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Role of damage on the flow and fracture of particulate reinforced alloys and metal matrix composites
Acta muter. Vol. 45, No. 12, pp. 5261-5274,
Pergamon
Published
PII: S1359-6454(97)00147-X
1997 #C 1997 Acta Metallurgica Inc. by Elsevier Science Ltd. All rights reserved Printed in Great Britain 13%6454/97 $17.00 + 0.00
ROLE OF DAMAGE ON THE FLOW AND FRACTURE OF PARTICULATE REINFORCED ALLOYS AND METAL MATRIX COMPOSITES E. MAIRE’,‘t,
D. S. WILKINSON,’ J. D. EMBURY’ and R. FOUGERES
‘Department of Materials Science and Engineering, McMaster University, Hamilton, ON L8S4L7, Canada and ‘GEMPPM UMR CNRS no 5510, Insa de Lyon, 20 Av. A. Einstein, 69621 Villeurbanne Cedex. France (Received 10 February 1997; accepted 8 April 1997) Abstract-A large number of models exist to predict the tensile behaviour of two-phase materials. They fail, however, to predict the behaviour at high strain because they do not properly account for the role which damage accumulation plays during plastic flow. A new model is proposed here which incorporates damage. It uses an incremental self-consistent method developed previously. In the new application, the model treats damaged and undamaged regions as distinct continuous “phases” and allows the relative volume fraction of each phase to vary during deformation. The stress redistribution due to damage during the evolution is carefully analysed. The model considers the influence of particle size Fia a Weibull analysis and incorporates the results of existing FEM calculations. The model agrees well with existing experimental data and provides a new method to evaluate the role of microstructural parameters. 0 1997 Acta Metallurgica Inc. R&urn&Un grand nombre de modkles existent pour prtdire le comportement en traction de mattriaux biphasks. Ces modiles sont en disaccord avec le comportement exptrimental aux grandes valeurs de la d&formation plastique parce qu’ils ne prennent pas en compte l’effet de l’endommagement sur I’koulement plastique. Un nouveau modkle incluant l’endommagement est propok. I1 traite les zones endommagkes et non endommagies comme deux “phases” distinctes et continues. La fraction volumique de chacune des phases peut varier pendant la dtformation. La redistribution des contraintes au tours de l’endommagement est ktudite en d&ail. Le modkle prend en compte I’effet de la taille des particules grke B I’analyse de Weibull et il incorpore des rtsultats existants de calculs par la mkthode des kltments finis. Le modtle est en bon accord avec des donnies expkimentales existantes et constitue une nouvelle mkthode pour &valuer le r81e de certains paramttres microstructuraux.
1. INTRODUCTION
This paper addresses the deformation of metallic alloys containing second phase particles which are heterogeneous as regards their thermomechanical properties. All the materials of this class exhibit common trends (e.g. as far as damage initiation and propagation mechanisms are concerned) and much attention, both experimental and theoretical, has been paid to their behaviour. These studies are motivated by industrial interest in some materials belonging to this class, such as steels containing inclusions, aluminium casting alloys and, more recently, metal matrix composites. The elastoplastic behaviour of these materials can be predicted by a range of numerical [l, 21 and analytical [3%7] models, founded in the latter case on the Eshelby analysis [8]. In general, damage occurs during deformation because of particle cracking or particle/ matrix interface decohesion. The microcracks thus formed reduce the ductility of the material. Studies
TTo whom all correspondence
should
be addressed. 5261
have been attempted to account for non trivial problems such as damage appearance in the material [7,9-141. Usually damage is incorporated using a Kachanov approach involving a damage parameter [9, lo], or by simulating the effect of the microcracks represented as penny shape voids [7,1 I]. A third investigated way consists in calculating the local behaviour of damaged and undamaged cells and then associating these cells according to a classical rule of mixture assuming that the strain is equal in the two regions [12-141. In the present paper, we present a new model based on this third kind of approach to account for the role of damage in the calculation of the mechanical behaviour of two-phase materials. The treatment of the global combination of the damaged and undamaged constituents is improved by the use of an existing selfconsistent analysis previously developed for (undamaged) flow in a heterogeneous material [3]. Damage is incorporated using the Weibull approach to particle fracture. The fraction of damaged particles is allowed to increase with the stress in the undamaged regions. The model also contains a cal-
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culation for stress redistribution due to damage. After presentation of the model, comparisons with experimental data are shown. Finally the effect of some of the microstructural parameters used in the calculation is predicted.
2. MODEL DESCRIPTION The analysis we have developed ling the behaviour of the material scales:
involves modelat two separate
?? at
a small scale we calculate the local behaviour of an elementary unit cell represented by a single particle surrounded by a shell of matrix the thickness of which depends on the overall volume fraction of reinforcement. The particle can be intact or broken. These two extreme cases will be hereafter referred as undamaged (UD) and fully damaged (FD), respectively. In principle either analytical or numerical approaches can be used to treat the problem. However, the case of broken particles is difficult to treat analytically, and we use existing FEM results to assess the local behaviour. ?? at a larger scale, the material is supposed to consist of a random mixture of unit cells, each one exhibiting one of the two above mentioned behaviours (i.e. damaged or undamaged). In order to determine the macroscopic tensile behaviour, we use an incremental self-consistent model. In the following development we first describe the behaviour of an assemblage of unit cells at large scale. We then proceed to develop the fine scale constitutive laws that are used in the analysis. 2.1. Calculation homogenisation)
of the global behaviour
only, the results of this model have been found to agree favourably with the results of FEM calculations for homogeneous two-phase materials [3]. The morphological hypothesis underlying this approach assumes an interpenetrating network of the two phases in question. This permits one to treat the case of high volume fraction of reinforcement and should be distinguished from the “generself-consistent approach proposed by alized” Christensen and Lo which assumes a matrix/inclusion morphology. It has been shown in [3] that if the Poisson’s ratio of the two constituents are taken equal to 0.5, the modulus of the composite EC can be calculated from the respective moduli El, E2 and volume fractions vI, v2 of the two constituents by solving the quadratic equation: 3E:, + [(2 - Sv,)El + (2 - 5v2)E2]Ec - 2E, E2 = 0 (2) which is also equivalent to the equation given Hill for the Young’s modulus of an isotropic persion of spheres [15]. The stresses in the two phases o1 and o2 are lated to the overall applied stress 0 (then to and to the elastic part of the total deformation by expressions of the type:
by disreEC) .a,~
5Ei
5Ei
Ecu
ai=2Ei+3Eca=2Ei+3Ec
(3)
If the stress differentials are now calculated the full stress-strain curve can be calculated incrementally through a piece-wise linear approximation. In this case the moduli of interest are the tangent moduli (i.e. the instantaneous work hardening rates) and the strain in equation (3) is now the total strain. Thus
(large scale dai =
2.1.1. Existing incremental self-consistent analysis. A previously developed self-consistent analysis [3] enables a calculation of the 0-c curve of a mixture of two phases provided that the properties of these phases are known separately. It considers two constituents, each exhibiting elastoplastic behaviour that can be approximated by the Ramberg-Osgood expression:
k (1) The yield stress oo, the Young’s modulus EO and the hardening coefficient N are constants for a given phase. In the case of brittle elastic particles such as Sic, Si or Al&u, the only parameter which is necessary to describe the behaviour is the elastic modulus Eo. The method proposed is based on the using an self-consistent analysis “classical” approach similar to that of Hill for the mixing of spheres [15]. Despite the fact that this approach utilises the normal component of the stress tensor
SE,’ 2ET + 3.67
dc =
5ET 2ET f 3q
EcTdE
(4)
where the superscript T indicates that the current tangent modulus is used. 2.1.2. Varying fraction of the two constituents. We now wish to adapt this model to treat damage accumulation in the form of progressive fracture of particles as illustrated in Fig. 1. In this context, the above described self-consistent incremental analysis can still be used with subscript i referring to damaged or undamaged regions of the material. However, to represent the experimental features, it must be reformulated to enable the relative volume fraction of the two constituents (i.e. undamaged and damaged cells) to vary with strain. This is complicated by the fact that deformation and damage occur simultaneously and interdependently. If the strain increments are kept small this problem is alleviated, since each increment can be divided into two sequential steps-deformation at constant damage followed by damage at constant strain. In the first, a strain increment de is applied and the
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I
SMALL SCALE HOMOGENISATION
T
0
= Undamaged
0
cells :
= Fully damaged cells :
Fig. 1. A schematic diagram illustrating how the actual material is idealized in the present model. The spatial distribution is uniform and the material is composed of cells containing either an intact (undamaged cells) or a broken particle (fully damaged cells).
stress increments for each constituent are calculated (according to the tenets of the self-consistent model). Following each strain increment, during which the stress in the undamaged region has increased, a certain fraction of material dvFD transforms from undamaged to damaged. dvFD is always positive as the level of damage increases monotonically with stress. It is also irreversible. Since
(2) The stresses in the two constituents must be updated to satisfy the constant overall strain condition. By differentiating equation (3) one can calculate the stress redistribution due to damage in the constituent i dgi,dam:
VFDf VUD= 1
dbUD,dam=%JD&~ - 3gUD dEc
(5)
dVFD
dVFD
thus dvuD -=-ds’ de
d”FD
(6)
Each increment of damage is accompanied by a redistribution of stress, which is assumed to occur instantaneously (i.e. at constant total strain). The transformation of a fraction of material from undamaged to damaged is accompanied by two modifications: (1) The modulus of the composite varies. By differentiating equation (2) one finds that: d& -= hw
-=
~(EFD - hm)& 6-k + (5rFD - 3)&D + (2 - SVFD)EFD
From this one can also calculate increment:
the overall
(7)
stress
Note that the second term is equal to zero since a change in elastic strain occurs only due to a change in elastic modulus.
=FD&I - 3gFD d& ~EFD + 3Ec dVFD
2-6j~ + 3Ec
dVFD
(9)
(10)
In each constituent, these variations are governed by Young’s modulus if they correspond to a decrease (elastic unloading), or by the tangent modulus if they correspond to an increase of the stress (elastoplastic loading). One can show that at constant Strain, both dbFD&,, and dbUD,dam are negative as the modulus EFD is smaller than the modulus EUD. In the incremental procedure described here each increment da of the overall strain of the composite is accompanied by an incremental loading of the two phases. As the two constituents unload due to damage, we must account for the fact that they first have to reload elastically up to the flow stress reached in the preceding step before plastic flow can recommence. The composite behaviour of each constituent is thus first governed by Young’s modulus, Once the yield points have been re-reached the flow behaviour of each constituent is assumed to be governed by its tangent modulus until the end of the step. From a numerical point of view the reloading after a damage event is done using small
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strain increments (namely d&/2000) which cient to fully describe this complex process. 2.2. Calculation homogenisation)
of the local behaviour
FRACTURE
is suffi-
(small scale
f=
vP,k (11) VP.k
+
VM,k
where f is the overall volume fraction of second phase particles. Once damage commences we suppose that the material consists of a mixture of two kinds of cell each with a particle (volume VP,,) embedded in a matrix region (volume VM,J. These are: particle ?? damaged
At a given strain E, the stress in the fully damaged cells is related to the stress in the unreinforced material CUR by the expression: gFD(&)
We assume for the sake of simplicity that the spatial distribution of the particles is uniform, i.e. a volume VP,k is sureach particle k occupying rounded by a volume of matrix V,,, satisfying:
?? undamaged
OF COMPOSITES
cells,
each
containing
cells, each containing
an unbroken
a broken
particle.
The properties of the undamaged cell can be calculated by any of a large number of models previously developed. According to the definitions given above, the fraction of damaged cells VFD is equal to the fraction of broken particles. The properties of the damaged cell are related to those of the unreinforced matrix using results developed by FEM calculations 112-141. This marriage of FEM and self-consistent models follows naturally in that both are essentially single cell calculations. The boundary conditions in one case are governed by periodicity and in the other by embedding the material in an equivalent continuum phase. Nonetheless the results of specific FEM calculations (such as the stress ratio between an undamaged and a damaged cell under the same conditions) can be used as inputs into self-consistent models. We consider this one of the strengths of the current approach. To calculate the behaviour of the undamaged cells we use the classical self-consistent method for a mixture with a volume fraction f of elastic particles in an elastoplastic matrix [3, 151. This approach may not be the most appropriate for the considered system, i.e. an unbroken particle surrounded by a matrix. However, it has been shown to give a realistic approximation of the tensile behaviour of metal matrix composites in the absence of damage, and also to be reasonably close to the equivalent FEM prediction [3]. Moreover, it will be seen later that this approach leads to an excellent fit with the initial loading behaviour (i.e. in the absence of damage) of the experimental tensile curves of the materials considered in this study. Finally it is extremely easy to compute. As a consequence it was chosen to predict the behaviour of the undamaged material. The resulting curve is fitted by a Ramberg-Osgood expression, the parameters of which being EUD, gUD and Noo.
=
gD(hJR(&).
(12)
Using FEM calculations, Bao [12] and more recently Brockenbrough and Zok [13] have shown that go depends on E for low strains (transient response) but tends to become constant quite rapidly when E increases to a few times the elastic strain (steady state response) especially when the ratio between particle and matrix moduli is high (as in the case of SIC particles in an aluminum matrix). In the current model we assume that gD is constant over the whole deformation range. Note that in some materials (e.g. metal matrix composites), damage initiation can occur by two coexisting mechanisms: particle cracking and particle/matrix interface decohesion [14]. The cells damaged according to these two kinds of mechanisms may have a different behaviour (i.e. a different value of go) but unfortunately no simple expressions of go have yet been proposed for the cells damaged by decohesion. Besides, our observations in [16] show that the proportion of particle decohesion is quantitatively small. As a consequence, we neglect the difference between these two mechanisms in what follows. As the variations of go with E can be neglected, the behaviour of the fully damaged cells is described by a modified RambergOsgood expression:
(13) In the actual partially damaged material, the damaged cells are located at random in the whole volume, such that the damaged and undamaged regions form an interpenetrating network. This is exactly the kind of morphological hypothesis that lies behind the self-consistent analysis and especially behind the classical model adopted here. Thus, we use this model to mix both damaged and undamaged cells. As a consequence the incompressibility assumption stated in the original model is applied to the undamaged phase, which is certainly not valid. Fortunately, it has been verified through preliminary calculations that the value of the Poisson’s ratio of each phase has only a weak effect on the predicted behaviour in the plastic regime. Clearly the approach to damage just outlined includes only damage initiation (i.e. particle cracking) and neglects growth and coalescence of damage elements. We justify this by noting that particle cracking is the only damage observed experimentally in the materials of interest until rather large strains, close to final failure [17]. In addition, the initial stages of damage growth involve the increase of the crack width during deformation, which is captured in the FEM calculations for the fully
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0.035 --
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-W
0.03 --
-
Vfd(a); op= 1100 MPa
-
Vfd(a); oP=
900 MPa
0.025 -2 5 5
0.02 --
2 IL 0.015 --
0.00
20.00
40.00
80.00
60.00
100.00
Particle size (pm) Fig. 2. f(a) and v&a) for two values of the stress in the particles, respectively 900 and 1100 MPa. The other parameters used arem = 3.8, UO= 1700 MPa, cru=580 MPa, a~= 11.5 pm and p = 0.51.
damaged discussed
material. The effect of crack coalescence in more detail below.
is W(ap,a) = 1 -,.exp[-(E)3(F)‘“]
2.3. Calculation of the fraction of damaged material We now have to calculate the evolution of the fraction of damaged material taking into account both mechanical and microstructural considerations. The experimentally determined size distribution 4(a) of the second phase particles is usually sufficiently well represented by a log-normal distribution given by:
where cru is the threshold stress below which no rupture is observed and 00 the representative stress, while m is the Weibull modulus. The parameter al is an arbitrarily chosen representative dimension. The fraction of damaged material for a given value of op can finally be calculated as: vFD(gP)
= I
da) = &exp[-i
(?)‘I
(14)
(18)
Urn/ (a)W(ap, a)da.
(19)
Note finally that --oo
where a0 is the mean size and p is representative the width of the distribution. Note that:
of
+m cp(a)da = 1. J’0
(15)
This size distribution must be transformed into a “volume fraction” distribution j(a), the fraction of material associated with the small particles being smaller than for the big particles: cp(a)a3
f(a)=
s,‘” cp(a)a3da ’
(16)
Here also +W .I’ 0 f(a)da At a given particle particles of size expression:
= l.
(17)
stress op, the fraction of broken a is given by the Weibull
vFD(bp)dgp = 1. J’0
(20)
The stress in the undamaged constituent oUD, and the stress acting on the particles of this constituent op are related by an expression of the form: nP(&)
=
gP(&hJD(&)
(21)
where gp can be easily calculated from the initial model [3] in the undamaged case. As with gD (see equation (12), gp exhibits a transient followed by a steady state [3, 12, 131. The variations of gp during the test have been shown by these authors to have a small amplitude. In the present case, we neglect these variations, and use for gp the average value of this parameter over the range of deformation considered. At a given strain level, we then calculate the average stress in the particles of the undamaged phase, from a knowledge of the average stress in this phase. In this Eshelby type calculation, this stress does not depend on the size of the particles.
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We assume also that at a given overall strain, the stress in each particle is the same (which also implies that they have the same aspect ratio). An illustrative plot of VFD vs the size of the particles is given in Fig. 2 for two values of the stress in the particles (900 and 1100 MPa, respectively). The size distribution &a) has been measured by means of a computerized surface analysis system on a 2618T4/ Sic composite (see the next section) assuming that the ratio between the dimensions measured in surface and the actual dimensions was equal to 417~as suggested in [7]. The parameter chosen to represent the size of the elongated particles was the equivalent diameter, defined as the diameter of the circle of the same surface area S as that of the particle. This distribution was found to be well described by a lognormal law with ao= 11.5 pm and p = 0.51. The choice of the Weibull parameters needed for this kind of calculation will be explained later. The total VdUe Of VFD for a given stress is equal to the area below the corresponding curves. Also, while VFD increases with the particle stress, the curves are asymmetric since the cracking of the larger particles occurs at lower stress. Note that this implies that the average size of the particles in the damaged and undamaged phases of the materials varies during the test although this is not taken into account in the present model. To do so it would be necessary to know whether gD depends on the size of the particle.
3. RESULTS
3.1. Comparison
with experiments
We first develop the model in the context of a specific set of experimental results. The chosen data is concerned with MMCs. The material is a 2618T4 Al alloy reinforced with 10 and 18 vol.% of SIC particles. Experimental results are reported elsewhere [ 171. The modulus of the SIC has been measured by means of nanoindentation experiments to be about 400 GPa [16]. The tensile behaviour of the unreinforced matrix under the same heat treatment condition was measured simultaneously and is used as input in the model. The Ramberg-Osgood coefficients of the unreinforced material were found to be: ao=200 MPa, E = 74.4 GPa, and N = 0.16. However, it has been shown by several authors [17201, that the mechanical behaviour of the matrix in the composite may be different than in the corresponding unreinforced alloy. The microstructure and dislocation density are modified during the heat treatment and the mechanical loading because of the presence of the particles which act as heterogeneities as regards their thermomechanical properties. The main microstructural modification induced tFor a more complete discussion of this approach Ref. [23].
see
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by the presence of the particles is the increase of the dislocation density in the matrix, especially in the T4 temper. The particles also affect the work hardening of the surrounding matrix during the deformation and this matrix/inclusion morphology is not fully accounted for in the classical self-consistent scheme proposed by Hill. In order to model the effect of particles on strength over a wide range of particle size, both dislocation and continuum effects need to be included. Thus, the dislocation terms can be used to modify the yield stress of the matrix prior to the development of a continuum level selfconsistent model. In the present comparison we account for these effects in an approximate wayt using the work of Nan and Clarke [11], i.e. we incorporate in the Ramberg-Osgood expression an additional contribution to the yield stress of the matrix in the composite aM,C such that: aM,C = OUR +h~R,p.
(22)
Expressions related to A0UR.P have been summarized by Nan and Clarke. The main contribution to AaUR,p in our case is associated with the dislocations generated during the quench of the material. The microstructural parameters required to implement this calculation are: the average diameter of the particles d = 11 pm; the temperature drop during the quench AT = 500”; and the difference between the coefficients of thermal expansion of Al and SIC Act = 20 x 10e6 K-‘. The other parameters characterizing the aluminum matrix were chosen equal to those used by Nan and Clarke who dealt with the same kind of material. One of the main interests of this approach is to permit the description of the effect of the size of the particles in the prediction of the behaviour of the undamaged material. This is illustrated in Fig. 3 where we for the curves calculated show flow 2618T4 + 10 vol.%SiC composites with particles of different size. The strengthening increases with smaller particles (i.e. below about 10 pm), consistent with many experimental studies [21]. We will consider in the following calculation that the tensile behaviour of a material reinforced with particles having a distribution of sizes is equivalent to the behaviour of a composite reinforced with a single family of particles exhibiting the average size. This assumption implies the global compensation of the contributions of the big and of the small particles on the global behaviour. It is an approximation of the actual behaviour, though it appears to be sufficient compared to the purpose of the present paper which deals with global effects. The size distributions studied hereafter will then be considered to affect the calculation of the fraction of damaged material only. Another important effect on the strengthening could be the heterogeneity of the spatial distribution of the particles. It has been shown in [3] that
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Ol
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Ao”~,~: a = 1 pm
-
u
-
~u~+Aou~,~:a=11.5pm
-
oUR+AoUR,P:a=lOOpm
-
uLIR
uR
+
/
I
I
I
I
1
2
3
4
5
/
0
6
I
7
--I
8
Strain (%) Fig. 3. Effect of the particle size on the strengthening of an aluminium alloy (N = 0.16, E = 74.4 GPa and (TO=200 MPa) reinforced with 10 vol.% of Sic particles (E = 400 GPa) calculated with a self-consistent analysis [3] for an undamaged material, along with Nan and Clarke’s correction [ll] for dislocation effects. a clustered distribution could result in a higher strengthening. The respective contribution of the effect of the modification of the microstructure of the matrix, and of the spatial distribution heterogeneity are not perfectly clear so far, and the actual strengthening is probably a combination of these two mechanisms but for the sake of simplicity we accounted only for the first effect in the present paper. The particles are not spherical and the materials in question have been extruded before the heat treatment. The particles are then aligned along the tensile axis and exhibit an aspect ratio of about 2 [17]. To estimate gD, we used Bao’s FEM results for this aspect ratio:
model for both the 10 and 18 vol.% composites. We perform the calculation for the 18% composite using the Weibull parameters deduced as described above from the case of the 10% composite. Figure 5 represents the tensile curves calculated with the present model, and compared with the experimental curves, and with calculations performed for undamaged materials. The agreement with the experimental results will be discussed in detail later. The squares on the figure show the resulting curve that would be calculated by the simple rule of mixtures initially used in [12-141, which implies that the strain is homogeneous in the two phases: rr =
VFD~FD
+
VU@IJD
with EFD = EUD = EC. (24)
3.2. Influence of the deferent parameters g,, = 0.958 + (f - 0.202)2.
(23)
It now remains to measure the Weibull characteristics of the SIC particles. For the 10 vol.% composite the evolution of rrD with strain was measured elsewhere [17] by means of SEM observations during in situ tensile experiments. We used this evolution to choose the correct Weibull parameters. Figure 4 shows the comparison between the experimental evolution, and the evolution calculated with the model using the following set of Weibull parameters: m = 3.8, oo= 1700 MPa and ou = 580 MPa. This set of characteristics is realistic for fine particulate Sic. These parameters depend on the value of al which can be chosen arbitrarily (it is kept equal to 5 pm in the present case). Having compiled all of the necessary parameters characterizing the microstructure and the damage development, it is now possible to evaluate the
In this section the predictive capability of the model is used to illustrate the effect of microstructural parameters on the tensile behaviour of a damaging material. These illustrations have been made using material parameters appropriate to a 10% composite. 3.2.1. Particle size distribution. Figure 6 illustrates the calculated tensile curves and damage evolution for several values of the mean particle size ao. The reduction in strength with increased particle size is primarily due to damage (see Fig. 6 inset). In particular, the near zero rate of work hardening in the 100 pm particle size material follows from the rapid onset of damage. Such behaviour would result in low ductility due to global instability as predicted by the Considere criterion. The same calculations have been performed for several values of the distribution width /I. They show that for a given average
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0.6
0.5
0.4
$J 0.3 n Experimental 0.2
-
Weibull Darameter fit
0.1
0
1
2
3
4
5
6
7
8
9
Plastic strain (%) Fig. 4. Experimental evolution fo the fraction of damaged material measured for a 2618T4 + 10 vol.% BC composite by means of SEM observations during in situ tensile tests (squares). The line represents a Weibull fit calculated with the present model using m = 3.8, (~a= 1700 MPa, 0” = 580 MPa.
particle size, fl has a smaller effect than a0 with narrower distributions contributing to greater strength. 3.2.2. Weibull parameters. The effect of Weibull modulus is illustrated in Fig. 7. In this particular case, we have chosen a set of Weibull parameters permitting to illustrate in a more visual way the special effect of m. Especially we assumed that the
width of the size distribution p was equal to 0. Under these special conditions, as m increases, damage is confined in a narrower strength range, the work hardening drops and can even become negative. The effect of the increase of the Weibull modulus calculated with the realistic set of parameters used in all the other calculations presented
500
-Model
Rule of mixture
100
0
- damage
J 0
/ 2
4
I
6
8
1
I
10
12
Strain (%) Fig. 5. Comparison between the tensile curves calculated with the present model (intermediately thin curves) and the experiments (thick curves) for 2618T4 + 10 and 18 vol.% Sic composites. The behaviour of the undamaged material calculated with [3] (thinner curves) and the prediction of the mixture rule with homogeneous strain (squares) is also shown.
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T
lop
,...___._ ...“.....
2
0
__.....--1
-.-----*-
4
1
2
I
3
4
5
Strain (%) Fig. 6. Tensile curves calculated with the model for three values of the mean particle size. Evolution of vFD with strain (inset). here
is more
damaged
simply
particles.
to
reduce
In addition,
or ou has the expected
of damage the
on the tensile
of the
particle
strengthening
fraction
increasing
8 shows
behaviour
volume
associated
with
either go
with increased
stresses
volume
4.1. Agreement
the effect
for different
fraction.
Much
high
of
volume
developed
fraction,
This in the
as can be
seen in Fig. 9.
440 m=lOO
/
4. DISCUSSION
of
the over-
is lost due to the effect of damage.
effect is due to the higher particles
amount
effect of increasing
all strength of the composite. 3.2.3. Volume fraction. Figure values
the
between theory and experiments
Previous models for the strength of two-phase materials provide reasonable predictions at low strains. However, they do not incorporate effects due to damage at larger strain. Such damage has a significant softening effect. The present model takes account of this softening. For the case of the 10 vol.% material, in which the fraction of damaged material has been measured experimentally and fitted by means of the Weibull characteristics of the particles, the agreement between the model and the
c :. ‘.
420
a
400
% 380 Z 2 380
320 --
_i 0
2
4
8
8
10
12
14
18
Strain (X) Fig. 7. Tensile curves calculated with the model for four values of the Weibull’s modulus of the particles. Evolution of VFDwith strain (inset).
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600
0
I
0
2
4
I
6
I
I
6
10
1 12
Strain (x) Fig. 8. Tensile curves calculated with the model for three values of the volume fraction The undamaged
behaviour
calculated
with [3] is also given in thin lines. Evolution (inset).
experimental curve is very good. For the 18 vol.% case, in which the same Weibull data is used, the agreement is also excellent. There are several sources of uncertainty which should be acknowledged, such as: (i) The transient state of gD and gp with E has been neglected; (ii) The self-consistent analysis on which the model is based was developed for spherical particles. The actual composite is reinforced with slightly elongated particles which may induce a higher strengthening; (iii) The data on damage evolution has been derived from surface observations of the 10 vol.% composite. This might be different than the actual evolution in the bulk of the material.
Despite these differences, this method gives a good description of the phenomena occurring during damage, such that the predictions of the present model offer a close match with the experimental results. Figure 5 also shows that a simple homogeneous strain assumption for combining damaged and undamaged regions may lead to the underestimation of the behaviour especially in the case of high volume fraction of reinforcement (see the curves for the 18% composite), i.e. the hardening of the matrix in the damaged regions of the composite
iStrictly, the Considere condition applies only in the absence of dilatation (i.e. without damage). However, since the volume change induced by particle cracking is small, we continue to use it here.
of the particles. of vro with strain
depends then on its strain history. Calculations based on homogeneous strain have been compared with experiments on Al/S alloys [22]. These authors also concluded that while providing a simple way to account for damage, this approach underestimates the real properties of the damaging material. In the case of MMCs, the tensile instability has been shown in [22] to be successfully predicted by the Considere criterion expressed in true stress and true strain?: da Z=O The comparison of the true stress and the strain hardening rate for our experiments is shown in Fig. 10. The Considhe criterion also gives here a fair description of the tensile instability of these samples. This criterion is very easy to implement in the present approach, which is based on an incremental algorithm. However, the ductilities we calculate using this criterion are much higher than those measured experimentally, except for some rare cases in which the work hardening rate drops very rapidly because of damage (see the points indicated with a letter C on the curves of Figs 6 and 7). Martinez et a/. [14] however, have used this criterion to predict the ductility of MMCs exhibiting a low hardening exponent (0.1 as compared to 0.2 in our case), and found a good agreement. Their model is based on FEM calculations of the behaviour of damaged and undamaged cells, but they use a homogeneous strain assumption to calculate the behaviour of the damaging composite. According to the Considere criterion, two contributions can promote the rupture of the sample -
MAIRE
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FRACTURE
OF COMPOSITES
5271
800
Q
700
% ';;600 ii : 500 & iz 400 d 5 300 5 *E 200 !I 2 100
0 0
2
4
6
8
10
Strain of the composite (%) Fig. 9. Evolution
of the stress in the undamaged volume fractions
phase vs the strain studied in Fig. 8.
an increase of the stress and a drop in the work hardening rate. In Fig. 1 l(a) and (b) the experimental and calculated work hardening rates are compared for the 10% and 18% composites, respectively. While the predicted behaviour is much closer than the models without damage, the experimental work hardening rates show an additional drop at higher strain. The model fails at predicting this drop, which may be due to the linkage of
in the composite
for the three
damage through the cracks in the matrix indicating the beginning of a percolation process that leads to the rupture of the samples. This effect would appear to be more pronounced for the higher volume fraction of particles where the interparticle spacing is smaller thus facilitating linking of cracks. Multiple cracking of the particles which is seen in these materials, but not accounted in the present approach, can also be invoked to explain this drop. As a con-
2500
s p 2000
II Tangent modulus IO %
9 E s 1500 sl 5 I3
1000
3 ii 0 2 +
500
0 0
1
2
3
4
5
6
7
8
9
True strain (%) Fig.
10. Strain
hardening rate and true stress vs the true strain in the experimental point of instability is consistent with the Considere criterion.
composites.
The
MAIRE et al.:
5272
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OF COMPOSITES
(a) 6000
s
1
5000
E tangent. Model - no damage
-
E tangent. Model - damage n E tangent. Experimental
ii 2
-
-Stress.
4000
Model-damage
x % E 3000 ? 0 1 2000
i
1000
c
2
0
6
4
8
IO
True strain ("9)
8000 q7000 B ‘:6000
04
T ---
-Stress.
Model-damage
5 c 5000 -E F gJ 4000 -2 m 3000 -8 2 2000 --
03 0
1
3
2
4
5
6
Truestrain
Fig. 11. Comparison between the experimental and the calculated evolution of the work hardening rate vs the true strain in (a) the 10% composite and (b) the 18% composite. sequence the Consid&e criterion calculated with the present model would give only an upper limit of the rupture stress.
(ii) The fraction
of damaged
4.2. Parametric analysis of the model
effect is much more important.
Among the parameters studied, the mean particle size has the strongest effect on the calculated tensile curves. This can be attributed to two mechanisms:
lead to tensile
(i) The damage-free strengthening calculated with smaller particles is higher (see Fig. 3) due to dislocation effects.
suggests
material
for a given
stress is higher with larger particles. For particles
larger
instabilities
one can deduce sponding
to
than a few microns,
at very low strains,
100 pm
in Fig.
is smaller.
6. The
of broken
if, for a given mean
the distribution
can as
from the shape of the curve corre-
that the fraction
be smaller
the second
Large particles
model
particles
also should
size, the width
of
MAIRE
et al.:
FRACTURE
The effect of the Weibull characteristics of the particles can be understood as follows. Increasing both the representative stress cro, associated with the intrinsic fracture resistance, and the threshold stress gu 9 increases the strength of the material, essentially by reducing the rate of damage evolution. Both go and uu are sensitive to the nature of the reinforcing material, but also to the size of the defects. They could therefore be increased by improving the fabrication process for the particles, or even that of the composite (since some of the defects can be induced at this stage). ou may also be increased through the development of internal compressive stresses in the particles. These stresses represent the unrelaxed part of the thermal stress field arising during cooling, due to mismatch in the coefficients of thermal expansion. They could therefore be increased by increasing the temperature drop, the quench speed, or the strength exhibited by the matrix before the quench. Perhaps the most interesting effect is that of the Weibull modulus. Under special calculation conditions, increasing m has the effect of reducing the range of strength over which damage occurs, leading to an unstable behaviour referred to as the Considkre criterion. The relatively low value of the Weibull modulus required to fit the experimental data in the case of Sic in the present study as well as others [14] is probably due to a number of factors (distribution of the aspect ratio, clustering of the particles, introduction of defects during the elaboration process, etc.) which will tend to spread the particle fracture stress over a wide range. Thus, we should regard this Weibull data as representing the equivalent fracture response of these particles embedded in a deforming matrix, rather than the inherent fracture behaviour of the particles themselves. The volume fraction has a positive influence on the strengthening, but this influence is limited by the higher damage accumulation rate that results from the development of higher particle stresses. Moreover as mentioned previously the materials reinforced with a high volume fraction of particles may develop many linking cracks in the matrix that may induce earlier tensile instabilities. 5. CONCLUSION
The effect of damage on the tensile deformation of metal based alloys containing brittle particles has been modelled by modifying an incremental selfconsistent model previously used to predict the behaviour of heterogeneous materials in the absence of damage. The material was visualized as consisting of two constituents, one representing the undamaged zones, and the other describing the damaged zones. Attention has been focussed on the stress redistribution required when a given quantity of material transforms from undamaged to damaged during a given strain increment. The predicted flow
OF COMPOSITES
5273
curves are in good agreement with the experimental results and the method offers an improvement compared to the rule of mixtures with a homogeneous strain assumption used up to now. This is because the strain hardening in a matrix region surrounding a damaged particle is considerably less than that surrounding an undamaged particle. The self-consistent approach used here accounts for this. Despite this, the model over-predicts the ductility of these materials, indicating that additional processes, such as microcrack linkage, play a role in the final failure. The model has been used to predict the effect of important microstructural parameters on the spread of damage in the material. This demonstrates several approaches towards optimizing microstructures to limit the influence of damage. These are related to the Weibull properties, the size distribution characteristics and the volume fraction of the particles. Some effects related to the size distribution of the particles have been neglected in the present study which is focussed on the global effect of damage on the tensile behaviour. These effects include the evolution of the mean particle size in the damaged and undamaged phases during the deformation of the material and the effect of the size distribution on the behaviour of the undamaged material. Acknowledgements-The authors wish Brkchet for fruitful discussions. The during E. Maire’s PhD. were provided Society which is also fully acknowledged. provided in part by the Natural Sciences Research Council of Canada.
to thank Yves materials tested by the Pechiney Funding was and Engineering
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9. 10.
11. 12. 13. 14.
Shen, Y. L., Finot, M., Needelman, A. and Suresh, S., Acta merail. mater., 1995, 43(4), 1701. Zhan, D. B. and McMeeking, R. M., Acta metall. mater., 1991, 39(6), 1117. Corbin, S. F. and Wilkinson, D. S., Acta metall. mater., 1994, 42(4), 1311. Christensen, R. M. and Lo, K. H. J., Mech. Phys. Solids, 1979, 27, 315. Hervk, E., Zaoui, E. and A, Int. J. Engng. Sci., 1993, 31, 1. Bornert, M., Hervk, E., Stolz, C. and Zaoui, A., Appl. Mech. Rev., 1994, 42(11), 3865. Bourgeois N. Ph. D. Thesis. Ecole Centrale de Paris, France. June 1994. 179 p. Eshelby, J. D., in Progress in Solid Mechanics, (I. N. Sneddon and R. Hill ed.), 1961, pp. 87-140, NorthHolland, Amsterdam. Corbin, S. F. and Wilkinson, D. S., Acta metall. mater., 1994, 42(4), 1329. Prangnell, P. B., Downes, T., Withers, P. J. and Stobbs, W. M., Acta metall. mater., 1994, 42(10), 3431. Nan, C. W. and Clarke, D. R., Acta metall. mater., 1996, 44, 3801. Bao, G., Acta metall. mater., 1992, 40(10), 2547. Brockenbrough, J. R. and Zok, F. W., Acta metall. mater., 1995, 43(l), 11. Martinez, J. L., Llorca, J. and Elites, M., Reinforcement fracture and ductility in sphere-re-
5214
15. 16. 17. 18. 19.
MAIRE
et al.:
FRACTURE
inforced Al-matrix composites. IUTAM Symposium on Micromechanics of Plasticity and damage of multiphase materials. Paris 29 August-1 September 1995. Hill, R., J. Mech. Phys. Sol., 1965, 13, 213. Maire, E., Verdu, C., Lormand, G. and Fougeres, R., Mater. Sci. Engng., 1995, A196, 135. Maire E. 25 January 1995. Ph. D. Thesis. Insa de Lyon, France. No 95 ISAL0007. 186 p. Taya, M., Lulay, K. E., Wakashima, K. and Lloyd, D. .I.. Mat. Sci. Enana. A. 1990. A24. 103. Arsenault, R. J. and_Shi; N., hater. Sci. Engng. A, 1986, 81, 175.
20.
OF COMPOSITES Vogelsang,
M., Arsenault,
R. J. and Fisher,
R. M.,
Metall. Trans. A, 1986, 17A(3), 379. 21.
Kamat,
S. V., Rollett,
A. D. and Hirth,
J. P., Scripta
metall. mater., 1991, 25, 27. 22.
Kiser, M. T., Zok, F. W. and Wilkinson,
D. S., Acta
metall. mater, 1996, 44(9), 3465.
-1 LJ’
Wilkinson,
D. S., Maire,
E. and Embury,
J. D., The
Role of Heterogeneity
and the Flow and Fracture
Two-Phase
Mater.
press.
Materials.
of
Sci. Engrzg, 1997, in
Pergamon
Published
PII: S1359-6454(97)00147-X
1997 #C 1997 Acta Metallurgica Inc. by Elsevier Science Ltd. All rights reserved Printed in Great Britain 13%6454/97 $17.00 + 0.00
ROLE OF DAMAGE ON THE FLOW AND FRACTURE OF PARTICULATE REINFORCED ALLOYS AND METAL MATRIX COMPOSITES E. MAIRE’,‘t,
D. S. WILKINSON,’ J. D. EMBURY’ and R. FOUGERES
‘Department of Materials Science and Engineering, McMaster University, Hamilton, ON L8S4L7, Canada and ‘GEMPPM UMR CNRS no 5510, Insa de Lyon, 20 Av. A. Einstein, 69621 Villeurbanne Cedex. France (Received 10 February 1997; accepted 8 April 1997) Abstract-A large number of models exist to predict the tensile behaviour of two-phase materials. They fail, however, to predict the behaviour at high strain because they do not properly account for the role which damage accumulation plays during plastic flow. A new model is proposed here which incorporates damage. It uses an incremental self-consistent method developed previously. In the new application, the model treats damaged and undamaged regions as distinct continuous “phases” and allows the relative volume fraction of each phase to vary during deformation. The stress redistribution due to damage during the evolution is carefully analysed. The model considers the influence of particle size Fia a Weibull analysis and incorporates the results of existing FEM calculations. The model agrees well with existing experimental data and provides a new method to evaluate the role of microstructural parameters. 0 1997 Acta Metallurgica Inc. R&urn&Un grand nombre de modkles existent pour prtdire le comportement en traction de mattriaux biphasks. Ces modiles sont en disaccord avec le comportement exptrimental aux grandes valeurs de la d&formation plastique parce qu’ils ne prennent pas en compte l’effet de l’endommagement sur I’koulement plastique. Un nouveau modkle incluant l’endommagement est propok. I1 traite les zones endommagkes et non endommagies comme deux “phases” distinctes et continues. La fraction volumique de chacune des phases peut varier pendant la dtformation. La redistribution des contraintes au tours de l’endommagement est ktudite en d&ail. Le modkle prend en compte I’effet de la taille des particules grke B I’analyse de Weibull et il incorpore des rtsultats existants de calculs par la mkthode des kltments finis. Le modtle est en bon accord avec des donnies expkimentales existantes et constitue une nouvelle mkthode pour &valuer le r81e de certains paramttres microstructuraux.
1. INTRODUCTION
This paper addresses the deformation of metallic alloys containing second phase particles which are heterogeneous as regards their thermomechanical properties. All the materials of this class exhibit common trends (e.g. as far as damage initiation and propagation mechanisms are concerned) and much attention, both experimental and theoretical, has been paid to their behaviour. These studies are motivated by industrial interest in some materials belonging to this class, such as steels containing inclusions, aluminium casting alloys and, more recently, metal matrix composites. The elastoplastic behaviour of these materials can be predicted by a range of numerical [l, 21 and analytical [3%7] models, founded in the latter case on the Eshelby analysis [8]. In general, damage occurs during deformation because of particle cracking or particle/ matrix interface decohesion. The microcracks thus formed reduce the ductility of the material. Studies
TTo whom all correspondence
should
be addressed. 5261
have been attempted to account for non trivial problems such as damage appearance in the material [7,9-141. Usually damage is incorporated using a Kachanov approach involving a damage parameter [9, lo], or by simulating the effect of the microcracks represented as penny shape voids [7,1 I]. A third investigated way consists in calculating the local behaviour of damaged and undamaged cells and then associating these cells according to a classical rule of mixture assuming that the strain is equal in the two regions [12-141. In the present paper, we present a new model based on this third kind of approach to account for the role of damage in the calculation of the mechanical behaviour of two-phase materials. The treatment of the global combination of the damaged and undamaged constituents is improved by the use of an existing selfconsistent analysis previously developed for (undamaged) flow in a heterogeneous material [3]. Damage is incorporated using the Weibull approach to particle fracture. The fraction of damaged particles is allowed to increase with the stress in the undamaged regions. The model also contains a cal-
MAIRE
5262
et al.:
FRACTURE OF COMPOSITES
culation for stress redistribution due to damage. After presentation of the model, comparisons with experimental data are shown. Finally the effect of some of the microstructural parameters used in the calculation is predicted.
2. MODEL DESCRIPTION The analysis we have developed ling the behaviour of the material scales:
involves modelat two separate
?? at
a small scale we calculate the local behaviour of an elementary unit cell represented by a single particle surrounded by a shell of matrix the thickness of which depends on the overall volume fraction of reinforcement. The particle can be intact or broken. These two extreme cases will be hereafter referred as undamaged (UD) and fully damaged (FD), respectively. In principle either analytical or numerical approaches can be used to treat the problem. However, the case of broken particles is difficult to treat analytically, and we use existing FEM results to assess the local behaviour. ?? at a larger scale, the material is supposed to consist of a random mixture of unit cells, each one exhibiting one of the two above mentioned behaviours (i.e. damaged or undamaged). In order to determine the macroscopic tensile behaviour, we use an incremental self-consistent model. In the following development we first describe the behaviour of an assemblage of unit cells at large scale. We then proceed to develop the fine scale constitutive laws that are used in the analysis. 2.1. Calculation homogenisation)
of the global behaviour
only, the results of this model have been found to agree favourably with the results of FEM calculations for homogeneous two-phase materials [3]. The morphological hypothesis underlying this approach assumes an interpenetrating network of the two phases in question. This permits one to treat the case of high volume fraction of reinforcement and should be distinguished from the “generself-consistent approach proposed by alized” Christensen and Lo which assumes a matrix/inclusion morphology. It has been shown in [3] that if the Poisson’s ratio of the two constituents are taken equal to 0.5, the modulus of the composite EC can be calculated from the respective moduli El, E2 and volume fractions vI, v2 of the two constituents by solving the quadratic equation: 3E:, + [(2 - Sv,)El + (2 - 5v2)E2]Ec - 2E, E2 = 0 (2) which is also equivalent to the equation given Hill for the Young’s modulus of an isotropic persion of spheres [15]. The stresses in the two phases o1 and o2 are lated to the overall applied stress 0 (then to and to the elastic part of the total deformation by expressions of the type:
by disreEC) .a,~
5Ei
5Ei
Ecu
ai=2Ei+3Eca=2Ei+3Ec
(3)
If the stress differentials are now calculated the full stress-strain curve can be calculated incrementally through a piece-wise linear approximation. In this case the moduli of interest are the tangent moduli (i.e. the instantaneous work hardening rates) and the strain in equation (3) is now the total strain. Thus
(large scale dai =
2.1.1. Existing incremental self-consistent analysis. A previously developed self-consistent analysis [3] enables a calculation of the 0-c curve of a mixture of two phases provided that the properties of these phases are known separately. It considers two constituents, each exhibiting elastoplastic behaviour that can be approximated by the Ramberg-Osgood expression:
k (1) The yield stress oo, the Young’s modulus EO and the hardening coefficient N are constants for a given phase. In the case of brittle elastic particles such as Sic, Si or Al&u, the only parameter which is necessary to describe the behaviour is the elastic modulus Eo. The method proposed is based on the using an self-consistent analysis “classical” approach similar to that of Hill for the mixing of spheres [15]. Despite the fact that this approach utilises the normal component of the stress tensor
SE,’ 2ET + 3.67
dc =
5ET 2ET f 3q
EcTdE
(4)
where the superscript T indicates that the current tangent modulus is used. 2.1.2. Varying fraction of the two constituents. We now wish to adapt this model to treat damage accumulation in the form of progressive fracture of particles as illustrated in Fig. 1. In this context, the above described self-consistent incremental analysis can still be used with subscript i referring to damaged or undamaged regions of the material. However, to represent the experimental features, it must be reformulated to enable the relative volume fraction of the two constituents (i.e. undamaged and damaged cells) to vary with strain. This is complicated by the fact that deformation and damage occur simultaneously and interdependently. If the strain increments are kept small this problem is alleviated, since each increment can be divided into two sequential steps-deformation at constant damage followed by damage at constant strain. In the first, a strain increment de is applied and the
MAIRE
LARGE SCALE HOMOGENISATION
et al.:
FRACTURE
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OF COMPOSITES
I
SMALL SCALE HOMOGENISATION
T
0
= Undamaged
0
cells :
= Fully damaged cells :
Fig. 1. A schematic diagram illustrating how the actual material is idealized in the present model. The spatial distribution is uniform and the material is composed of cells containing either an intact (undamaged cells) or a broken particle (fully damaged cells).
stress increments for each constituent are calculated (according to the tenets of the self-consistent model). Following each strain increment, during which the stress in the undamaged region has increased, a certain fraction of material dvFD transforms from undamaged to damaged. dvFD is always positive as the level of damage increases monotonically with stress. It is also irreversible. Since
(2) The stresses in the two constituents must be updated to satisfy the constant overall strain condition. By differentiating equation (3) one can calculate the stress redistribution due to damage in the constituent i dgi,dam:
VFDf VUD= 1
dbUD,dam=%JD&~ - 3gUD dEc
(5)
dVFD
dVFD
thus dvuD -=-ds’ de
d”FD
(6)
Each increment of damage is accompanied by a redistribution of stress, which is assumed to occur instantaneously (i.e. at constant total strain). The transformation of a fraction of material from undamaged to damaged is accompanied by two modifications: (1) The modulus of the composite varies. By differentiating equation (2) one finds that: d& -= hw
-=
~(EFD - hm)& 6-k + (5rFD - 3)&D + (2 - SVFD)EFD
From this one can also calculate increment:
the overall
(7)
stress
Note that the second term is equal to zero since a change in elastic strain occurs only due to a change in elastic modulus.
=FD&I - 3gFD d& ~EFD + 3Ec dVFD
2-6j~ + 3Ec
dVFD
(9)
(10)
In each constituent, these variations are governed by Young’s modulus if they correspond to a decrease (elastic unloading), or by the tangent modulus if they correspond to an increase of the stress (elastoplastic loading). One can show that at constant Strain, both dbFD&,, and dbUD,dam are negative as the modulus EFD is smaller than the modulus EUD. In the incremental procedure described here each increment da of the overall strain of the composite is accompanied by an incremental loading of the two phases. As the two constituents unload due to damage, we must account for the fact that they first have to reload elastically up to the flow stress reached in the preceding step before plastic flow can recommence. The composite behaviour of each constituent is thus first governed by Young’s modulus, Once the yield points have been re-reached the flow behaviour of each constituent is assumed to be governed by its tangent modulus until the end of the step. From a numerical point of view the reloading after a damage event is done using small
5264
MAIRE
et al.:
strain increments (namely d&/2000) which cient to fully describe this complex process. 2.2. Calculation homogenisation)
of the local behaviour
FRACTURE
is suffi-
(small scale
f=
vP,k (11) VP.k
+
VM,k
where f is the overall volume fraction of second phase particles. Once damage commences we suppose that the material consists of a mixture of two kinds of cell each with a particle (volume VP,,) embedded in a matrix region (volume VM,J. These are: particle ?? damaged
At a given strain E, the stress in the fully damaged cells is related to the stress in the unreinforced material CUR by the expression: gFD(&)
We assume for the sake of simplicity that the spatial distribution of the particles is uniform, i.e. a volume VP,k is sureach particle k occupying rounded by a volume of matrix V,,, satisfying:
?? undamaged
OF COMPOSITES
cells,
each
containing
cells, each containing
an unbroken
a broken
particle.
The properties of the undamaged cell can be calculated by any of a large number of models previously developed. According to the definitions given above, the fraction of damaged cells VFD is equal to the fraction of broken particles. The properties of the damaged cell are related to those of the unreinforced matrix using results developed by FEM calculations 112-141. This marriage of FEM and self-consistent models follows naturally in that both are essentially single cell calculations. The boundary conditions in one case are governed by periodicity and in the other by embedding the material in an equivalent continuum phase. Nonetheless the results of specific FEM calculations (such as the stress ratio between an undamaged and a damaged cell under the same conditions) can be used as inputs into self-consistent models. We consider this one of the strengths of the current approach. To calculate the behaviour of the undamaged cells we use the classical self-consistent method for a mixture with a volume fraction f of elastic particles in an elastoplastic matrix [3, 151. This approach may not be the most appropriate for the considered system, i.e. an unbroken particle surrounded by a matrix. However, it has been shown to give a realistic approximation of the tensile behaviour of metal matrix composites in the absence of damage, and also to be reasonably close to the equivalent FEM prediction [3]. Moreover, it will be seen later that this approach leads to an excellent fit with the initial loading behaviour (i.e. in the absence of damage) of the experimental tensile curves of the materials considered in this study. Finally it is extremely easy to compute. As a consequence it was chosen to predict the behaviour of the undamaged material. The resulting curve is fitted by a Ramberg-Osgood expression, the parameters of which being EUD, gUD and Noo.
=
gD(hJR(&).
(12)
Using FEM calculations, Bao [12] and more recently Brockenbrough and Zok [13] have shown that go depends on E for low strains (transient response) but tends to become constant quite rapidly when E increases to a few times the elastic strain (steady state response) especially when the ratio between particle and matrix moduli is high (as in the case of SIC particles in an aluminum matrix). In the current model we assume that gD is constant over the whole deformation range. Note that in some materials (e.g. metal matrix composites), damage initiation can occur by two coexisting mechanisms: particle cracking and particle/matrix interface decohesion [14]. The cells damaged according to these two kinds of mechanisms may have a different behaviour (i.e. a different value of go) but unfortunately no simple expressions of go have yet been proposed for the cells damaged by decohesion. Besides, our observations in [16] show that the proportion of particle decohesion is quantitatively small. As a consequence, we neglect the difference between these two mechanisms in what follows. As the variations of go with E can be neglected, the behaviour of the fully damaged cells is described by a modified RambergOsgood expression:
(13) In the actual partially damaged material, the damaged cells are located at random in the whole volume, such that the damaged and undamaged regions form an interpenetrating network. This is exactly the kind of morphological hypothesis that lies behind the self-consistent analysis and especially behind the classical model adopted here. Thus, we use this model to mix both damaged and undamaged cells. As a consequence the incompressibility assumption stated in the original model is applied to the undamaged phase, which is certainly not valid. Fortunately, it has been verified through preliminary calculations that the value of the Poisson’s ratio of each phase has only a weak effect on the predicted behaviour in the plastic regime. Clearly the approach to damage just outlined includes only damage initiation (i.e. particle cracking) and neglects growth and coalescence of damage elements. We justify this by noting that particle cracking is the only damage observed experimentally in the materials of interest until rather large strains, close to final failure [17]. In addition, the initial stages of damage growth involve the increase of the crack width during deformation, which is captured in the FEM calculations for the fully
MAIRE et al.:
FRACTURE
0.035 --
5265
OF COMPOSITES
-W
0.03 --
-
Vfd(a); op= 1100 MPa
-
Vfd(a); oP=
900 MPa
0.025 -2 5 5
0.02 --
2 IL 0.015 --
0.00
20.00
40.00
80.00
60.00
100.00
Particle size (pm) Fig. 2. f(a) and v&a) for two values of the stress in the particles, respectively 900 and 1100 MPa. The other parameters used arem = 3.8, UO= 1700 MPa, cru=580 MPa, a~= 11.5 pm and p = 0.51.
damaged discussed
material. The effect of crack coalescence in more detail below.
is W(ap,a) = 1 -,.exp[-(E)3(F)‘“]
2.3. Calculation of the fraction of damaged material We now have to calculate the evolution of the fraction of damaged material taking into account both mechanical and microstructural considerations. The experimentally determined size distribution 4(a) of the second phase particles is usually sufficiently well represented by a log-normal distribution given by:
where cru is the threshold stress below which no rupture is observed and 00 the representative stress, while m is the Weibull modulus. The parameter al is an arbitrarily chosen representative dimension. The fraction of damaged material for a given value of op can finally be calculated as: vFD(gP)
= I
da) = &exp[-i
(?)‘I
(14)
(18)
Urn/ (a)W(ap, a)da.
(19)
Note finally that --oo
where a0 is the mean size and p is representative the width of the distribution. Note that:
of
+m cp(a)da = 1. J’0
(15)
This size distribution must be transformed into a “volume fraction” distribution j(a), the fraction of material associated with the small particles being smaller than for the big particles: cp(a)a3
f(a)=
s,‘” cp(a)a3da ’
(16)
Here also +W .I’ 0 f(a)da At a given particle particles of size expression:
= l.
(17)
stress op, the fraction of broken a is given by the Weibull
vFD(bp)dgp = 1. J’0
(20)
The stress in the undamaged constituent oUD, and the stress acting on the particles of this constituent op are related by an expression of the form: nP(&)
=
gP(&hJD(&)
(21)
where gp can be easily calculated from the initial model [3] in the undamaged case. As with gD (see equation (12), gp exhibits a transient followed by a steady state [3, 12, 131. The variations of gp during the test have been shown by these authors to have a small amplitude. In the present case, we neglect these variations, and use for gp the average value of this parameter over the range of deformation considered. At a given strain level, we then calculate the average stress in the particles of the undamaged phase, from a knowledge of the average stress in this phase. In this Eshelby type calculation, this stress does not depend on the size of the particles.
MAIRE
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FRACTURE
We assume also that at a given overall strain, the stress in each particle is the same (which also implies that they have the same aspect ratio). An illustrative plot of VFD vs the size of the particles is given in Fig. 2 for two values of the stress in the particles (900 and 1100 MPa, respectively). The size distribution &a) has been measured by means of a computerized surface analysis system on a 2618T4/ Sic composite (see the next section) assuming that the ratio between the dimensions measured in surface and the actual dimensions was equal to 417~as suggested in [7]. The parameter chosen to represent the size of the elongated particles was the equivalent diameter, defined as the diameter of the circle of the same surface area S as that of the particle. This distribution was found to be well described by a lognormal law with ao= 11.5 pm and p = 0.51. The choice of the Weibull parameters needed for this kind of calculation will be explained later. The total VdUe Of VFD for a given stress is equal to the area below the corresponding curves. Also, while VFD increases with the particle stress, the curves are asymmetric since the cracking of the larger particles occurs at lower stress. Note that this implies that the average size of the particles in the damaged and undamaged phases of the materials varies during the test although this is not taken into account in the present model. To do so it would be necessary to know whether gD depends on the size of the particle.
3. RESULTS
3.1. Comparison
with experiments
We first develop the model in the context of a specific set of experimental results. The chosen data is concerned with MMCs. The material is a 2618T4 Al alloy reinforced with 10 and 18 vol.% of SIC particles. Experimental results are reported elsewhere [ 171. The modulus of the SIC has been measured by means of nanoindentation experiments to be about 400 GPa [16]. The tensile behaviour of the unreinforced matrix under the same heat treatment condition was measured simultaneously and is used as input in the model. The Ramberg-Osgood coefficients of the unreinforced material were found to be: ao=200 MPa, E = 74.4 GPa, and N = 0.16. However, it has been shown by several authors [17201, that the mechanical behaviour of the matrix in the composite may be different than in the corresponding unreinforced alloy. The microstructure and dislocation density are modified during the heat treatment and the mechanical loading because of the presence of the particles which act as heterogeneities as regards their thermomechanical properties. The main microstructural modification induced tFor a more complete discussion of this approach Ref. [23].
see
OF COMPOSITES
by the presence of the particles is the increase of the dislocation density in the matrix, especially in the T4 temper. The particles also affect the work hardening of the surrounding matrix during the deformation and this matrix/inclusion morphology is not fully accounted for in the classical self-consistent scheme proposed by Hill. In order to model the effect of particles on strength over a wide range of particle size, both dislocation and continuum effects need to be included. Thus, the dislocation terms can be used to modify the yield stress of the matrix prior to the development of a continuum level selfconsistent model. In the present comparison we account for these effects in an approximate wayt using the work of Nan and Clarke [11], i.e. we incorporate in the Ramberg-Osgood expression an additional contribution to the yield stress of the matrix in the composite aM,C such that: aM,C = OUR +h~R,p.
(22)
Expressions related to A0UR.P have been summarized by Nan and Clarke. The main contribution to AaUR,p in our case is associated with the dislocations generated during the quench of the material. The microstructural parameters required to implement this calculation are: the average diameter of the particles d = 11 pm; the temperature drop during the quench AT = 500”; and the difference between the coefficients of thermal expansion of Al and SIC Act = 20 x 10e6 K-‘. The other parameters characterizing the aluminum matrix were chosen equal to those used by Nan and Clarke who dealt with the same kind of material. One of the main interests of this approach is to permit the description of the effect of the size of the particles in the prediction of the behaviour of the undamaged material. This is illustrated in Fig. 3 where we for the curves calculated show flow 2618T4 + 10 vol.%SiC composites with particles of different size. The strengthening increases with smaller particles (i.e. below about 10 pm), consistent with many experimental studies [21]. We will consider in the following calculation that the tensile behaviour of a material reinforced with particles having a distribution of sizes is equivalent to the behaviour of a composite reinforced with a single family of particles exhibiting the average size. This assumption implies the global compensation of the contributions of the big and of the small particles on the global behaviour. It is an approximation of the actual behaviour, though it appears to be sufficient compared to the purpose of the present paper which deals with global effects. The size distributions studied hereafter will then be considered to affect the calculation of the fraction of damaged material only. Another important effect on the strengthening could be the heterogeneity of the spatial distribution of the particles. It has been shown in [3] that
MAIRE et al.:
Ol
FRACTURE
OF COMPOSITES
Ao”~,~: a = 1 pm
-
u
-
~u~+Aou~,~:a=11.5pm
-
oUR+AoUR,P:a=lOOpm
-
uLIR
uR
+
/
I
I
I
I
1
2
3
4
5
/
0
6
I
7
--I
8
Strain (%) Fig. 3. Effect of the particle size on the strengthening of an aluminium alloy (N = 0.16, E = 74.4 GPa and (TO=200 MPa) reinforced with 10 vol.% of Sic particles (E = 400 GPa) calculated with a self-consistent analysis [3] for an undamaged material, along with Nan and Clarke’s correction [ll] for dislocation effects. a clustered distribution could result in a higher strengthening. The respective contribution of the effect of the modification of the microstructure of the matrix, and of the spatial distribution heterogeneity are not perfectly clear so far, and the actual strengthening is probably a combination of these two mechanisms but for the sake of simplicity we accounted only for the first effect in the present paper. The particles are not spherical and the materials in question have been extruded before the heat treatment. The particles are then aligned along the tensile axis and exhibit an aspect ratio of about 2 [17]. To estimate gD, we used Bao’s FEM results for this aspect ratio:
model for both the 10 and 18 vol.% composites. We perform the calculation for the 18% composite using the Weibull parameters deduced as described above from the case of the 10% composite. Figure 5 represents the tensile curves calculated with the present model, and compared with the experimental curves, and with calculations performed for undamaged materials. The agreement with the experimental results will be discussed in detail later. The squares on the figure show the resulting curve that would be calculated by the simple rule of mixtures initially used in [12-141, which implies that the strain is homogeneous in the two phases: rr =
VFD~FD
+
VU@IJD
with EFD = EUD = EC. (24)
3.2. Influence of the deferent parameters g,, = 0.958 + (f - 0.202)2.
(23)
It now remains to measure the Weibull characteristics of the SIC particles. For the 10 vol.% composite the evolution of rrD with strain was measured elsewhere [17] by means of SEM observations during in situ tensile experiments. We used this evolution to choose the correct Weibull parameters. Figure 4 shows the comparison between the experimental evolution, and the evolution calculated with the model using the following set of Weibull parameters: m = 3.8, oo= 1700 MPa and ou = 580 MPa. This set of characteristics is realistic for fine particulate Sic. These parameters depend on the value of al which can be chosen arbitrarily (it is kept equal to 5 pm in the present case). Having compiled all of the necessary parameters characterizing the microstructure and the damage development, it is now possible to evaluate the
In this section the predictive capability of the model is used to illustrate the effect of microstructural parameters on the tensile behaviour of a damaging material. These illustrations have been made using material parameters appropriate to a 10% composite. 3.2.1. Particle size distribution. Figure 6 illustrates the calculated tensile curves and damage evolution for several values of the mean particle size ao. The reduction in strength with increased particle size is primarily due to damage (see Fig. 6 inset). In particular, the near zero rate of work hardening in the 100 pm particle size material follows from the rapid onset of damage. Such behaviour would result in low ductility due to global instability as predicted by the Considere criterion. The same calculations have been performed for several values of the distribution width /I. They show that for a given average
5268
MAIRE et al.:
FRACTURE
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0.6
0.5
0.4
$J 0.3 n Experimental 0.2
-
Weibull Darameter fit
0.1
0
1
2
3
4
5
6
7
8
9
Plastic strain (%) Fig. 4. Experimental evolution fo the fraction of damaged material measured for a 2618T4 + 10 vol.% BC composite by means of SEM observations during in situ tensile tests (squares). The line represents a Weibull fit calculated with the present model using m = 3.8, (~a= 1700 MPa, 0” = 580 MPa.
particle size, fl has a smaller effect than a0 with narrower distributions contributing to greater strength. 3.2.2. Weibull parameters. The effect of Weibull modulus is illustrated in Fig. 7. In this particular case, we have chosen a set of Weibull parameters permitting to illustrate in a more visual way the special effect of m. Especially we assumed that the
width of the size distribution p was equal to 0. Under these special conditions, as m increases, damage is confined in a narrower strength range, the work hardening drops and can even become negative. The effect of the increase of the Weibull modulus calculated with the realistic set of parameters used in all the other calculations presented
500
-Model
Rule of mixture
100
0
- damage
J 0
/ 2
4
I
6
8
1
I
10
12
Strain (%) Fig. 5. Comparison between the tensile curves calculated with the present model (intermediately thin curves) and the experiments (thick curves) for 2618T4 + 10 and 18 vol.% Sic composites. The behaviour of the undamaged material calculated with [3] (thinner curves) and the prediction of the mixture rule with homogeneous strain (squares) is also shown.
MAIRE et al.:
500
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5269
T
lop
,...___._ ...“.....
2
0
__.....--1
-.-----*-
4
1
2
I
3
4
5
Strain (%) Fig. 6. Tensile curves calculated with the model for three values of the mean particle size. Evolution of vFD with strain (inset). here
is more
damaged
simply
particles.
to
reduce
In addition,
or ou has the expected
of damage the
on the tensile
of the
particle
strengthening
fraction
increasing
8 shows
behaviour
volume
associated
with
either go
with increased
stresses
volume
4.1. Agreement
the effect
for different
fraction.
Much
high
of
volume
developed
fraction,
This in the
as can be
seen in Fig. 9.
440 m=lOO
/
4. DISCUSSION
of
the over-
is lost due to the effect of damage.
effect is due to the higher particles
amount
effect of increasing
all strength of the composite. 3.2.3. Volume fraction. Figure values
the
between theory and experiments
Previous models for the strength of two-phase materials provide reasonable predictions at low strains. However, they do not incorporate effects due to damage at larger strain. Such damage has a significant softening effect. The present model takes account of this softening. For the case of the 10 vol.% material, in which the fraction of damaged material has been measured experimentally and fitted by means of the Weibull characteristics of the particles, the agreement between the model and the
c :. ‘.
420
a
400
% 380 Z 2 380
320 --
_i 0
2
4
8
8
10
12
14
18
Strain (X) Fig. 7. Tensile curves calculated with the model for four values of the Weibull’s modulus of the particles. Evolution of VFDwith strain (inset).
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600
0
I
0
2
4
I
6
I
I
6
10
1 12
Strain (x) Fig. 8. Tensile curves calculated with the model for three values of the volume fraction The undamaged
behaviour
calculated
with [3] is also given in thin lines. Evolution (inset).
experimental curve is very good. For the 18 vol.% case, in which the same Weibull data is used, the agreement is also excellent. There are several sources of uncertainty which should be acknowledged, such as: (i) The transient state of gD and gp with E has been neglected; (ii) The self-consistent analysis on which the model is based was developed for spherical particles. The actual composite is reinforced with slightly elongated particles which may induce a higher strengthening; (iii) The data on damage evolution has been derived from surface observations of the 10 vol.% composite. This might be different than the actual evolution in the bulk of the material.
Despite these differences, this method gives a good description of the phenomena occurring during damage, such that the predictions of the present model offer a close match with the experimental results. Figure 5 also shows that a simple homogeneous strain assumption for combining damaged and undamaged regions may lead to the underestimation of the behaviour especially in the case of high volume fraction of reinforcement (see the curves for the 18% composite), i.e. the hardening of the matrix in the damaged regions of the composite
iStrictly, the Considere condition applies only in the absence of dilatation (i.e. without damage). However, since the volume change induced by particle cracking is small, we continue to use it here.
of the particles. of vro with strain
depends then on its strain history. Calculations based on homogeneous strain have been compared with experiments on Al/S alloys [22]. These authors also concluded that while providing a simple way to account for damage, this approach underestimates the real properties of the damaging material. In the case of MMCs, the tensile instability has been shown in [22] to be successfully predicted by the Considere criterion expressed in true stress and true strain?: da Z=O The comparison of the true stress and the strain hardening rate for our experiments is shown in Fig. 10. The Considhe criterion also gives here a fair description of the tensile instability of these samples. This criterion is very easy to implement in the present approach, which is based on an incremental algorithm. However, the ductilities we calculate using this criterion are much higher than those measured experimentally, except for some rare cases in which the work hardening rate drops very rapidly because of damage (see the points indicated with a letter C on the curves of Figs 6 and 7). Martinez et a/. [14] however, have used this criterion to predict the ductility of MMCs exhibiting a low hardening exponent (0.1 as compared to 0.2 in our case), and found a good agreement. Their model is based on FEM calculations of the behaviour of damaged and undamaged cells, but they use a homogeneous strain assumption to calculate the behaviour of the damaging composite. According to the Considere criterion, two contributions can promote the rupture of the sample -
MAIRE
et al.:
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5271
800
Q
700
% ';;600 ii : 500 & iz 400 d 5 300 5 *E 200 !I 2 100
0 0
2
4
6
8
10
Strain of the composite (%) Fig. 9. Evolution
of the stress in the undamaged volume fractions
phase vs the strain studied in Fig. 8.
an increase of the stress and a drop in the work hardening rate. In Fig. 1 l(a) and (b) the experimental and calculated work hardening rates are compared for the 10% and 18% composites, respectively. While the predicted behaviour is much closer than the models without damage, the experimental work hardening rates show an additional drop at higher strain. The model fails at predicting this drop, which may be due to the linkage of
in the composite
for the three
damage through the cracks in the matrix indicating the beginning of a percolation process that leads to the rupture of the samples. This effect would appear to be more pronounced for the higher volume fraction of particles where the interparticle spacing is smaller thus facilitating linking of cracks. Multiple cracking of the particles which is seen in these materials, but not accounted in the present approach, can also be invoked to explain this drop. As a con-
2500
s p 2000
II Tangent modulus IO %
9 E s 1500 sl 5 I3
1000
3 ii 0 2 +
500
0 0
1
2
3
4
5
6
7
8
9
True strain (%) Fig.
10. Strain
hardening rate and true stress vs the true strain in the experimental point of instability is consistent with the Considere criterion.
composites.
The
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(a) 6000
s
1
5000
E tangent. Model - no damage
-
E tangent. Model - damage n E tangent. Experimental
ii 2
-
-Stress.
4000
Model-damage
x % E 3000 ? 0 1 2000
i
1000
c
2
0
6
4
8
IO
True strain ("9)
8000 q7000 B ‘:6000
04
T ---
-Stress.
Model-damage
5 c 5000 -E F gJ 4000 -2 m 3000 -8 2 2000 --
03 0
1
3
2
4
5
6
Truestrain
Fig. 11. Comparison between the experimental and the calculated evolution of the work hardening rate vs the true strain in (a) the 10% composite and (b) the 18% composite. sequence the Consid&e criterion calculated with the present model would give only an upper limit of the rupture stress.
(ii) The fraction
of damaged
4.2. Parametric analysis of the model
effect is much more important.
Among the parameters studied, the mean particle size has the strongest effect on the calculated tensile curves. This can be attributed to two mechanisms:
lead to tensile
(i) The damage-free strengthening calculated with smaller particles is higher (see Fig. 3) due to dislocation effects.
suggests
material
for a given
stress is higher with larger particles. For particles
larger
instabilities
one can deduce sponding
to
than a few microns,
at very low strains,
100 pm
in Fig.
is smaller.
6. The
of broken
if, for a given mean
the distribution
can as
from the shape of the curve corre-
that the fraction
be smaller
the second
Large particles
model
particles
also should
size, the width
of
MAIRE
et al.:
FRACTURE
The effect of the Weibull characteristics of the particles can be understood as follows. Increasing both the representative stress cro, associated with the intrinsic fracture resistance, and the threshold stress gu 9 increases the strength of the material, essentially by reducing the rate of damage evolution. Both go and uu are sensitive to the nature of the reinforcing material, but also to the size of the defects. They could therefore be increased by improving the fabrication process for the particles, or even that of the composite (since some of the defects can be induced at this stage). ou may also be increased through the development of internal compressive stresses in the particles. These stresses represent the unrelaxed part of the thermal stress field arising during cooling, due to mismatch in the coefficients of thermal expansion. They could therefore be increased by increasing the temperature drop, the quench speed, or the strength exhibited by the matrix before the quench. Perhaps the most interesting effect is that of the Weibull modulus. Under special calculation conditions, increasing m has the effect of reducing the range of strength over which damage occurs, leading to an unstable behaviour referred to as the Considkre criterion. The relatively low value of the Weibull modulus required to fit the experimental data in the case of Sic in the present study as well as others [14] is probably due to a number of factors (distribution of the aspect ratio, clustering of the particles, introduction of defects during the elaboration process, etc.) which will tend to spread the particle fracture stress over a wide range. Thus, we should regard this Weibull data as representing the equivalent fracture response of these particles embedded in a deforming matrix, rather than the inherent fracture behaviour of the particles themselves. The volume fraction has a positive influence on the strengthening, but this influence is limited by the higher damage accumulation rate that results from the development of higher particle stresses. Moreover as mentioned previously the materials reinforced with a high volume fraction of particles may develop many linking cracks in the matrix that may induce earlier tensile instabilities. 5. CONCLUSION
The effect of damage on the tensile deformation of metal based alloys containing brittle particles has been modelled by modifying an incremental selfconsistent model previously used to predict the behaviour of heterogeneous materials in the absence of damage. The material was visualized as consisting of two constituents, one representing the undamaged zones, and the other describing the damaged zones. Attention has been focussed on the stress redistribution required when a given quantity of material transforms from undamaged to damaged during a given strain increment. The predicted flow
OF COMPOSITES
5273
curves are in good agreement with the experimental results and the method offers an improvement compared to the rule of mixtures with a homogeneous strain assumption used up to now. This is because the strain hardening in a matrix region surrounding a damaged particle is considerably less than that surrounding an undamaged particle. The self-consistent approach used here accounts for this. Despite this, the model over-predicts the ductility of these materials, indicating that additional processes, such as microcrack linkage, play a role in the final failure. The model has been used to predict the effect of important microstructural parameters on the spread of damage in the material. This demonstrates several approaches towards optimizing microstructures to limit the influence of damage. These are related to the Weibull properties, the size distribution characteristics and the volume fraction of the particles. Some effects related to the size distribution of the particles have been neglected in the present study which is focussed on the global effect of damage on the tensile behaviour. These effects include the evolution of the mean particle size in the damaged and undamaged phases during the deformation of the material and the effect of the size distribution on the behaviour of the undamaged material. Acknowledgements-The authors wish Brkchet for fruitful discussions. The during E. Maire’s PhD. were provided Society which is also fully acknowledged. provided in part by the Natural Sciences Research Council of Canada.
to thank Yves materials tested by the Pechiney Funding was and Engineering
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Wilkinson,
D. S., Maire,
E. and Embury,
J. D., The
Role of Heterogeneity
and the Flow and Fracture
Two-Phase
Mater.
press.
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