Prediction of Flow Behavior in the Heterogeneously Dispersed Al-10 vol%sic Composites

Transcription

1 Materials Transactions, Vol. 49, No. 3 (2008) pp. 671 to 680 #2008 The Japan Institute of Metals EXPRESS REGULAR ARTICLE Prediction of Flow Behavior in the Heterogeneously Dispersed Al-10 vol%sic Composites Di Zhang*, Kenjiro Sugio, Kazuyuki Sakai*, Hiroshi Fukushima and Osamu Yanagisawa Mechanical System Engineering, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima , Japan A newly developed model was proposed to predict the flow and strain-hardening behavior of the heterogeneously dispersed Al-10 vol%sic composites with particle damage in tensile deformation. The frequency of local number of particles was decomposed into several Poisson distributions and the frequency of local number of damaged particles was approximated by one Poisson distribution. Then, the flow stress was obtained by summation of strain-hardening rate for each decomposed distribution and/or relieved stress for the distribution of damaged particles. This model gave a good agreement with the experimental results. [doi: /matertrans.mer ] (Received November 12, 2007; Accepted January 11, 2008; Published February 14, 2008) Keywords: heterogeneous composite, random distribution, decomposition, strain-hardening behavior 1. Introduction It is well established that the deformation behavior of the particulate reinforced metal matrix composites (PMMCs) are sensitive to the spatial arrangement of the reinforcing particles. The distribution of local stress and strain in the matrix and the particles must be considered to understand the overall deformation behavior of the composites, when the particles are heterogeneously distributed in it. The highest strain and stress concentrations occur in regions where particles are closely packed. 1 3) In general, the local concentration of particle distribution is found to have an adverse effect on the ductility and flow stress of the composites 4 8) mostly due to the enhanced damage initiation and development within the clusters. Thus the previous model, 9,10) considering only the combination of strain-hardening and corresponding damage accumulation behavior, is not enough to understand the effect of spatial distribution of the second phase particles in the composites. During the past few years, most of the numerical studies based on the finite element analysis 11,12) and self-consistent model 13,14) dedicated from Eshelby s equivalent inclusion method have been carried out to understand the effect of particle clustering on the flow behavior of the composites. The particle-clustering tendency of the composites is evaluated by considering that the composite is consisted with different local volume fraction of reinforcing particles. Thus, the flow stress of the composite has been made up of the flow stress of each local area incorporating the damage behavior of the particles determined by the traditional Weibull function. 15,16) However, these kinds of research are limited in the field of numerical investigations and are not correlated with the experimental studies. In order to understand better the deformation behavior, various methods have been developed to quantify the spatial distribution of the second-phase particles in the heterogeneously dispersed composites ) Here, a 2-dimensional local number, LN2D, is introduced to quantitatively evaluate the clustering tendency of the composite, detailed in our previous paper. 21,22) *Graduate Student, Hiroshima University In our previous paper, 23) a simple model was developed to explain the flow behavior of the homogeneous composites having different volume fraction, especially considering the spatial distribution of the damaged particles and gave a good fit to the experimental results. Here, based on our previous research, we have systematically evaluated the flow and strain-hardening behavior of the heterogeneously dispersed Al-10 vol%sic composites. The frequency distribution of the local number (LN), defined by us, of all and damaged particles were decomposed into several Poisson distributions. The overall flow behavior of the composite was quantitatively determined by the composition of the damage accumulation and the strain-hardening behavior of each random Poisson distribution. 2. Experimental Procedure 2.1 Material preparation The studies were performed on the pure Al powder with the average particle size of about 1 mm, 3mm, 20mm and 30 mm containing 10 vol% of SiC particles having a mean particle size of 2 3 mm. The current composites were fabricated by a spark sintering technique. The mixed powders were holding under a pressure of 33.2 MPa at K for about 100 min in a vacuum system. Five kinds of composites with different clustering tendency were mixed based on the geometrical dependent clustering when there was a difference between the size of the matrix and the reinforcement particles. Pure Al matrix samples with different particle size were also prepared to investigate the effect of matrix particle size on the deformation behavior. The particle size, average matrix particle size, d p, relative particle size (RPS) ratio defined as average particle size ratio between the matrix and particles, and relative density of the specimens are listed in Table 1. The Al and the SiC powders were mixed in ethyl alcohol using ultrasonic vibration for 60 min and then wet blended for 24 hours for Sample 1. The other 7 samples were dry mixed thoroughly for 24 hours. The cylindrical tensile samples were loaded at room temperature using a strain rate of 0.002/s to obtain the basic tensile properties of the composite. A SEM was used to examine the spatial distribution of all particles measured

2 672 D. Zhang, K. Sugio, K. Sakai, H. Fukushima and O. Yanagisawa Table 1 Particle size, average matrix particle size, d p, relative particle size (RPS) ratio and relative density of samples. Sample Al 1 mm Al 3 mm Volume fraction of the particle Al 20 mm Al 30 mm SiC 2 3 mm d p (mm) RPS Ratio Relative density (%) P ðvolaldalþ d p ¼ P VolAl RPS ratio ¼ dp dsic d Al : Particle size of each kind of Al particles. Vol Al : Volume fraction of each kind of Al particles. d p : Average particle size of each kind of Al particles. d SiC : Average particle size of SiC particles. before deformation and the delaminated particles during tensile deformation. 2.2 Definition of Poisson decomposition Let us assume gravity centers (GCs) of second phase particles are spatially distributed in the 2-dimensional space according to random distribution (Poisson point field) and the clustering distribution, schematically shown in Figs. 1(a) and (b). Measuring circle is defined so that the number density with 7 GCs in it equals to that of a whole. Putting the center of the measuring circle on the GC of a noticed particle, in the case of spatially uniform-random arrangement, probability of GC, P, is represented by the following equation and shown in Fig. 1(a 0 ). 21,22) PðLN2D ¼ k þ 1Þ ¼ 7k expð 7Þ k ¼ 0; 1; 2 ð1þ k! Here, average LN2D, LN2D av, equals to 8. On the other hand, when the GCs are distributed to be clustering, larger LN is achieved and probability histogram of LN must shift from the uniform-random state to the larger LN, which is shown in Fig. 1(b 0 ). Then, the probability of LN2D is expressed by the following equation with >7, if it can be represented by one Poisson distribution (PD). PðLN2D ¼ k þ 1Þ ¼ k expð Þ k ¼ 0; 1; 2 ð2þ k! In that case, LN2D av equals to þ 1 and the particle number density ratio, c, to the uniform-random distribution is given by c ¼ LN2D av : ð3þ 8 When the frequency of LN2D of that distribution cannot be represented or approximated only by one PD, we assume that the probability curve of a clustering spatial distribution is made up of M random distributions. Each i-th distribution possesses its own number fraction (NF) of particles,, and the average particle LN in the measuring circle, i þ 1. In that case, the probability of LN2D for each decomposed distribution is reconstructed and takes a form as k i PðLN2D ¼ k þ 1Þ ¼ k! expð iþ k ¼ 0; 1; 2 : ð4þ Then the probability of clustering distribution is expressed as the sum of the decomposed random distributions. PðLN2D ¼ k þ 1Þ ¼ XM k i k! expð iþ k ¼ 0; 1; 2 The probabilities of the LN distribution for all particles and the several decomposed random distributions are schematically shown in Fig. 1(b 0 ). The particle number density ratio of each decomposed random distribution is given by the following equation in the same form as eq. (3). ¼ i þ 1 ð6þ 8 It is important to point out that, if the clustering distribution of second phase particles is approximately decomposed into several random distributions by using the descriptor of LN2D, we may estimate an overall property by composing properties in the regions of particle groups belonging to the decomposed Poisson distributions. 2.3 Process of Poisson decomposition For estimating and i of each random distribution, an error function was introduced by comparison the expected probability, Pðk þ 1Þ, with the experimental probability, P exp ðk þ 1Þ, in the following equation. E ¼ XN ½P exp ðk þ 1Þ Pðk þ 1ÞŠ 2 k ¼ 0; 1; 2 ð7þ k¼0 where N is the largest value of k. The quantitative characterization could be approached by minimization of E using the conjugate gradient method, 24) and the gradients were calculated by ð5þ

3 Prediction of Flow Behavior in the Heterogeneously Dispersed Al-10 vol%sic Composites 673 Fig. 1 2-dimensional arrangement types. (a) uniform random and (b) clustering arrangement of the gravity center (GC) of particles. (a 0 ) probability distribution of local number (LN) for (a) and (b 0 ) that for (b) with its decomposed Poisson distributions. PD: Poisson distribution. Table 2 Average of LN2D, LN2D av, and variance of LN2D, LN2D var,of each sample. Sample LN2D av LN2D var ¼ 2 XN k¼0 k ¼ 0; 1; X N ¼ i ½P exp ðk þ 1Þ Pðk þ 1ÞŠ k i k! expð iþ k¼0 ½P exp ðk þ 1Þ Pðk þ 1ÞŠ k i k! expð k iþ 1 i k ¼ 0; 1; 2 : Minimization of the error function was performed until the variation of E from a previous iteration was less than 10 20, and then several sets of and i were obtained. 3. Results and Discussion 3.1 Distribution of microstructure Table 2 lists the average of LN2D, LN2D av, and variance of LN2D, LN2D var,(ln2d var ¼ LN2Dstd 2, LN2D std: standard deviation of LN2D) calculated directly from the experimental results for the 5 reinforced samples. The LN2D av and ð8þ ð9þ LN2D var increase from Sample 1 to 5 with increasing RPS ratio between the matrix and the reinforcements. This means that the distribution of the particles becomes more heterogeneous from Sample 1 to 5. Although Sample 1 has LN2D av close to 8 for the uniform-random distribution, the LN2D var is much smaller than 7 for the uniform-random distribution shown by eq. (1). For Samples 2 and 3, the LN2D var are also smaller than LN2D av 1, expected by the eq. (2). This means that the experimental relative frequency curve of Sample 1 to 3 is narrower than the theoretically expected frequency of Poisson point field. On the other hand, for the last two samples, LN2D var are larger than LN2D av 1. It was shown in our previous study that many particles were failed by a particle/matrix delamination 23) in deformation process. Figures 2 and 3 show that the average of LN2D of delaminated particles, LN2D d;av, and the NF of delaminated particles, f nd, increase with increasing plastic strain. Both of them could be well approximated by the empirical equations proposed in the previous study 23) by fitting the experimental results, LN2D d;av ð"þ ¼a" þ b and ð10þ f nd ð"þ ¼ f nd0 expðp"þ; ð11þ where a, b, f nd0 (initial number fraction of voids) and p are four fitting parameters shown in these figures. Figures 4(a) to (e) plot the approximated relative frequency (RF) distributions of LN2D of all particles and delaminated particles during deformation, comparing with the experimental results. The approximated distributions are obtained by direct calculation by eq. (2) using the exper-

4 674 D. Zhang, K. Sugio, K. Sakai, H. Fukushima and O. Yanagisawa Fig. 2 Relation between plastic strain and average LN2D of the delaminated particles, LN2D d;av. Fig. 3 Relation between plastic strain and number fraction (NF) of delaminated particles, f nd. imental LN2D av for all particles in Table 2, and experimental LN2D d;av during deformation for delaminated particles shown in Fig. 2. As shown in these figures, these RF distributions of delaminated particles could be approximated by only one PD. It is shown that the RF distribution of the delaminated particles during tensile deformation concentrates at the lower-right corner of the RF distribution of all particles. 3.2 Decomposed random distributions When a RF distribution of LN is decomposed into several random distributions of Poisson point field, the regions containing the dispersed particles belonging to each decomposed distribution, are schematically shown in Fig. 5. If N 0 represents the number of all particles, N 0 is the number of particles belonging to the i-th decomposed distribution. Supposing that the number density of the i-th decomposed distribution cover a whole, the particle number of a whole is N 0. Thus, the area fraction occupied by the particles belonging to the i-th decomposed distribution, A i, can be expressed by the following equation. A i ¼ N 0 N 0 ¼ ð12þ Fig. 4 Relative frequency of LN2D of all and delaminated particles. (a) to (e) corresponds to Sample 1 to 5. The area fraction of the empty area, having no particles, are shown in Fig. 5 and can be described by the following equation.

5 Prediction of Flow Behavior in the Heterogeneously Dispersed Al-10 vol%sic Composites 675 Fig. 5 Schematic of decomposed random distributions of the GC for all particles in the 2-D arrangement. A emp ¼ 1 XM ð13þ It is important to point that the particle number densities of the decomposed random spatial distribution in 2-dimension corresponds to those in 3-dimension and the area fractions, A i or A emp, are equal to the corresponding volume fractions in 3- dimension. 19) Figures 6(a) to (e) show the approximately decomposed random distributions of the 5 reinforced samples. Table 3 shows the value i,,, A i and A emp of decomposed PDs for each sample. The numbers of decomposed distribution of Sample 1 to 3 are one. In the case of Sample 1 and 2, the distribution of LN2D is narrower than the theoretically expected distribution of point field (see also Table 2). However, Sample 3 gets a relatively good fit to the experimental results, because the variance of this sample is closer to the random distribution than Sample 1 and 2. Although there was the difference between the variances of experiment and approximation for Sample 1 to 3, the average of LN2D, LN2D av, obtained by the approximation were used for the analysis. Both Samples 4 and 5, having higher clustering tendency, can be decomposed into 3 and 2 PDs respectively and the sum of these distributions calculated by eq. (5) agree with the experimental result. The empty area of each sample increases with increasing clustering tendency, as shown in Table 3. As previously mentioned, the distribution of delaminated particles was assumed to be only one PD not by decomposing, as shown in Figs. 4(a) to (e), since the measuring number of the delaminated particles (about ) was much less than 2000 of the critical value for Poisson decomposition, which would bring large error of decomposing. 3.3 Flow and strain-hardening behavior Flow stress without particle damage The overall flow stress of the composite, c, is represented by the following equation using a simple rule of mixture, c ¼ V p p þð1 V p Þm c ð14þ where V p, m c and p are the volume fraction of the particles, the stress of the matrix and the stress of the particles, respectively. As can be seen previously, the spatial distributions of SiC particles of some of the samples had clustering tendency. In the samples with heterogeneously distributed particles, they had particle-dense (higher volume fraction) and -poor (lower volume fraction) regions coexisted with each other. Then it is necessary to take into account their effect on the deformation stress. The flow stress, m, of the pure matrix material can be expressed by a power law equation of the plastic strain, ", shown as follows: m ¼ 0 þ C" n ð15þ where 0, C and n are the yield stress of the pure matrix and two fitting parameters, respectively. The three kinds of pure Al samples using different Al powder particle size, Sample 6, 7 and 8, were prepared to verify the effect of matrix particle size on the flow stress. Figure 7 shows the tensile 0.2% yield stress and ultimate tensile strength (UTS) of three sintered pure Al samples. As shown in this figure, the yield stress decreases with increasing average Al powder particle size, and can be approximated by a linear function. 0 ¼ 0:22d p þ 102:9 ð16þ where d p is the average size of the pure Al powder particles and was determined by the equation used in Table 1. Arrows in Fig. 7 shows d p of Sample 1 to 5 for comparison. On the other hand, C and n in eq. (15) were almost the same in the three kinds of pure Al samples by fitting the experimental flow curves, and were 148 MPa and 0.37 respectively. In the present research, since the SiC particles were distributed heterogeneously in the composite, the volume fraction of the particles for the i-th decomposed distribution is replaced by V p. Thus, a general expression of the matrix stress, mi c, in the i-th decomposed regions is reconstructed and takes a form as mi c ¼ " n 0 þ C : ð17þ 1 V p It has been shown by the finite element calculations 25) that after a transition of 1 2% strain, the apparent stress in the particles for the i-th particle group is simply a constant multiple of the matrix stress. pi ¼ k i mi c ð18þ where k i is the stress amplification factor depended on the volume fraction of the particles for the i-th particle group, V p, the strain-hardening exponent, n, and the elastic/plastic mismatch between the matrix and the particles, E p =E m. The value of k i is considered to include effects of elastic stress in particles and locally concentrated plastic deformation stress in higher strain hardened regions adjacent to the strengthening particles. It was determined in our previous study 23) that the relationship between k and V p was approximated by the following linear function. k ¼ 5:424V p þ 2:577 ð19þ The stress amplification factor for each decomposed distribution, k i, was calculated by eq. (19), where V p is replaced by V p. The value of k i for each decomposed

6 676 D. Zhang, K. Sugio, K. Sakai, H. Fukushima and O. Yanagisawa Fig. 6 Relative frequency distribution of all particles and their decomposed Poisson distributions. Table 3, a i,, A i and A emp of decomposed random distribution for the five samples. Sample PD i A i A emp Total Total random distribution increased with the increasing clustering tendency, as shown in Table 4. Using eqs. (14) to (19), flow stress in the case of inhomogeneous distribution of reinforced particles is represented by the following equation, c ¼ XM þ V p pi þ XM 1 XM! m ð1 V p Þ c mi ð20þ where V p = and ð1 V p Þ = represent the area fraction of particles and matrix in the i-th decomposed particle group, respectively. When the volume of particleempty region appears, the third term of the above equation is available using the eq. (15) for the matrix stress m in particle-empty region. The strain-hardening rate, d c =d", of the composite

7 Prediction of Flow Behavior in the Heterogeneously Dispersed Al-10 vol%sic Composites 677 Fig. 7 Experimental tensile properties for the samples with different average Al powder particle size, d p. Table 4 Stress amplification factor, k, of all of the samples. Sample PD V p k i without damage then can be calculated by the derivative of eq. (20) respect to strain. d c d" ¼ XM d pi V p c i d" þ XM ( ) þ 1 XM dm d" ð1 V p Þ dmi c d" ð21þ Figures 8(a) to (c) show the effect of (a) the Al powder particle size, (b) the particle clustering and (c) both of them on the flow stress without damage by the above equations for the present samples. The flow stress of Sample 1 in these figures corresponds to that for M ¼ 1, f n ¼ 1, c ¼ 1 and d p ¼ 2:6 mm in the eq. (20), in which the effect of spatial distribution was not considered. In Fig. 8(a), different average powder particle sizes, d p, were taken from Table 2 and the yield stress was calculated using empirical eq. (16) while maintaining c ¼ 1 for Sample 2 to 5. On the other hand, in Fig. 8(b), the value of and k i was taken from Tables 3 and 4 for Sample 2 to 5. In that case, the value of d p of the Sample 1, 2.6 mm, was assumed to be equal to that for all of the samples. In Fig. 8(c),, k i, and d p are directly taken from Tables 2, 3 and 4 for sample 2 to 5. It is clear that the flow stresses of current samples in the case of no damage of reinforcement particles, are predicted as the Fig. 8(c), affected by the Al powder size as the Fig. 8(a), in which the increased size decreases the flow stress, and the clustering Fig. 8 Tensile flow curves calculated by the eq. (20) with (15), (17), (18) and (19), in the case of no particles damage. Effect of (a) Al powder particle size, (b) SiC particle clustering and (c) both of the two effects on the flow stress. tendency as the Fig. 8(b), in which the increased clustering tendency increases the flow stress, for the Samples 1 to 5, respectively. It is an interesting prediction that clustering of reinforcement particles enhances the deformation stress as shown by the Fig. 8(b). The particle clustering regions (high volume fraction of particles), which increase flow stress, always bring the particle dilute regions (low volume fraction of particles), which decrease flow stress. The increased flow stress in the Fig. 8(b) with increasing clustering tendency is due to that the enhancing effect is larger than the lowering effect on flow stresses. This expected tendency is coincident with the previous report. 12,26) Flow stress accompanying with particle damage The flow stress of the composite considering the damage of

8 678 D. Zhang, K. Sugio, K. Sakai, H. Fukushima and O. Yanagisawa Fig. 9 Tensile flow curves by experiment and calculation by the eq. (22) with eqs. (15), (17), (18), (19), (24) and (25) comparing with the predicted curves for no particles damage (eq. 20) and no clustering tendency (eq. 27). (a) to (e) corresponds to Sample 1 to 5. the second phase particles, c, can be described by a simple rule of mixture, adding Dð"Þ pd to the eq. (20), c ¼ XM þ V p pi þ XM 1 XM ð1 V p Þ c mi! m Dð"Þ pd ; ð22þ where Dð"Þ and pd are the damage parameter and the stress carried by the particles to be delaminated, respectively. If the frequency distribution of damaged particles is decomposed to several distributions, Dð"Þ pd in eq. (22) must be replaced by the following equation. Dð"Þ pd ¼ XM d D j ð"þ pdj ð23þ j¼1 where M d is the number of the decomposed distributions for delaminated particles. However, as described in the subsection 3.2, the distributions of delaminated particles were approximated by only one PD, and Dð"Þ pd was used for the negative damage effect in the present work. Dð"Þ is expressed by the following equation, changing i to d in the eq. (12) and multiplying the stress-relief parameter q. Dð"Þ ¼A d qc d V p ¼ f nd qv p ð24þ The parameter, q, is the apparent volume ratio of stressrelieved region to that of delaminated particle. If all stresses in a particle to be delaminated and locally concentrated stress in the matrix due to the existing of particle are relieved, q ¼ 1. If a part of them is relieved, q < 1. The value of q 0:7 was used for cv p 0:1 according to the previous analysis. 23) The stress carried by the particles to be delaminated, pd, in the eq. (22) can be expressed by the following equation pd ¼ k d md c ; ð25þ replacing i to d in the eq. (18). The coefficient k d was calculated by eq. (19) with V p replaced by c d V p, where

9 Prediction of Flow Behavior in the Heterogeneously Dispersed Al-10 vol%sic Composites 679 Fig. 10 Strain-hardening rate curves by experiment and calculation by the eq. (26) with (15), (17), (18), (19), (24) and (25) comparing with the predicted curves for no particles damage (eq. 21) and no clustering tendency (eq. 28). (a) to (e) corresponds to Sample 1 to 5. c d is the particle number density ratio of the delaminated particles. The composite flow stress is consisted of the stress of coexisting particles and matrix, the decomposed pure Al matrix without reinforced particles and the negative effect of the delaminated particles. The strain-hardening rate, d c =d", then is calculated by the derivative of eq. (22) respect to strain. d c d" ¼ XM d pi V p c i d" þ XM ( ) þ 1 XM dm ð1 V p Þ dmi c d" d" Dð"Þ d pd d" pd ddð"þ d" ð26þ The first three terms of the equation show the strainhardening rate in the absence of damage, while the forth and the fifth term express the negative effect of damage and damage accumulation rate on the strain-hardening behavior. When the clustering tendency of all and delaminated particles is not considered, the flow stress in the eq. (22) can be simplified as c ¼ V p p þð1 V p Þm c Dð"Þ pd " n ¼ðV p k þ 1 V p Þ 0 þ C 1 V p " n f nd V p qk 0 þ C : ð27þ 1 V p In this equation, the flow stress for all of the samples, Sample 1 to 5, without considering clustering tendency corresponds to c ¼ c d ¼ 1, f nd0 ¼ 2:35 and p ¼ 15:3 (see Fig. 3) of Sample 1. Then the values of k, k d and f nd were directly calculated from eqs. (19) and (11), respectively. Here, the flow stresses of the five composites were affected only by the different average powder particle size. Then, the strainhardening rate in the uniform-random state can be calculated by the derivative of eq. (27) respect to strain. d c d" ¼ V d p p d" þð1 V pþ dc m d"

10 680 D. Zhang, K. Sugio, K. Sakai, H. Fukushima and O. Yanagisawa Dð"Þ d pd d" ddð"þ pd ð28þ d" Figures 9(a) to (e) present the flow stress in the presence of particle damage and particle clustering tendency calculated by the eq. (22) with eqs. (15), (17), (18), (19), (24) and (25) based on the decomposed model, that in the absence of particle damage calculated by eq. (20) and that in the absence of particle clustering tendency calculated by eq. (27). It is shown in these figures that the calculated flow stress curves considering the presence of particle damage and particleclustering tendency are close to the experimental results. This figure also shows that the overall relieved stress increases with increasing clustering tendency, Sample 1 to 5, comparing with the expected flow stress in the case of no particle damage. Figures 10(a) to (e) show the strain-hardening rate calculated by the eq. (26) with the eqs. (15), (17), (18), (19), (24) and (25), together with the strain-hardening rate calculated by eq. (21) without particle damage and by eq. (28) without considering clustering tendency. It is seen that the calculated strain-hardening rate using eq. (26) is a better approximation of the experimental one than eqs. (21) and (28). Since the condition of macroscopinstability is d c =d" ¼ c, the intersection of the strain-hardening rate curve and the flow stress curve represents the beginning of the calculated tensile instability. It is seen that the calculated instability strain with particle damage and particle-clustering tendency is a better approximation of the experimental strain than the non-damaged and non-clustering model. This means that tensile strength and elongation were affected by clustering tendency of the particles 6,7) and their flow and strainhardening behavior were approximated quantitatively by our analysis based on decomposing LN distribution. 4. Conclusions The flow and strain-hardening behavior of the heterogeneously dispersed Al-10 vol%sic composites were predicted using a newly developed model, based on a definition of the spatial distribution of SiC particles. The frequency distribution of the particle local number, LN2D, was decomposed into several PDs of random point field with different average LN2D. At first, flow stress in the absence of particle damage was quantitatively predicted by considering the effect of Al powder particle size and the clustering tendency of SiC particles. In this process, it was predicted that clustering tendency increases flow stress. Then, the overall flow behavior of the composite in the presence of particle damage was determined additionally with the stress-relief by particle delamination considering clustering tendency of damaged particles. It was shown that the flow behavior in this model could explain the effect of reinforcement particle clustering and particle damage on the flow stress and the beginning of tensile instability. REFERENCES 1) P. Ganguly and W. J. Poole: J. Mech. Phys. Solids. 52 (2004) ) H. Shen and C. J. Lissenden: Metall. Mater. Trans. A 36 (1998) ) R. J. Arsenault, N. Shi, C. R. Feng and L. Wang: Mat. Sci. Engng. A 131 (1991) ) J. Segruado and J. LLorca: Mech. Mater. 38 (2006) ) M. Geni and M. Kikuchi: Acta Mater. 46 (1998) ) A. Slipenyuk, V. Kuprin, Yu. Milman, J. E. Spowart and D. B. Miracle: Mater. Sci. Eng. A 381 (2004) ) A. M. Murphy, S. J. Howard and T. W. Clyne: Mater. Sci. Technol. 14 (1998) ) T. Oki, K. Matsugi, K. Shimizu and O. Yanagisawa: J. of Japan Institute of Light Metals 52 (2002) (in Japanese). 9) W. J. Poole and N. Charras: Mater. Sci. Eng. A 406 (2005) ) F. Zhou, J. N. Wang and J. S. Lian: Mater. Sci. Eng. A 332 (2002) ) J. Segruado, C. Gonzalez and J. LLorca: Acta Mater. 51 (2003) ) D. S. Wilkson, E. Maire and J. D. Embury: Mater. Sci. Eng. A 233 (1997) ) D. S. Wilkson, W. Pompe and M. Oeschner: Prog. Mater. Sci. 46 (2001) ) S. F. Corbin and D. S. Wilkson: Acta Metall. Mater. 42 (1994) ) C. A. Lewis and P. J. Withers: Acta Metall. Mater. 43 (1995) ) J. Llorca: Acta Metall. Mater. 43 (1995) ) S. Yotte, J. Riss, D. Breysse and S. Ghosh: CMES-Comp. Model. Eng. Sci. 5 (2004) ) P. Louis and A. M. Gokhale: Metall. Maters. Trans. A 26 (1995) ) P. Louis and A. M. Gokhale: Acta Mater. 44 (1996) ) M. Li, S. Ghosh, O. Richmond, H. Weiland and T. N. Rouns: Mater. Sci. Eng. A 265 (1999) ) K. Sugio, Y. Momota, D. Zhang, H. Fukushima and O. Yanagisawa: Mater. Trans. 48 (2007) ) K. Sugio, Y. Momota, D. Zhang, H. Fukushima and O. Yanagisawa: Mater. Trans. 48 (2007) ) D. Zhang, K. Sugio, K. Sakai, H. Fukushima and O. Yanagisawa: Mater. Trans. 49 (2008) (doi: /matertrans.mer ). 24) W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery: Numerical Recipes in Fortran 77, (Cambridge University Press) pp ) J. R. Brochenbrough and F. W. Zok: Acta Metall. Mater. 43 (1995) ) D. S. Wilkson, E. Maire and R. Fougeres: Mater. Sci. Eng. A 262 (1999)