 - specimen, 282 friction, 127, 298, 416 frictional slipping, 180 Froude number, 5 fuel, 3 -
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687
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s
s
gas corrosion, 610 gas turbine, 4, 29, 71 - blade, 12~-13 gaseous isostat, 496, 498 glass, 4 glass fibre reinforced polymers, 34 glass matrix composites, 512 - cracking in, 181, 217, 219, 22 l, 223, 225, 227, 229, 231 graphite/boron-glass 513 mechanical properties, 513 Gordon's crack, 190 grain size, 27 graphite, 107 graphite/aluminium composites, 66, 244, 447, 455, 515, 571 - creep-rupture, 575 drawing, 500-501 elastic constants, 574 fabrication, 571 failure surface, 67 - high temperature strength, 575 - hot pressing, 445, 447, 457, 475, 513, 515 447, 478, 515 liquid infiltration, 456, 571 plasma spraying, 447 pressure casting, 571 - strength, 572, 575, 577, 579, 581 parameters, 571 - Young's modulus, 574 graphite-fibre/carbide-matrix composites, 107, 537 elastic constants, 104, 107 electrical conductivity, 540 - fracture surface, 539 - fracture toughness, 538 behaviour, 545 stress/strain curve, 107 tangent modulus, 109 technological parameters, 540 - Young's modulus, 81, 96, 99, 103-107 grass, 410 Griffith condition, 390 Griffith crack, 23, 148, 185, 209 -
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Hagen-Poiseuille formula, 549 hardening coefficient, 319
Subject index
688
Hashin and Shtrikman bounds, 89. 99, 107, 109 heat resistant structures, 9 heat-resistant alloys, 29 Hedgepeth model, 134 high-temperature alloys, 4 high-temperature super conductive layer, 423 Hook's law. 78-79, 85, 99, 116, 125, 204 hot pressing, 346, 396, 475, 513, 515, 533 dynamic, 496 technological parameters, 476, 480, 492, 495, 497, 513 hydrostatic pressure, 392, 396, 507 -
-
ideal solid body, 21 shear strength of, 23 - strength of, 20-21, 23, 25 ideal thermal cycle. 9 infiltration, 467 infiltration mechanics, 547, 549, 551,553, 555, 557, 559 influence zones in matrix, 467, 506 alumina-based-fibre/Ti-50% Al-matrix, 467 - boron/aluminium, 238, 242, 506 SiC-fibre/titanium matrix, 467 initial imperfection, 374, 378, 410 interface, 191, 197, 199, 204, 315, 320, 334. 347348, 352, 366, 370, 416, 419. 421,423,425. 427, 429, 431,433. 441,445, 454 bonding, 416 chemical reactions at, 442-443. 445, 447, 449, 45 I. 453 coating/fibre, 196 coherent, 416, 418 of, 189. 191, 193. 195, 197, 199, 201, 203 - debonding, 169, 193, 199. 21 !, 367, 369 - defects of, 419 dislocation at, 3 ! 5 - energy of, 419, 425 incoherent, 3 ! 5 - layer, 316, 419, 421,423, 425, 427, 429, 431,433 matrix/coating, 196 - non-homogeneous, 211 partially debonded. 125 - roughness of, 416 shear strength of, 458 - strength of, 367, 419 structure of, 416-4 i 7 - weak, 189, 347, 352, 510, 522, 525 - zone, 317, 353, 545 intermetallics. 27. 295 intermetallic based composites, 449, 581 - c r e e p rupture 581 -
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- fibre/matrix interaction, 562, 581 matrix strength, 581 -microstructure, 556, 581 reaction zone, 450 - s t r e n g t h . 572, 575. 577. 579. 581 - vacuum infiltration. 562. 577, 581 internal boriding, 505 internal crystallization method, 587-588, 590, 592, 594, 596, 598. 600. 602. 604, 606, 608, 610, 612, 614. 616, 618. 620. 622. 624. 626, 628, 630, 632, 634, 636, 638 - blotting paper technology, 630-631,633, 635 - fibres produced by. 636-637 - with pre-made fibres, 622-623, 625, 627, 629 internal oxidation, 27. 505 inviscid melt spinning. 58 .Irwin's relationship, ! 53. 180. 203. 231 -
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J-integral, joints, 217, 219. 22 !. 223, 225, 227, 229, 231 bearing, 209 - cracking in, 18 I, 217. 2 ! 9, 22 I, 223, 225, 227, 229, 231 - overlapping, 217, 228 - pin-loaded hole, 217, 222 - shearing-out, 225 K-calibration, 284 kink, 374 Kirkendall-Frenkel effect, 442 K opiev-Ovchinsky's model, 248 Kozeny-Carman constant, 550 Krylov functions, 357 1
5
7
,
1
5
9
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c
r
a
c
k
i
n
g
Lamc's coefficients. 81 laminate, 95. 223 boron/aluminium, 224 - K i r c h h o f f hypothesis, 96 Leonov-Panasyvk's model, 157 linear elastic fracture mechanics, 156 liquid infiltration, 425, 427, 431,547 -capillary effects, 547, 562 -ceramic-matrix composites, 582-583, 585 - compocasting. 567 --composite wire, 569 - fibre bundle, 420 - fibre distribution, 566 + fibre preferm, 547, 559 -infiltration pressure, 567 infiltration time, 567 - matrix solidification, 552 -- oxide-based-fibre/Ni3AI-matrix, 563 pressure infiltration, 564 pressureless infiltration, 562 rapid infiltration, 568 -
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-
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-
Subject index Saffil 6-alumina-fibre/aluminium-alloy matrix, 566 - short-fibre/aluminium-matrix, 567 squeeze casting, 565 -technological parameters 562 thermal effects in, 551 - vacuum infiltration, 562, 577, 581 Love-Kirchhoff hypotheses, 405 -
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-
Mach number, 12 macrocrack, 147, 169, 213, 280, 287 matrix cracking, 177, 181,298, 353, 369, 513 matrix plastic flow, 169 matrix yielding, 199 metal alloys, 69 metal laminate, 333, 335 - perforation, 339, 341,371 metal-matrix laminates, 369 metals, 3, 18, 26, 29 microcrack, 32-33, 147, 170, 173, 236, 286, 290, 353, 464 ring-like, 289-290 micro-rope, 58 Monte Carlo, 249, 276 motor car, 5 multiple cracking, 168, 178, 464
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Nabarro-Herring creep, 29, 472, 487 nanocrystalline materials, 27 natural frequency, 347, 356 natural-frequency/cycle-number curves, 358 necking, 293 nickel, 420 nickel aluminide, 31 nickel based composites, 611 - Al203-FP-fibre/Ni3Al-matrix, 564 - A1203-FP-fibre/NiAl-matrix, 564 - alumina fibres, 522 - c a r b i d e whiskers, 522 - c r e e p rupture, 320, 324, 328 Ni3Al-alloy matrix, 564 technological parameters, 476, 480, 492, 495, 497, 513 nickel superalloys, 30 niobium, 18 nitridation, 522 non-homogeneous medium, 92 dispersive characteristics, 92 effective moduli, 92 Poisson's ratio, 82, 99, 107 Voigt-Reuss boundaries, 89 - Young's modulus, 81, 96, 99, 103-107 non-homogeneous materials, 147 -
-
-
-
-
-
689
off-axis loading, 329 orthotropic material, 189, 236 orthotropic plate, 156 oxidation, 12 oxide/oxide composites, 543 anisotropic oxides at the interface in, 544 elastic constants, 622 - sapphire-fibre/YAG-matrix, 543 - shear strength, 544 parameters, 544 oxidizing environment, 370 -
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t
e
c
h
n
o
l
o
g
i
c
a
l
Papkovich's formula, 394 -modification of, 397 Paris law, 358, 364, 371-372 particle-strengthening, 27 particulate composites, 27 Peierls-Nabarro stress, 25 permeability, 548 - effective, 557 plane strain conditions, 154 plasma-sprayed matrix, 480, 488, 491 plasma-sprayed tape, 475, 491,493 plastic dissipation, 455 plastic zone, 157, 199 plates made of composites, 495 plugging, 339 Poisson distribution, 40 Poisson's ratio, 23, 325, 383, 416 polyethylene, 4 polymer matrix laminates, 339 porosity, 476 powder cloth, 479 powder metallurgy, 28, 107, 454, 470, 476-477, 532, 542 injection molding, 521 - reaction-bonding, 521,526 - shaping, 520 - sintering, 521,527, 533, 541 slip-casting, 521 precipitation-hardening, 27 pressure infiltration, 427, 452 projectile, 337, 371 propulsive coefficient, 10 pseudo-macrocrack, 184, 290 - penny-shaped, 186, 188 pull-out, 123, 164, 180, 286, 300, 304, 369, 371, 417, 418, 458, 513, 545 push-out, 123, 418 push-out tests, 544 pyrolysis, 47, 422, 526 pyrolysis of matrix precursor, 523, 525 -
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690
Subject index
radiation, 12 reaction zone, 461,524 - graphite-fibres/aluminium-matrix, 569 recrystallization, 460 residual stress, 290, 595 Reynolds number, 5, 548 rheocasting, 567 rocket, 6, 29 ballistic, 7 - payload, 8 rods, 373, 375, 377, 379, 409 rolling, 567 rule of mixtures, 262 rupture stress, 512 - SCS-6 SiC/(Ti3AI + Nb), 510 - SiC-fibre/Ti-24AI-6Al-matrix, 509 titanium alloy matrix, 509 -
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- powder technology fabrication route to produce, 527 - strength, 527, 532, 538 SiC-fibre/aluminium (6061) matrix, 368 SiC-fbre/titanium-matrix, 105 SiC/A1, 244 SiC/Ti, 193 Sih's criterion, 161 silicon-carbide/silicon-carbide composites -CVI-process, 524, 544 fatigue, 545 - fibre/matrix interface, 525, 527 fracture toughness, 72 - good oxidation resistance, 546 high temperature behaviour, 545 - interface, 522, 525 strength, 37, 39, 41, 43, 45, 51, 72 stress/strain behaviour, 545 - thermal shock, 546 silicon-carbide/titanium composite, 67, 114, 448, 508-509, 511 debonding, 510, 513 - effective surface energy, 510 elastic-plastic behaviour, 114 fracture toughness, 510 - interaction zone, 449-450 - reaction zone, 509 silicon-carbide/titanium-matrix, 114, 479 silicon-carbide-fibre/aluminium-matrix, 366 silicon-nitride-matrix composites, 521 continuous-silicon-carbide(SCS-6)-fibre, 521 fracture toughness, 72 strength, 37, 39, 41, 43, 45, 51, 72 single-fibre specimen, 266 singularity, 24, 155, 157, 191, 199, 231 - oscillatory, 194, 199 sintering, 458, 470-471,473,475, 480, 487 slip casting, 477 slurry impregnation, 477 solution-hardening, 27 Soviet Union, 70 space plane, 12 space shuttle, 263 spalling, 339 specific fuel consumption, 11 specific surface energy, 186 splitting, 381 spray casting, 28 squeeze casting, 28, 427 statistical homogeneity, 41 steel, 4 straight jet engine, 10 strain energy release rate, 281 -
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sapphire-fibre/Ti3Al + NI-matrix, 452 satellite, 4 launching, 8 orbit, 8 SCS-6 SiC-fibre/Ti3Al + Nb-matrix, 487 secant modulus, 248, 395, 398 service conditions, 454 corrosive environments, 454 high temperatures, 454 oxidation, 458 sessile drop method, 426 shear band, 379 shear modulus, 248 shear strain, 374 shear-lag analysis, 124, 134-135, 137, 139, 141, 143, 145, 247, 416, 464 shell, 13, 16 buckling, 13, 16 shells, 392-393, 395, 397, 399, 401,403, 405, 407, 410-411,497 - aluminium, 393 boron/aluminium, 381,396, 411 -critical pressure, 396, 402, 411 - cylindrical, 392 elastic, 394 metal, 394 - metal-matrix-composite, 392 - polymer matrix composite, 392 - thick-walled, 405 ship, 5 short fibre, 307, 309, 311,313, 315, 317, 319, 321 short-fibre composites, 304, 526-527 -creep, 307, 323 - c r e e p rupture, 320, 324, 328 -mechanical properties, 542 -
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-
-
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-
-
-
-
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Subject index strength criteria, 194 strength scatter, 259 strengthening coefficient, 324 stress intensity factor, 24, 151, 154, 156-157, 173, 175, 182, 187, 231,286, 289, 305, 369, 462, 464 stress/strain behaviour, 369, 392 stress/strain curve, 178, 216, 248, 266, 292, 302, 375, 395, 400 brittle matrix composite, 178 structural materials, 26-27, 29, 31, 33 submarine, 14 superconductive oxides, 630 superconductive wire, 423 surface energy, 21, 157, 424, 427, 471,473 surface roughness, 334, 427, 481 synergism, 291
transformation toughening, 32 Tsai-Hill criterion, 122 tubes made of composites, 379, 381,383,410, 496 - bamboo structure, 411 - boron/aluminium, 381,384, 396, 411 fitting ends, 508 -
ultimate compressive stress, 384 ultimate strain, 320 ultrasonic pulse method, 575 ultrasonic technique, 100
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tangent modulus, 376, 395 tantalum, 420 target, 337 - perforation, 339, 341,371 tensor, 78 moduli, 321 stiffness, 78 strain, 78 stress, 78 - symmetry, 78 thermal performance coefficient, 10 Timoshenko's method, 373 titanium, 4 - c r e e p parameters, 484-485 titanium alloys, 29 titanium aluminide, 30 titanium-aluminide matrix, 465, 509 titanium based composite, 352, 448,451,532, 569, 580 - c a r b o n fibre/titanium matrix, 580, 622 - fibre/matrix reactions, 569 hot-pressing, 580 - interface zone, 452 - powder metallurgy, 580 - reaction zone, 450 - SCS-6 SiC-fibre/Ti3Al + Nb-matrix, 487 - short carbon fibres, 532 - short fibres, 532 - technological parameters, 476, 480, 492, 495, 497, 513 titanium alloy, 508 - Young's modulus, 534 toughening, 34 -
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e
l
a
s
t
i
c
691
van der Waals' bonds, 418 vapour condensation, 478 viscous liquid, 315 von Mises' criterion, 84, 197 wave propagation, 101 - velocity, 17 Weibull distribution, 37, 41, 45, 168,218,233,250, 252, 267, 277, 323, 325, 367, 389, 520, 528 strength, 38 standard deviation, 38 variation coefficient, 38 wetting, 424, 547, 562, 567 conditions, 425 - HfC by Cu, 427 improving, 455 whisker coating, 542 whisker/ceramic matrix composites, 519, 541, 562 mixing process, 542 - fracture toughness, 72, 519 - powder metallurgy processes, 60, 72, 542 - SiC/AI203, 541 - SiC/Si3N4, 542 -sintering, 521,527, 533, 541 strength, 37, 39, 41, 43, 45, 51, 72 whiskers, 62, 99, 369 carcinogenicity of, 62 - silicon carbide, 42, 52, 55, 60, 62 wood, 4 work hardening of matrix, 468 -
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m
e
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n
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c
h
e
m
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Young's modulus, 81, 96, 99, 103-107, 108, 206, 369
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zirconia, 32
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