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Interphase effect on the strengthening behavior of particle-reinforced metal matrix composites
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Computational Materials Science 41 (2007) 145–155 www.elsevier.com/locate/commatsci
Interphase effect on the strengthening behavior of particle-reinforced metal matrix composites W.X. Zhang, L.X. Li, T.J. Wang
*
MOE Key Laboratory for Strength and Vibration, Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an 710049, China Received 12 January 2007; received in revised form 12 March 2007; accepted 15 March 2007 Available online 15 May 2007
Abstract Interphase effect on the strengthening behavior of particle-reinforced metal matrix composites is numerically investigated, in which the stiffness, thickness and debonding location of interphase are considered. Embedded cell model and finite element method are used in analysis. The numerical results indicate that hard and soft interphases result in the increase and significant decrease of the strength of composite, respectively. Location of debonding has quite different effects. Pole debonding results in a remarkable decrease of the overall strength of composite, while equator debonding results in a slight decrease of the strength of composite. Debonding between the interphase and matrix has more significant effect than that between the interphase and particle. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Metal matrix composites (MMCs); Interphase; Strength; Plastic deformation; Finite element analysis (FEA)
1. Introduction Interface and/or interphase are important for composites, which affects the microscopic fields and then the overall properties of composites significantly [1–6]. However, interface is only an idealized mode and extensive work has been carried out. Interphase is really the third phase between the reinforced phase and matrix of composites and exists inevitably, which is due to the coated material to enhance the bond between matrix and inclusion in manufacture process, the chemical diffusion between inclusion and matrix, special design (e.g. functionally graded interphase, adsorbed contaminants on the surface of fiber/ particle results in a third material phase) and others. Extensive analytical and numerical work has been devoted to the elastic solutions of the interphase problems, including homogeneous [7–10] and inhomogeneous [6,11– 16] interphases. Up to now, relatively less work has been
*
Corresponding author. Tel./fax: +86 029 82665168. E-mail address: [email protected] (T.J. Wang).
0927-0256/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.03.011
carried out on the inelastic solutions of the interphase problem. Veazie and Qu [17] developed a spring-layer model without thickness to describe the interphase and then studied the interphase effect on the transverse deformation behavior of unidirectional fiber reinforced metal matrix composites (MMCs). Wang and Yang [18] employed unit cell to numerically study the energy dissipation of particle reinforced MMCs with ductile interphase. Su et al. [19] studied the effects of interphase strength on the damage modes and mechanical behavior of MMCs. Interphase may be harder or softer than the matrix [20]. Based on the secant modulus approach, Ding and Weng [21] theoretically studied the effect of ductile interphase on the overall stress–strain curves of particle-reinforced composites, in which both hard and soft interphases are considered. However, due to the limitation of their theoretical approach, the initial yield stress could not be predicted exactly and the local deformation field could not be presented. Zhang and Wang [22] carried out the 3D numerical analysis to study the effect of ductile interphase on the deformation behavior of MMCs with regularly distributed particles.
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Interphase debonding is another important factor to affect the mechanical behavior of composites. Haddadi and Teodosiu [23] studied the effect of debonding process on the plastic behavior of two-phase composites through the finite element analysis with a 3D unit cell. Ghassemieh [24] developed a finite element model to simulate the stress– strain behavior of particulate composites with an interfacial crack. Basically, the ductile interphase may be harder or softer than the matrix [20], and debonding may have different locations. However, the effects of ductile interphase and the effects of debonding modes, especially when interphase exits, on the overall mechanical behavior of composites with randomly distributed particles are not well understood up to now. The previous analytical method is only suitable for some ideal cases and is difficult to predict the overall mechanical property of composites with partial debonding interphase. The purpose of this work is to numerically investigate the effect of ductile interphase on the strengthening behavior of particle-reinforced metal matrix composites (PMMCs), in which the stiffness, thickness and debonding location of interphase are considered. Embedded cell model is employed to consider the random distribution of particles. In Section 2, the embedded unit cell model and the calculation procedures are presented. Effects of the stiffness and thickness of interphase and the location of debonding on the overall mechanical behavior of composites are presented and discussed in Section 3. Finally, concluding remarks are presented in Section 4. 2. Micromechanics approach 2.1. Embedded cell model and constitutive equations For random particle reinforced composites, one can choose a sufficiently large representative volume element (RVE) to describe the features of microstructures as many as possible [25,26] and then to numerically obtain the overall mechanical behavior of composites, which needs a great number of calculations and becomes increasingly feasible with development of the computer technology. An alternative approach is using the generalized self-consistent model (GSCM), which proposed by Christensen and Lo [27] in predicting the elastic moduli of composites, as shown in Fig. 1a. Chen et al. [16], Veazie and Qu [17], and Dong and Schmauder [28] have extended the original idea of GSCM to numerically study the elastoplastic deformation of composites, which is the so-called embedded cell model. The numerical results obtained by embedded cell model agree very well with experiments. This approach has high accuracy and saves calculations, and can be used to study the mechanical behavior of composites with much more complex microstructures, such as imperfect interphase [16], damage [19], voids [29], etc. In what follows, an axisymmetric embedded cell model is employed to study the effect of ductile interphase on
the overall mechanical properties of particle reinforced MMCs. The coordinate system is shown in Fig. 1b in which outer size L of the model is large enough so that the external boundary conditions have no influence on the deformation of the embedded cell. R1, R2 and R3 are radius of the particle, outer radius of the interphase and radius of the cell, respectively. Dong and Schmauder [28] indicated that if L/R3 P 5, the boundary effect can be neglected. In calculation, we take L/R3 = 10 and fix R3. R1 and R2 are changed to obtain various volume fractions of particle by fp ¼ R31 =R33 . Small deformation assumption is adopted in analysis. The inclusion, e.g. SiC particle, is assumed to be linear isotropic and to obey Hook’s law. The matrix, e.g. aluminum, is assumed to be isotropic and to obey von Mises yield criterion and the following stress–strain relation: 8 e 6 e0 ; < Em e; N r¼ ð1Þ : r0 ee ; e > e0 ; 0 where r and e are the uniaxial stress and strain in matrix. Em, r0 and e0 are the Young’s modulus, initial yield stress and yield strain of matrix, respectively, and have the relation of e0 = r0/Em. N is the strain hardening exponent of matrix. It is assumed that the interphase has the same strain hardening exponent and Poisson ratio, and obeys the following stress–strain relation: 8 e 6 e0 ; < bEm e; N r¼ ð2Þ : br0 ee ; e > e0 ; 0 where b is a constant. Obviously, b = 1 corresponds to the case of no interphase, and b < 1 and b > 1 correspond to the soft and hard interphases, respectively. The smaller the value of b is, the softer the interphase is, and vice verse. ij and strain eij in the composites can The overall stress r be calculated as Z 1 ij ¼ r rij dV ; i; j ¼ 1; 2; 3; ð3Þ V V Z 1 eij ¼ eij dV ; i; j ¼ 1; 2; 3; ð4Þ V V where rij and eij are the local stress and strain, respectively, and V volume of the embedded cell. According to [28], the v and strain ev of the embedded equivalent overall stress r cell can be calculated as rffiffiffiffiffiffiffiffiffiffiffiffi 3 v ¼ sijsij ; r ð5aÞ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 ev ¼ eijeij ; ð5bÞ 1 þ m 2 where sij and eij are overall deviator stress and strain tensors, respectively, m is the apparent Poisson ratio of nonlin-
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Fig. 1. The embedded unit cell model. (a) Material model and (b) axisymmetric computational model and coordinate system.
ear deformation which is defined as the negative ratio of the strain normal to the loading direction to the strain along the loading direction. m increases gradually from the elastic Poisson ratio to 0.5 with the increase of deformation from the elastic to plastic states [28]. 2.2. Numerical implementation Uniaxial loading is applied to the upside of the axisymmetric embedded unit cell, in z-direction as shown in Fig. 1b. For the present problem, there are only three nonzero components of overall stress and strain tensors, i.e. the z and ez , the axial r r and er , and the circumferential radial r h and eh . From the deformation theory of plasticity, one r can obtain the Poisson ratio m by using the following stress–strain relation:
8 9 2 38 9 z > > = 1 1 m m > = < ez > > ; E ; : > : > eh h m m 1 r
ð6Þ
where E is the secant modulus of nonlinear deformation of the embedded cell. When the deformation of composite is elastic, E and m reduce to the overall Young’s modulus and Poisson ratio of composite. To the authors’ knowledge, the explicit formula Eq. (6) was not seen in literatures. Practically, the three equations in Eq. (6) are linearly dependent for E and m. So, any two of them can be used to determine E and m in calculation. After calcular and r h vanish because the composite tion converges, r subjected to a uniaxial tensile loading. For the present problem, we have the following boundary conditions:
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Fig. 2. Finite element meshes. (a) Partially enlarged and (b) finite element meshes.
uz ¼ 0 at z ¼ 0; uz ¼ U at z ¼ L;
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ð7aÞ ð7bÞ ð7cÞ
where ur and uz are the displacements in r- and z-directions, respectively. Boundary of the model at r = L is enforced to hold straight during deformation. ANSYS software is used for calculation. The finite element meshes are shown in Fig. 2. The material parameters are Em = 69 GPa, rm = 43 MPa and mm = 0.33 for matrix, and Ep = 440 GPa and mp = 0.25 for particle. Five different values of b, i.e. 0.01, 0.1, 1, 10 and 100 are considered to study the effects of soft (b < 1) and hard (b > 1) interphases. Let Nite denotes the number of iteration. The iteration procedures are as follows: Step 1: Initiation. Nite = 0; Let the property of the effective composite be that of the matrix. Step 2: Calculation. (1) Apply the boundary conditions to obtain the local fields by using the ANSYS software, and further calculate the average stress and strain of the embedded cell by using Eqs. (3) and (4). (2) Calculate the overall Young’s modulus and Poisson ratio of composite by using Eq. (6) and then obtain the equivalent stress–strain curve by Eq. (5). Step 3: Judge. Nite = Nite + 1; Calculate the absolute value of maximum difference of the equivalent stress at the same equivalent strain between the Nite step and the Nite-1 step. If the relative difference is less than 0.5%, then stop, otherwise, go to Step 4. Step 4: Iteration. Replace the overall property of the effective composite with the newly obtained one, and then go to Step 2. Fig. 3a and b shows two examples of the convergence procedures for the composites with different volume fractions of particle (fp = 10% and =50%) and without inter-
150 125
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at r ¼ 0;
Nite=0 Nite=1 Nite=2 Nite=3 Nite=4 Nite=5
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Strain Fig. 3. Examples of iterative computations of the overall stress–strain curves of composites without interphase (i.e. b = 1). (a) fp = 10% and (b) fp = 50%. (Nite – the number of iteration).
phase (b = 1). It is seen that for a lower value of fp, the stress–strain curve converges more rapidly (2 steps) than that for a higher value of fp (at least 5 steps). So, the convergence rate of calculation depends on the volume fraction of particle.
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3. Results and discussion The effect of ductile interphase on the strengthening and overall deformation behavior of particle reinforced MMCs is discussed in this section. Three kinds of interphases are
considered, i.e. hard (b = 10 and 100), soft (b = 0.01 and 0.1) and no (b = 1) interphases. Thickness of the interphase can be calculated as h = tR3. In Sections 3.1 and 3.2, the 4.0 pure matrix fp=10% fp=20% fp=30% fp=40% fp=50%
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Fig. 6. Effect of interphases stiffness on the strengthening of composites with different particle volume fractions (t = 0.055).
Strain Fig. 4. Overall stress–strain curves of composites with different particle volume fractions without interphase (i.e. b = 1).
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Strain Fig. 5. Overall stress–strain curves of composites with different particle volume fractions and interphases stiffnesses (t = 0.055). (a) fp = 10% and (b) fp = 50%.
0.02
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thickness t Fig. 7. Thickness effect of the hard interphase on the overall stress–strain curves (a) and the strengthening (b) of composite with particle volume fraction fp = 30% and b = 10. (a) Overall stress–strain curves and (b) rc/ rm vs thickness t.
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particle, interphase, matrix and composite are perfectly bonded each other. In Section 3.3, we will also consider four different locations of debonding, i.e. pole debonding between the interphase and the matrix (PDIM), pole debonding between the interphase and particle (PDIP), equator debonding between the interphase and matrix (EDIM) and equator between the interphase and particle (EDIP). Five values of particle volume fraction, i.e. fp = 10%, 20%, 30%, 40% and 50% are considered in analysis, which were considered by Dong and Schmauder [28] to study the particle volume fraction effect on the strengthening of composites. The overall stress–strain curves obtained herein for the composites with different particle volume fractions and without interphase are shown in Fig. 4, which agree very well with the results of Dong and Schmauder [28]. According to Bao et al. [30], Dong and Schmauder [28] and Zhang et al. [31], rc/rm, the ratio of overall stress rc of composite to the strength rm of pure matrix for the same overall strain, does not significantly change for composite when the deformation approaches certain critical value. So, the ratio rc/rm can be used to characterize the strengthening level of composites.
3.1. Effect of interphase stiffness The stiffness of interphase is described by the value of b as discussed in Section 2.1. Here, t = 0.055 for calculations. Effects of interphase stiffness (b = 100, 10, 1, 0.1 and 0.01) on the overall stress–strain curves of composites with different particle volume fractions (fp = 10% and 50%) are shown in Fig. 5a and b, respectively. It is seen that hard interphase (i.e. b > 1) results in the enhancement of com-
400 350
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t=0 t=0.029 t=0.055 t=0.079 t=0.103
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2.5 strengthening curve fp=30% β=0.1
σc/σm
2.0
1.5
1.0
0.00
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0.04 0.06 thickness t
0.08
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Fig. 8. Thickness effect of the soft interphase on the overall stress–strain curves (a) and the strengthening (b) of composite with particle volume fraction fp = 30% and b = 0.1. (a) Overall stress–strain curves and (b) rc/ rm vs thickness t.
Fig. 9. Distributions of microscopic von Mises strain in the composites with different interphases (fp = 30%, t = 0.055). (a) Without interphase (b = 1), (b) hard interphase (b = 10), (c) soft interphase (b = 0.01).
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posite strengthening and further enhancement becomes unclear while b > 10. The soft interphase (i.e. when b < 1) deteriorates the strengthening effect. If the interphase is very soft, e.g. b = 0.01, the particle has no reinforce effect and behaves as a void. Fig. 6 shows the variations of the ratio rc/rm with the value of b, from which one can clearly see the effects of interphase stiffness and particle volume fraction on the strengthening behavior of composites. It is interesting to note that there is a fixing point near b = 0.1 for the composites with different volume fraction of particle. This indicates that the average property of hard particle and soft interphase with proper thickness and stiffness is just identical to the matrix. It can be concluded that there exists specific stiffness for the soft interphase, with which the hard particles have no reinforcement to the matrix even though the volume fraction of particle is high. 3.2. Effect of interphase thickness Here, the volume fraction of particle is fixed at f = 30%, and b = 10 and 0.1 for hard and soft interphases, respectively. Effect of the thickness of hard interphase on the
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overall stress–strain curves of composites is shown in Fig. 7a. It is readily seen that the reinforcement of particle becomes more significant with the increase of interphase thickness. Fig. 7b shows the corresponding strengthening effect for different interphase thicknesses, which can be approximately expressed by a cubic polynomial function of the interphase thickness t as rc ¼ 2:454 þ 11:114t þ 105:513t2 þ 1019:395t3 : rm
ð8Þ
For soft interphase, the effect of interphase thickness on the overall stress–strain curves of composites and the variation of rc/rm with interphase thickness t are shown in Fig. 8a and b, respectively. It is seen that if there is a soft interphase between hard particle and matrix, the hard particle will have no reinforcement to the matrix. Even the very thin soft layer between particle and matrix can result in a significant decrease of the overall strengthen of composites. This conforms the conclusion obtained in Section 3.1 once again. Comparisons of the distribution of local von Mises strain in the composites without and with hard and soft
Fig. 10. Locations of the debonding and finite element meshes. (a) PDIM, (b) PDIP, (c) EDIM, and (d) EDIP.
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300 250
stress (MPa)
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Fig. 11. Overall stress–strain curves of the composites with debonding at pole (fp = 30%, t = 0.103). (a) PDIM, hard interphase (b = 10), (b) PDIP, hard interphase (b = 10), (c) PDIM, soft interphase (b = 0.1), (d) PDIP, soft interphase (b = 0.1).
interphases are shown in Fig. 9a–c, respectively. It is readily seen that if there is no interphase (i.e. b = 1), the local deformation concentrates mainly in the matrix, as shown in Fig. 9a. For the case of hard interphase (b = 10), the situation is similar to the case of no interphase, that is, the deformation is also concentrates mainly in the matrix, as shown in Fig. 9b. However, the situation is very different for the case of soft interphase, the deformation concentrates mainly in the interphase, which can explain why the soft interphase can not make the composite strengthening. So, one should pay much more attention to the soft interphase when studying the overall behavior of composites. 3.3. Effect of debonding locations Some investigations were carried out to study the effect of interfacial debonding process on the overall mechanical properties of composites without interphase, e.g. Haddadi and Teodosiu [23] and Ghassemieh [24], etc. Here, we further consider the effect of debonding location on the overall stress–strain curves and strengthening of composites with interphase for more detail. We assume four debonding
modes (PDIM, PDIP, EDIM and EDIP) mentioned at the beginning of Section 3 before calculations. The locations of debonding and finite element meshes are shown in Fig. 10. The thickness of interphase and the volume fraction of particle are fixed in calculations, e.g. the interphase thickness parameter t = 0.103 and the particle volume fraction fp = 30%. Here, we use fd = Sd/S to describe the degree of debonding, in which Sd and S are the debonding area and the whole interface area, respectively. The ratio fd can be interpreted as the average debonding ratio of all the interphase. For the case of pole debonding, we consider fd = 4.9%, 10.9% and 29.3% for both hard and soft interphases. The effects of pole debonding on the overall stress–strain curves of the composites with hard and soft interphases are shown in Fig. 11a–d. For the case of hard interphase, debonding between the interphase and matrix has more significant effect on the overall deformation of composites than that between the interphase and particle, as shown in Fig. 11a and b, respectively. For the case of soft interphase, debonding between interphase/matrix and/or interphase/particle has similar effect on the overall deformation of composites, as shown in Fig. 11c and d. It should be mentioned that the
W.X. Zhang et al. / Computational Materials Science 41 (2007) 145–155
b
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Fig. 12. Overall stress–strain curves of the composites with debonding at equator (fp = 30%, t = 0.103). (a) EDIM, hard interphase (b = 10), (b) EDIP, hard interphase (b = 10), (c) EDIM, soft interphase (b = 0.1), (d) EDIP, soft interphase (b = 0.1).
finite element analysis used for interfacial debonding did not adopt the singular element and due to the difficulty in numerical convergence, the overall stress–strain curve can only be obtained in a small range of strain for the case of soft interphase. For the case of equator debonding, we consider fd = 30%, 45% and 59% for both hard and soft interphases.
The effect of equator debonding on the overall stress–strain curves of the composites with hard and soft interphases are shown in Fig. 12a–d. It is seen that the equator debonding has much smaller effect compared to the pole debonding. For the case of hard interphase, EDIM results in the decrease of overall strength of composites, but EDIP has almost no effect on the overall strength of composites, as
Fig. 13. Distributions of microscopic von Mises stress in the composites without debonding (fp = 30%, t = 0.103, macroscopic strain ez ¼ 0:01). (a) Hard interphase (b = 10), (b) soft interphase (b = 0.1).
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Fig. 14. Distributions of microscopic von Mises stress in the composites with pole debonding (fp = 30%, t = 0.103, fd = 29.3%, macroscopic strain ez ¼ 0:01). (a) PDIM, hard interphase (b = 10), (b) PDIP, hard interphase (b = 10), (c) PDIM, soft interphase (b = 0.1), (d) PDIP, soft interphase (b = 0.1).
Fig. 15. Distributions of microscopic von Mises stress in the composites with equator debonding (fp = 30%, t = 0.103, fd = 30%, macroscopic strain ez ¼ 0:01). (a) EDIM, hard interphase (b = 10), (b) EDIP, hard interphase (b = 10), (c) EDIM, soft interphase (b = 0.1), (d) EDIP, soft interphase (b = 0.1).
shown in Fig. 12a and b, which is quite different from the case of pole debonding. For the case of soft interphase,
the effects of EDIM and EDIP are almost the same even for a very large debonding ratio, as shown in Fig. 12c and d.
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Distributions of the microscopic von Mises stress in the composites with or without debonding are shown in Figs. 13–15 for different locations of debonding at macroscopic strain ez ¼ 0:01. For the case of hard interphase, debonding between the interphase and matrix has more significant effect on the distribution of microscopic stress than the debonding between the interphase and particle, as shown in Fig. 14a and b and Fig. 15a and b. For the case of soft interphase, (see Fig. 13b, Fig. 14c and d and Fig. 15c and d), debonding locations have considerably small effects on the distribution of microscopic stress in the composites, as shown in Fig. 14c and d and Fig. 15c and d.
4. Conclusions Embedded cell model has been used to numerically study the effect of interphase on the strengthening behavior of particle reinforced metal matrix composites. The stiffness, thickness and debonding locations of interphase were considered in analysis. It can be concluded that the hard interphase between the particle and matrix enhances the reinforcement of particle to the matrix, but the soft interphase deteriorates the strengthening effect of PMMC significantly. A very soft interphase makes the particle behave as a void, so the overall stress–strain curve of composite decreases with the particle volume fraction. Also, debonding modes has different effects on the strengthening of composites. Pole debonding has more significant effect on the overall mechanical behavior of composites than equator debonding. For the case of hard interphase, debonding between the interphase and matrix has clearer effect on the overall strengthening of composites than the debonding between the interphase and particle.
Acknowledgements This work was supported by NSFC (10125212, 10572109 and 10672129), MOE and XJTU Doctoral Foundations.
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Computational Materials Science 41 (2007) 145–155 www.elsevier.com/locate/commatsci
Interphase effect on the strengthening behavior of particle-reinforced metal matrix composites W.X. Zhang, L.X. Li, T.J. Wang
*
MOE Key Laboratory for Strength and Vibration, Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an 710049, China Received 12 January 2007; received in revised form 12 March 2007; accepted 15 March 2007 Available online 15 May 2007
Abstract Interphase effect on the strengthening behavior of particle-reinforced metal matrix composites is numerically investigated, in which the stiffness, thickness and debonding location of interphase are considered. Embedded cell model and finite element method are used in analysis. The numerical results indicate that hard and soft interphases result in the increase and significant decrease of the strength of composite, respectively. Location of debonding has quite different effects. Pole debonding results in a remarkable decrease of the overall strength of composite, while equator debonding results in a slight decrease of the strength of composite. Debonding between the interphase and matrix has more significant effect than that between the interphase and particle. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Metal matrix composites (MMCs); Interphase; Strength; Plastic deformation; Finite element analysis (FEA)
1. Introduction Interface and/or interphase are important for composites, which affects the microscopic fields and then the overall properties of composites significantly [1–6]. However, interface is only an idealized mode and extensive work has been carried out. Interphase is really the third phase between the reinforced phase and matrix of composites and exists inevitably, which is due to the coated material to enhance the bond between matrix and inclusion in manufacture process, the chemical diffusion between inclusion and matrix, special design (e.g. functionally graded interphase, adsorbed contaminants on the surface of fiber/ particle results in a third material phase) and others. Extensive analytical and numerical work has been devoted to the elastic solutions of the interphase problems, including homogeneous [7–10] and inhomogeneous [6,11– 16] interphases. Up to now, relatively less work has been
*
Corresponding author. Tel./fax: +86 029 82665168. E-mail address: [email protected] (T.J. Wang).
0927-0256/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.03.011
carried out on the inelastic solutions of the interphase problem. Veazie and Qu [17] developed a spring-layer model without thickness to describe the interphase and then studied the interphase effect on the transverse deformation behavior of unidirectional fiber reinforced metal matrix composites (MMCs). Wang and Yang [18] employed unit cell to numerically study the energy dissipation of particle reinforced MMCs with ductile interphase. Su et al. [19] studied the effects of interphase strength on the damage modes and mechanical behavior of MMCs. Interphase may be harder or softer than the matrix [20]. Based on the secant modulus approach, Ding and Weng [21] theoretically studied the effect of ductile interphase on the overall stress–strain curves of particle-reinforced composites, in which both hard and soft interphases are considered. However, due to the limitation of their theoretical approach, the initial yield stress could not be predicted exactly and the local deformation field could not be presented. Zhang and Wang [22] carried out the 3D numerical analysis to study the effect of ductile interphase on the deformation behavior of MMCs with regularly distributed particles.
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Interphase debonding is another important factor to affect the mechanical behavior of composites. Haddadi and Teodosiu [23] studied the effect of debonding process on the plastic behavior of two-phase composites through the finite element analysis with a 3D unit cell. Ghassemieh [24] developed a finite element model to simulate the stress– strain behavior of particulate composites with an interfacial crack. Basically, the ductile interphase may be harder or softer than the matrix [20], and debonding may have different locations. However, the effects of ductile interphase and the effects of debonding modes, especially when interphase exits, on the overall mechanical behavior of composites with randomly distributed particles are not well understood up to now. The previous analytical method is only suitable for some ideal cases and is difficult to predict the overall mechanical property of composites with partial debonding interphase. The purpose of this work is to numerically investigate the effect of ductile interphase on the strengthening behavior of particle-reinforced metal matrix composites (PMMCs), in which the stiffness, thickness and debonding location of interphase are considered. Embedded cell model is employed to consider the random distribution of particles. In Section 2, the embedded unit cell model and the calculation procedures are presented. Effects of the stiffness and thickness of interphase and the location of debonding on the overall mechanical behavior of composites are presented and discussed in Section 3. Finally, concluding remarks are presented in Section 4. 2. Micromechanics approach 2.1. Embedded cell model and constitutive equations For random particle reinforced composites, one can choose a sufficiently large representative volume element (RVE) to describe the features of microstructures as many as possible [25,26] and then to numerically obtain the overall mechanical behavior of composites, which needs a great number of calculations and becomes increasingly feasible with development of the computer technology. An alternative approach is using the generalized self-consistent model (GSCM), which proposed by Christensen and Lo [27] in predicting the elastic moduli of composites, as shown in Fig. 1a. Chen et al. [16], Veazie and Qu [17], and Dong and Schmauder [28] have extended the original idea of GSCM to numerically study the elastoplastic deformation of composites, which is the so-called embedded cell model. The numerical results obtained by embedded cell model agree very well with experiments. This approach has high accuracy and saves calculations, and can be used to study the mechanical behavior of composites with much more complex microstructures, such as imperfect interphase [16], damage [19], voids [29], etc. In what follows, an axisymmetric embedded cell model is employed to study the effect of ductile interphase on
the overall mechanical properties of particle reinforced MMCs. The coordinate system is shown in Fig. 1b in which outer size L of the model is large enough so that the external boundary conditions have no influence on the deformation of the embedded cell. R1, R2 and R3 are radius of the particle, outer radius of the interphase and radius of the cell, respectively. Dong and Schmauder [28] indicated that if L/R3 P 5, the boundary effect can be neglected. In calculation, we take L/R3 = 10 and fix R3. R1 and R2 are changed to obtain various volume fractions of particle by fp ¼ R31 =R33 . Small deformation assumption is adopted in analysis. The inclusion, e.g. SiC particle, is assumed to be linear isotropic and to obey Hook’s law. The matrix, e.g. aluminum, is assumed to be isotropic and to obey von Mises yield criterion and the following stress–strain relation: 8 e 6 e0 ; < Em e; N r¼ ð1Þ : r0 ee ; e > e0 ; 0 where r and e are the uniaxial stress and strain in matrix. Em, r0 and e0 are the Young’s modulus, initial yield stress and yield strain of matrix, respectively, and have the relation of e0 = r0/Em. N is the strain hardening exponent of matrix. It is assumed that the interphase has the same strain hardening exponent and Poisson ratio, and obeys the following stress–strain relation: 8 e 6 e0 ; < bEm e; N r¼ ð2Þ : br0 ee ; e > e0 ; 0 where b is a constant. Obviously, b = 1 corresponds to the case of no interphase, and b < 1 and b > 1 correspond to the soft and hard interphases, respectively. The smaller the value of b is, the softer the interphase is, and vice verse. ij and strain eij in the composites can The overall stress r be calculated as Z 1 ij ¼ r rij dV ; i; j ¼ 1; 2; 3; ð3Þ V V Z 1 eij ¼ eij dV ; i; j ¼ 1; 2; 3; ð4Þ V V where rij and eij are the local stress and strain, respectively, and V volume of the embedded cell. According to [28], the v and strain ev of the embedded equivalent overall stress r cell can be calculated as rffiffiffiffiffiffiffiffiffiffiffiffi 3 v ¼ sijsij ; r ð5aÞ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 ev ¼ eijeij ; ð5bÞ 1 þ m 2 where sij and eij are overall deviator stress and strain tensors, respectively, m is the apparent Poisson ratio of nonlin-
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Fig. 1. The embedded unit cell model. (a) Material model and (b) axisymmetric computational model and coordinate system.
ear deformation which is defined as the negative ratio of the strain normal to the loading direction to the strain along the loading direction. m increases gradually from the elastic Poisson ratio to 0.5 with the increase of deformation from the elastic to plastic states [28]. 2.2. Numerical implementation Uniaxial loading is applied to the upside of the axisymmetric embedded unit cell, in z-direction as shown in Fig. 1b. For the present problem, there are only three nonzero components of overall stress and strain tensors, i.e. the z and ez , the axial r r and er , and the circumferential radial r h and eh . From the deformation theory of plasticity, one r can obtain the Poisson ratio m by using the following stress–strain relation:
8 9 2 38 9 z > > = 1 1 m m > = < ez >
ð6Þ
where E is the secant modulus of nonlinear deformation of the embedded cell. When the deformation of composite is elastic, E and m reduce to the overall Young’s modulus and Poisson ratio of composite. To the authors’ knowledge, the explicit formula Eq. (6) was not seen in literatures. Practically, the three equations in Eq. (6) are linearly dependent for E and m. So, any two of them can be used to determine E and m in calculation. After calcular and r h vanish because the composite tion converges, r subjected to a uniaxial tensile loading. For the present problem, we have the following boundary conditions:
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Fig. 2. Finite element meshes. (a) Partially enlarged and (b) finite element meshes.
uz ¼ 0 at z ¼ 0; uz ¼ U at z ¼ L;
175
ð7aÞ ð7bÞ ð7cÞ
where ur and uz are the displacements in r- and z-directions, respectively. Boundary of the model at r = L is enforced to hold straight during deformation. ANSYS software is used for calculation. The finite element meshes are shown in Fig. 2. The material parameters are Em = 69 GPa, rm = 43 MPa and mm = 0.33 for matrix, and Ep = 440 GPa and mp = 0.25 for particle. Five different values of b, i.e. 0.01, 0.1, 1, 10 and 100 are considered to study the effects of soft (b < 1) and hard (b > 1) interphases. Let Nite denotes the number of iteration. The iteration procedures are as follows: Step 1: Initiation. Nite = 0; Let the property of the effective composite be that of the matrix. Step 2: Calculation. (1) Apply the boundary conditions to obtain the local fields by using the ANSYS software, and further calculate the average stress and strain of the embedded cell by using Eqs. (3) and (4). (2) Calculate the overall Young’s modulus and Poisson ratio of composite by using Eq. (6) and then obtain the equivalent stress–strain curve by Eq. (5). Step 3: Judge. Nite = Nite + 1; Calculate the absolute value of maximum difference of the equivalent stress at the same equivalent strain between the Nite step and the Nite-1 step. If the relative difference is less than 0.5%, then stop, otherwise, go to Step 4. Step 4: Iteration. Replace the overall property of the effective composite with the newly obtained one, and then go to Step 2. Fig. 3a and b shows two examples of the convergence procedures for the composites with different volume fractions of particle (fp = 10% and =50%) and without inter-
150 125
Stress (MPa)
at r ¼ 0;
Nite=0 Nite=1 Nite=2 Nite=3 Nite=4 Nite=5
100 75 50 25 0 0.000
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150 100 50 0 0.000
0.005
0.010
0.015
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Strain Fig. 3. Examples of iterative computations of the overall stress–strain curves of composites without interphase (i.e. b = 1). (a) fp = 10% and (b) fp = 50%. (Nite – the number of iteration).
phase (b = 1). It is seen that for a lower value of fp, the stress–strain curve converges more rapidly (2 steps) than that for a higher value of fp (at least 5 steps). So, the convergence rate of calculation depends on the volume fraction of particle.
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3. Results and discussion The effect of ductile interphase on the strengthening and overall deformation behavior of particle reinforced MMCs is discussed in this section. Three kinds of interphases are
considered, i.e. hard (b = 10 and 100), soft (b = 0.01 and 0.1) and no (b = 1) interphases. Thickness of the interphase can be calculated as h = tR3. In Sections 3.1 and 3.2, the 4.0 pure matrix fp=10% fp=20% fp=30% fp=40% fp=50%
3.5
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50 0 0.000
149
-2
-1
0
1
2
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0.010
0.015
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Fig. 6. Effect of interphases stiffness on the strengthening of composites with different particle volume fractions (t = 0.055).
Strain Fig. 4. Overall stress–strain curves of composites with different particle volume fractions without interphase (i.e. b = 1).
400
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β=100 β=10 β=1
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f=10%
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pure matrix β=0.1 β=0.01
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Numerical Results Polynomial Fit Curve
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β =100 β =10 β =1
2.0 0.00 0.005
0.010
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Strain Fig. 5. Overall stress–strain curves of composites with different particle volume fractions and interphases stiffnesses (t = 0.055). (a) fp = 10% and (b) fp = 50%.
0.02
0.04
0.06
0.08
0.10
thickness t Fig. 7. Thickness effect of the hard interphase on the overall stress–strain curves (a) and the strengthening (b) of composite with particle volume fraction fp = 30% and b = 10. (a) Overall stress–strain curves and (b) rc/ rm vs thickness t.
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particle, interphase, matrix and composite are perfectly bonded each other. In Section 3.3, we will also consider four different locations of debonding, i.e. pole debonding between the interphase and the matrix (PDIM), pole debonding between the interphase and particle (PDIP), equator debonding between the interphase and matrix (EDIM) and equator between the interphase and particle (EDIP). Five values of particle volume fraction, i.e. fp = 10%, 20%, 30%, 40% and 50% are considered in analysis, which were considered by Dong and Schmauder [28] to study the particle volume fraction effect on the strengthening of composites. The overall stress–strain curves obtained herein for the composites with different particle volume fractions and without interphase are shown in Fig. 4, which agree very well with the results of Dong and Schmauder [28]. According to Bao et al. [30], Dong and Schmauder [28] and Zhang et al. [31], rc/rm, the ratio of overall stress rc of composite to the strength rm of pure matrix for the same overall strain, does not significantly change for composite when the deformation approaches certain critical value. So, the ratio rc/rm can be used to characterize the strengthening level of composites.
3.1. Effect of interphase stiffness The stiffness of interphase is described by the value of b as discussed in Section 2.1. Here, t = 0.055 for calculations. Effects of interphase stiffness (b = 100, 10, 1, 0.1 and 0.01) on the overall stress–strain curves of composites with different particle volume fractions (fp = 10% and 50%) are shown in Fig. 5a and b, respectively. It is seen that hard interphase (i.e. b > 1) results in the enhancement of com-
400 350
Stress (MPa)
300 250
t=0 t=0.029 t=0.055 t=0.079 t=0.103
200 150 100 50 0 0.000
0.005
0.010
0.015
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Strain
2.5 strengthening curve fp=30% β=0.1
σc/σm
2.0
1.5
1.0
0.00
0.02
0.04 0.06 thickness t
0.08
0.10
Fig. 8. Thickness effect of the soft interphase on the overall stress–strain curves (a) and the strengthening (b) of composite with particle volume fraction fp = 30% and b = 0.1. (a) Overall stress–strain curves and (b) rc/ rm vs thickness t.
Fig. 9. Distributions of microscopic von Mises strain in the composites with different interphases (fp = 30%, t = 0.055). (a) Without interphase (b = 1), (b) hard interphase (b = 10), (c) soft interphase (b = 0.01).
W.X. Zhang et al. / Computational Materials Science 41 (2007) 145–155
posite strengthening and further enhancement becomes unclear while b > 10. The soft interphase (i.e. when b < 1) deteriorates the strengthening effect. If the interphase is very soft, e.g. b = 0.01, the particle has no reinforce effect and behaves as a void. Fig. 6 shows the variations of the ratio rc/rm with the value of b, from which one can clearly see the effects of interphase stiffness and particle volume fraction on the strengthening behavior of composites. It is interesting to note that there is a fixing point near b = 0.1 for the composites with different volume fraction of particle. This indicates that the average property of hard particle and soft interphase with proper thickness and stiffness is just identical to the matrix. It can be concluded that there exists specific stiffness for the soft interphase, with which the hard particles have no reinforcement to the matrix even though the volume fraction of particle is high. 3.2. Effect of interphase thickness Here, the volume fraction of particle is fixed at f = 30%, and b = 10 and 0.1 for hard and soft interphases, respectively. Effect of the thickness of hard interphase on the
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overall stress–strain curves of composites is shown in Fig. 7a. It is readily seen that the reinforcement of particle becomes more significant with the increase of interphase thickness. Fig. 7b shows the corresponding strengthening effect for different interphase thicknesses, which can be approximately expressed by a cubic polynomial function of the interphase thickness t as rc ¼ 2:454 þ 11:114t þ 105:513t2 þ 1019:395t3 : rm
ð8Þ
For soft interphase, the effect of interphase thickness on the overall stress–strain curves of composites and the variation of rc/rm with interphase thickness t are shown in Fig. 8a and b, respectively. It is seen that if there is a soft interphase between hard particle and matrix, the hard particle will have no reinforcement to the matrix. Even the very thin soft layer between particle and matrix can result in a significant decrease of the overall strengthen of composites. This conforms the conclusion obtained in Section 3.1 once again. Comparisons of the distribution of local von Mises strain in the composites without and with hard and soft
Fig. 10. Locations of the debonding and finite element meshes. (a) PDIM, (b) PDIP, (c) EDIM, and (d) EDIP.
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300 250
stress (MPa)
b
350 No debonding fd=4.9% fd=10.9% fd=29.3%
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stress (MPa)
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60 40 20
20 0 0.0000
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stress (MPa)
c 120
0.010
strain
strain
0.0025
0.0050
0.0075
0.0100
strain
0 0.0000
0.0025
0.0050
strain
Fig. 11. Overall stress–strain curves of the composites with debonding at pole (fp = 30%, t = 0.103). (a) PDIM, hard interphase (b = 10), (b) PDIP, hard interphase (b = 10), (c) PDIM, soft interphase (b = 0.1), (d) PDIP, soft interphase (b = 0.1).
interphases are shown in Fig. 9a–c, respectively. It is readily seen that if there is no interphase (i.e. b = 1), the local deformation concentrates mainly in the matrix, as shown in Fig. 9a. For the case of hard interphase (b = 10), the situation is similar to the case of no interphase, that is, the deformation is also concentrates mainly in the matrix, as shown in Fig. 9b. However, the situation is very different for the case of soft interphase, the deformation concentrates mainly in the interphase, which can explain why the soft interphase can not make the composite strengthening. So, one should pay much more attention to the soft interphase when studying the overall behavior of composites. 3.3. Effect of debonding locations Some investigations were carried out to study the effect of interfacial debonding process on the overall mechanical properties of composites without interphase, e.g. Haddadi and Teodosiu [23] and Ghassemieh [24], etc. Here, we further consider the effect of debonding location on the overall stress–strain curves and strengthening of composites with interphase for more detail. We assume four debonding
modes (PDIM, PDIP, EDIM and EDIP) mentioned at the beginning of Section 3 before calculations. The locations of debonding and finite element meshes are shown in Fig. 10. The thickness of interphase and the volume fraction of particle are fixed in calculations, e.g. the interphase thickness parameter t = 0.103 and the particle volume fraction fp = 30%. Here, we use fd = Sd/S to describe the degree of debonding, in which Sd and S are the debonding area and the whole interface area, respectively. The ratio fd can be interpreted as the average debonding ratio of all the interphase. For the case of pole debonding, we consider fd = 4.9%, 10.9% and 29.3% for both hard and soft interphases. The effects of pole debonding on the overall stress–strain curves of the composites with hard and soft interphases are shown in Fig. 11a–d. For the case of hard interphase, debonding between the interphase and matrix has more significant effect on the overall deformation of composites than that between the interphase and particle, as shown in Fig. 11a and b, respectively. For the case of soft interphase, debonding between interphase/matrix and/or interphase/particle has similar effect on the overall deformation of composites, as shown in Fig. 11c and d. It should be mentioned that the
W.X. Zhang et al. / Computational Materials Science 41 (2007) 145–155
b
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350
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stress (MPa)
stress (MPa)
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stress (MPa)
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strain
strain
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60 No debonding fd=30% fd=45% fd=59%
40 20
0.020
strain
0 0.000
0.005
0.010
0.015
0.020
strain
Fig. 12. Overall stress–strain curves of the composites with debonding at equator (fp = 30%, t = 0.103). (a) EDIM, hard interphase (b = 10), (b) EDIP, hard interphase (b = 10), (c) EDIM, soft interphase (b = 0.1), (d) EDIP, soft interphase (b = 0.1).
finite element analysis used for interfacial debonding did not adopt the singular element and due to the difficulty in numerical convergence, the overall stress–strain curve can only be obtained in a small range of strain for the case of soft interphase. For the case of equator debonding, we consider fd = 30%, 45% and 59% for both hard and soft interphases.
The effect of equator debonding on the overall stress–strain curves of the composites with hard and soft interphases are shown in Fig. 12a–d. It is seen that the equator debonding has much smaller effect compared to the pole debonding. For the case of hard interphase, EDIM results in the decrease of overall strength of composites, but EDIP has almost no effect on the overall strength of composites, as
Fig. 13. Distributions of microscopic von Mises stress in the composites without debonding (fp = 30%, t = 0.103, macroscopic strain ez ¼ 0:01). (a) Hard interphase (b = 10), (b) soft interphase (b = 0.1).
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Fig. 14. Distributions of microscopic von Mises stress in the composites with pole debonding (fp = 30%, t = 0.103, fd = 29.3%, macroscopic strain ez ¼ 0:01). (a) PDIM, hard interphase (b = 10), (b) PDIP, hard interphase (b = 10), (c) PDIM, soft interphase (b = 0.1), (d) PDIP, soft interphase (b = 0.1).
Fig. 15. Distributions of microscopic von Mises stress in the composites with equator debonding (fp = 30%, t = 0.103, fd = 30%, macroscopic strain ez ¼ 0:01). (a) EDIM, hard interphase (b = 10), (b) EDIP, hard interphase (b = 10), (c) EDIM, soft interphase (b = 0.1), (d) EDIP, soft interphase (b = 0.1).
shown in Fig. 12a and b, which is quite different from the case of pole debonding. For the case of soft interphase,
the effects of EDIM and EDIP are almost the same even for a very large debonding ratio, as shown in Fig. 12c and d.
W.X. Zhang et al. / Computational Materials Science 41 (2007) 145–155
Distributions of the microscopic von Mises stress in the composites with or without debonding are shown in Figs. 13–15 for different locations of debonding at macroscopic strain ez ¼ 0:01. For the case of hard interphase, debonding between the interphase and matrix has more significant effect on the distribution of microscopic stress than the debonding between the interphase and particle, as shown in Fig. 14a and b and Fig. 15a and b. For the case of soft interphase, (see Fig. 13b, Fig. 14c and d and Fig. 15c and d), debonding locations have considerably small effects on the distribution of microscopic stress in the composites, as shown in Fig. 14c and d and Fig. 15c and d.
4. Conclusions Embedded cell model has been used to numerically study the effect of interphase on the strengthening behavior of particle reinforced metal matrix composites. The stiffness, thickness and debonding locations of interphase were considered in analysis. It can be concluded that the hard interphase between the particle and matrix enhances the reinforcement of particle to the matrix, but the soft interphase deteriorates the strengthening effect of PMMC significantly. A very soft interphase makes the particle behave as a void, so the overall stress–strain curve of composite decreases with the particle volume fraction. Also, debonding modes has different effects on the strengthening of composites. Pole debonding has more significant effect on the overall mechanical behavior of composites than equator debonding. For the case of hard interphase, debonding between the interphase and matrix has clearer effect on the overall strengthening of composites than the debonding between the interphase and particle.
Acknowledgements This work was supported by NSFC (10125212, 10572109 and 10672129), MOE and XJTU Doctoral Foundations.
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