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The effects of particle clustering on the mechanical behavior of particle reinforced composites
Composites Part B 29B (1998) 299-308
PII: S 1359-8368(97)00031-0
ELSEVIER
© 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00
The effects of particle clustering on the mechanical behavior of particle reinforced composites
T. C. Tszeng* Department of Mechanical Engineering, University of Nebraska, Lincoln, NE 685880656, USA (Received 25 April 1996; accepted 9 June 1997) Particle clustering is thought to be one of the detrimental factors to the performance of particle reinforced composites. In this paper, the effects of particle clustering on the macroscopic mechanical behavior and the micromechanics of particle-reinforced metal matrix composites (MMCs) were predicted theoretically by using the equivalent inclusion method. Two types of particle clustering were examined. The calculation results are focused on the effects of clustering on (1) the macroscopic behavior of the composite, (2) stress concentration, and (3) void nucleation. © 1998 Elsevier Science Limited. All rights reserved (Keywords: particle clustering; mechanical behaviour; void nucleation)
INTRODUCTION The reinforcing particles are more or less non-uniformly dispersed in metal matrix composites (MMCs). Particle clustering is originated from the manufacturing processes. In cast composites, the fine particles may be pushed by the advancing solidification front and eventually trapped in the interdendritic regions. It is not unusual to find a large area of matrix which is free of particles [1]. The tendency of particle clustering depends on the solidification rate of the meltS'z; lower solidification rate leads to an increase in particle clustering. In the powder metallurgy route of making composites, the fine particles may remain in the form of algomerate in the blending and mixing stages, and formed into clusters in the composites 3 5. In literature, the terminology of particle clustering was used to describe two types of uneven local concentration of particle distribution. In the first type, particles are quasirandomly distributed, i.e., the statistical distribution of interparticle distance is invariant with respect to the position in space. For particle distributions in true randomness, the distance between two neighboring particles follows a Poisson's distribution. Thus, there are situations where the interparticle distance between two particles can be much smaller than the mean distance. In the strictest sense, particle distribution of this type is a product of the nature, instead of being a defect in manufacturing. The regular clustering used by the FEM analyses 6,v is an approximate to * Formerly with Scientific Forming Technologies Corporation, Columbus, OH 43202, USA.
this type of distribution, in which the distribution function of interparticle distance is discrete, yet invariant in space. In the second type of particle clustering, the mean value of the statistical distribution of interparticle distance in some regions is higher than the mean of the surrounding region. This type of distribution includes the particle clustering caused by the pushing of solidification front. In this paper, we will name the first type of aforementioned particle clustering as local inhomogeneity, and reserve the term 'clustering' for the second type of clustering. The effects of clustering and local inhomogeneity of particles on the mechanical properties of composites were the topic of several prior studies. One of the major concerns associated with particle clustering is the promoted tendency of void nucleation/growth/coalescence from the clustered regions. Some earlier studies in this respect were on the particle-strengthened alloys, for example, spheroidized low carbon steels 8-1°. For these materials with second-phase particles at a relatively low volume fraction compared with the usual MMCs, a local region that has a high volume fraction of particles would serve as the site of void nucleation at the lowest strain. This was attributed to the increased stress concentration at the particle-matrix interface in the regions of higher particle volume fraction 4'8 10 The works conducted by Fisher and Gurland 9 and Argon e t al. 1° on the particle-strengthened alloys pointed out the adverse effects of particle local inhomogeneity. Recently, several theoretical treatments of the effects of 'clustering' (particle local inhomogeneity, in our term) were addressed specifically for M M C s 6'7"11. The finite element methodbased works of Llorca et al. 6 and Christman et al. 7 are
299
Particle clustering: T. C. Tszeng involved with deformation analysis of a unit cell which comprises matrix and reinforcement phase in a certain geometrical arrangement in order to represent particle local inhomogeneity in composite systems, i.e., the distances between two or three adjacent reinforcements are shortened. Because of the geometrical arrangement, the 'clustering' is restricted to the interactions between at most two or three particles/short fibers. Repetition of unit cell in space implies that the composite would consist uniformly distributed 'clusters' each of which only contains two or three particles. Therefore, this approach does not properly represent the clustering of a group of particles and the associated mechanical responses. Everett ~2 studied the implications of local inhomogeneity of fiber distribution by using a large number of tuples. The axial strength of a bundle of continuous fibers is shown to decrease with the increase in standard deviation of volume fraction. In that study, the fiber distribution is established by the mathematical model. While the distribution is more realistic than those used by Llorca et al. 6 and Christman et al. 7, it certainly is more complicated for a mathematical treatment of the associated mechanics. Komenda and Henderson 13 experimentally determined the creep rupture strength of shortalumina fibers reinforced aluminum. They attributed the differences in creep rupture strength to the different fiber distributions. They first used a scheme to determine the fiber-free zone in the specimens. Correlation was found between the creep rupture strength and the coefficient of variation of zone size which is defined as the SD of zone size divided by the mean zone size. For heat treated SiC-A1 composites, Ferry et al. 14 found that the grains of aluminum matrix is finer in the clustered areas. This was caused by the inhibition on grain growth imposed by the reinforcing particles with shorter interparticle distance. Davidson ~5 experimentally observed the crack initiation in the region with particle local inhomogeneity. It is thought that the voids in the area of local inhomogeneity seem to originate from the interface. Thus, smaller space between particles tends to promote crack initiation. This observation was attributed to the stress concentration in the region of local inhomogeneity4:4. It is also suspected that the interfacial strength in the area of higher local particle fraction is lower than that in the homogeneous, unclustered region ]5. Lewandowski et al. 16 conducted a similar experimental study to determine the effects of particle local inhomogeneity on crack initiation. According to these experimental studies, there seems to exist a positive correlation between the tendency of crack initiation and the particle local inhomogeneity. Shi et al. ~~ seemed to carry out the first study using a twostep model to examine the effects of particle clustering. In the first step, the elastic moduli associated with the clustered and unclustered regions are determined by the Eshelby's inclusion method. Next, a finite element method is utilized to calculate the macroscopic response. In that study, only elastic response and initial yielding are addressed. According to the above brief review, it is still not very clear how clustering affects the mechanics and mechanical
300
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Figure 1 The patterns of particle clustering. (a) Pattern I, (b) Pattern lI
behavior. There is a need to further investigate the mechanics of particle reinforced composites when particle clustering appears. In particular, the interfacial stress in the clustered regions when the composite is subjected to a macroscopic stress is one of the most important subjects. In this paper, we try to provide some insight into the effects of particle clustering by theoretical calculation. The macroscopic as well as the microscopic response of the clustered composite systems are to be examined theoretically. In this connection, we have to note that the composites with particle clustering are a special type of functional gradient materials (FGMs). Extensive studies were conducted recently. A technical issue related to the modeling of FGMs is the high inclusion volume fraction. The effective mechanical moduli of the material with high inclusion volume fraction can be predicted satisfactorily by the Differential scheme ~7, the Generalized Self-Consistent Method is, the Generalized Inclusion Method 19-28 and the modification by Kwon and Dharan 29. In this study, the composites with particle clustering are treated as a hybrid composite material, i.e., the clusters are considered to be the generalized inclusion while the unclustered region is the generalized matrix material. The approach uses a two-stage analysis based on the equivalent inclusion method. The applicability of an equivalent inclusion method to the concerned problem will be discussed in the next section. The calculation results are to be focused on the effects of clustering on: (1) the macroscopic behavior of the composite; (2) stress concentration; and (3) void nucleation.
THEORETICAL TREATMENT The shape and distribution of particle clusters in particle reinforced MMCs is very irregular, depending on the processing method and history. To simplify the analysis, particle clustering is characterized by: (1) the pattern of the clusters; (2) the shape of the clusters; (3) the volume fraction of the clusters in the matrix; and (4) the particle volume fraction in the clusters. In this work, three patterns of particle distribution are studied. The reference system has particles homogeneously dispersed in the matrix, i.e., no particle clustering or local inhomogeneity. The two clustering patterns are: Pattern I--segregated particle-rich (clusters) and particle-free regions ( F i g u r e l a ) , and Pattern
Particle clustering: 7". C. Tszeng II--uniformly dispersed particles with few isolated clusters (Figure lb). In clustering Pattern II, the composite is considered to comprise clusters in a generalized matrix material. Any region in the composite is categorized into clustered or unclustered. The clusters are regions in which the particle volume fraction is higher than the nominal particle volume fraction. The particle volume fraction in the unclustered region is lower than the nominal volume fraction. In this paper, matrix simply denotes the plain matrix material surrounding the reinforcement particles. The clusters are assumed to be in the shape of an ellipsoid. Reinforcing particles are spherical. To facilitate our discussion, three independent measures of volume fraction are used. First, the nominal volume fraction of the particle in the composite, denoted byf. This is the volume fraction usually referred to. Second, the volume fraction of particle within the clusters, denoted byfc. Third, the volume fraction of cluster in the composites, denoted by F. A fourth, dependent, measure is the volume fraction of particles in the unclustered region, denoted byfo. A relation between these measures of volume fraction is: fcF + (1 --F)fo= f .
(1)
To estimate the volume fraction of clusters, F, in a clustered composite, we will consider that the mean diameter of the spherical clusters is d and the mean distance between a cluster and its nearest neighbors is D. It can be shown by using the equation of Kao et al. 30 that the volume fraction of distributed clusters is F=
2 \2A/
"
(2)
With a given f, fc and F, the fourth measure, fo, is calculated by using equation (1). However, if an estimate of fo is readily available, either F orf~ can be obtained by equation (1). Note that equation (2) is only applicable to the cases when the clusters are spherical in shape. The microstructure of A356-15vol.%SiC composite reported by Lloyd ~ is one of the available examples showing particle clustering in MMCs. At an estimated cooling rate of I°C s 1 (figure l l d in Lloyd1), the mean cluster diameter, d, is about 50 gm and the mean distance, D, between a cluster and its nearest neighbor is about 120/xm. Note that these data are very crude and by no means represent a good estimation. Substituting these data in equation (2) gives the volume fraction of clusters, F, as 0.2. In the aforementioned picture of Lloyd', it shows that the volume fraction fo of particles in the matrix outside the clusters is essentially zero. Equation (1) is then reduced to f~F = f, and the volume fraction fc of particle within the cluster is therefore equal to 0.75. Note that the volume fraction of a random dense packing of equal-sized rigid, spherical particles is 0.6429"31. Since the particles are relatively undeformable, a volume fraction higher than 0.64 is possible only by mixing particles of a spectrum of size 29. The deformation of a composite containing elastic particles in an elastic or elastoplastic matrix is the subject of many recent studies 19'29'32. The present author has
Table 1
Volume fractions in composites containing clusters
Particle Distribution
f
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0.1 0.1 0.1
. . 0.125 0.0001
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modeled the multiaxial elastoplastic constitutive behavior of short fiber/whisker/particulate-reinforced MMCs based on the formulation of Eshelby inclusion method 23 incorporated with Mori-Tanaka mean field theory 24. The full formulation can be found in an array of papers by the author 19-22, and will not be described here. The formulation reported in these studies is intended for composite systems with uniformly dispersed reinforcement in the matrix. A special issue does emerge for the prediction of material responses at high inclusion volume fraction. First, the equivalent inclusion method gives a less reliable solution at a higher particle volume fraction 26'27. At the considered particle volume fraction of about 0.8 inside the clusters, the equivalent inclusion method may not give a very reasonable prediction of the composite's elastic response 24'25'29. As discussed in Kwon and Dharan 29, the equivalent inclusion method may underestimate the Young's modulus in composites with high particle volume fractions. The incorporation of phase continuity as a weighting function in the Mori-Tanaka method only improve the predictions slightly 29. Further, no investigation was conducted to assess the reliability of the equivalent inclusion method for the present case in which the matrix metal in the cluster may deform plastically. In the present study, we will use the equivalent inclusion method to calculate the elastoplastic behavior. Further study will be needed to improve the computational technique for composites with high particle volume fractions. Specifically, we will need to compare with the predictions using the Differential scheme 17 and the Generalized Self-Consistent Method t8 in the future. At a given fc and fo, the constitutive behaviors in terms of the stress-strain relations of both the clustered and unclustered regions are first calculated by using the equivalent inclusion method. Secant moduli were utilized when plastic deformation was incurred in the matrix. Since the reinforcing particulates are assumed to be spherical in shape, the behavior of both clustered and unclustered regions are isotropic. Further, as indicated by Tszeng 22, the macroscopic plastic behavior of a homogeneous composite reinforced with spherical particulates can be characterized by the yon Mises stress. That is, the uniaxial stress-strain curve is actually the effective stress-effective strain curve. For convenience, the calculated effective stress-plastic strain curve is to be represented by the modified Ludwick relation through the least-square best fitting. In the second step, the clustered composite system is subjected to a macroscopic stress; the clustered regions are regarded as inclusions while the unclustered region is the matrix. The computational procedure then repeats to determine the constitutive relations. However, as will be seen later in this paper, the clusters tend to deform plastically at a rather low level of macroscopic stress,
301
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Figure 2 Macroscopic uniaxial stress-strain curves of clustered region (f = 0.8) and the generalized matrix (f = 0.1 )
even before the unclustered region does so. The original formulation for determining the constitutive response was limited to the composite systems whose reinforcing phase is elastic 19-22. Thus, a minor modification to the original formulation is needed for us to analyze the deformation of a composite containing plastically deformable inclusions (clusters). Again, secant modulus was utilized when plastic deformation was incurred in the inclusions. The average stress in the cluster, ~rc, can thus be calculated when the composite is subjected to a macroscopic stress, a. As a consequence of the present approach, the stress, o c, is uniform in the clustered region. In the third step, a new set of calculation is needed for obtaining the internal stress in the reinforcing particles. Again, the computational procedure is exactly the same as that in determining the constitutive response, except now the clusters are subjected to the stress a c. A m o n g others, the normal stress at the p a r t i c l e - m a t r i x interface is of particular importance, as was discussed earlier in the Introduction. The formulation of interfacial stress can be found in Tszeng 21. In summary, the computational procedure consists of the following three steps when the composite with particle clustering is subjected to a macroscopic stress o: (1) Determine the constitutive behavior of both the clustered and unclustered regions. (2) Determine the macroscopic response of the clustered composites and the average stress, o c, in the clustered region. (3) Determine the internal stress state in the cluster subject to the average stress, crc.
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C A L C U L A T E D RESULTS
Material properties
The model composite system is 10 vol.% SiCp reinforced 2124 aluminum alloy. This composite system has received intensive study recently 6'7"1m9 22,25. The general observations to be derived from the theoretical calculations are somewhat generic and therefore should be applicable to other composite systems. The elastic moduli are: E M = 60 GPa, M = 0.3, E e = 342.6 GPa, u e = 0.21, where the superscripts M and P represent the plain matrix (2124 aluminum alloy) and the particle (SIC), respectively. Flow stress of the aluminum matrix is a function of the effective plastic strain, as given by the modified Ludwick relation: Oy = Y + h(~P) n
(3)
with Y = 300 MPa, h = 442.1 MPa, and n = 0.478. As described earlier, three patterns of particle distribution are considered in this study. The data pertaining to each type of particle distribution are summarized in T a b l e 1. For each of the two clustering patterns, variations in c l u s t e r aspect ratio, a, are considered. Note that the effects of volume fraction fc will not be examined in the present study. We also restrict our consideration to the situations that the elongated clusters are aligned in the direction of the major macroscopic stress. The elastic moduli and effective stress-plastic strain relations for the clustered and unclustered regions are obtained by using the inclusion method mentioned earlier. F i g u r e 2 shows the uniaxial stress-strain curves for both the cluster and the generalized matrix (f = 0.1). For
Particle clustering: 7-. C. Tszeng Table 2 clusters
Calculated material properties in the generalized matrix and
Volume Fraction
E (GPa)
u
ao (MPa)
h
n
0.1 0.8
69.2 214.78
0.293 0.237
323 488.7
537. 2106.
0.444 0.388
Table 3 Calculated macroscopic e, u, and initial yield stress for clustered composites Clustering
Aspect ratio, ~
Stress for initial yielding (MPa) matrix/cluster
E (GPa)
~,
Pattern I
1 2 3 1 2 3
323/323 339/272 351/243 323/310 323/253 323/221
69.2 71.5 73.2 --
0.292 0.294 0.294 --
--
--
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--
Pattern II
convenience, the calculated effective stress-plastic strain relations were expressed in terms of the Ludwick equation by least-square fitting. Specifically, we need first to obtain the elastic moduli and associated parameters, Oo (initial yield), h, and n, in the Ludwick relations for homogeneous composite with volume fraction f of 0.1 for the unclustered region, and 0.8 for the clustered region. The obtained data are shown in Table 2.
Mechanical properties The macroscopic behavior of the composite subjected to uniaxial tensile stress in the direction of the 3-axis is considered in this section. The calculated Young's moduli and Poisson's ratios are shown in Table 3 for each type of particle cluster with different aspect ratios. It was found that particle clustering has virtually no effects on the composite Young's moduli and Poisson's ratio, regardless of the
cluster aspect ratio. For composites with spherical clusters in Pattern I, the calculations show that both the clustered and unclustered regions (the latter is actually the plain aluminum matrix) would start yielding at the same macroscopic stress, 033, of 323 MPa. For composites with particle clustering Pattern I and the particles and clusters are all in spherical shape, we have shown rigorously (see Appendix 1) that the macroscopic stresses which lead to the initial yielding in the clustered and unclustered regions are the same. This statement is true regardless the particle volume fraction, fc, in the clusters. However, in the cases that the reinforcing phase is not spherical in shape but with a certain aspect ratio, the earlier statement is not valid anymore. On the other hand, in composites containing spherical clusters in Pattern II, the clustered regions yield at a macroscopic stress o33, of 310 MPa. That is, the clustered region would start yielding at a lower macroscopic stress during uniaxial tension. As one might expect, the macroscopic stress which leads to the initial yielding in the clustered and unclustered regions should be strongly dependent on the cluster aspect ratio, a. The dependency is shown in Table 3 for both clustering patterns. The macroscopic uniaxial stress 033 which leads to the initial yielding in the clustered region decreases with the increasing aspect ratio in both types of clustering patterns. Conversely, the macroscopic initial yield stress in the matrix increases with the increasing aspect ratio in composites with clustering Pattern I. The dependence of macroscopic uniaxial stress, o33, for initial yielding is again shown in Figure 3. It can be concluded that the clustered region would start yielding at an equal or lower macroscopic stress than that in the unclustered region. Further, the composites with clustering Pattern II are relatively more affected. The uniaxial stress-strain curves for selected composite systems with different particle distribution are shown in
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Figure 5
The interfacial normal stress in clusters with a range of aspect ratios. The macroscopic uniaxial stress, a33 = 1 MPa. Circles: cluster aspect ratio = 1 ; Squares: cluster aspect ratio = 2; Triangles: cluster aspect ratio = 3. Solid lines: cluster pattern I; Broken lines: cluster pattern II
Figure 4. Void nucleation/growth/coalescence are assumed not to occur. According to the studies by FEM calculations for the simplified clustering 6"7, the influence of reinforcement clustering on the stress-strain response is not
304
significant at all for composites reinforced with particulates of unity aspect ratio, especially in the range of elastic and small plastic strain. The present calculations basically lead to the same conclusion. When the clusters are not spherical
Particle clustering: 7. C. Tszeng N
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Figure 6 The interfacial normal stress in clusters with a range of aspect ratio. The macroscopic uniaxial stress, 033 400 MPa. Circles: cluster aspect ratio = 1; Squares: cluster aspect ratio = 2; Triangles: cluster aspect ratio = 3. Solid lines: cluster pattern I; Broken lines: cluster pattern II =
in shape, the stress-strain response of the composite systems will be strongly dependent on the aspect ratio of clusters, as is also shown in Figure 4.
Stress concentration The internal stresses in the clustered region are of particular importance in assessing the occurrence of interface failure. In order to determine these stresses, one has to first calculate the average stress in the cluster. For a uniaxial macroscopic stress state, the stress incurred in the aligned clusters has a major component in the direction of the macroscopic stress and two minor stresses in the transverse direction. In this section, major concerns will be on the stress state at the interface between the particle and the matrix metal in the clustered region. For the model composites with spherical clusters of Pattern I subjected to a uniaxial macroscopic stress 033 1 MPa (elastic deformation), the average stress, 0"c, appearing in the clustered region is ( - 0 . 0 8 , - 0 . 0 8 , 1.427) MPa, which seems to show stress concentration. The average stress is applied to the cluster which has a particle volume fraction of 0.8. The calculated distributions of interfacial normal stress on a particle in the clustered region are plotted in Figure 5 for various cluster aspect ratios. For this type of macroscopic stress state, the interfacial normal stress has its m a x i m u m at the pole of the spherical particle. The severity of stress concentration is represented by the ratio of the peak interfacial normal stress to the macorscopic stress 0"33- It is found that the interfacial normal stress on particles in the spherical clustered region (aspect ratio a = 1) is the same as ~---
that in the unclustered region, despite the very high particle volume fraction in the clusters. Hence, within the assumption and limitation of the present computational method, the expected stress concentration caused by particle clustering is not seen when the clusters are in spherical shape and subjected to uniaxial macroscopic stress. On the other hand, the influence of the cluster aspect ratio is obvious in Figure 5. The intensity of stress concentration in the clustered region increases with the cluster aspect ratio. Similar trends can also be found in composites with clustering Pattern II. For composites with the same cluster aspect ratios and being subjected to the same uniaxial macroscopic stress, the peak interfacial normal stress on reinforcing particles is marginally higher in composites with clustering Pattern II. When a composite system with or without particles clustering deformed plastically caused by a uniaxial macroscopic stress, the normal stress at the m a t r i x - p a r t i c l e interface again has its m a x i m u m at the pole on the spherical particle. At macroscopic stress 0.33 = 400 MPa, the m a x i m u m interfacial normal stresses on particles in the clustered region are plotted in Figure 6. The general trends are the same as those with composites undergoing elastic deformation, i.e.: (1) stress concentration on the reinforcing particles caused by particle clustering is not significant when the clusters are in spherical shape; (2) tremendous influence of the cluster aspect ratio on the maximum normal stress at the interface; and (3) the peak interfacial normal stress is higher in composites for particles with clustering Pattern II. A comparison between Figures 5 and 6 also indicates that the stress concentration is more severe when the matrix is deformed plastically.
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611 = c~22 (MPa) Figure 7 A diagram showing the comparison of the calculated nucleation loci for axisymmetric deformation of composites. Bonding strength of 500 (empty symbols) and 1000 MPa (solid symbols) are considered
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MEAN STRESS (MPa) Figure 8 A plot of the square root of nucleation strain as a function of the macroscopic mean stress for axisymmetric deformation. Bonding strength of 500 (empty symbols) and 1000 MPa (solid symbols) are considered
Void nucleation
In the following, we will address on the process of void nucleation from the matrix-particle interface in the clustered regions. It was shown earlier that the shape of clusters seems to be the most influential factor in void nucleation. In this work, composites with spherical clusters (a = 1) will be considered while the quantitative influence of cluster aspect ratio will be addressed in a separate paper. As theorized by many investigators, there are two prevailing criteria for void nucleation from the interface, namely, energy criterion and stress criterion 9"1°'z°. In the considered SiCp-reinforced composites, the size of SiC
306
particle is in the order of 10/xm 1'3'2°. According to most of the previous studies, the energy criterion for void nucleation on a particle of this size is automatically satisfied. That is, the stress criterion solely governs the occurrence of void nucleation. In this respect, the criterion states that the tensile normal stress at the inclusion-matrix interface has to exceed the bonding strength of the interface. In this respect, a reasonable value of the bonding strength in a SiCaluminum alloy system is in the range between 400 and 900 MPa 21. It is further understood that the bonding strength is strongly dependent on the processing historyL In this respect, bonding strengths of 500 and 1000 MPa are used in this study to examine the influences of particle clustering on
Particle clustering: T. C. Tszeng the void nucleation and therefore on the ductility of composites. The nucleation loci after LeRoy et al. 8 is used here to indicate the occurrence of void nucleation in a multiaxial stress state. The axisymmetric macroscopic stress state is assumed to take the form: O l l = 0"22 = 00"33
(4)
where 0 < p < 1. By changing the value of P, a range of the mean stress and deviatoric/effective stress can be obtained. The same procedure was used in Tszeng 2°, pertaining to void nucleation in spheriodized steels. The nucleation loci for different particle distributions are shown in Figure 7. The same figure also shows the loci corresponding to initial yield and hydrostatic stress state. The composite with homogeneous particle distribution has the same resistance to void nucleation. The composite with particle clustering Pattern II shows less resistance to void nucleation. Also note that void nucleation occurs approximately at the elastic limit during uniaxial loading (oll = 022 ~-- 0 ) . The nucleation strains in square root are plotted as a function of the mean stress am in Figure 8. A higher mean (tensile) stress leads to a smaller nucleation strain. Again, the tremendous influence of bonding strength is obvious. With the higher bonding strength, the influences of particle clustering are now more evident than that indicated in Figure 7. For example, the nucleation strain in composites with clustering Pattern II subjected to uniaxial loading (am ~ 175 MPa) is about 65% of that corresponding to the homogeneous particle distribution. Several experimental studies cited in the Introduction indicated the increased tendency of failure initialized in the regions with smaller spacing between particles. This type of particle clustering is different from what we have discussed in this paper. In the present approach, no account was furnished to address the closeness between two particles. Strictly speaking, what was considered in the present study is the mean effect of local particle concentration. From the standpoint of void nucleation, the most detrimental region is very possibly between two particles with the minimum spacing. Thus, the present calculated nucleation loci are on the conservative side.
CONCLUDING REMARKS This paper reports our study of the influences of particle clustering on the mechanical properties in particle reinforced composites. There are several issues of concern when applying the equivalent inclusion method to the present problem of particle clustering. First, the equivalent inclusion method gives a less reliable solution at a higher particle volume fraction 26 '27 "29 . The particle volume fraction can be as highh as 0.8 inside the clusters. Although it is not the major objective of this study to investigate or develop a computation method for composites with high inclusion volume fraction, we do believe that there is a strong need to further study this issue. Further, no study was conducted to assess the reliability of the equivalent inclusion method for
the present case in which the matrix metal in the cluster may deform plastically. This was one of the reasons preventing us from studying the effects of particle volume fraction, fc, in clusters. Second, the present formulation dropped the higher order terms of the stress/strain fields such that they became uniform in the inclusions. This presumption stands well when only elastic deformation is involved in both the inclusion and the matrix. Apparently, further study is needed to shed some light onto the situations when both phases can deform plastically. As far as void nucleation in the matrix is concerned, the local plastic strain is one of the controlling factors 6. Since the matrix stress/strain in the context of equivalent inclusion method is the volume averaged measure of the short range stress/strain fields around the inclusions, it is not very meaningful to relate this measure to the void nucleation in the matrix. Within the assumptions and limitation of the analysis method based upon the equivalent inclusion method, the present study reached the following concluding remarks: (1) Particle clustering has no essential effects on the macroscopic elastic moduli of the composites with different particle distributions. (2) When subjected to the same macroscopic stress, the clustered region would start yielding at an equal or lower macroscopic stress than that in the unclustered region. (3) The influence of reinforcement clustering on the stressstrain response is not significant at all for composites reinforced with particulates of unity aspect ratio, especially in the range of elastic and small plastic strain. When the clusters are elongated in the major loading direction, the stress-strain response of the composite systems will be strongly dependent on the aspect ratio of clusters. (4) Stress concentration on the reinforcing particles caused by particle clustering is not significant when the clusters are spherical in shape. We observed the tremendous influence of the cluster aspect ratio on the maximum normal stress at the interface. (5) The theoretical calculations did show the adverse effects of clustering on the resistance to void nucleation for composite systems with a higher interfacial bonding strength.
APPENDIX In this brief account, we will prove that the clustered and unclustered region would yield at the same macroscopic stress when both the particles and clusters are spherical in shape. To proceed, the explicit expressions developed by Weng 2s for spherical inclusion embedded in matrix is probably easier to use than the one developed in Tszengl9 22. Therefore, we will liberally use the equations as well as most of the notations in Weng 28. In the following, subscripts 0, and 1 denote variables for the plain matrix and the particle, respectively.
307
Particle clustering: T. C. Tszeng Let the initial yield stress in the plain matrix be Y. The initial yield stress of a composite reinforced by spherical particles of volume fraction f is28: Y cr = - bo
(A1)
By using equations (A2) and (A5) in (A12), it is straightforward to show that o-m and Oc in equations (A11) and (A12) are equivalent. Thus, we have proved that the clustered and unclustered region would yield at the same macroscopic stress when both the particles and clusters are spherical in shape.
where bo = (flo(#l -/~o) + q = (f
+
#o)/q
(A2)
'
f fio)(#~ - #o) + #o.
(A3)
In these expressions, /z is the Lame constant, f = 1 - f , and/30 is related to the Eshelby's transformation tensor, S, i.e.,
1. 2. 3. 4.
2 (4 - 5Vo)
flo --
REFERENCES
(A4)
15 (1 - Vo)
5.
where v is the Poisson's ratio. For the present case, we actually have the initial yield stress of the clustered region with f = fc in equation (A1). The corresponding Lame constant for the cluster is:
6. 7. 8.
tt = tto [1 + /
f (tq -/Xo)
f,
]
flo(/g'l --/'to) "]-/-toJ "
(A5)
For the hybrid composite with spherical clusters in the plain matrix, the volume fraction of the generalized inclusion (cluster) is F. The unclustered region (plain matrix) will yield at the stress 2s. O'm
Y b
(A6)
11. 12. 13. 14. 15. 16.
where b = (flo(tt -- tto) + #o)/qc
(A7)
17. 18.
qc = (F + F '/30)(# - / t o ) + #o-
(A8)
19. 20. 21.
In these equations, F' = 1 - F, and t~ is given by equation (A5). On the other hand, the cluster will yield at the stress Y
crc- bob1
(A9)
where b i -- --./~ qc
(A10)
qcY O-m
=
Bo(# - G#o) + #o"
(A11)
24. 25. 26.
28. 29. 30. 31.
Combining equations (A9) and (A10), we have
qcY Oc = #b o.
22. 23.
27.
In equation (A10),/~ and qc are given by equations (A5) and (A8), respectively. Substituting equation (A7) in equations (A6) gives
308
9. 10.
(A12)
32.
Lloyd, D.J., Composites Science and Technology, 1989, 35, 159. Mortensen, A., Cornie, J. A. and Flemings, M. C., Journal of Metals, 39, 1988, February, 12. Weng, B.J., Chang, S.T. and Shiau, J.S., Scripta Metallurgica et Materialia, 1992, 27, 1127. Hunt Jr, W. H., Richmond, O. and Young, R. D., ICCM/VI, 1989, 2.209. Rack, H. J. and Ratnaparkhi, Journal of Metals, 39, 1988, November, 55. Llorca, J., Needleman, A. and Suresh, S., Acta Metallurgica et Materialia, 1991, 39, 2317. Christman, T., Needleman, A. and Suresh, S., Acta Metallurgica, 1989, 37, 3029. LeRoy, G., Embury, J.D., Edward, G. and Ashby, M.F., Acta Metallurgica, 1981, 29, 1509. Fisher, J. R. and Gurland, J., Metal Science, 15, 1981, 193. Argon, A.S., Im, J. and Safoglu, R., Metallurgical Transactions, 1975, 6A, 825. Shi, N., Wilner, B. and Arsenault, R.J., Acta Metallurgica et Materialia, 1992, 40, 2841. Everett, R.K., Scripta Metallurgica et Materialia, 1993, 28, 663. Komenda, J. and Henderson, P.J., Scripta Metallurgica et Materialia, 1993, 28, 553. Ferry, M., Munroe, P.R. and Crosky, A., Scripta Metallurgica et Materialia, 1993, 28, 1235. Davidson, D., Metallurgical Transactions, 1991, 22A, 113. Lewandowski, J.J., Liu, C. and Hunt, W.H. Jr, Materials Science and Engineering, 1989, A107, 241. Norris, A.N., Mechanics of Materials, 1985, 4, 1. Christensen, R.M. and Lo, K.H., Mechanics and Physics of Solids, 1979, 27, 315. Tszeng, T.C., Composites Science and Technology, 1994, 51, 75. Tszeng, T.C., Scripta Metallurgica et Materialia, 1993, 28, 1520. Tszeng, T. C., ASME J. Engineering Materials and Technology, 1997, in review. Tszeng, T.C., Journal of Composite Materials, 1994, 28, 800. Eshelby, J.D., Proceeding of the Royal Society, London, 1957, A241, 376. Mori, T. and Tanaka, K., Acta Metallurgica, 1973, 21, 571. Arsenault, R.J., Wang, L. and Feng, C.R., Acta Metallurgica et Materialia, 1991, 39, 47. Christensen, R.M., Journal of Mechanics and Physics of Solids, 1990, 38, 379. Christensen, R., Schantz, H. and Shapiro, J., Journal of Mechanic and Physics of Solids, 1992, 40, 69. Weng, G.J., Journal of Mechanics and Physics of Solids, 1990, 38, 419. Kwon, P. and Dharan, C.K.H., Acta Metallurgica Materiala, 1995, 43, 1141. Kao, A.S., Kuhn, H.A., Richmond, O. and Spitzig, W.A., Metallurgical Transactions, 1989, 20A, 1735. Tszeng, T. C., Fabrication of Particulates Reinforced Metal Composites, ed. J. Masounave and F. G. Hamel. ASM International, 1990, 53. Arnold, S.M., Pindera, M.-J. and Wilt, T.E., International Journal of Plasticity, 1996, 12, 507.
PII: S 1359-8368(97)00031-0
ELSEVIER
© 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00
The effects of particle clustering on the mechanical behavior of particle reinforced composites
T. C. Tszeng* Department of Mechanical Engineering, University of Nebraska, Lincoln, NE 685880656, USA (Received 25 April 1996; accepted 9 June 1997) Particle clustering is thought to be one of the detrimental factors to the performance of particle reinforced composites. In this paper, the effects of particle clustering on the macroscopic mechanical behavior and the micromechanics of particle-reinforced metal matrix composites (MMCs) were predicted theoretically by using the equivalent inclusion method. Two types of particle clustering were examined. The calculation results are focused on the effects of clustering on (1) the macroscopic behavior of the composite, (2) stress concentration, and (3) void nucleation. © 1998 Elsevier Science Limited. All rights reserved (Keywords: particle clustering; mechanical behaviour; void nucleation)
INTRODUCTION The reinforcing particles are more or less non-uniformly dispersed in metal matrix composites (MMCs). Particle clustering is originated from the manufacturing processes. In cast composites, the fine particles may be pushed by the advancing solidification front and eventually trapped in the interdendritic regions. It is not unusual to find a large area of matrix which is free of particles [1]. The tendency of particle clustering depends on the solidification rate of the meltS'z; lower solidification rate leads to an increase in particle clustering. In the powder metallurgy route of making composites, the fine particles may remain in the form of algomerate in the blending and mixing stages, and formed into clusters in the composites 3 5. In literature, the terminology of particle clustering was used to describe two types of uneven local concentration of particle distribution. In the first type, particles are quasirandomly distributed, i.e., the statistical distribution of interparticle distance is invariant with respect to the position in space. For particle distributions in true randomness, the distance between two neighboring particles follows a Poisson's distribution. Thus, there are situations where the interparticle distance between two particles can be much smaller than the mean distance. In the strictest sense, particle distribution of this type is a product of the nature, instead of being a defect in manufacturing. The regular clustering used by the FEM analyses 6,v is an approximate to * Formerly with Scientific Forming Technologies Corporation, Columbus, OH 43202, USA.
this type of distribution, in which the distribution function of interparticle distance is discrete, yet invariant in space. In the second type of particle clustering, the mean value of the statistical distribution of interparticle distance in some regions is higher than the mean of the surrounding region. This type of distribution includes the particle clustering caused by the pushing of solidification front. In this paper, we will name the first type of aforementioned particle clustering as local inhomogeneity, and reserve the term 'clustering' for the second type of clustering. The effects of clustering and local inhomogeneity of particles on the mechanical properties of composites were the topic of several prior studies. One of the major concerns associated with particle clustering is the promoted tendency of void nucleation/growth/coalescence from the clustered regions. Some earlier studies in this respect were on the particle-strengthened alloys, for example, spheroidized low carbon steels 8-1°. For these materials with second-phase particles at a relatively low volume fraction compared with the usual MMCs, a local region that has a high volume fraction of particles would serve as the site of void nucleation at the lowest strain. This was attributed to the increased stress concentration at the particle-matrix interface in the regions of higher particle volume fraction 4'8 10 The works conducted by Fisher and Gurland 9 and Argon e t al. 1° on the particle-strengthened alloys pointed out the adverse effects of particle local inhomogeneity. Recently, several theoretical treatments of the effects of 'clustering' (particle local inhomogeneity, in our term) were addressed specifically for M M C s 6'7"11. The finite element methodbased works of Llorca et al. 6 and Christman et al. 7 are
299
Particle clustering: T. C. Tszeng involved with deformation analysis of a unit cell which comprises matrix and reinforcement phase in a certain geometrical arrangement in order to represent particle local inhomogeneity in composite systems, i.e., the distances between two or three adjacent reinforcements are shortened. Because of the geometrical arrangement, the 'clustering' is restricted to the interactions between at most two or three particles/short fibers. Repetition of unit cell in space implies that the composite would consist uniformly distributed 'clusters' each of which only contains two or three particles. Therefore, this approach does not properly represent the clustering of a group of particles and the associated mechanical responses. Everett ~2 studied the implications of local inhomogeneity of fiber distribution by using a large number of tuples. The axial strength of a bundle of continuous fibers is shown to decrease with the increase in standard deviation of volume fraction. In that study, the fiber distribution is established by the mathematical model. While the distribution is more realistic than those used by Llorca et al. 6 and Christman et al. 7, it certainly is more complicated for a mathematical treatment of the associated mechanics. Komenda and Henderson 13 experimentally determined the creep rupture strength of shortalumina fibers reinforced aluminum. They attributed the differences in creep rupture strength to the different fiber distributions. They first used a scheme to determine the fiber-free zone in the specimens. Correlation was found between the creep rupture strength and the coefficient of variation of zone size which is defined as the SD of zone size divided by the mean zone size. For heat treated SiC-A1 composites, Ferry et al. 14 found that the grains of aluminum matrix is finer in the clustered areas. This was caused by the inhibition on grain growth imposed by the reinforcing particles with shorter interparticle distance. Davidson ~5 experimentally observed the crack initiation in the region with particle local inhomogeneity. It is thought that the voids in the area of local inhomogeneity seem to originate from the interface. Thus, smaller space between particles tends to promote crack initiation. This observation was attributed to the stress concentration in the region of local inhomogeneity4:4. It is also suspected that the interfacial strength in the area of higher local particle fraction is lower than that in the homogeneous, unclustered region ]5. Lewandowski et al. 16 conducted a similar experimental study to determine the effects of particle local inhomogeneity on crack initiation. According to these experimental studies, there seems to exist a positive correlation between the tendency of crack initiation and the particle local inhomogeneity. Shi et al. ~~ seemed to carry out the first study using a twostep model to examine the effects of particle clustering. In the first step, the elastic moduli associated with the clustered and unclustered regions are determined by the Eshelby's inclusion method. Next, a finite element method is utilized to calculate the macroscopic response. In that study, only elastic response and initial yielding are addressed. According to the above brief review, it is still not very clear how clustering affects the mechanics and mechanical
300
CLUSTERS 0 0 o 00000 o
°go% 000
000
O0
0
%°0Oo
0
--
J
--"-~
/
--
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/
/'
/
o Oo 000%°
ogooo
o Oo
/
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0
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o
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000%0
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o
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00
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o
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0
o
o
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0
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o
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[al
0
~ %~%~o
o o o%°°;°o° o ooo°°° 0
o ~ o° o
0
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o 0
0
[bl
Figure 1 The patterns of particle clustering. (a) Pattern I, (b) Pattern lI
behavior. There is a need to further investigate the mechanics of particle reinforced composites when particle clustering appears. In particular, the interfacial stress in the clustered regions when the composite is subjected to a macroscopic stress is one of the most important subjects. In this paper, we try to provide some insight into the effects of particle clustering by theoretical calculation. The macroscopic as well as the microscopic response of the clustered composite systems are to be examined theoretically. In this connection, we have to note that the composites with particle clustering are a special type of functional gradient materials (FGMs). Extensive studies were conducted recently. A technical issue related to the modeling of FGMs is the high inclusion volume fraction. The effective mechanical moduli of the material with high inclusion volume fraction can be predicted satisfactorily by the Differential scheme ~7, the Generalized Self-Consistent Method is, the Generalized Inclusion Method 19-28 and the modification by Kwon and Dharan 29. In this study, the composites with particle clustering are treated as a hybrid composite material, i.e., the clusters are considered to be the generalized inclusion while the unclustered region is the generalized matrix material. The approach uses a two-stage analysis based on the equivalent inclusion method. The applicability of an equivalent inclusion method to the concerned problem will be discussed in the next section. The calculation results are to be focused on the effects of clustering on: (1) the macroscopic behavior of the composite; (2) stress concentration; and (3) void nucleation.
THEORETICAL TREATMENT The shape and distribution of particle clusters in particle reinforced MMCs is very irregular, depending on the processing method and history. To simplify the analysis, particle clustering is characterized by: (1) the pattern of the clusters; (2) the shape of the clusters; (3) the volume fraction of the clusters in the matrix; and (4) the particle volume fraction in the clusters. In this work, three patterns of particle distribution are studied. The reference system has particles homogeneously dispersed in the matrix, i.e., no particle clustering or local inhomogeneity. The two clustering patterns are: Pattern I--segregated particle-rich (clusters) and particle-free regions ( F i g u r e l a ) , and Pattern
Particle clustering: 7". C. Tszeng II--uniformly dispersed particles with few isolated clusters (Figure lb). In clustering Pattern II, the composite is considered to comprise clusters in a generalized matrix material. Any region in the composite is categorized into clustered or unclustered. The clusters are regions in which the particle volume fraction is higher than the nominal particle volume fraction. The particle volume fraction in the unclustered region is lower than the nominal volume fraction. In this paper, matrix simply denotes the plain matrix material surrounding the reinforcement particles. The clusters are assumed to be in the shape of an ellipsoid. Reinforcing particles are spherical. To facilitate our discussion, three independent measures of volume fraction are used. First, the nominal volume fraction of the particle in the composite, denoted byf. This is the volume fraction usually referred to. Second, the volume fraction of particle within the clusters, denoted byfc. Third, the volume fraction of cluster in the composites, denoted by F. A fourth, dependent, measure is the volume fraction of particles in the unclustered region, denoted byfo. A relation between these measures of volume fraction is: fcF + (1 --F)fo= f .
(1)
To estimate the volume fraction of clusters, F, in a clustered composite, we will consider that the mean diameter of the spherical clusters is d and the mean distance between a cluster and its nearest neighbors is D. It can be shown by using the equation of Kao et al. 30 that the volume fraction of distributed clusters is F=
2 \2A/
"
(2)
With a given f, fc and F, the fourth measure, fo, is calculated by using equation (1). However, if an estimate of fo is readily available, either F orf~ can be obtained by equation (1). Note that equation (2) is only applicable to the cases when the clusters are spherical in shape. The microstructure of A356-15vol.%SiC composite reported by Lloyd ~ is one of the available examples showing particle clustering in MMCs. At an estimated cooling rate of I°C s 1 (figure l l d in Lloyd1), the mean cluster diameter, d, is about 50 gm and the mean distance, D, between a cluster and its nearest neighbor is about 120/xm. Note that these data are very crude and by no means represent a good estimation. Substituting these data in equation (2) gives the volume fraction of clusters, F, as 0.2. In the aforementioned picture of Lloyd', it shows that the volume fraction fo of particles in the matrix outside the clusters is essentially zero. Equation (1) is then reduced to f~F = f, and the volume fraction fc of particle within the cluster is therefore equal to 0.75. Note that the volume fraction of a random dense packing of equal-sized rigid, spherical particles is 0.6429"31. Since the particles are relatively undeformable, a volume fraction higher than 0.64 is possible only by mixing particles of a spectrum of size 29. The deformation of a composite containing elastic particles in an elastic or elastoplastic matrix is the subject of many recent studies 19'29'32. The present author has
Table 1
Volume fractions in composites containing clusters
Particle Distribution
f
F
Homogeneous Clustering Pattern I Clustering Pattern II
0.1 0.1 0.1
. . 0.125 0.0001
f~ . 0.8 0.8
fo
e~
. 0 0.1
1-3 1-3
modeled the multiaxial elastoplastic constitutive behavior of short fiber/whisker/particulate-reinforced MMCs based on the formulation of Eshelby inclusion method 23 incorporated with Mori-Tanaka mean field theory 24. The full formulation can be found in an array of papers by the author 19-22, and will not be described here. The formulation reported in these studies is intended for composite systems with uniformly dispersed reinforcement in the matrix. A special issue does emerge for the prediction of material responses at high inclusion volume fraction. First, the equivalent inclusion method gives a less reliable solution at a higher particle volume fraction 26'27. At the considered particle volume fraction of about 0.8 inside the clusters, the equivalent inclusion method may not give a very reasonable prediction of the composite's elastic response 24'25'29. As discussed in Kwon and Dharan 29, the equivalent inclusion method may underestimate the Young's modulus in composites with high particle volume fractions. The incorporation of phase continuity as a weighting function in the Mori-Tanaka method only improve the predictions slightly 29. Further, no investigation was conducted to assess the reliability of the equivalent inclusion method for the present case in which the matrix metal in the cluster may deform plastically. In the present study, we will use the equivalent inclusion method to calculate the elastoplastic behavior. Further study will be needed to improve the computational technique for composites with high particle volume fractions. Specifically, we will need to compare with the predictions using the Differential scheme 17 and the Generalized Self-Consistent Method t8 in the future. At a given fc and fo, the constitutive behaviors in terms of the stress-strain relations of both the clustered and unclustered regions are first calculated by using the equivalent inclusion method. Secant moduli were utilized when plastic deformation was incurred in the matrix. Since the reinforcing particulates are assumed to be spherical in shape, the behavior of both clustered and unclustered regions are isotropic. Further, as indicated by Tszeng 22, the macroscopic plastic behavior of a homogeneous composite reinforced with spherical particulates can be characterized by the yon Mises stress. That is, the uniaxial stress-strain curve is actually the effective stress-effective strain curve. For convenience, the calculated effective stress-plastic strain curve is to be represented by the modified Ludwick relation through the least-square best fitting. In the second step, the clustered composite system is subjected to a macroscopic stress; the clustered regions are regarded as inclusions while the unclustered region is the matrix. The computational procedure then repeats to determine the constitutive relations. However, as will be seen later in this paper, the clusters tend to deform plastically at a rather low level of macroscopic stress,
301
Particle clustering: T. C. Tszeng F coo L 0 7OO W S T R E S S
f-0.8
f-O. I
600
500
400
300
( M P a
}
200
100
I
0.00
0.05
I
I
O.i0
0.t5
PLASTIC
I
0.20
I
I
0.25
0.30
STRAIN
Figure 2 Macroscopic uniaxial stress-strain curves of clustered region (f = 0.8) and the generalized matrix (f = 0.1 )
even before the unclustered region does so. The original formulation for determining the constitutive response was limited to the composite systems whose reinforcing phase is elastic 19-22. Thus, a minor modification to the original formulation is needed for us to analyze the deformation of a composite containing plastically deformable inclusions (clusters). Again, secant modulus was utilized when plastic deformation was incurred in the inclusions. The average stress in the cluster, ~rc, can thus be calculated when the composite is subjected to a macroscopic stress, a. As a consequence of the present approach, the stress, o c, is uniform in the clustered region. In the third step, a new set of calculation is needed for obtaining the internal stress in the reinforcing particles. Again, the computational procedure is exactly the same as that in determining the constitutive response, except now the clusters are subjected to the stress a c. A m o n g others, the normal stress at the p a r t i c l e - m a t r i x interface is of particular importance, as was discussed earlier in the Introduction. The formulation of interfacial stress can be found in Tszeng 21. In summary, the computational procedure consists of the following three steps when the composite with particle clustering is subjected to a macroscopic stress o: (1) Determine the constitutive behavior of both the clustered and unclustered regions. (2) Determine the macroscopic response of the clustered composites and the average stress, o c, in the clustered region. (3) Determine the internal stress state in the cluster subject to the average stress, crc.
302
C A L C U L A T E D RESULTS
Material properties
The model composite system is 10 vol.% SiCp reinforced 2124 aluminum alloy. This composite system has received intensive study recently 6'7"1m9 22,25. The general observations to be derived from the theoretical calculations are somewhat generic and therefore should be applicable to other composite systems. The elastic moduli are: E M = 60 GPa, M = 0.3, E e = 342.6 GPa, u e = 0.21, where the superscripts M and P represent the plain matrix (2124 aluminum alloy) and the particle (SIC), respectively. Flow stress of the aluminum matrix is a function of the effective plastic strain, as given by the modified Ludwick relation: Oy = Y + h(~P) n
(3)
with Y = 300 MPa, h = 442.1 MPa, and n = 0.478. As described earlier, three patterns of particle distribution are considered in this study. The data pertaining to each type of particle distribution are summarized in T a b l e 1. For each of the two clustering patterns, variations in c l u s t e r aspect ratio, a, are considered. Note that the effects of volume fraction fc will not be examined in the present study. We also restrict our consideration to the situations that the elongated clusters are aligned in the direction of the major macroscopic stress. The elastic moduli and effective stress-plastic strain relations for the clustered and unclustered regions are obtained by using the inclusion method mentioned earlier. F i g u r e 2 shows the uniaxial stress-strain curves for both the cluster and the generalized matrix (f = 0.1). For
Particle clustering: 7-. C. Tszeng Table 2 clusters
Calculated material properties in the generalized matrix and
Volume Fraction
E (GPa)
u
ao (MPa)
h
n
0.1 0.8
69.2 214.78
0.293 0.237
323 488.7
537. 2106.
0.444 0.388
Table 3 Calculated macroscopic e, u, and initial yield stress for clustered composites Clustering
Aspect ratio, ~
Stress for initial yielding (MPa) matrix/cluster
E (GPa)
~,
Pattern I
1 2 3 1 2 3
323/323 339/272 351/243 323/310 323/253 323/221
69.2 71.5 73.2 --
0.292 0.294 0.294 --
--
--
--
--
Pattern II
convenience, the calculated effective stress-plastic strain relations were expressed in terms of the Ludwick equation by least-square fitting. Specifically, we need first to obtain the elastic moduli and associated parameters, Oo (initial yield), h, and n, in the Ludwick relations for homogeneous composite with volume fraction f of 0.1 for the unclustered region, and 0.8 for the clustered region. The obtained data are shown in Table 2.
Mechanical properties The macroscopic behavior of the composite subjected to uniaxial tensile stress in the direction of the 3-axis is considered in this section. The calculated Young's moduli and Poisson's ratios are shown in Table 3 for each type of particle cluster with different aspect ratios. It was found that particle clustering has virtually no effects on the composite Young's moduli and Poisson's ratio, regardless of the
cluster aspect ratio. For composites with spherical clusters in Pattern I, the calculations show that both the clustered and unclustered regions (the latter is actually the plain aluminum matrix) would start yielding at the same macroscopic stress, 033, of 323 MPa. For composites with particle clustering Pattern I and the particles and clusters are all in spherical shape, we have shown rigorously (see Appendix 1) that the macroscopic stresses which lead to the initial yielding in the clustered and unclustered regions are the same. This statement is true regardless the particle volume fraction, fc, in the clusters. However, in the cases that the reinforcing phase is not spherical in shape but with a certain aspect ratio, the earlier statement is not valid anymore. On the other hand, in composites containing spherical clusters in Pattern II, the clustered regions yield at a macroscopic stress o33, of 310 MPa. That is, the clustered region would start yielding at a lower macroscopic stress during uniaxial tension. As one might expect, the macroscopic stress which leads to the initial yielding in the clustered and unclustered regions should be strongly dependent on the cluster aspect ratio, a. The dependency is shown in Table 3 for both clustering patterns. The macroscopic uniaxial stress 033 which leads to the initial yielding in the clustered region decreases with the increasing aspect ratio in both types of clustering patterns. Conversely, the macroscopic initial yield stress in the matrix increases with the increasing aspect ratio in composites with clustering Pattern I. The dependence of macroscopic uniaxial stress, o33, for initial yielding is again shown in Figure 3. It can be concluded that the clustered region would start yielding at an equal or lower macroscopic stress than that in the unclustered region. Further, the composites with clustering Pattern II are relatively more affected. The uniaxial stress-strain curves for selected composite systems with different particle distribution are shown in
4OO 375
S 3
350
R I X .
PATTERN I
-~_
325 30O
M
250 225
2 0 0
2
CLUSTER Figure 3
i
i
!
ASPECT
3
RATIO
The macroscopic uniaxial stress, a3~, for initial yielding in the clustered and unclustered regions
303
Particle clustering: T. C. Tszeng F L 0 W
coo
S
600
T R
500
E S
400
700
a-3
il-2
8-t
S 300
( M P
200
8
10o
) I
0.00
I
0.05
I
0.I0
I
0.15
PLASTIC Figure 4
I
0.20
I
0.25
0.30
STRAIN
Macroscopic uniaxial stress-strain curves of composites containing clusters of Pattern II with a range of cluster aspect ratio
N 0
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R M A
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L 1.5
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M P a
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-0.5
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15
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30
ANGLE
I
45
FROM
I
60
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75
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90
APEX
Figure 5
The interfacial normal stress in clusters with a range of aspect ratios. The macroscopic uniaxial stress, a33 = 1 MPa. Circles: cluster aspect ratio = 1 ; Squares: cluster aspect ratio = 2; Triangles: cluster aspect ratio = 3. Solid lines: cluster pattern I; Broken lines: cluster pattern II
Figure 4. Void nucleation/growth/coalescence are assumed not to occur. According to the studies by FEM calculations for the simplified clustering 6"7, the influence of reinforcement clustering on the stress-strain response is not
304
significant at all for composites reinforced with particulates of unity aspect ratio, especially in the range of elastic and small plastic strain. The present calculations basically lead to the same conclusion. When the clusters are not spherical
Particle clustering: 7. C. Tszeng N
0
1500
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A k 1000
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\ 750
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45
FROM
60
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75
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APEX
Figure 6 The interfacial normal stress in clusters with a range of aspect ratio. The macroscopic uniaxial stress, 033 400 MPa. Circles: cluster aspect ratio = 1; Squares: cluster aspect ratio = 2; Triangles: cluster aspect ratio = 3. Solid lines: cluster pattern I; Broken lines: cluster pattern II =
in shape, the stress-strain response of the composite systems will be strongly dependent on the aspect ratio of clusters, as is also shown in Figure 4.
Stress concentration The internal stresses in the clustered region are of particular importance in assessing the occurrence of interface failure. In order to determine these stresses, one has to first calculate the average stress in the cluster. For a uniaxial macroscopic stress state, the stress incurred in the aligned clusters has a major component in the direction of the macroscopic stress and two minor stresses in the transverse direction. In this section, major concerns will be on the stress state at the interface between the particle and the matrix metal in the clustered region. For the model composites with spherical clusters of Pattern I subjected to a uniaxial macroscopic stress 033 1 MPa (elastic deformation), the average stress, 0"c, appearing in the clustered region is ( - 0 . 0 8 , - 0 . 0 8 , 1.427) MPa, which seems to show stress concentration. The average stress is applied to the cluster which has a particle volume fraction of 0.8. The calculated distributions of interfacial normal stress on a particle in the clustered region are plotted in Figure 5 for various cluster aspect ratios. For this type of macroscopic stress state, the interfacial normal stress has its m a x i m u m at the pole of the spherical particle. The severity of stress concentration is represented by the ratio of the peak interfacial normal stress to the macorscopic stress 0"33- It is found that the interfacial normal stress on particles in the spherical clustered region (aspect ratio a = 1) is the same as ~---
that in the unclustered region, despite the very high particle volume fraction in the clusters. Hence, within the assumption and limitation of the present computational method, the expected stress concentration caused by particle clustering is not seen when the clusters are in spherical shape and subjected to uniaxial macroscopic stress. On the other hand, the influence of the cluster aspect ratio is obvious in Figure 5. The intensity of stress concentration in the clustered region increases with the cluster aspect ratio. Similar trends can also be found in composites with clustering Pattern II. For composites with the same cluster aspect ratios and being subjected to the same uniaxial macroscopic stress, the peak interfacial normal stress on reinforcing particles is marginally higher in composites with clustering Pattern II. When a composite system with or without particles clustering deformed plastically caused by a uniaxial macroscopic stress, the normal stress at the m a t r i x - p a r t i c l e interface again has its m a x i m u m at the pole on the spherical particle. At macroscopic stress 0.33 = 400 MPa, the m a x i m u m interfacial normal stresses on particles in the clustered region are plotted in Figure 6. The general trends are the same as those with composites undergoing elastic deformation, i.e.: (1) stress concentration on the reinforcing particles caused by particle clustering is not significant when the clusters are in spherical shape; (2) tremendous influence of the cluster aspect ratio on the maximum normal stress at the interface; and (3) the peak interfacial normal stress is higher in composites for particles with clustering Pattern II. A comparison between Figures 5 and 6 also indicates that the stress concentration is more severe when the matrix is deformed plastically.
305
Particle clustering: 7-. C. Tszeng
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200
400
600
800
611 = c~22 (MPa) Figure 7 A diagram showing the comparison of the calculated nucleation loci for axisymmetric deformation of composites. Bonding strength of 500 (empty symbols) and 1000 MPa (solid symbols) are considered
0.7
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0
200
400
600
800
MEAN STRESS (MPa) Figure 8 A plot of the square root of nucleation strain as a function of the macroscopic mean stress for axisymmetric deformation. Bonding strength of 500 (empty symbols) and 1000 MPa (solid symbols) are considered
Void nucleation
In the following, we will address on the process of void nucleation from the matrix-particle interface in the clustered regions. It was shown earlier that the shape of clusters seems to be the most influential factor in void nucleation. In this work, composites with spherical clusters (a = 1) will be considered while the quantitative influence of cluster aspect ratio will be addressed in a separate paper. As theorized by many investigators, there are two prevailing criteria for void nucleation from the interface, namely, energy criterion and stress criterion 9"1°'z°. In the considered SiCp-reinforced composites, the size of SiC
306
particle is in the order of 10/xm 1'3'2°. According to most of the previous studies, the energy criterion for void nucleation on a particle of this size is automatically satisfied. That is, the stress criterion solely governs the occurrence of void nucleation. In this respect, the criterion states that the tensile normal stress at the inclusion-matrix interface has to exceed the bonding strength of the interface. In this respect, a reasonable value of the bonding strength in a SiCaluminum alloy system is in the range between 400 and 900 MPa 21. It is further understood that the bonding strength is strongly dependent on the processing historyL In this respect, bonding strengths of 500 and 1000 MPa are used in this study to examine the influences of particle clustering on
Particle clustering: T. C. Tszeng the void nucleation and therefore on the ductility of composites. The nucleation loci after LeRoy et al. 8 is used here to indicate the occurrence of void nucleation in a multiaxial stress state. The axisymmetric macroscopic stress state is assumed to take the form: O l l = 0"22 = 00"33
(4)
where 0 < p < 1. By changing the value of P, a range of the mean stress and deviatoric/effective stress can be obtained. The same procedure was used in Tszeng 2°, pertaining to void nucleation in spheriodized steels. The nucleation loci for different particle distributions are shown in Figure 7. The same figure also shows the loci corresponding to initial yield and hydrostatic stress state. The composite with homogeneous particle distribution has the same resistance to void nucleation. The composite with particle clustering Pattern II shows less resistance to void nucleation. Also note that void nucleation occurs approximately at the elastic limit during uniaxial loading (oll = 022 ~-- 0 ) . The nucleation strains in square root are plotted as a function of the mean stress am in Figure 8. A higher mean (tensile) stress leads to a smaller nucleation strain. Again, the tremendous influence of bonding strength is obvious. With the higher bonding strength, the influences of particle clustering are now more evident than that indicated in Figure 7. For example, the nucleation strain in composites with clustering Pattern II subjected to uniaxial loading (am ~ 175 MPa) is about 65% of that corresponding to the homogeneous particle distribution. Several experimental studies cited in the Introduction indicated the increased tendency of failure initialized in the regions with smaller spacing between particles. This type of particle clustering is different from what we have discussed in this paper. In the present approach, no account was furnished to address the closeness between two particles. Strictly speaking, what was considered in the present study is the mean effect of local particle concentration. From the standpoint of void nucleation, the most detrimental region is very possibly between two particles with the minimum spacing. Thus, the present calculated nucleation loci are on the conservative side.
CONCLUDING REMARKS This paper reports our study of the influences of particle clustering on the mechanical properties in particle reinforced composites. There are several issues of concern when applying the equivalent inclusion method to the present problem of particle clustering. First, the equivalent inclusion method gives a less reliable solution at a higher particle volume fraction 26 '27 "29 . The particle volume fraction can be as highh as 0.8 inside the clusters. Although it is not the major objective of this study to investigate or develop a computation method for composites with high inclusion volume fraction, we do believe that there is a strong need to further study this issue. Further, no study was conducted to assess the reliability of the equivalent inclusion method for
the present case in which the matrix metal in the cluster may deform plastically. This was one of the reasons preventing us from studying the effects of particle volume fraction, fc, in clusters. Second, the present formulation dropped the higher order terms of the stress/strain fields such that they became uniform in the inclusions. This presumption stands well when only elastic deformation is involved in both the inclusion and the matrix. Apparently, further study is needed to shed some light onto the situations when both phases can deform plastically. As far as void nucleation in the matrix is concerned, the local plastic strain is one of the controlling factors 6. Since the matrix stress/strain in the context of equivalent inclusion method is the volume averaged measure of the short range stress/strain fields around the inclusions, it is not very meaningful to relate this measure to the void nucleation in the matrix. Within the assumptions and limitation of the analysis method based upon the equivalent inclusion method, the present study reached the following concluding remarks: (1) Particle clustering has no essential effects on the macroscopic elastic moduli of the composites with different particle distributions. (2) When subjected to the same macroscopic stress, the clustered region would start yielding at an equal or lower macroscopic stress than that in the unclustered region. (3) The influence of reinforcement clustering on the stressstrain response is not significant at all for composites reinforced with particulates of unity aspect ratio, especially in the range of elastic and small plastic strain. When the clusters are elongated in the major loading direction, the stress-strain response of the composite systems will be strongly dependent on the aspect ratio of clusters. (4) Stress concentration on the reinforcing particles caused by particle clustering is not significant when the clusters are spherical in shape. We observed the tremendous influence of the cluster aspect ratio on the maximum normal stress at the interface. (5) The theoretical calculations did show the adverse effects of clustering on the resistance to void nucleation for composite systems with a higher interfacial bonding strength.
APPENDIX In this brief account, we will prove that the clustered and unclustered region would yield at the same macroscopic stress when both the particles and clusters are spherical in shape. To proceed, the explicit expressions developed by Weng 2s for spherical inclusion embedded in matrix is probably easier to use than the one developed in Tszengl9 22. Therefore, we will liberally use the equations as well as most of the notations in Weng 28. In the following, subscripts 0, and 1 denote variables for the plain matrix and the particle, respectively.
307
Particle clustering: T. C. Tszeng Let the initial yield stress in the plain matrix be Y. The initial yield stress of a composite reinforced by spherical particles of volume fraction f is28: Y cr = - bo
(A1)
By using equations (A2) and (A5) in (A12), it is straightforward to show that o-m and Oc in equations (A11) and (A12) are equivalent. Thus, we have proved that the clustered and unclustered region would yield at the same macroscopic stress when both the particles and clusters are spherical in shape.
where bo = (flo(#l -/~o) + q = (f
+
#o)/q
(A2)
'
f fio)(#~ - #o) + #o.
(A3)
In these expressions, /z is the Lame constant, f = 1 - f , and/30 is related to the Eshelby's transformation tensor, S, i.e.,
1. 2. 3. 4.
2 (4 - 5Vo)
flo --
REFERENCES
(A4)
15 (1 - Vo)
5.
where v is the Poisson's ratio. For the present case, we actually have the initial yield stress of the clustered region with f = fc in equation (A1). The corresponding Lame constant for the cluster is:
6. 7. 8.
tt = tto [1 + /
f (tq -/Xo)
f,
]
flo(/g'l --/'to) "]-/-toJ "
(A5)
For the hybrid composite with spherical clusters in the plain matrix, the volume fraction of the generalized inclusion (cluster) is F. The unclustered region (plain matrix) will yield at the stress 2s. O'm
Y b
(A6)
11. 12. 13. 14. 15. 16.
where b = (flo(tt -- tto) + #o)/qc
(A7)
17. 18.
qc = (F + F '/30)(# - / t o ) + #o-
(A8)
19. 20. 21.
In these equations, F' = 1 - F, and t~ is given by equation (A5). On the other hand, the cluster will yield at the stress Y
crc- bob1
(A9)
where b i -- --./~ qc
(A10)
qcY O-m
=
Bo(# - G#o) + #o"
(A11)
24. 25. 26.
28. 29. 30. 31.
Combining equations (A9) and (A10), we have
qcY Oc = #b o.
22. 23.
27.
In equation (A10),/~ and qc are given by equations (A5) and (A8), respectively. Substituting equation (A7) in equations (A6) gives
308
9. 10.
(A12)
32.
Lloyd, D.J., Composites Science and Technology, 1989, 35, 159. Mortensen, A., Cornie, J. A. and Flemings, M. C., Journal of Metals, 39, 1988, February, 12. Weng, B.J., Chang, S.T. and Shiau, J.S., Scripta Metallurgica et Materialia, 1992, 27, 1127. Hunt Jr, W. H., Richmond, O. and Young, R. D., ICCM/VI, 1989, 2.209. Rack, H. J. and Ratnaparkhi, Journal of Metals, 39, 1988, November, 55. Llorca, J., Needleman, A. and Suresh, S., Acta Metallurgica et Materialia, 1991, 39, 2317. Christman, T., Needleman, A. and Suresh, S., Acta Metallurgica, 1989, 37, 3029. LeRoy, G., Embury, J.D., Edward, G. and Ashby, M.F., Acta Metallurgica, 1981, 29, 1509. Fisher, J. R. and Gurland, J., Metal Science, 15, 1981, 193. Argon, A.S., Im, J. and Safoglu, R., Metallurgical Transactions, 1975, 6A, 825. Shi, N., Wilner, B. and Arsenault, R.J., Acta Metallurgica et Materialia, 1992, 40, 2841. Everett, R.K., Scripta Metallurgica et Materialia, 1993, 28, 663. Komenda, J. and Henderson, P.J., Scripta Metallurgica et Materialia, 1993, 28, 553. Ferry, M., Munroe, P.R. and Crosky, A., Scripta Metallurgica et Materialia, 1993, 28, 1235. Davidson, D., Metallurgical Transactions, 1991, 22A, 113. Lewandowski, J.J., Liu, C. and Hunt, W.H. Jr, Materials Science and Engineering, 1989, A107, 241. Norris, A.N., Mechanics of Materials, 1985, 4, 1. Christensen, R.M. and Lo, K.H., Mechanics and Physics of Solids, 1979, 27, 315. Tszeng, T.C., Composites Science and Technology, 1994, 51, 75. Tszeng, T.C., Scripta Metallurgica et Materialia, 1993, 28, 1520. Tszeng, T. C., ASME J. Engineering Materials and Technology, 1997, in review. Tszeng, T.C., Journal of Composite Materials, 1994, 28, 800. Eshelby, J.D., Proceeding of the Royal Society, London, 1957, A241, 376. Mori, T. and Tanaka, K., Acta Metallurgica, 1973, 21, 571. Arsenault, R.J., Wang, L. and Feng, C.R., Acta Metallurgica et Materialia, 1991, 39, 47. Christensen, R.M., Journal of Mechanics and Physics of Solids, 1990, 38, 379. Christensen, R., Schantz, H. and Shapiro, J., Journal of Mechanic and Physics of Solids, 1992, 40, 69. Weng, G.J., Journal of Mechanics and Physics of Solids, 1990, 38, 419. Kwon, P. and Dharan, C.K.H., Acta Metallurgica Materiala, 1995, 43, 1141. Kao, A.S., Kuhn, H.A., Richmond, O. and Spitzig, W.A., Metallurgical Transactions, 1989, 20A, 1735. Tszeng, T. C., Fabrication of Particulates Reinforced Metal Composites, ed. J. Masounave and F. G. Hamel. ASM International, 1990, 53. Arnold, S.M., Pindera, M.-J. and Wilt, T.E., International Journal of Plasticity, 1996, 12, 507.