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Deformation of crack-bridging ductile reinforcements in toughened brittle materials
Acta metall. Vol. 37, No. 12, pp. 3349-3359, 1989 Printed in Great Britain. All rights reserved
0001-6160/89 $3.00 + 0.00 Copyright ~ 1989 Pergamon Press plc
D E F O R M A T I O N OF CRACK-BRIDGING DUCTILE REINFORCEMENTS IN T O U G H E N E D BRITTLE MATERIALS P. A. M A T A G A Materials Department, College of Engineering, University of California, Santa Barbara, CA 93106, U,S.A. (Received 20 March 1989)
Abstract--The addition of a dispersed ductile phase to a brittle material can lead to significant increases in fracture resistance compared to the untoughened matrix material. Often the important mechanism appears to be bridging by intact ductile ligaments behind the advancing crack tip. Although a framework for predicting toughness enhancements from bridging mechanisms exists, the required detailed model of ligament deformation which would provide the load-extension relation for a typical ligament has not been available. In this paper, numerical modeling of a plastically deforming ligament constrained by surrounding elastic matrix material is performed and the relevant toughness enhancement information extracted. Comparison is made to model experiments as needed to investigate such deformation processes as well as to toughnesses measured for technologically important composites. The results suggest that debonding along the interface between the ligament and the matrix may enhance the toughening effect of a ductile phase. R6sum6--L'addition d'une phase ductile dispers6e dans un mat6riau fragile peut accroitre de faqon significative sa r6sistance 5. la rupture par rapport 5. la matrice qui n'a pas 6t6 consolid6e. Le m6canisme important semble 6tre souvent la liaison entre ligaments ductiles intacts laiss6s derri6re l'extr6mit6 des fissures qui avancent. Bien qu'il existe un sch6ma pour pr6dire l'augmentation de durcissement 5. partir des m6canismes de liaison, on ne dispose pas d'un mod61e d6taill6 de la d6formation des ligaments qui fournirait la relation charge-allongement pour un ligament donn6. Nous pr6sentons dans cet article la mod61isation num6rique d'un ligament qui se d6forme plastiquement quand il est contraint par la matrice 61astique qui l'entoure, et nous en d6duisons l'augmentation de r6sistance qui en r6sulte. Nous comparons ces r6sultats 5. des exp6riences mod61es, ce que est n6cessaire pour &udier de tels m6canismes de dbformation, ainsi qu'aux r6sistances mesur6es sur des composites importants d'un point de vue technologique. Les r6sultats sugg6rent que la d6coh6sion ~, l'interface entre le ligament et la matrice peut accro~tre l'effet de durcissement par une phase ductile. Zusammenfassung--Zugabe einer verteilten duktilen Phase zu einem spr6den Material kann zu einer bedeutenden Verbesserung des Bruchwiderstandes verglichen, mit dem unge~inderten Matrixmaterial, fiihren. Hfiufig scheint der wichtige Mechanismus das ~berbriicken des Risses dutch intakte duktile Bfinder hinter der fortschreitenden RiBspitze zu sein. Zwar gibt as einen Rahmen, in dem die Verbesserung der Z~ihigkeit durch den Oberbriickungsmechanismus vorausgesagt werden kann, aber das ben6tigte ausffihrliche Modell der Verformung der B~inder, mit dem man einen Zusammenhang zwischen Last und Dehnung fiir einen typischen Band erhalten k6nnte, ist noch nicht verftigbar. In dieser Arbeit wird die plastische Verformung eines durch die umgebende, elastisch sich verhaltende, Matrix eingeschrfinkten Bandes numerisch berechnet; daraus wird die Verbesserung der Zii.higkeit extrahiert. Die Ergebnisse werden mit Modellexperimenten, wie sie notwendig sind zur Untersuchung einer solchen Verformung, und mit gemessenen ZS.higkeitswerten yon technologisch wichtigen Werkstoffen verglichen. Die Ergebnisse weisen darauf bin, dab die Abl6sung entlang der Grenzfl~iche zwischen Band und Matrix zur Z~ihigkeitsverbessern durch eine duktile Phase welter beitragen kann.
1. INTRODUCTION T h e a d d i t i o n of a dispersed ductile phase to a brittle material can lead to significant increases in fracture resistance c o m p a r e d to the u n t o u g h e n e d matrix material. Examples of current or potential technological significance which have been studied are tungsten carbide reinforced with cobalt [1], alumina reinforced with a l u m i n u m [2,3] a n d a brittle t i t a n i u m a l u m i n u m alloy reinforced with n i o b i u m [4]. The first two systems are interconnecting networks, while in the third the n i o b i u m phase is in the form of oriented thin disks. Successful t o u g h e n i n g has also been observed in a model system consisting of large alu-
m i n u m particles in glass [5]. The p r i m a r y m e c h a n i s m responsible for the e n h a n c e d toughnesses appears to be bridging by intact ligaments of the ductile phase b e h i n d the a d v a n c i n g crack tip (Fig. 1), a l t h o u g h
IFig. 1. Schematic of crack-bridging behavior of ductile particles in a brittle matrix.
3349
3350
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS phase (particulate or network) may be reduced to that of determining the load-separation relation for a "typical" bridging ligament, whence the fracture energy increase may be written as
g
AG = G - Gm = f
u*/ao u/a o
Fig. 2. Non-dimensional stress vs stretch behavior of a typical ligament and associated area w. other effects such as crack deflection and crack trapping by the ductile phase also contribute. One of the aims of the modeling presented here is to determine the extent to which bridging effects due to deformation of ligaments can account for the observed toughness increases. The presence of ductile ligaments does not necessarily imply a significant contribution to toughness from ligament deformation. For weakly bonded metallic fibers (nickel, iron, cobalt) in magnesium oxide [6], the major toughening mechanism appears to be pullout. Furthermore, the literature dealing with toughening of brittle polymers by rubber particles [7, 8] indicates that bridging is not the dominant mechanism [9, 10], although synergism with other effects is probably important. The contribution to fracture energy from bridging has been modeled in various contexts: examples are ligaments due to ledge formation in quasi-cleavage of steels [I1, 12] and fiber reinforcements of various kinds [6, 13, 14]. Recently, bridging models based on distributions of nonlinear springs have been considered in some detail [15, 16]. In general, the enhancement of steady-state (i.e. resistance curve plateau) fracture toughness of a brittle material due to crack bridging effects may be estimated [17] if a relationship is established between the distance of separation of the faces of a crack in the composite and the average stress supported across the crack faces by the bridging ligaments at that separation. Given this framework, the problem of estimating the toughness enhancement due to the presence of a bridging ductile tThe two entries for the TiA1/Nb material represent different orientations of the reinforcements, which in this case are disks of niobium (a0 corresponding to half the thickness of the disks).
f:
a du -fGm
(1)
where G is the fracture energy of the composite, Gm is the matrix fracture energy, ~r is the nominal stress supported across a bridging ligament at stretch u (cr falling to zero at u = u *), a n d f i s the volume fraction of ductile phase (assumed to be the same as the area fraction intercepted by the crack). The final term on the right hand side is introduced [16] to account for the fact that the area of matrix fractured is reduced by the presence of a bridging phase. This relationship may be written in a form scaled by the properties of the bridging ligament as
G =f%aow + (1 - f ) G m
(2)
where a0 is the radius of the ligament, ~r0 is the initial yield stress of the ductile phase and w is a number defined by w=
(3) J0
*0
~0'
The scalar w may be interpreted as containing information about the geometric constraint experienced during the deformation of the ligament, about strain-hardening, and about the failure of the ligament. The problem thus becomes that of determining w by obtaining the (a/a0) vs (u/ao) relation for some representative configuration and computing the area under the curve (Fig. 2). The preceding formulation is well-established. However, there is a lack of detailed modeling of the a - u relation (and hence of the bridging contribution to toughening). Measurements of composite fracture toughness may be employed to infer values of w via the inverted relationship
[
1 ( 1 - v2)K 2 w = fao a----o E
(1-f)(l_-vm)K~. ] Em
-] (4)
where the energy release rates haye been expressed in term of plane strain fracture toughness and elastic properties for the matrix (Kin, Em, Vm) and composite (K, E, v). Values of w inferred in this way for several systems of interest are displayed in Table 1.t The wide variation in w for the various systems remains to be explained. Note that yield properties for pure, bulk ligament material have been used here.
Table 1. Matrix, ligament and composite properties and inferred values of w for some materials of interest System Glass/Al AI20 3/Al
WC/Co TiAl/Nb(L) TiAI/Nb(H)
Km
E,~
(MPa~/m) 0.8 3 8
(GPa) 70 420 710
12.5 12.5
200 200
a0
ao
E
vm 0.20 0.20 0.21
f 0.2 0.2 0.2
(MPa) 70 70 450
(rum) 55 2 0.75
(GPa) 70 310 570
v 0.23 0.22 0.22
K
0.3 0.3
0.2 0.2
300 300
12.5 12.5
175 175
0.31 0.31
(MPa,v/m) 6.5 8.4 16 20 25
w 0.73 7.2 5,3 2.0 3.6
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS Table 2, Values of cohesive stress in Dugdale model inferred in Ref. [16] from observed bridging zone lengths, and corresponding failure stretches System
w
Crcohesiv¢/or0
u */ao
Glass/Al AI;O3/AI WC/Co
0,73 7,2 5.3
3.6 22 22
0.20 0.32 0.24
Some recent model experiments on lead fibers surrounded by glass [18] provide direct measurements of cr - u relations for constrained ligaments, as distinct from comparison to values of w inferred from toughness measurements. The insights gained from these tests will be discussed below. Alternatively, experimental observations of bridging zone length L have been compared to the predictions of a cohesive zone model with an assumed shape of the a - u relation and used to infer the peak load supported by the ligament [16]. In one limit of this model, the bridging zone is a constant stress (Dugdale) zone. The (acoho~i~e/ao) level then required to explain the observed toughnesses are shown in Table 2. The levels are anomalously high in both the WC/Co and AI203/AI materials. The predicted failure stretches conjugate to these stress levels are shown, and are seen to be quite small. 2. OBJECTIVES AND IDEALIZED PROBLEM
The work described in the present article is an initial attempt to model numerically the behavior of a single ductile bridging particle suitably constrained by a stiffer, elastic matrix, and to determine whether the contribution to toughness thereby predicted via the bridging model is sufficient to explain experimental observations. The problems to be considered concern the response of a single elastic/plastic particle surrounded by a cylinder of an elastic material with a considerably higher modulus. The cylinder is fully cracked in the equatorial plane of the particle perpendicular to the axis of the cylinder (Fig. 3) as needed to model circumstances in which the matrix crack has left the intact particle in its wake. The numerical study consists of deforming the ensemble axially (with boundary conditions appropriate to an array of such particles) and determining the a - u characteristics. The constraint experienced by the particle is expected to be similar to that experienced by a ligament which is part of a fibrous network of ductile material, since only the region near the cracks faces can deform plastically. Objections can be raised to considering the geometry described above to be a "typical" bridging particle. One is that the symmetry of the loading rules out failure mechanisms of a shear nature. However, since the overall relative motion of the crack faces is an opening and the matrix is relatively stiff, these effects are expected to be minimal except for strongly deviated cracks. For example, in the TiA1/Nb material configurations are observed such that the cracks between disks are parallel but
3351
t Elastic Matrix
rackFces
Fig. 3. Idealized axisymmetric configuration for analysis of behavior of a single particle. Inset shows initial bluntness of crack tip at particle equator.
not in a single plane [4]. Nevertheless, the niobium particles still tend to fail by necking to a point rather than by shear failure. The idealized model considered here closely corresponds to the model experiments [18], the results of which are schematically reproduced in Fig. 4. The first set of results [Fig. 4(a)] involve circumstances in which there was no debonding of the lead from the glass. The second set of results [Fig. 4(b)] involve substantial matrix cracking near the lead-glass interface adjacent to the crack tip. Significant differences are apparent between the two sets of results. In particular, the results in (a) indicate an average w of about 1.6, while those in (b) indicate a range of values with an average of about 4.5 and an extreme of around 6. 3. REGIMES OF LIGAMENT BEHAVIOR
Fractographic examination of toughened composites and the model experiments show that several
u/ao
2
Fig. 4. Sketch of results of model experiments of Ref. [18]. (a) Without debonding between ligament and matrix (b) With debonding,
3352
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
II Elastic
(al
(bl
(C)
(d)
Fig. 5. Regimes of ligament behavior. (a) Small-scale yielding. (b) Constrained plasticity. (c) Plastic necking. (d) Necking with internal cavitation.
distinct types of behavior may be expected for the constrained bridging particle. Indeed, analysis of the deformation history may be distilled into the circumstances wherein the transitions between regimes take place. Under very low loads, the particle should be largely elastic, with plasticity confined to a region close to the tip of the crack, which has a blunted tip in the particle [Fig. 5(a)]. The detailed stress fields will not be classical elastic/plastic small-scale yielding fields, due to the material inhomogeneity around the tip, but should have many of the same features. When the entire particle has yielded, one possibility is that constraint levels remain high enough to produce a deformation field resembling a perfectly-plastic plane strain slip-line solution [17] [Fig. 5(b) and Fig. 6]. Under these circumstances, the stress distribution within the particle would be expected to be very sensitive to events at the blunt tip and would tend to involve relatively high levels of hydrostatic stress. The load supported by the particle would be correspondingly high (note that no effect of strain-hardening has been included in Fig. 6). An alternative for the fully yielded particle is that the constraint of the stiffer surrounding matrix is relieved sufficiently by crack-tip deformation and/or interfacial debonding or sliding to allow the bridging ligament to begin to neck [Fig. 5(c)]. The necking process involves lower stress levels within the particle; moreover, lower nominal stress levels tr arise because of the decreased load-bearing area. Constraint by the surrounding stiff matrix will have a significant effect on the load history since the rate at which necking takes place is strongly influenced. This regime must dominate the later stages of deformation when the ligament fails in a highly ductile manner.
A different deformation regime for bridging ligaments involves the nucleation and growth of a single large void (or a small number of large voids) within the ligament [Fig. 5(d)]. In the model experiments, a majority of the tests involved the cavitation of a single internal void [18]. This is also occasionally observed on the AI203/AI fracture surface [19]. This phenomenon will have the effect of lowering the constraint on the ligament. In this context, it is interesting to note that hollow rubber particles, created in situ by chemical means, have been successfully used in toughening of polymers [20]. The details of ligament failure will determine the extent of the tr - u curve, in particular the failure stretch u*. Fully ductile failure of the ligaments occurs for the AI203/A1 and TiA1/Nb materials and for polycrystalline lead in glass: the ligament necks in a ductile fashion to a point or chisel edge. Conversely, the cobalt ligaments in WC/Co fail by microvoid nucleation, growth and coalescence, leading to fracture surfaces reminiscent of ductile fracture of the bulk material. This mechanism involves smaller failure separation distances than fully ductile failure, although extensive plasticity is involved in necking down of the ligaments between the voids [1]. Similar behavior is occasionally observed in the larger niobium ligaments in the TiAI/Nb material, albeit after a significant amount of ductile necking. 4. FACTORS INFLUENCING LIGAMENT BEHAVIOR
A number of material- and processing-dependent factors which might affect the load-displacement behavior of a ductile ligament are now considered in the context of the deformation regimes delineated above. The stress-strain behavior of the ligament
Brittle-Ductile Interface I~
I
Deforming
.
Crack I,
a0
.
.
.
.
.
~t
.
~,
b) 10
0
I
t
i
i
O, 5
0.10-
u/a o
Fig. 6. Plane strain slipline solution o f R¢f. [I 7]. (a) Slipline geometry for ao/u = 1150. (b) ~ - u relation.
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS material has been taken to be that of the bulk material, with incompressible, pressure-insensitive plastic deformation assumed. It appears to be a matter of controversy as to whether the ligaments observed in the materials studied are of sufficiently small dimensiont that size effects could result in yield stresses in the ligament much higher than those of the bulk material (note that the largest values of w in Table 1 correspond to the smallest dimension ligaments). Such an argument was employed in a model for WC/Co toughening [1], in which a Hall-Petch effect was incorporated into the material model. Microstructural studies of the AI phase in the A1203/A1 material [19] suggest that dispersion strengthening might significantly raise the yield stress of the ligaments, perhaps by as much as a factor of three, although no quantitative hardness data is as yet available. The toughening effectiveness of a ductile phase will in general be enhanced by strain-hardening or an increased yield stress, unless an associated lack of ductility leads to failure at significantly decreased separation distance. Any increase in a 0 will lead to a proportionate decrease in the value of w (Table 1) and in the constraint factors emerging from the analysis of Ref. [16] (Table 2) which employed properties appropriate to pure bulk materials. It is initially assumed that the interfacial bond is strong enough that no debonding takes place. However, under circumstances where the particle remains highly constrained for a significant portion of its history, features near the crack tip may control stresses throughout the particle. Should this be the case, small scale events such as debonding of the interface near the tip affect the load history of the ligament in the early stages of deformation. In general, debonding will tend to lower constraint levels and hence lead to lower peak loads. However, the influence of debonding on the necking regime is also very strong, since the dimensions of the necking ligament are thereby determined. Debonds which are significant fractions of the ligament radius in extent are observed both in model experiments [18] and in detailed fractographic examination of the A1203/A1 material [19]. No attempt has been made in the present work to incorporate internal cavitation, either as explicit individual large voids or via a constitutive law for the ligament involving microvoidage and failure. However, the stress and strain history of the ligament provides information relevant to nucleation of internal voids. The modeling of void growth during ligament deformation is an important area of future investigation. The qualitative observation may be made that constraint conditions which raise the hydrostatic stress levels in the bridging ligament in the initial stages will increase the peak load, but also serve to encourage void growth. The relative ease of
tOf the order of 1/~m in WC/Co and AI203/AI.
(Y
~
~
3353
With Thermal Effects Effeets
u
Fig. 7. Putative effect of thermal residual stresses on a - u relation.
void nucleation in the ligament material will thus strongly influence the mode of final failure of the ligament. The elastic (or plastic) behavior of the surrounding matrix will in general affect the constraint levels of the particle. For all the cases considered here, the matrix has considerably higher elastic modulus than the particle and is effectively rigid, particularly once substantial plastic deformation of the particle occurs. Matrix elasticity may have some effect on the detailed near-tip deformation of the ductile material, but this aspect has not been thoroughly explored. Interaction between bridging ligaments is likely to be an important effect when the volume fraction of ductile phase is significant. The cylinder model employed here incorporates this interaction to lowest order by having an outer radius to the matrix jacket which reflects the volume fraction, and appropriate "unit cell" boundary conditions: straight but unconstrained boundaries. Again, with a stiff matrix, these effects are small. Composite fabrication typically involves high temperature processing, with thermal stresses arising in the reinforcing phase and matrix on cooling to ambient. In many of the systems of interest, this leads to tensile stresses in the ligaments which contribute to the load necessary to separate the crack surfaces. It has been suggested that this could lead to a a - u relation of the form sketched in Fig. 7. This effect and its coupling with nonlinear response of a bridging phase has been studied in the context of fiberreinforced brittle materials [21]. One of the aims of the present study is to determine the extent to which thermal residual strains contribute for the whole range of ligament stretch. 5. NUMERICAL MODELING The numerical modeling of this problem was implemented using the finite element package ABAQUS [22] on a CONVEX minisupercomputer. The computations employed finite strain plasticity in order to account for geometry changes. The upper boundary of the matrix material was extended gradually (per-
3354
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
5.1. Constrained regime
,,-
_~,, Ar
I.
.I
ao
Fig. 8. Geometry for initial deformation near crack tip in ductile phase.
haps after a thermal step involving compression across the crack face) in order to generate data cr - u curves for the ligament. Separation u is measured as the axial displacement of the matrix point at the tip of the crack, while tr is computed from the total force exerted on the upper boundary, normalized by the original ligament area. The aim at the outset was to describe the initial stages and also the necking regime as far as possible. However, serious difficulty was encountered in extending the cr - u curves to large u. Indeed, the results never extend into a regime of clearly decreasing total load on the ligament. The breakdown is attributed to the intense and non-uniform deformation in the tip-interface region. The reason for the intense deformation may be seen from the fact that the large-scale deformation of the ligament material is essentially volume-conserving. The behavior is exemplified by considering the volume contained in that part of the stretched ligament which extends below its original midplane (Fig. 8). This volume must be accounted for by the "void space" contained in the blunting cracktip. If a small amount of necking Ar has occurred, but large enough that elastic volume changes are not important, the volume to be accounted for is of order na2u. The corresponding volume of the toroidal "void" is of order naoArAh. Volume conservation then requires that
The majority of the results were obtained for material parameters characteristic of the (bulk) properties of A1203 and AI: the matrix Young's modulus is six times larger than that of the inclusion; the inclusion initial yield stress is ~-~ l of its Young's modulus and the material strain-hardens with an exponent n = 0.2 until the flow stress saturates at three times initial yield. A typical tr - u curve for these material parameters is shown in Fig. 9. The small-scale yielding regime exists only for nominal stress levels below tr ~ a0, corresponding to openings u which are extremely small. The particle is fully plastic at tr ~ 4tr0. The constrained fully plastic regime appears to produce a peak stress level tr ~ 6a0, by which time a macroscopic softening due to the onset of necking has begun, even though all of the material in the particle is still slowly strain-hardening. Figure 9 also includes results in which no strain-hardening is incorporated. The constraint magnification factor is seen to be similar in either case. A typical failure stretch, whether observed experimentally or predicted on the basis of necking calculations, corresponds to (u/ao) on the order of 1 when failure occurs by necking to a point, so that the results displayed here cover only a tiny range of the expected total load-displacement relation for the ligament. The solid curve in Fig. 9 was generated using an initial tip raidus r0 such that (ro/ao) had a value of 0.05. Even when this rather large initial radius is used, the computations fail at a small stretch. The results suggest that necking becomes important rather rapidly, although additional results obtained using smaller initial radii (dashed curves in Fig. 9) indicate that the characteristics described above may be sensitive to the choice of initial tip radius. Finally, a plane strain computation (Fig. 10) analogous to Fig. 9 reveals that the initial behavior is very similar when an appropriate nondimensionalization is used.
5 w
Ah u
--
a0 Ar
~--.
(5)
That is, a small crack-tip "void" is driven in a direction normal to the crack plane more rapidly than the opening rate of the tip itself. Without debonding or sliding, the constraint of the stiff surrounding matrix produces intense shear deformation. When no debonding occurs, this pattern of deformation leads to a "hollowing out" of the ductile phase near the matrix crack tip. Such behavior is observed experimentally. To produce results relevant to larger stretches, a finite radius blunt tip was included from the initial stages, and results for different radii compared for consistency.
5-
= / / ~
4
2 I
/
/~
1
ro/ao=0.05
- - - ro/ao=o
n=o
F
0
0.0
I
0.005
I
0.010 u/ao
I
0.015
I
0.020
Fig. 9. I n i t i a l s t a g e finite e l e m e n t r e s u l t s for a - u r e l a t i o n
for different choices of work-hardening exponent n and of initial root radius r0 (bonded interface).
MATAGA:
DUCTILE REINFORCEMENTS
IN BRITTLE MATERIALS
3355
6
L
2
1
/~/ ¢P
0 0,0
- - --Plane Strain
I 0.005
I 0,010
I 0.015
I 0.020
i
~'ISN-"x~
"o/ao = 0.S
.... I 0.5
0.0
....... I 1.0
----
I 1.5
I 2.0
u/a 0
u/a o Fig. 10. Finite element plane strain vs axisymmetric comparison for initial stages (bonded interface).
Constrained Ends Unconstrained Ends
---
Fig. 12. Constrained necking bar finite element results for various "debond" lengths h0. Also shown for comparison are corresponding uniform deformation (unconstrained ends) results. Material parameters are for a lead ligament.
5.2. Necking regime Since the numerical approach described in the previous section was not successful in providing results extending into a clearly necking regime, the behavior of constrained necking plastic bars was next studied. The axial deformation of plastic bars was modeled, with the ends of the bars constrained from shrinking radially (Fig. 11). This is intended to model a ligament for which the constraint of the surrounding stiff matrix has been partially relaxed by debonding/hollowing of the ligament near the crack tip. A finite element model was employed to discover how the initial bar height affects the subsequent load~lisplacement history. The stretch u is now measured as the extension of the bar. Such a study is clearly appropriate for the case when debonding occurs in the very early stages of deformation. A set of results (Fig. 12) with the material properties appropriate to the A1203/AI material for initial heights ranging from (ho/ao)=0.1 to (ho/ao)= 0.5 indicate that the necking process is strongly accelerated by the constraint when the bar is short, so that the load decreases more rapidly with increasing ligament stretch than for the long bar. Some results for analogous plane strain cases (Fig. 13) reveal that additional constraint is thereby imposed.
6. G E O M E T R I C N E C K I N G M O D E L S
Approximate models of the necking process in constrained ligaments may be constructed along the lines of a model presented in [2]. These models may be described as "geometric" necking models in that each assumes some form for the ligament shape during deformation, parameterized in a convenient way. The volume-conserving nature of the large-scale plastic deformation is employed to relate various features of the shape. In this way, in terms of the chosen parameter, the separation u is calculated and the nominal axial stress ~ is estimated, thereby generating a ~r - u curve for the ligament. For example, the geometry sketched in Fig. 11 can be used to model the axisymmetric necking bars. If the bar is assumed to be a paraboloid of rotation, volume constancy leads to the relation h a0
h0 l a01 _ ]4p + ~p2
5 \ \\
C~
al
Axisymmetric Plane Strain
4
I
I
2h
2a o
----
(6a)
2h = 2ho+ u
L
2ao bl
Fig. 11. Constrained necking bar geometry designed to simulate the large deformation of a ligament with debonding.
0 0.0
I 0.5
( 1.0
I 1.5
I 2.0
u/a 0 Fig. 13. Plane strain vs axisymmetric c o m p a r i s o n for constrained necking bars for various " d e b o n d " lengths h 0.
3356
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
8
(a)
7~
a 0
(7 0
where
6
o
o" = (1 - p)2 aB
ho/a 0 = 0.1
FiniteElement - - - Model of Eqn.(6) .... - Model of Ref [18]
~ "~1~,
5
~4 U
h2 aB ,,o =
1+
:(1---:)a
xIn[l
3
- J L\ L(l-o) lhh J/
(7)
2
and where the uniaxial (true) stress-strain behavior is described by
1
o 0.0
0,5
1.0
1.5
2.0
au.ia~al F(E).
il
(b)
O"o
h0/a0 = 0.25 Finite Element Model of Eqn. (6) . . . . . Model of Ref [18] ---
.~
o
i
0.0
0.5
1.0
1,5
2.0
u/a 0
8
•~ 4 io
(C)
h0/a0 : 0.5
i
(8)
=
u/a0
i'L'/i
- - - Model of Eqn.(6) .... - Modelof Ref[18]
3 2
.......................
Using material parameters as directly measured in the glass/Pb experiments [18] (power-law hardening with n = 0.25), this model is compared with finite element results in Fig. 14. Also shown are the predictions from a geometric model employed in Ref. [18], where the assumed shape of the neck is cylindrical and a virtual work balance is used to determine the stress supported. There are qualitative similarities in overall behavior, but a significant difference in the final necking behavior. The geometric model introduced here agrees reasonably well with the finite element results. Geometric models have also been employed to obtain results for the full range of ligament deformation when debonding does not occur, but a necking ligament is formed by hollowing out of the material near the crack tip. However, it would appear that the results are rather sensitive to the choice of geometry. This is demonstrated in Fig. 16, where three models are compared: (a) a model described in Ref. [2It which employs the geometry of Fig. 15(a) for the whole of the range of deformation, (b) the model of Ref. [2] with fewer approximations in the geometrical description, (c) a model similar to that of Ref. [2] but
1 o 0.o
I 0.5
I 1.o
I 1.5
I 2.0
u/ao
(a)
I I I I I I I
Fig. 14. Results of finite element and several necking models for necking ligament regime, for various debond lengths. (a) (ho/ao)=0.1. (b) (ho/ao)=0.25. (c) (ho/ao)=0.5. where the necking parameter is
.
.
.
.
Ligament
Matrix
L i g ~
Matrix
.
a
p = 1 --
(6b) a0
The nominal stress may be estimated from the Bridgman result for a necking bar [23] (using the local radius of curvature at the ligament mid-plane) as
fNote that there is an error in equation (I0) in Ref. [2] which should read 2u X 2 ...~. _ _ _ g a o"
c._ (b)
"~
I u_2 Fig. 15. Two plausible geometric models for full range of ligament deformation. (a) From Ref. [2]. (b) Alternative geometry.
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
ii 7 i
~
0
0.0
by equations (6) and (7) for various bar lengths. The material parameters are again appropriate for the glass/Pb model experiments. Interpreting these as w values when failure suddenly occurs at stretch uf due to some mechanism not incorporated in the fully ductile model, it would appear that increased debonding increases w if the failure stretch uf is greater than 0.5a0. This suggests that for sufficiently ductile failure debonding leads to larger values of w. The large debond, large failure stretch case yields a value for w of 3 as a conservative prediction.
Model of Fig. 15(a)as in Ref [2]
. . . . . Model of Fig. 15(a) reeomputed Model of Fig. 15(b) - - - Model of Ref [18]
i •
-
0.5
-
. . . . . . . . . i- . . . . . . . . . l 1.0 1.5 2.0 u/a o
with a different, albeit plausible, choice of necking geometry [Fig. 15(b)] and (d) a model described in Ref. [18] which employed a dislocation pileup calculation to give an initial neck height. Material parameters appropriate to glass/Pb are used. Again, the predictions are widely different. For a real ligament, some failure mechanism may truncate the ¢ r - u relation. The contribution to w from the necking regime is in general more significant than that from the initial highly constrained regime, due to the fact that the ligament stretches which occur in the latter regime are tiny compared to the necking stretches. Ignoring the amount by which the necking calculations underestimate that contribution, Fig. 17 shows cumulative w values, defined as
wf(Uf~.~.fo'f/a°G.)~-d(U a0
7. THERMAL STRESSES
7.1. Constrained regime
Fig. 16. Predictions of various necking models for full range of ligament deformation.
\a0/
3357
a0
(9)
obtained from the geometric necking model defined
tPerhaps even compared to material ultimate strengths!
Finite element solutions for the initial behavior of a bridging particle incorporating thermal straining have been obtained. The material properties are appropriate to a spherical AI particle in A1203 matrix, here subjected to a 500°C cooling. It is assumed that the matrix is very easily cracked. Separation of the crack faces both at the crack tip and at the remote boundary of the unit cell are plotted in Fig. 18. At the point at which the crack tip begins to open (the residual compression across the faces having been relieved as the load increases), the average traction across the particle is in fact slightly larger than the residual hydrostatic stress in the particle in the uncracked composite, erR, given by the elastic result [24] EpACtAT
OR =
(10)
1 - 2Vp + (1 + Vm)~ m where subscript p refers to particle elastic properties and subscript m to those of the matrix, and A~AT is the thermal strain mismatch. The small increase may be predicted on the basis of a purely elastic analysis of the load at which the tip begins to open [25]. Since the ratio (aR/tro) for this combination of materials and this geometry is relatively large, this represents a high initial stress supported by the ligament, even compared to constrained deformation.t Moreover,
20-
Tip
ho/a o = o . s / /
15 ~ ' ~
o
1 f
0.1
lO;, l
/
/
/
/
--AT=0 ---AT=500K
/ /~'~'Remote
5
0L~P" 0.0
I 0.5
I 1.0
I 1.5
I 2.0
uda0
Fig. 17. Cumulative contribution to w, wr, as a function of "sudden failure stretch" uf as predicted by necking model of equation (6) with various amounts of debonding.
0.0
0.005
I 0.010
I 0.015
I 0.020
u/a 0
Fig. 18. Initial stage finite element results showing effect of thermal residual stresses (no debonding).
3358
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
6\ 41~ / a 5
\
21
---
,~
o o.o
AT=0 A T = 500K
o=O.S I
I
0.5
1 .o
u/a o
Fig. 19. Constrained necking bar finite element results showing effect of thermal residual stresses. the load curve has a subsequent slope which is of the same order of magnitude as when thermal strains are absent.
7.2. Necking regime Necking calculations were carried out in a similar fashion to those described previously, with the addition of thermal strains. The same material and temperature parameters were chosen as above. Some sample results are shown in Fig. 19, where c r - u records for strain-hardening bars of three different lengths are displayed, with and without thermal strains. The results indicate that the constrained thermal stresses which are evident in the initial stages of deformation do not remain in the necking regime. Indeed, the net effect of the thermal strains appears to be to reduce the load supported by the ligament, presumably because the shrinkage encourages necking. 8. DISCUSSION Comparison of the various models described above to the experiments of Ref. [18] shows qualitative agreement, but convincing quantitative predictions are yet to be obtained. For the initial, constrained stages of deformation, the numerical results indicate constraint factors in the requisite range. A priority of future work will be to extend the constrained results into a clearly necking regime. For the case of non-debonding ductile failure of the ligament, a previously used geometric model [2] apparently gave good agreement with the experiments for the necking regime. However, it is shown here that plausible alternative geometrical assumptions produce quantitatively different results. This strongly suggests that detailed modeling of the deformation, with the interaction between the material properties and the geometry evolution incorporated, is necessary if accurate predictions are to be obtained. For debonded ligaments, the correlation between
increased debond length and increased w is predicted by the models. The experimental behavior of the debonded ligaments is, however, somewhat different in character to the necking bar models in that the stress remains elevated for significant stretches and then falls rather abruptly [18]. The measured w values with debonding are approached only by the model described in Ref. [18]. Constrained bar calculations somewhat underestimate the failure stretch due to the accelerated necking which they entail. The toughness enhancement due to bridging is dominated by the contribution from the necking regime, since the high stresses in the constrained plasticity regime persist only briefly. However, this conclusion does not exclude the early stages of deformation as irrelevant, since the initial stresses and near-tip deformation may control whether debonding or internal cavitation occurs. Thus thermal mismatches, for example, although not appearing to be responsible for a significant direct contribution to toughening by ductile ligaments, may have significant indirect influence via the resultant stresses in the early stages of deformation. Clearly the significance of such effects will depend strongly on the material and interfacial properties. In addition, the evolution of the necking ligament geometry may depend on the details of the initial deformation even when no debonding takes place. It is thus important to model the whole range of ligament deformation accurately. The inclusion of phenomena such as debonding and internal cavitation would be desirable, but may be difficult to achieve in a convincing way. From the materials design viewpoint, perhaps the most interesting aspect of the results is the suggestion that the ductile phase need not be completely bonded to the matrix during deformation to provide toughening. Limited debonding may well be beneficial in terms of toughness enhancement. From this point of view, there are clear advantages associated with a reinforcing phase which is a network rather than a distribution of equiaxed particles, since complete interface failure and particle pullout is avoided. In terms of the underlying aim of the investigation, namely to assess the extent to which bridging can explain the observed toughnesses of brittle materials reinforced with ductile phases, the results are not completely conclusive. The model glass/A1 material and the TiAI/Nb materials fall within the range of the model predictions; on the other hand, the w values predicted by the models and the model experiments require large-scale debonding to explain the toughness enhancement in the WC/Co and A1203/A1 materials. Preliminary investigations [19] suggest that observed debonding and significant dispersion hardening of the aluminum may be sufficient to explain the latter case, but the WC/Co material remains an anomaly in terms of the ideas presented here, unless the properties of the ligament significantly depart from those of bulk cobalt. These aspects are subject to further experimental study.
MATAGA:
DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
Acknowledgements--The author was fortunate enough to have been able to discuss various aspects of this work with A. G. Evans, R. M. McMeeking, L. S. Sigl, M. F. Ashby, B. Budiansky, and F. W. Zok. Financial support by the Defense Advanced Research Projects Agency under O.N.R. contract N00014-86-K-0753 is gratefully acknowledged. REFERENCES
1. L. S. Sigl and H. E. Exner, Metall. Trans. A 18A, 1299 (1987). 2. L. S. Sigl, P. A. Mataga, B. J. Dalgleish, R. M. McMeeking and A. G. Evans, Acta metall. 36, 945 (1988). 3. B. J. Dalgleish and A. G. Evans. To appear. 4. C. K. Elliot, G. R. Odette, G. E. Lucas and J. W. Sheckherd, Proc. M.R.S. Meeting, Reno (1988). 5. V. D. Krstic, Phil. Mag. 48, 695 (1983). 6. P. Hing and G. W. Groves, J. Mater. Sci. 1,427 (1972). 7. C. B. Bucknall, Toughened Plastics. Applied Science, New York (1977). 8. A. G. Evans, Z. B. Ahmad, D. G. Gilbert and P. W. R. Beaumont, Acta metall. 34, 79 (1986). 9. S. Kunz-Douglas, P. W. R. Beaumont and M. F. Ashby, J. Mater. Sci. 15, 1109 (1986). 10. Z. B. Ahmad, M. F. Ashby and P. W. R. Beaumont, Scripta metall. 20, 843 (1986).
A.M 3 7 / 1 ~ P
3359
II. R. G. Hoagland, A. P. Rosenfield and G. T. Hahn, Metall. Trans. 3, 123 (1972). 12. W. W. Gerberich and E. Kurman, Scripta metall. 19, 295 (1985). 13. G. A. Cooper and A. Kelly, J. Mech. Phys. Solids 15, 279 (1967). 14. W. W. Gerberich, J. Mech. Phys. Solids 19, 71 (1971). 15. L. R. F. Rose, J. Mech. Phys. Solids 35, 383 (1987). 16. B. Budiansky, J. C. Amazigo and A. G. Evans, J. Mech. Phys. Solids 36, 167 (1988). 17. A. G. Evans and R. M. McMeeking, Acta metall. 34, 2435 (1986). 18. M. F. Ashby, F. J. Blunt and M. Bannister, Cambridge Univ. Engng Dept. Tech. Rep. 19. B. D. Flinn, M. R/ihle and A. G. Evans. To be published. 20. M. J. Doyle and I. J. Gardner, Proc. 38th Annual Conf. Soc. Plastics Industry (1983). 21. D. B. Marshall and A. G. Evans, Mater. Forum 11,304 (1988). 22. Hibbitt, Karlsson and Sorensen, Inc., Providence, R.I. 23. P. W. Bridgman, Studies of Large Plastic Flow and Fracture. Harvard Univ. Press (1964). 24. S. P. Timoshenko and J. N. Goodier, Theorv of Elasticity. McGraw-Hill, New York (1959). 25. P. A. Mataga, unpublished work.
0001-6160/89 $3.00 + 0.00 Copyright ~ 1989 Pergamon Press plc
D E F O R M A T I O N OF CRACK-BRIDGING DUCTILE REINFORCEMENTS IN T O U G H E N E D BRITTLE MATERIALS P. A. M A T A G A Materials Department, College of Engineering, University of California, Santa Barbara, CA 93106, U,S.A. (Received 20 March 1989)
Abstract--The addition of a dispersed ductile phase to a brittle material can lead to significant increases in fracture resistance compared to the untoughened matrix material. Often the important mechanism appears to be bridging by intact ductile ligaments behind the advancing crack tip. Although a framework for predicting toughness enhancements from bridging mechanisms exists, the required detailed model of ligament deformation which would provide the load-extension relation for a typical ligament has not been available. In this paper, numerical modeling of a plastically deforming ligament constrained by surrounding elastic matrix material is performed and the relevant toughness enhancement information extracted. Comparison is made to model experiments as needed to investigate such deformation processes as well as to toughnesses measured for technologically important composites. The results suggest that debonding along the interface between the ligament and the matrix may enhance the toughening effect of a ductile phase. R6sum6--L'addition d'une phase ductile dispers6e dans un mat6riau fragile peut accroitre de faqon significative sa r6sistance 5. la rupture par rapport 5. la matrice qui n'a pas 6t6 consolid6e. Le m6canisme important semble 6tre souvent la liaison entre ligaments ductiles intacts laiss6s derri6re l'extr6mit6 des fissures qui avancent. Bien qu'il existe un sch6ma pour pr6dire l'augmentation de durcissement 5. partir des m6canismes de liaison, on ne dispose pas d'un mod61e d6taill6 de la d6formation des ligaments qui fournirait la relation charge-allongement pour un ligament donn6. Nous pr6sentons dans cet article la mod61isation num6rique d'un ligament qui se d6forme plastiquement quand il est contraint par la matrice 61astique qui l'entoure, et nous en d6duisons l'augmentation de r6sistance qui en r6sulte. Nous comparons ces r6sultats 5. des exp6riences mod61es, ce que est n6cessaire pour &udier de tels m6canismes de dbformation, ainsi qu'aux r6sistances mesur6es sur des composites importants d'un point de vue technologique. Les r6sultats sugg6rent que la d6coh6sion ~, l'interface entre le ligament et la matrice peut accro~tre l'effet de durcissement par une phase ductile. Zusammenfassung--Zugabe einer verteilten duktilen Phase zu einem spr6den Material kann zu einer bedeutenden Verbesserung des Bruchwiderstandes verglichen, mit dem unge~inderten Matrixmaterial, fiihren. Hfiufig scheint der wichtige Mechanismus das ~berbriicken des Risses dutch intakte duktile Bfinder hinter der fortschreitenden RiBspitze zu sein. Zwar gibt as einen Rahmen, in dem die Verbesserung der Z~ihigkeit durch den Oberbriickungsmechanismus vorausgesagt werden kann, aber das ben6tigte ausffihrliche Modell der Verformung der B~inder, mit dem man einen Zusammenhang zwischen Last und Dehnung fiir einen typischen Band erhalten k6nnte, ist noch nicht verftigbar. In dieser Arbeit wird die plastische Verformung eines durch die umgebende, elastisch sich verhaltende, Matrix eingeschrfinkten Bandes numerisch berechnet; daraus wird die Verbesserung der Zii.higkeit extrahiert. Die Ergebnisse werden mit Modellexperimenten, wie sie notwendig sind zur Untersuchung einer solchen Verformung, und mit gemessenen ZS.higkeitswerten yon technologisch wichtigen Werkstoffen verglichen. Die Ergebnisse weisen darauf bin, dab die Abl6sung entlang der Grenzfl~iche zwischen Band und Matrix zur Z~ihigkeitsverbessern durch eine duktile Phase welter beitragen kann.
1. INTRODUCTION T h e a d d i t i o n of a dispersed ductile phase to a brittle material can lead to significant increases in fracture resistance c o m p a r e d to the u n t o u g h e n e d matrix material. Examples of current or potential technological significance which have been studied are tungsten carbide reinforced with cobalt [1], alumina reinforced with a l u m i n u m [2,3] a n d a brittle t i t a n i u m a l u m i n u m alloy reinforced with n i o b i u m [4]. The first two systems are interconnecting networks, while in the third the n i o b i u m phase is in the form of oriented thin disks. Successful t o u g h e n i n g has also been observed in a model system consisting of large alu-
m i n u m particles in glass [5]. The p r i m a r y m e c h a n i s m responsible for the e n h a n c e d toughnesses appears to be bridging by intact ligaments of the ductile phase b e h i n d the a d v a n c i n g crack tip (Fig. 1), a l t h o u g h
IFig. 1. Schematic of crack-bridging behavior of ductile particles in a brittle matrix.
3349
3350
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS phase (particulate or network) may be reduced to that of determining the load-separation relation for a "typical" bridging ligament, whence the fracture energy increase may be written as
g
AG = G - Gm = f
u*/ao u/a o
Fig. 2. Non-dimensional stress vs stretch behavior of a typical ligament and associated area w. other effects such as crack deflection and crack trapping by the ductile phase also contribute. One of the aims of the modeling presented here is to determine the extent to which bridging effects due to deformation of ligaments can account for the observed toughness increases. The presence of ductile ligaments does not necessarily imply a significant contribution to toughness from ligament deformation. For weakly bonded metallic fibers (nickel, iron, cobalt) in magnesium oxide [6], the major toughening mechanism appears to be pullout. Furthermore, the literature dealing with toughening of brittle polymers by rubber particles [7, 8] indicates that bridging is not the dominant mechanism [9, 10], although synergism with other effects is probably important. The contribution to fracture energy from bridging has been modeled in various contexts: examples are ligaments due to ledge formation in quasi-cleavage of steels [I1, 12] and fiber reinforcements of various kinds [6, 13, 14]. Recently, bridging models based on distributions of nonlinear springs have been considered in some detail [15, 16]. In general, the enhancement of steady-state (i.e. resistance curve plateau) fracture toughness of a brittle material due to crack bridging effects may be estimated [17] if a relationship is established between the distance of separation of the faces of a crack in the composite and the average stress supported across the crack faces by the bridging ligaments at that separation. Given this framework, the problem of estimating the toughness enhancement due to the presence of a bridging ductile tThe two entries for the TiA1/Nb material represent different orientations of the reinforcements, which in this case are disks of niobium (a0 corresponding to half the thickness of the disks).
f:
a du -fGm
(1)
where G is the fracture energy of the composite, Gm is the matrix fracture energy, ~r is the nominal stress supported across a bridging ligament at stretch u (cr falling to zero at u = u *), a n d f i s the volume fraction of ductile phase (assumed to be the same as the area fraction intercepted by the crack). The final term on the right hand side is introduced [16] to account for the fact that the area of matrix fractured is reduced by the presence of a bridging phase. This relationship may be written in a form scaled by the properties of the bridging ligament as
G =f%aow + (1 - f ) G m
(2)
where a0 is the radius of the ligament, ~r0 is the initial yield stress of the ductile phase and w is a number defined by w=
(3) J0
*0
~0'
The scalar w may be interpreted as containing information about the geometric constraint experienced during the deformation of the ligament, about strain-hardening, and about the failure of the ligament. The problem thus becomes that of determining w by obtaining the (a/a0) vs (u/ao) relation for some representative configuration and computing the area under the curve (Fig. 2). The preceding formulation is well-established. However, there is a lack of detailed modeling of the a - u relation (and hence of the bridging contribution to toughening). Measurements of composite fracture toughness may be employed to infer values of w via the inverted relationship
[
1 ( 1 - v2)K 2 w = fao a----o E
(1-f)(l_-vm)K~. ] Em
-] (4)
where the energy release rates haye been expressed in term of plane strain fracture toughness and elastic properties for the matrix (Kin, Em, Vm) and composite (K, E, v). Values of w inferred in this way for several systems of interest are displayed in Table 1.t The wide variation in w for the various systems remains to be explained. Note that yield properties for pure, bulk ligament material have been used here.
Table 1. Matrix, ligament and composite properties and inferred values of w for some materials of interest System Glass/Al AI20 3/Al
WC/Co TiAl/Nb(L) TiAI/Nb(H)
Km
E,~
(MPa~/m) 0.8 3 8
(GPa) 70 420 710
12.5 12.5
200 200
a0
ao
E
vm 0.20 0.20 0.21
f 0.2 0.2 0.2
(MPa) 70 70 450
(rum) 55 2 0.75
(GPa) 70 310 570
v 0.23 0.22 0.22
K
0.3 0.3
0.2 0.2
300 300
12.5 12.5
175 175
0.31 0.31
(MPa,v/m) 6.5 8.4 16 20 25
w 0.73 7.2 5,3 2.0 3.6
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS Table 2, Values of cohesive stress in Dugdale model inferred in Ref. [16] from observed bridging zone lengths, and corresponding failure stretches System
w
Crcohesiv¢/or0
u */ao
Glass/Al AI;O3/AI WC/Co
0,73 7,2 5.3
3.6 22 22
0.20 0.32 0.24
Some recent model experiments on lead fibers surrounded by glass [18] provide direct measurements of cr - u relations for constrained ligaments, as distinct from comparison to values of w inferred from toughness measurements. The insights gained from these tests will be discussed below. Alternatively, experimental observations of bridging zone length L have been compared to the predictions of a cohesive zone model with an assumed shape of the a - u relation and used to infer the peak load supported by the ligament [16]. In one limit of this model, the bridging zone is a constant stress (Dugdale) zone. The (acoho~i~e/ao) level then required to explain the observed toughnesses are shown in Table 2. The levels are anomalously high in both the WC/Co and AI203/AI materials. The predicted failure stretches conjugate to these stress levels are shown, and are seen to be quite small. 2. OBJECTIVES AND IDEALIZED PROBLEM
The work described in the present article is an initial attempt to model numerically the behavior of a single ductile bridging particle suitably constrained by a stiffer, elastic matrix, and to determine whether the contribution to toughness thereby predicted via the bridging model is sufficient to explain experimental observations. The problems to be considered concern the response of a single elastic/plastic particle surrounded by a cylinder of an elastic material with a considerably higher modulus. The cylinder is fully cracked in the equatorial plane of the particle perpendicular to the axis of the cylinder (Fig. 3) as needed to model circumstances in which the matrix crack has left the intact particle in its wake. The numerical study consists of deforming the ensemble axially (with boundary conditions appropriate to an array of such particles) and determining the a - u characteristics. The constraint experienced by the particle is expected to be similar to that experienced by a ligament which is part of a fibrous network of ductile material, since only the region near the cracks faces can deform plastically. Objections can be raised to considering the geometry described above to be a "typical" bridging particle. One is that the symmetry of the loading rules out failure mechanisms of a shear nature. However, since the overall relative motion of the crack faces is an opening and the matrix is relatively stiff, these effects are expected to be minimal except for strongly deviated cracks. For example, in the TiA1/Nb material configurations are observed such that the cracks between disks are parallel but
3351
t Elastic Matrix
rackFces
Fig. 3. Idealized axisymmetric configuration for analysis of behavior of a single particle. Inset shows initial bluntness of crack tip at particle equator.
not in a single plane [4]. Nevertheless, the niobium particles still tend to fail by necking to a point rather than by shear failure. The idealized model considered here closely corresponds to the model experiments [18], the results of which are schematically reproduced in Fig. 4. The first set of results [Fig. 4(a)] involve circumstances in which there was no debonding of the lead from the glass. The second set of results [Fig. 4(b)] involve substantial matrix cracking near the lead-glass interface adjacent to the crack tip. Significant differences are apparent between the two sets of results. In particular, the results in (a) indicate an average w of about 1.6, while those in (b) indicate a range of values with an average of about 4.5 and an extreme of around 6. 3. REGIMES OF LIGAMENT BEHAVIOR
Fractographic examination of toughened composites and the model experiments show that several
u/ao
2
Fig. 4. Sketch of results of model experiments of Ref. [18]. (a) Without debonding between ligament and matrix (b) With debonding,
3352
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
II Elastic
(al
(bl
(C)
(d)
Fig. 5. Regimes of ligament behavior. (a) Small-scale yielding. (b) Constrained plasticity. (c) Plastic necking. (d) Necking with internal cavitation.
distinct types of behavior may be expected for the constrained bridging particle. Indeed, analysis of the deformation history may be distilled into the circumstances wherein the transitions between regimes take place. Under very low loads, the particle should be largely elastic, with plasticity confined to a region close to the tip of the crack, which has a blunted tip in the particle [Fig. 5(a)]. The detailed stress fields will not be classical elastic/plastic small-scale yielding fields, due to the material inhomogeneity around the tip, but should have many of the same features. When the entire particle has yielded, one possibility is that constraint levels remain high enough to produce a deformation field resembling a perfectly-plastic plane strain slip-line solution [17] [Fig. 5(b) and Fig. 6]. Under these circumstances, the stress distribution within the particle would be expected to be very sensitive to events at the blunt tip and would tend to involve relatively high levels of hydrostatic stress. The load supported by the particle would be correspondingly high (note that no effect of strain-hardening has been included in Fig. 6). An alternative for the fully yielded particle is that the constraint of the stiffer surrounding matrix is relieved sufficiently by crack-tip deformation and/or interfacial debonding or sliding to allow the bridging ligament to begin to neck [Fig. 5(c)]. The necking process involves lower stress levels within the particle; moreover, lower nominal stress levels tr arise because of the decreased load-bearing area. Constraint by the surrounding stiff matrix will have a significant effect on the load history since the rate at which necking takes place is strongly influenced. This regime must dominate the later stages of deformation when the ligament fails in a highly ductile manner.
A different deformation regime for bridging ligaments involves the nucleation and growth of a single large void (or a small number of large voids) within the ligament [Fig. 5(d)]. In the model experiments, a majority of the tests involved the cavitation of a single internal void [18]. This is also occasionally observed on the AI203/AI fracture surface [19]. This phenomenon will have the effect of lowering the constraint on the ligament. In this context, it is interesting to note that hollow rubber particles, created in situ by chemical means, have been successfully used in toughening of polymers [20]. The details of ligament failure will determine the extent of the tr - u curve, in particular the failure stretch u*. Fully ductile failure of the ligaments occurs for the AI203/A1 and TiA1/Nb materials and for polycrystalline lead in glass: the ligament necks in a ductile fashion to a point or chisel edge. Conversely, the cobalt ligaments in WC/Co fail by microvoid nucleation, growth and coalescence, leading to fracture surfaces reminiscent of ductile fracture of the bulk material. This mechanism involves smaller failure separation distances than fully ductile failure, although extensive plasticity is involved in necking down of the ligaments between the voids [1]. Similar behavior is occasionally observed in the larger niobium ligaments in the TiAI/Nb material, albeit after a significant amount of ductile necking. 4. FACTORS INFLUENCING LIGAMENT BEHAVIOR
A number of material- and processing-dependent factors which might affect the load-displacement behavior of a ductile ligament are now considered in the context of the deformation regimes delineated above. The stress-strain behavior of the ligament
Brittle-Ductile Interface I~
I
Deforming
.
Crack I,
a0
.
.
.
.
.
~t
.
~,
b) 10
0
I
t
i
i
O, 5
0.10-
u/a o
Fig. 6. Plane strain slipline solution o f R¢f. [I 7]. (a) Slipline geometry for ao/u = 1150. (b) ~ - u relation.
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS material has been taken to be that of the bulk material, with incompressible, pressure-insensitive plastic deformation assumed. It appears to be a matter of controversy as to whether the ligaments observed in the materials studied are of sufficiently small dimensiont that size effects could result in yield stresses in the ligament much higher than those of the bulk material (note that the largest values of w in Table 1 correspond to the smallest dimension ligaments). Such an argument was employed in a model for WC/Co toughening [1], in which a Hall-Petch effect was incorporated into the material model. Microstructural studies of the AI phase in the A1203/A1 material [19] suggest that dispersion strengthening might significantly raise the yield stress of the ligaments, perhaps by as much as a factor of three, although no quantitative hardness data is as yet available. The toughening effectiveness of a ductile phase will in general be enhanced by strain-hardening or an increased yield stress, unless an associated lack of ductility leads to failure at significantly decreased separation distance. Any increase in a 0 will lead to a proportionate decrease in the value of w (Table 1) and in the constraint factors emerging from the analysis of Ref. [16] (Table 2) which employed properties appropriate to pure bulk materials. It is initially assumed that the interfacial bond is strong enough that no debonding takes place. However, under circumstances where the particle remains highly constrained for a significant portion of its history, features near the crack tip may control stresses throughout the particle. Should this be the case, small scale events such as debonding of the interface near the tip affect the load history of the ligament in the early stages of deformation. In general, debonding will tend to lower constraint levels and hence lead to lower peak loads. However, the influence of debonding on the necking regime is also very strong, since the dimensions of the necking ligament are thereby determined. Debonds which are significant fractions of the ligament radius in extent are observed both in model experiments [18] and in detailed fractographic examination of the A1203/A1 material [19]. No attempt has been made in the present work to incorporate internal cavitation, either as explicit individual large voids or via a constitutive law for the ligament involving microvoidage and failure. However, the stress and strain history of the ligament provides information relevant to nucleation of internal voids. The modeling of void growth during ligament deformation is an important area of future investigation. The qualitative observation may be made that constraint conditions which raise the hydrostatic stress levels in the bridging ligament in the initial stages will increase the peak load, but also serve to encourage void growth. The relative ease of
tOf the order of 1/~m in WC/Co and AI203/AI.
(Y
~
~
3353
With Thermal Effects Effeets
u
Fig. 7. Putative effect of thermal residual stresses on a - u relation.
void nucleation in the ligament material will thus strongly influence the mode of final failure of the ligament. The elastic (or plastic) behavior of the surrounding matrix will in general affect the constraint levels of the particle. For all the cases considered here, the matrix has considerably higher elastic modulus than the particle and is effectively rigid, particularly once substantial plastic deformation of the particle occurs. Matrix elasticity may have some effect on the detailed near-tip deformation of the ductile material, but this aspect has not been thoroughly explored. Interaction between bridging ligaments is likely to be an important effect when the volume fraction of ductile phase is significant. The cylinder model employed here incorporates this interaction to lowest order by having an outer radius to the matrix jacket which reflects the volume fraction, and appropriate "unit cell" boundary conditions: straight but unconstrained boundaries. Again, with a stiff matrix, these effects are small. Composite fabrication typically involves high temperature processing, with thermal stresses arising in the reinforcing phase and matrix on cooling to ambient. In many of the systems of interest, this leads to tensile stresses in the ligaments which contribute to the load necessary to separate the crack surfaces. It has been suggested that this could lead to a a - u relation of the form sketched in Fig. 7. This effect and its coupling with nonlinear response of a bridging phase has been studied in the context of fiberreinforced brittle materials [21]. One of the aims of the present study is to determine the extent to which thermal residual strains contribute for the whole range of ligament stretch. 5. NUMERICAL MODELING The numerical modeling of this problem was implemented using the finite element package ABAQUS [22] on a CONVEX minisupercomputer. The computations employed finite strain plasticity in order to account for geometry changes. The upper boundary of the matrix material was extended gradually (per-
3354
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
5.1. Constrained regime
,,-
_~,, Ar
I.
.I
ao
Fig. 8. Geometry for initial deformation near crack tip in ductile phase.
haps after a thermal step involving compression across the crack face) in order to generate data cr - u curves for the ligament. Separation u is measured as the axial displacement of the matrix point at the tip of the crack, while tr is computed from the total force exerted on the upper boundary, normalized by the original ligament area. The aim at the outset was to describe the initial stages and also the necking regime as far as possible. However, serious difficulty was encountered in extending the cr - u curves to large u. Indeed, the results never extend into a regime of clearly decreasing total load on the ligament. The breakdown is attributed to the intense and non-uniform deformation in the tip-interface region. The reason for the intense deformation may be seen from the fact that the large-scale deformation of the ligament material is essentially volume-conserving. The behavior is exemplified by considering the volume contained in that part of the stretched ligament which extends below its original midplane (Fig. 8). This volume must be accounted for by the "void space" contained in the blunting cracktip. If a small amount of necking Ar has occurred, but large enough that elastic volume changes are not important, the volume to be accounted for is of order na2u. The corresponding volume of the toroidal "void" is of order naoArAh. Volume conservation then requires that
The majority of the results were obtained for material parameters characteristic of the (bulk) properties of A1203 and AI: the matrix Young's modulus is six times larger than that of the inclusion; the inclusion initial yield stress is ~-~ l of its Young's modulus and the material strain-hardens with an exponent n = 0.2 until the flow stress saturates at three times initial yield. A typical tr - u curve for these material parameters is shown in Fig. 9. The small-scale yielding regime exists only for nominal stress levels below tr ~ a0, corresponding to openings u which are extremely small. The particle is fully plastic at tr ~ 4tr0. The constrained fully plastic regime appears to produce a peak stress level tr ~ 6a0, by which time a macroscopic softening due to the onset of necking has begun, even though all of the material in the particle is still slowly strain-hardening. Figure 9 also includes results in which no strain-hardening is incorporated. The constraint magnification factor is seen to be similar in either case. A typical failure stretch, whether observed experimentally or predicted on the basis of necking calculations, corresponds to (u/ao) on the order of 1 when failure occurs by necking to a point, so that the results displayed here cover only a tiny range of the expected total load-displacement relation for the ligament. The solid curve in Fig. 9 was generated using an initial tip raidus r0 such that (ro/ao) had a value of 0.05. Even when this rather large initial radius is used, the computations fail at a small stretch. The results suggest that necking becomes important rather rapidly, although additional results obtained using smaller initial radii (dashed curves in Fig. 9) indicate that the characteristics described above may be sensitive to the choice of initial tip radius. Finally, a plane strain computation (Fig. 10) analogous to Fig. 9 reveals that the initial behavior is very similar when an appropriate nondimensionalization is used.
5 w
Ah u
--
a0 Ar
~--.
(5)
That is, a small crack-tip "void" is driven in a direction normal to the crack plane more rapidly than the opening rate of the tip itself. Without debonding or sliding, the constraint of the stiff surrounding matrix produces intense shear deformation. When no debonding occurs, this pattern of deformation leads to a "hollowing out" of the ductile phase near the matrix crack tip. Such behavior is observed experimentally. To produce results relevant to larger stretches, a finite radius blunt tip was included from the initial stages, and results for different radii compared for consistency.
5-
= / / ~
4
2 I
/
/~
1
ro/ao=0.05
- - - ro/ao=o
n=o
F
0
0.0
I
0.005
I
0.010 u/ao
I
0.015
I
0.020
Fig. 9. I n i t i a l s t a g e finite e l e m e n t r e s u l t s for a - u r e l a t i o n
for different choices of work-hardening exponent n and of initial root radius r0 (bonded interface).
MATAGA:
DUCTILE REINFORCEMENTS
IN BRITTLE MATERIALS
3355
6
L
2
1
/~/ ¢P
0 0,0
- - --Plane Strain
I 0.005
I 0,010
I 0.015
I 0.020
i
~'ISN-"x~
"o/ao = 0.S
.... I 0.5
0.0
....... I 1.0
----
I 1.5
I 2.0
u/a 0
u/a o Fig. 10. Finite element plane strain vs axisymmetric comparison for initial stages (bonded interface).
Constrained Ends Unconstrained Ends
---
Fig. 12. Constrained necking bar finite element results for various "debond" lengths h0. Also shown for comparison are corresponding uniform deformation (unconstrained ends) results. Material parameters are for a lead ligament.
5.2. Necking regime Since the numerical approach described in the previous section was not successful in providing results extending into a clearly necking regime, the behavior of constrained necking plastic bars was next studied. The axial deformation of plastic bars was modeled, with the ends of the bars constrained from shrinking radially (Fig. 11). This is intended to model a ligament for which the constraint of the surrounding stiff matrix has been partially relaxed by debonding/hollowing of the ligament near the crack tip. A finite element model was employed to discover how the initial bar height affects the subsequent load~lisplacement history. The stretch u is now measured as the extension of the bar. Such a study is clearly appropriate for the case when debonding occurs in the very early stages of deformation. A set of results (Fig. 12) with the material properties appropriate to the A1203/AI material for initial heights ranging from (ho/ao)=0.1 to (ho/ao)= 0.5 indicate that the necking process is strongly accelerated by the constraint when the bar is short, so that the load decreases more rapidly with increasing ligament stretch than for the long bar. Some results for analogous plane strain cases (Fig. 13) reveal that additional constraint is thereby imposed.
6. G E O M E T R I C N E C K I N G M O D E L S
Approximate models of the necking process in constrained ligaments may be constructed along the lines of a model presented in [2]. These models may be described as "geometric" necking models in that each assumes some form for the ligament shape during deformation, parameterized in a convenient way. The volume-conserving nature of the large-scale plastic deformation is employed to relate various features of the shape. In this way, in terms of the chosen parameter, the separation u is calculated and the nominal axial stress ~ is estimated, thereby generating a ~r - u curve for the ligament. For example, the geometry sketched in Fig. 11 can be used to model the axisymmetric necking bars. If the bar is assumed to be a paraboloid of rotation, volume constancy leads to the relation h a0
h0 l a01 _ ]4p + ~p2
5 \ \\
C~
al
Axisymmetric Plane Strain
4
I
I
2h
2a o
----
(6a)
2h = 2ho+ u
L
2ao bl
Fig. 11. Constrained necking bar geometry designed to simulate the large deformation of a ligament with debonding.
0 0.0
I 0.5
( 1.0
I 1.5
I 2.0
u/a 0 Fig. 13. Plane strain vs axisymmetric c o m p a r i s o n for constrained necking bars for various " d e b o n d " lengths h 0.
3356
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
8
(a)
7~
a 0
(7 0
where
6
o
o" = (1 - p)2 aB
ho/a 0 = 0.1
FiniteElement - - - Model of Eqn.(6) .... - Model of Ref [18]
~ "~1~,
5
~4 U
h2 aB ,,o =
1+
:(1---:)a
xIn[l
3
- J L\ L(l-o) lhh J/
(7)
2
and where the uniaxial (true) stress-strain behavior is described by
1
o 0.0
0,5
1.0
1.5
2.0
au.ia~al F(E).
il
(b)
O"o
h0/a0 = 0.25 Finite Element Model of Eqn. (6) . . . . . Model of Ref [18] ---
.~
o
i
0.0
0.5
1.0
1,5
2.0
u/a 0
8
•~ 4 io
(C)
h0/a0 : 0.5
i
(8)
=
u/a0
i'L'/i
- - - Model of Eqn.(6) .... - Modelof Ref[18]
3 2
.......................
Using material parameters as directly measured in the glass/Pb experiments [18] (power-law hardening with n = 0.25), this model is compared with finite element results in Fig. 14. Also shown are the predictions from a geometric model employed in Ref. [18], where the assumed shape of the neck is cylindrical and a virtual work balance is used to determine the stress supported. There are qualitative similarities in overall behavior, but a significant difference in the final necking behavior. The geometric model introduced here agrees reasonably well with the finite element results. Geometric models have also been employed to obtain results for the full range of ligament deformation when debonding does not occur, but a necking ligament is formed by hollowing out of the material near the crack tip. However, it would appear that the results are rather sensitive to the choice of geometry. This is demonstrated in Fig. 16, where three models are compared: (a) a model described in Ref. [2It which employs the geometry of Fig. 15(a) for the whole of the range of deformation, (b) the model of Ref. [2] with fewer approximations in the geometrical description, (c) a model similar to that of Ref. [2] but
1 o 0.o
I 0.5
I 1.o
I 1.5
I 2.0
u/ao
(a)
I I I I I I I
Fig. 14. Results of finite element and several necking models for necking ligament regime, for various debond lengths. (a) (ho/ao)=0.1. (b) (ho/ao)=0.25. (c) (ho/ao)=0.5. where the necking parameter is
.
.
.
.
Ligament
Matrix
L i g ~
Matrix
.
a
p = 1 --
(6b) a0
The nominal stress may be estimated from the Bridgman result for a necking bar [23] (using the local radius of curvature at the ligament mid-plane) as
fNote that there is an error in equation (I0) in Ref. [2] which should read 2u X 2 ...~. _ _ _ g a o"
c._ (b)
"~
I u_2 Fig. 15. Two plausible geometric models for full range of ligament deformation. (a) From Ref. [2]. (b) Alternative geometry.
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
ii 7 i
~
0
0.0
by equations (6) and (7) for various bar lengths. The material parameters are again appropriate for the glass/Pb model experiments. Interpreting these as w values when failure suddenly occurs at stretch uf due to some mechanism not incorporated in the fully ductile model, it would appear that increased debonding increases w if the failure stretch uf is greater than 0.5a0. This suggests that for sufficiently ductile failure debonding leads to larger values of w. The large debond, large failure stretch case yields a value for w of 3 as a conservative prediction.
Model of Fig. 15(a)as in Ref [2]
. . . . . Model of Fig. 15(a) reeomputed Model of Fig. 15(b) - - - Model of Ref [18]
i •
-
0.5
-
. . . . . . . . . i- . . . . . . . . . l 1.0 1.5 2.0 u/a o
with a different, albeit plausible, choice of necking geometry [Fig. 15(b)] and (d) a model described in Ref. [18] which employed a dislocation pileup calculation to give an initial neck height. Material parameters appropriate to glass/Pb are used. Again, the predictions are widely different. For a real ligament, some failure mechanism may truncate the ¢ r - u relation. The contribution to w from the necking regime is in general more significant than that from the initial highly constrained regime, due to the fact that the ligament stretches which occur in the latter regime are tiny compared to the necking stretches. Ignoring the amount by which the necking calculations underestimate that contribution, Fig. 17 shows cumulative w values, defined as
wf(Uf~.~.fo'f/a°G.)~-d(U a0
7. THERMAL STRESSES
7.1. Constrained regime
Fig. 16. Predictions of various necking models for full range of ligament deformation.
\a0/
3357
a0
(9)
obtained from the geometric necking model defined
tPerhaps even compared to material ultimate strengths!
Finite element solutions for the initial behavior of a bridging particle incorporating thermal straining have been obtained. The material properties are appropriate to a spherical AI particle in A1203 matrix, here subjected to a 500°C cooling. It is assumed that the matrix is very easily cracked. Separation of the crack faces both at the crack tip and at the remote boundary of the unit cell are plotted in Fig. 18. At the point at which the crack tip begins to open (the residual compression across the faces having been relieved as the load increases), the average traction across the particle is in fact slightly larger than the residual hydrostatic stress in the particle in the uncracked composite, erR, given by the elastic result [24] EpACtAT
OR =
(10)
1 - 2Vp + (1 + Vm)~ m where subscript p refers to particle elastic properties and subscript m to those of the matrix, and A~AT is the thermal strain mismatch. The small increase may be predicted on the basis of a purely elastic analysis of the load at which the tip begins to open [25]. Since the ratio (aR/tro) for this combination of materials and this geometry is relatively large, this represents a high initial stress supported by the ligament, even compared to constrained deformation.t Moreover,
20-
Tip
ho/a o = o . s / /
15 ~ ' ~
o
1 f
0.1
lO;, l
/
/
/
/
--AT=0 ---AT=500K
/ /~'~'Remote
5
0L~P" 0.0
I 0.5
I 1.0
I 1.5
I 2.0
uda0
Fig. 17. Cumulative contribution to w, wr, as a function of "sudden failure stretch" uf as predicted by necking model of equation (6) with various amounts of debonding.
0.0
0.005
I 0.010
I 0.015
I 0.020
u/a 0
Fig. 18. Initial stage finite element results showing effect of thermal residual stresses (no debonding).
3358
MATAGA: DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
6\ 41~ / a 5
\
21
---
,~
o o.o
AT=0 A T = 500K
o=O.S I
I
0.5
1 .o
u/a o
Fig. 19. Constrained necking bar finite element results showing effect of thermal residual stresses. the load curve has a subsequent slope which is of the same order of magnitude as when thermal strains are absent.
7.2. Necking regime Necking calculations were carried out in a similar fashion to those described previously, with the addition of thermal strains. The same material and temperature parameters were chosen as above. Some sample results are shown in Fig. 19, where c r - u records for strain-hardening bars of three different lengths are displayed, with and without thermal strains. The results indicate that the constrained thermal stresses which are evident in the initial stages of deformation do not remain in the necking regime. Indeed, the net effect of the thermal strains appears to be to reduce the load supported by the ligament, presumably because the shrinkage encourages necking. 8. DISCUSSION Comparison of the various models described above to the experiments of Ref. [18] shows qualitative agreement, but convincing quantitative predictions are yet to be obtained. For the initial, constrained stages of deformation, the numerical results indicate constraint factors in the requisite range. A priority of future work will be to extend the constrained results into a clearly necking regime. For the case of non-debonding ductile failure of the ligament, a previously used geometric model [2] apparently gave good agreement with the experiments for the necking regime. However, it is shown here that plausible alternative geometrical assumptions produce quantitatively different results. This strongly suggests that detailed modeling of the deformation, with the interaction between the material properties and the geometry evolution incorporated, is necessary if accurate predictions are to be obtained. For debonded ligaments, the correlation between
increased debond length and increased w is predicted by the models. The experimental behavior of the debonded ligaments is, however, somewhat different in character to the necking bar models in that the stress remains elevated for significant stretches and then falls rather abruptly [18]. The measured w values with debonding are approached only by the model described in Ref. [18]. Constrained bar calculations somewhat underestimate the failure stretch due to the accelerated necking which they entail. The toughness enhancement due to bridging is dominated by the contribution from the necking regime, since the high stresses in the constrained plasticity regime persist only briefly. However, this conclusion does not exclude the early stages of deformation as irrelevant, since the initial stresses and near-tip deformation may control whether debonding or internal cavitation occurs. Thus thermal mismatches, for example, although not appearing to be responsible for a significant direct contribution to toughening by ductile ligaments, may have significant indirect influence via the resultant stresses in the early stages of deformation. Clearly the significance of such effects will depend strongly on the material and interfacial properties. In addition, the evolution of the necking ligament geometry may depend on the details of the initial deformation even when no debonding takes place. It is thus important to model the whole range of ligament deformation accurately. The inclusion of phenomena such as debonding and internal cavitation would be desirable, but may be difficult to achieve in a convincing way. From the materials design viewpoint, perhaps the most interesting aspect of the results is the suggestion that the ductile phase need not be completely bonded to the matrix during deformation to provide toughening. Limited debonding may well be beneficial in terms of toughness enhancement. From this point of view, there are clear advantages associated with a reinforcing phase which is a network rather than a distribution of equiaxed particles, since complete interface failure and particle pullout is avoided. In terms of the underlying aim of the investigation, namely to assess the extent to which bridging can explain the observed toughnesses of brittle materials reinforced with ductile phases, the results are not completely conclusive. The model glass/A1 material and the TiAI/Nb materials fall within the range of the model predictions; on the other hand, the w values predicted by the models and the model experiments require large-scale debonding to explain the toughness enhancement in the WC/Co and A1203/A1 materials. Preliminary investigations [19] suggest that observed debonding and significant dispersion hardening of the aluminum may be sufficient to explain the latter case, but the WC/Co material remains an anomaly in terms of the ideas presented here, unless the properties of the ligament significantly depart from those of bulk cobalt. These aspects are subject to further experimental study.
MATAGA:
DUCTILE REINFORCEMENTS IN BRITTLE MATERIALS
Acknowledgements--The author was fortunate enough to have been able to discuss various aspects of this work with A. G. Evans, R. M. McMeeking, L. S. Sigl, M. F. Ashby, B. Budiansky, and F. W. Zok. Financial support by the Defense Advanced Research Projects Agency under O.N.R. contract N00014-86-K-0753 is gratefully acknowledged. REFERENCES
1. L. S. Sigl and H. E. Exner, Metall. Trans. A 18A, 1299 (1987). 2. L. S. Sigl, P. A. Mataga, B. J. Dalgleish, R. M. McMeeking and A. G. Evans, Acta metall. 36, 945 (1988). 3. B. J. Dalgleish and A. G. Evans. To appear. 4. C. K. Elliot, G. R. Odette, G. E. Lucas and J. W. Sheckherd, Proc. M.R.S. Meeting, Reno (1988). 5. V. D. Krstic, Phil. Mag. 48, 695 (1983). 6. P. Hing and G. W. Groves, J. Mater. Sci. 1,427 (1972). 7. C. B. Bucknall, Toughened Plastics. Applied Science, New York (1977). 8. A. G. Evans, Z. B. Ahmad, D. G. Gilbert and P. W. R. Beaumont, Acta metall. 34, 79 (1986). 9. S. Kunz-Douglas, P. W. R. Beaumont and M. F. Ashby, J. Mater. Sci. 15, 1109 (1986). 10. Z. B. Ahmad, M. F. Ashby and P. W. R. Beaumont, Scripta metall. 20, 843 (1986).
A.M 3 7 / 1 ~ P
3359
II. R. G. Hoagland, A. P. Rosenfield and G. T. Hahn, Metall. Trans. 3, 123 (1972). 12. W. W. Gerberich and E. Kurman, Scripta metall. 19, 295 (1985). 13. G. A. Cooper and A. Kelly, J. Mech. Phys. Solids 15, 279 (1967). 14. W. W. Gerberich, J. Mech. Phys. Solids 19, 71 (1971). 15. L. R. F. Rose, J. Mech. Phys. Solids 35, 383 (1987). 16. B. Budiansky, J. C. Amazigo and A. G. Evans, J. Mech. Phys. Solids 36, 167 (1988). 17. A. G. Evans and R. M. McMeeking, Acta metall. 34, 2435 (1986). 18. M. F. Ashby, F. J. Blunt and M. Bannister, Cambridge Univ. Engng Dept. Tech. Rep. 19. B. D. Flinn, M. R/ihle and A. G. Evans. To be published. 20. M. J. Doyle and I. J. Gardner, Proc. 38th Annual Conf. Soc. Plastics Industry (1983). 21. D. B. Marshall and A. G. Evans, Mater. Forum 11,304 (1988). 22. Hibbitt, Karlsson and Sorensen, Inc., Providence, R.I. 23. P. W. Bridgman, Studies of Large Plastic Flow and Fracture. Harvard Univ. Press (1964). 24. S. P. Timoshenko and J. N. Goodier, Theorv of Elasticity. McGraw-Hill, New York (1959). 25. P. A. Mataga, unpublished work.