Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces

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International Journal of Solids and Structures 44 (2007) 8390–8406 www.elsevier.com/locate/ijsolstr

Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces H.K. Lee *, S.H. Pyo Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejon 305-701, South Korea Received 2 March 2007; received in revised form 11 May 2007; accepted 18 June 2007 Available online 21 June 2007

Abstract A micromechanical framework is proposed to predict the effective elastic behavior and weakened interface evolution of particulate composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu, J., 1993a. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. Journal of Applied Mechanics 60 (4), 1048– 1050; Qu, J., 1993b. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mechanics of Materials 14, 269–281] is adopted to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to assess the potential of the present framework.  2007 Elsevier Ltd. All rights reserved. Keywords: Micromechanics; Imperfect interfaces; Weakened interface; Evolutionary damage

1. Introduction Interfacial bonding condition is one of the most important factors that control the local elastic fields and the overall properties of composites (Duan et al., 2007). The imperfect interfacial bonding may be due to damage in a very thin interfacial layer known as interphase or interface (Nie and Basaran, 2005). The term imperfect interface is used to characterize a situation in which the displacements are discontinuous at the interface between the inclusion and the matrix (Crouch and Mogilevskaya, 2006). The linear spring-layer of vanishing thickness, which is the simplest model of an imperfect interface, assumes that interfacial traction becomes *

Corresponding author. Tel.: +82 42 869 3623. E-mail address: [email protected] (H.K. Lee).

0020-7683/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2007.06.019

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continuous, but the displacements at the two sides of the interphase-layer become discontinuous (Crouch and Mogilevskaya, 2006). Refer to Benveniste (1985), Aboudi (1987), Achenbach and Zhu (1989, 1990), Hashin (1990, 1991), Zhu and Achenbach (1991), Qu (1993a,b), Sudak et al. (1999), and Mogilevskaya and Crouch (2002) for a further literature review on imperfect interface. Micromechanical damage models for composites considering interfacial debonding were proposed by many researchers (e.g., Zhao and Weng, 1997; Ju and Lee, 2000, 2001; Lee, 2001; Liang et al., 2006; Liu et al., 2006; Lee and Kim, 2007). They replaced the isotropic debonded inclusions by the perfectly debonded inclusions with yet unknown transversely isotropic properties to describe the loss of the load-transfer capacity of debonded interface between inclusion and matrix. Schjudt-Thomsen and Pyrz (2000) studied creep modeling of short fiber reinforced composites with weakened interfaces and complex fiber orientation. Their approach was based upon the Mori-Tanaka mean field theory and the modified Eshelby’s tensor for an inclusion with weakened interface proposed by Qu (1993a,b). A similar work was previously conducted by Jun and Haian (1997). Zhong et al. (2004) studied three-dimensional micromechanical modeling of particulate composites with imperfect interface. In their derivations, the imperfect interface was characterized by a spring-type model assuming that the traction continuity remains intact, while the displacement experienced a jump proportional to the interfacial traction. Recently, Ju and Ko (submitted for publication) proposed a new micromechanical elastoplastic progressive damage model where the partial debonding at the fiber interface was represented by the growing debonding angles of arc microcracks. A micromechanical framework is proposed in the present study to predict the effective elastic behavior and weakened interface evolution of particulate composites. The schematic of a particulate composite with weakened interfaces is illustrated in Fig. 1. In order to model spherical particles having imperfect interfaces in the composite, the Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a,b) is adopted. Based on the Eshelby’s micromechanics, the effective elastic moduli of three-phase particulate composites are derived. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to describe the varying probability of evolution of weakened interface between the inclusion and the matrix. The novelty in the present derivation is in the simplicity and flexibility of the model to represent the weakened interface evolution with limited number of model parameters. It is also straightforward to extend the proposed model to be used as a multi-level damage model, where some initially perfectly bonded particles may be transformed to particles with mildly weakened interface, some particles with mildly weakened interface may then be transformed to particles with severely weakened interface, and all particles may be transformed to completely debonded particles asymptotically. In our derivations, the particles are assumed to be elastic spheres that are randomly dispersed in an elastic matrix. All particles are assumed to be non-interacting and initially embedded firmly in the matrix with perfect

a

b

Perfectly bonded particle (phase 1) Matrix (phase 0)

Matrix (phase 0) Perfectly bonded particle (phase 1)

Particle with weakened interface (phase 2)

Fig. 1. Schematics of a particulate composite subjected to uniaxial tension: (a) the initial state; (b) the damaged state.

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interfaces. It is also assumed that the evolution of weakened interface is controlled by the average internal stresses of the particles as well as the Weibull parameters (Weibull, 1951). The proposed micromechanical elastic damage model is then applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Comparisons between the present predictions with other numerical and analytical predictions and available experimental data are conducted to further assess the potential of the present framework. The present paper is organized as follows. A micromechnics-based constitutive model for three-phase particulate composites containing perfectly bonded spherical particles and spherical particles with weakened interfaces is derived in Section 2. In Section 3, an evolutionary damage model for weakened interface is considered in accordance with the Weibull’s probabilistic function. The internal stresses of particles, which is the controlling factor of the Weibull function, are also derived in Section 3. Elastic stress–strain relations for the three-phase particulate composites are applied to the uniaxial, biaxial and triaxial loading conditions to predict the various stress–strain responses in Section 4. Finally, the present predictions are compared with other numerical (Ju and Lee, 2001) and analytical (Kerner, 1956; Halpin and Kardos, 1976) predictions as well as available experimental data (Zhou et al., 2004) to assess the potential of the present micromechanical framework in Section 5. 2. A micromechnics-based constitutive model for particulate composites with weakened interfaces 2.1. Recapitulation of the Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface Qu (1993a,b) derived the Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface in an elastic matrix of infinite extent. The weakened interface between the inclusion and the matrix was modeled by a spring layer of vanishing thickness. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a,b) is adopted to model spherical particles having imperfect interfaces in particulate composites and is incorporated into the micromechanical framework. The summary of the Eshelby’s tensor derived by Qu (1993a,b) is repeated here for completeness of the proposed elastic damage model. With the help of the Green’s function in the unbounded domain, the strain field inside an elliptical inclusion in a homogeneous linearly elastic solid can be represented by (Qu, 1993b) ij ðxÞ ¼ S ijkl kl þ T ijst ðxÞðI stkl  S stkl Þkl   SM ijkl ðxÞkl

ð1Þ

where S is the (original) Eshelby inclusion tensor, * is the uniform eigenstrain, I is the fourth-rank identity tensor and the fourth-rank tensor T is given by Z T ijpq ðxÞ ¼ Lklmn Lpqst gks Gijmn ðn  xÞnt nl dSðnÞ ð2Þ S

in which L is the fourth-rank elasticity tensor, G(x) is related to the Green’s function / given in Eq.(2.6a) and the Appendix of Qu (1993b), and n denotes the unit outward normal vector. In addition, the second-order tensor gij denoting the compliance of the interface spring layer is given by (Qu, 1993a) gij ¼ adij þ ðb  aÞni nj

ð3Þ

where a and b represent the compliance inRthe tangential and normal directions of the interface and dij signifies the Kronecker delta. Since S ijkl ¼ Lklmn X Gijmn ðn  xÞ dV ðnÞ, one arrive at (Qu, 1993b) Z 1 SM ¼ S ijkl ðxÞ dV ðxÞ ¼ S ijkl þ ðI ijpq  S ijpq ÞH pqrs Lrsmn ðI mnkl  S mnkl Þ ð4Þ ijkl X X where H pqrs

1 ¼ 4X

Z

ðgik nj nl þ gjk ni nl þ gil nj nk þ gjl ni nk Þ dS s

In the case of spherical inclusions, Hijkl can be simplified as (Qu, 1993b)

ð5Þ

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H ijkl ¼ aP ijkl þ ðb  aÞQijkl

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ð6Þ

in which 1 P ijkl ¼ I ijkl a 1 Qijkl ¼ ð2I ijkl þ dij dkl Þ 5a

ð7Þ ð8Þ

Here, a denotes the radius of inclusion. Further details of the derivation of the Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interfaces can be found in Qu (1993a,b). 2.2. Effective elastic moduli of three-phase particulate composites Let us consider an initially perfectly bonded two-phase composite consisting of an elastic matrix (phase 0) with bulk modulus j0 and shear modulus l0, and randomly dispersed elastic spherical particles (phase 1) with bulk modulus j1 and shear modulus l1. As loadings or deformations are applied, some particles may be weakened in their interfaces (phase 2). The effective stiffness tensor C* for the three-phase composite can be derived based on the governing equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions (Ju and Chen, 1994) as " # 2 X 1 1 1  C ¼ C0  I þ f/r ðAr þ Sr Þ  ½I  /r Sr  ðAr þ Sr Þ  g ð9Þ r¼1

where C0 is the elasticity tensor of the phase 0 (matrix), ‘‘Æ’’ is the tensor multiplication, /r denotes the volume fraction of the r-phase inclusion, and Sr signifies the Eshelby’s tensor for the r-phase. The fourth-rank tensor Ar is defined as Ar  ðCr  C0 Þ

1

 C0

ð10Þ

where Cr is the elasticity tensor of the r-phase. The components of Eshelby’s tensor S1 for a (perfectly bonded) spherical particle embedded in an isotropic linear elastic and infinite matrix can be derived as ðS 1 Þijkl ¼

1 fð5m0  1Þdij dkl þ ð4  5m0 Þðdik djl þ dil djk Þg 15ð1  m0 Þ

ð11Þ

where m0 denotes the Poisson’s ratio of the matrix. By carrying out the lengthy algebra, the Eshelby’s tensors S2 for a spherical particle with weakened interface embedded in an isotropic linear elastic and infinite matrix given in Eq. (4) can be rephrased as ðS 2 Þijkl ¼

1 1200að1  m0 Þ

2

½fð80 þ 480m0  400m20 Þa þ 500ð1  2m0 Þ2 k0 b

þ ð98 þ 140m0  50m20 Þl0 a þ ð268  1240m0 þ 1300m2o Þl0 bgdij dkl 2

þ fð320  720m0 þ 400m20 Þa þ l0 ð3a þ 2bÞð7  5m0 Þ gðdik djl þ dil djk Þ

ð12Þ

where k0 denotes the Lame constant of the matrix. Substituting Eqs. (11) and (12) into Eq. (9) yields the effective stiffness tensor C* for the three-phase particulate composite as C ¼ k dij dkl þ l ðdik djl þ dil djk Þ

ð13Þ

Accordingly, the effective Lame constant k* and the shear modulus l* for the three-phase composite can be expressed as k ¼ ð3k0 þ 2l0 ÞðK1 þ K3 Þ þ 2k0 ð12 þ K2 þ K4 Þ 

l ¼

2l0 ð12

þ K2 þ K4 Þ

where the components K1, . . . , K4 are listed in Appendix A.

ð14Þ ð15Þ

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Alternatively, the effective Young’s modulus E* and Poisson’s ratio m* of the three-phase particulate composite are easily obtained through the following relations l ð3k þ 2l Þ k  þ l k m ¼  2ðk þ l Þ

E ¼

ð16Þ ð17Þ

3. Damage modeling Imperfect interfaces between inclusion and matrix in particulate composites may occur as deformations or loadings continue to increase, and may affect the load-carrying capacity and the overall stress–strain behavior of the composites. For convenience, following Tohgo and Weng (1994) and Zhao and Weng (1995, 1996, 1997), the probability of weakened interface is modeled as a two-parameter Weibull process. Assuming that the Weibull (1951) statistics governs and the average internal stresses of particles are the controlling factor of the Weibull function, one can express the current volume fraction of particles with weakened interface /2 at a given level of ð r11 Þ1 (see also Ju and Lee, 2001) ( "  M #) ð r11 Þ1 /2 ¼ /P d ½ð r11 Þ1  ¼ / 1  exp   ð18Þ S0 where / is the original particle volume fraction, Pd denotes the cumulative probability distribution function of weakened interface for the uniaxial tensile loading (in the 1-direction), ð r11 Þ1 is the internal stress of particles  are the Weibull param(phase 1) in the 1-direction, the subscript (Æ)1 denotes the particle phase, and S0 and M eters. For the biaxial and hydrostatic tensile loadings, ð r11 Þ1 in Eq. (18) are replaced by ½ð r22 Þ1 þ ð r33 Þ1 =2 and ½ð r11 Þ1 þ ð r22 Þ1 þ ð r33 Þ1 =3, respectively. Within the context of the first-order (noninteracting) approximation, the stresses inside particles should be uniform. The internal stresses of particles required for the initiation of the weakened interface were explicitly derived by Ju and Lee (2000) as " #1 2 X 1 1 1 ¼ C1  ½I  S1  ðA1 þ S1 Þ   I  r /m Sm  ðAm þ Sm Þ :   U :  ð19Þ m¼1

By carrying out the lengthy algebra, the components of the positive definite fourth-rank tensor U for the present model are explicitly given by U ijkl ¼ U 1 dij dkl þ U 2 ðdik djl þ dil djk Þ

ð20Þ

where ð1  2n2  2n4 Þfn1 ð3k1 þ 2l1 Þ þ k1 ð/1  2n2 Þg þ 2l1 ðn1 þ n3 Þð/1  2n2 Þ /1 ð1  2n2  2n4 Þð1  3n1  2n2  3n3  2n4 Þ l1 ð/1  2n2 Þ U2 ¼ /1 ð1  2n2  2n4 Þ

U1 ¼

ð21Þ ð22Þ

where /1 is the current volume fraction of perfectly bonded particles, k1 is the Lame constant of particles, and the components n1, . . . , n4 are listed in Appendix A. In the case of tensile loading, the averaged internal stresses of particles can be obtained as follows (see also Ju and Lee, 2000): ð r11 Þ1 ¼ ðU 1 þ 2U 2 Þ11 þ U 122 þ U 133 ð r22 Þ1 ¼ U 111 þ ðU 1 þ 2U 2 Þ22 þ U 133 ð r33 Þ1 ¼ U 111 þ U 122 þ ðU 1 þ 2U 2 Þ33 where 11 , 22 and 33 are the total strains in the 1, 2 and 3 directions, respectively.

ð23Þ ð24Þ ð25Þ

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4. Numerical simulations To illustrate the proposed micromechanical elastic damage model for particulate composites, let us first  can be consider the uniaxial tension loading. In this case, the components of the macroscopic stress r 11 6¼ 0 and r ij ¼ 0 for all other stress components. According to the linear elasticity, the macexpressed as r roscopic incremental elastic strain for this case reads (see also Ju and Lee, 2001) 0

1 B D ¼ @ 0 0

0 m 0

1 0 r11 C D 0 A  E m

ð26Þ

For convenience, we adopt the material properties and volume fraction of particles for the 6061-T6 aluminum alloy matrix/silicon-carbide particle composites according to Ju and Lee (2001) and Zhao and Weng (1995): E0 = 68.3 GPa, m0 = 0.33, E1 = 490 GPa, m1 = 0.17, ry = 250 MPa; / = 0.2. The compliance parameters a and b given in Eq. (3), which are related to the components of Eshelby’s tensor S2 for spherical inclusions with weakened interfaces, are assumed to be a = 2.0 and b = 3.0. Moreover, to address the influence of the Weibull parameter S0 on the evolution of weakened interface and to evaluate the proposed elastic damage model sensitivity to the Weibull parameter S0 , three different sets of Weibull parameters are used:  ¼ 5; S0 ¼ 2:18  ry and M  ¼ 5; S0 ¼ 3:27  ry and M  ¼ 5. S0 ¼ 1:09  ry and M We plot the present predicted stress–strain responses of particulate composites with weakened interfaces (weakened interface model) under uniaxial tension with various S0 values in Fig. 2. We also plot the present predicted response of the perfectly bonded particulate composites (perfect bonding model) in the figure for comparison. Figs. 3–5 exhibit the evolutions of perfectly bonded particles and particles with weakened interface as functions of the uniaxial strains corresponding to Fig. 2. From Figs. 2–5, it is clear that lower S0 leads to faster evolution of weakened interface, indicating that most particles are weakened in their

800

700

Stress [σ11] (MPa)

600

500

400

300

200

perfect bonding perfectly bonded weakened y*3.27 interface ( S0 = 3.27*σ y) weakened y*2.18 interface ( S0 = 2.18*σ y) weakened y*1.09 interface ( S0 = 1.09*σ y)

100

0 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Strain [ε 11]

Fig. 2. The present predicted stress–strain responses of particulate composites with weakened interfaces under uniaxial tension with various S0 values.

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Fiber Volume Fraction

0.15

0.1

0.05

perfect bonding weakened interface 0 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Strain [ε11]

Fig. 3. The predicted evolution of volume fractions of perfectly bonded particles and particles with weakened interface corresponding to the prediction from the weakened interface model in Fig. 2 ðS0 ¼ 1:09  ry Þ. 0.2

Fiber Volume Fraction

0.15

perfect bonding 0.1

weakened interface

0.05

0 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Strain [ε11]

Fig. 4. The predicted evolution of volume fractions of perfectly bonded particles and particles with weakened interface corresponding to the prediction from the weakened interface model in Fig. 2 ðS0 ¼ 2:18  ry Þ.

interfaces in early stage with lower S0 and the composites show the nonlinear stress–strain behavior. It is also noted that the responses with the weakened interface are lower than the response without the weakened interface.

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0.2

Fiber Volume Fraction

0.15

0.1

perfect bonding weakened interface 0.05

0 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Strain [ε11]

Fig. 5. The predicted evolution of volume fractions of perfectly bonded particles and particles with weakened interface corresponding to the prediction from the weakened interface model in Fig. 2 ðS0 ¼ 3:27  ry Þ.

To further examine the effect of varying S0 values on the stress–strain responses of the composites under different loading cases, we consider biaxial and triaxial loading cases on next. In biaxial tensile loading case,  can be expressed as r 22 ¼ r 33 6¼ 0 and r ij ¼ 0 for all other comthe components of the macroscopic stress r ponents. From the linear elasticity, the macroscopic incremental elastic strain for a biaxial tensile loading is given by (see also Ju and Lee, 2001) 0 1 2m 0 0 r22 B C D D ¼ @ 0 1  m 0 A  ð27Þ E  0 0 1m  can be expressed as In triaxial tensile loading case, the components of the macroscopic stress r 22 ¼ r 33 ¼ C ij ¼ 0 for all other components, where C is the proportional loading ratio. r r11 6¼ 0 and all other r In particular, C = 1 indicates a pure hydrostatic loading. From the linear elasticity, the macroscopic incremental elastic strain for a triaxial tensile loading can be expressed as (see also Ju and Lee, 2001) 0 1 1  2Cm 0 0 r11 B C D D ¼ @ 0 C  ð1 þ CÞm 0 ð28Þ A  E  0 0 C  ð1 þ CÞm Figs. 6 and 8 show the present predicted stress–strain responses of particulate composites with weakened interfaces under biaxial and hydrostatic tension loadings with various S0 values, respectively. Figs. 7 and 9 exhibit the evolution of particles with weakened interface as functions of the biaxial and hydrostatic strains corresponding to Figs. 6 and 8, respectively. Similar to the uniaxial tensile loading case, lower S0 leads to faster evolution of weakened interface and lower stress–strain response. It is also observed that hydrostatic tension results in higher stress–strain behavior, which corresponds to findings from Ju and Lee’s (2001) observations. Lastly, to examine the effect of varying a and b values on the stress–strain response of the proposed micromechanical framework, we conduct a parametric analysis of a and b. Four different sets of the compliance parameters are used: a = 2.0 and b = 3.0; a = 2.0 · 104 and b = 3.0 · 104; a = 2.0 · 105 and

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600

Stress [σ22 (=σ33 )] (MPa)

500

400

300

200

perfect bonding perfectly bonded weakened interface ( S0 = 3.27*σy) 3.27 2.18 weakened interface ( S0 = 2.18*σy) 1.09 weakened interface ( S0 = 1.09*σy)

100

0 0.000

0.001

0.002

0.003

0.004

0.005

Strain [ε 22 (=ε33)]

Fig. 6. The present predicted stress–strain responses of particulate composites with weakened interfaces under biaxial tension with various S0 values. 0.2

Fiber Volume Fraction

0.15

0.1

0.05

S0 = 3.27*σy 3.27 S0 = 2.18*σy 2.18 S0 = 1.09*σy 1.09

0 0.000

0.001

0.002

0.003

0.004

0.005

Strain [ε22(=ε33)]

Fig. 7. The predicted evolution of volume fraction of particles with weakened interface corresponding to the predictions from the weakened interface model in Fig. 6.

b = 3.0 · 105; a = 2.0 · 106 and b = 3.0 · 106. Fig. 10 exhibit the present predicted stress–strain responses of particulate composites with weakened interfaces under uniaxial tension with various values of a and b. It is seen from the figure that as the values of a and b become higher, the effect of weakened interface on the stress–

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8399

1400

1200

Stress [σ11 (=σ22 =σ33)] (MPa)

1000

800

600

400

perfect bondin weakened interface ( S0 = 3.27*σ y) weakened interface ( S0 = 2.18*σ y) weakened interface ( S0 = 1.09*σ y)

200

0 0.000

0.001

0.002

0.003

0.004

0.005

Strain [ε11(=ε22=ε33)]

Fig. 8. The present predicted stress–strain responses of particulate composites with weakened interface under hydrostatic tension with various S0 values. 0.2

Fiber Volume Fraction

0.15

0.1

S0 = 3.27*σy 3.27 S0 = 2.18*σy 2.18 S0 = 1.09*σy 1.09

0.05

0 0.000

0.001

0.002

0.003

0.004

0.005

Strain [ε11(=ε22=ε 33)]

Fig. 9. The predicted evolution of volume fraction of particles with weakened interface corresponding to the predictions from the weakened interface model in Fig. 8.

strain behavior is more pronounced. Further, it is shown from the parametric analysis that the converged values of the parameters for the lower bound are a = 2.0 and b = 3.0. It is also worthy mentioning that if a = b = 0, the material shows a perfect bonding material characteristic.

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450

400

Stress [σ11] (MPa)

350

300

250

200

150

perfect bonding α =2.0, β =3.0 α=2.E-4, β =3.E-4 α=2.E-5, β =3.E-5 α=2.E-6, β =3.E-6

100

50

0 0.000

0.001

0.002

0.003

0.004

0.005

Strain [ε11]

Fig. 10. The present predicted elastic responses of particulate composites with weakened interfaces under uniaxial tension with various a and b values.

5. Comparisons with other numerical and analytical predictions and experiments To verify and further assess the proposed micromechanical framework, we compare the present predictions with other numerical (Ju and Lee, 2001) and analytical (Kerner, 1956; Halpin and Kardos, 1976) predictions and available experimental data (Zhou et al., 2004). We first compare the present prediction with the numerical prediction given in Ju and Lee (2001) for 6061-T6 aluminum alloy matrix/silicon-carbide particle composites under uniaxial tension in Fig. 11. Similar to Section 4, the elastic properties of the composites used in this simulation are (see Ju and Lee, 2001; Zhao and Weng, 1995; Arsenault, 1984; Nieh and Chellman, 1984) E0 = 68.3 GPa, m0 = 0.33, E1 = 490 GPa, m1 = 0.17, ry = 250 MPa; / = 0.2. The compliance parameters a and b are assumed to be a = 2.0 and b = 3.0 (lower bound) to clearly show the effect of weakened interface on the stress–strain behavior of weakened particulate composites. The Weibull  are assumed to be S0 ¼ 1:09  ry , M  ¼ 5. Clearly, the present weakened interface model parameters S0 and M shows slightly high stress values during the evolution of weakened interface in comparison with the partial debonding model (Ju and Lee, 2001). It should be noted that we do not display our predictions beyond the strain value 11 = 0.004, which corresponds to the yielding stress of the Ju and Lee’s (2001) predicted response, in the figure since the present model does not consider plastic formulations. Our analytical predictions are compared with Halpin-Tsai’s analytical solutions (Kerner, 1956; Halpin and Kardos, 1976) for particulate composites to validate the proposed micromechanical framework. The HalpinTsai’s analytical equations predict the elastic moduli of composites with different reinforcement geometries changing from a sphere (aspect ratio of one in the principal material direction) to a long fiber (aspect ratio approaching infinity) (Halpin and Kardos, 1976). The analytical equation for the Young’s modulus of particulate composites is given by (Kerner, 1956; Halpin and Kardos, 1976; Lee and Simunovic, 2001) 2

/f lf ð75mm Þlm þð810mm Þlf

/m þ 15ð1m mÞ

/f lm ð75mm Þlm þð810mm Þlf

/m þ 15ð1m fÞ

E ¼ 2ð1 þ m Þ4

3 5

ð29Þ

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8401

400

partial debonding (Ju and Lee, 2001) weakened interface (present prediction) perfect bonding (present prediction)

Stress [σ11] (MPa)

300

200

100

0 0.000

0.001

0.002

0.003

0.004

Strain [ε11]

Fig. 11. The comparison of predicted overall elastic uniaxial responses of 6061-T6 aluminum alloy matrix/silicon-carbide particle composites between the present perfect bonding and weakened interface damage models and Ju and Lee’s (2001) partial debonding model.

where subscripts m and f denote the matrix and particle phases, respectively, and the Poisson ratio m* is given by a rule of mixtures m ¼ mm /m þ mf /f

ð30Þ

The following constituent elastic properties for carbon fiber polymer matrix composites are considered for comparison: E0 = 3 GPa, m0 = 0.35, E1 = 380 GPa and m1 = 0.25 (Lee and Simunovic, 2001). Fig. 12 shows the effective (normalized) Young’s modulus versus volume fraction of particles predicted from the present perfect bonding and weakened interface models. In particular, various combinations of volume fractions of perfectly bonded particles (phase 1) and particles with weakened interface (phase 2) are considered in the weakened interface model as: /1//2 = 9/1, 7/3, 5/5, 0/10 (lower bound). The compliance parameters are assumed to be a = 2.0, b = 3.0. It is also assumed that the present weakened interface model is stationary (no damage evolution). The theoretical prediction based on the Halpin-Tsai’s analytical solution is also plotted in the figure. It is seen from Fig. 12 that the present prediction from the perfect bonding model (upper bound) is almost identical with the Halpin-Tsai’s analytical solution. The present predictions from the weakened interface model are well within the bounds. To further assess the potential of the present framework, we compare the present prediction with experimental data reported by Zhou et al. (2004) for the uniaxial stress–strain behavior of Mg–Al matrix with Al2O3 particulate composites. Here, we adopt the elastic properties according to Zhou et al. (2004) as follow: E0 = 73 GPa, m0 = 0.33, E1 = 400 GPa, m1 = 0.24, ry = 250 MPa, / = 0.48. It is emphasized that we need to  estimate the compliance parameters a and b given in Eq. (12) as well as the Weibull parameters S0 and M in Eq. (18) due Zhou et al. (2004) did not report those parameters. In accordance with the experimental data reported by Zhou et al. (2004), the above compliance and Weibull parameters are estimated to be: a = 2.0,  ¼ 5. b = 3.0; S0 ¼ 625 MPa; M Based on the above material properties and parameters, we depict our uniaxial stress–strain predictions on the composites with and without evolution of weakened interface against experimental data provided by Zhou et al. (2004) in Fig. 13. Overall, the present prediction from the weakened interface model and the experimen-

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H.K. Lee, S.H. Pyo / International Journal of Solids and Structures 44 (2007) 8390–8406 140

120

Halpin-Tsai’s analytical solution Kerner perfect perfectbonding bonding 9:1 weakened interface ( φ 1: φ 2 =9:1) 7:3 weakened interface ( φ 1: φ 2 =7:3) 5:5 weakened interface ( φ 1: φ 2 =5:5) 0:10 weakened interface ( φ 1: φ 2 =0:10)

100

E*/E0

80

60

40

20

0

0

0.2

0.4

0.6

0.8

1

Volume fraction of particle

Fig. 12. The comparison between the present predictions with various combinations of volume fractions of perfectly bonded particles and particles of weakened interface and the Halpin-Tsai’s theoretical prediction for effective (normalized) Young’s modulus versus volume fraction of particles.

400

Stress [σ11] (MPa)

300

200

100

experiment weakened interface perfect bonding 0 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Strain [ε 11]

Fig. 13. The comparison between the present prediction and experimental data (Zhou et al., 2004) for overall uniaxial tensile responses of Mg–Al matrix with Al2O3 particulate composites.

H.K. Lee, S.H. Pyo / International Journal of Solids and Structures 44 (2007) 8390–8406

8403

0.5

Fiber Volume Fraction

0.4

0.3

0.2

perfect bonding 0.1

weakened interface

0 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Strain [ε11]

Fig. 14. The predicted evolution of volume fractions of perfectly bonded particles and particles with weakened interface corresponding to the prediction from the weakened interface model in Fig. 13.

tal data match well. The predicted evolution of weakened interface corresponding to Fig. 13 is shown in Fig. 14. The good agreement between the present prediction and the experiment encourages for possible use of the proposed micromechanical framework to modeling the imperfect interface in (quasi-brittle) particulate composites. 6. Concluding remarks A micromechanical framework to predict the effective elastic behavior and weakened interface evolution of particulate composites is presented. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface (Qu, 1993a,b) is employed to model spherical particles having imperfect interfaces in the composites and is incorporated into the micromechanical framework. The effective elastic moduli of three-phase particulate composites are derived based on the Eshelby’s micromechanics. A damage model is subsequently considered in accordance with the Weibull’s probabilistic function to characterize the varying probability of evolution of weakened interface between the inclusion and the matrix. The proposed micromechanical elastic damage model is applied to the uniaxial, biaxial and triaxial tensile loadings to predict the various stress–strain responses. Further, the present predictions are compared with other numerical and analytical predictions and available experimental data to verify and assess the proposed micromechanical framework. The successful incorporation of the weakened interface model by a spring layer of vanishing thickness (Qu, 1993a,b) into governing micromechanical ensemble-volume averaged field equations (Ju and Chen, 1994) yields computational frameworks that enable us to predict the effective elastic behavior and weakened interface evolution of particulate composites, which can be portrayed as the first noninteracting, micromechanicsbased approximation for weakened interface modeling. It is observed from Figs. 2–10 that the effect of weakened interfaces on the effective stress–strain responses is more pronounced as the damage evolution becomes rapid. It is found from the parametric study on the compliance and Weibull parameters that higher a and b and lower S0 , which is related the interfacial strength of composites, lead to faster evolution of weakened interface. By contrast, if the a and b are low and the S0 is

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high, then the effects of weakened interface are not pronounced when compared with the perfect bonding model. The proposed micromechanical framework is assessed by comparing the present predictions with other numerical (Ju and Lee, 2001) and analytical (Kerner, 1956; Halpin and Kardos, 1976) predictions as well as experimental data (Zhou et al., 2004). The predicted stress–strain behavior of particulate composites featuring the current evolution in weakened interface is observed to be in good qualitative agreement with the experimental data as shown in Fig. 13. However, further comparisons between numerical predictions based on the proposed elastic damage model and other numerical simulations and experiments will be needed for  the calibration of the model parameters (a, b, S0 , M). In a forthcoming paper, a multi-step elastic damage model considering mildly, moderately, and severely weakened interfaces will be formulated for a more realistic simulation of weakened interface evolution in particulate composites. Acknowledgements The authors gratefully acknowledge the Ministry of Science and Technology, Korea for the financial support by a grant (NC36456, R11-2006-101-02004-0) to the Smart Infra-Structure Technology Center (SISTeC), KAIST. The second author would like to thank Mr. B.R. Kim at Korea Advanced Institute of Science and Technology (KAIST) for the advice on the Eshelby approach relevant to this research. Appendix A. Parameters K1, . . . , K4 in Eqs. (14), (15), (21), and (22)  K1 ¼ 15ð1  m0 Þ/1 K2 ¼

 x1 ð2n2  1Þ þ n1 ð3x1 þ 2w1 Þ 2w1 ð3x1 þ 2w1 Þð1  2n2 Þð1  3n1  2n2 Þ

15ð1  m0 Þ/1 4w1 ð1  2n2 Þ

ð31Þ ð32Þ



2

K3 ¼ 600að1  m0 Þ /2

 x2 ð2n4  1Þ þ n3 ð3x2 þ 2w2 Þ w2 ð3x2 þ 2w2 Þð1  2n4 Þð1  3n3  2n4 Þ

ð33Þ

2

K4 ¼ in which n1 ¼

300að1  m0 Þ /2 w2 ð1  2n4 Þ

ð34Þ

  /1 5x1 ð1 þ m0 Þ þ 5m0  1 3x1 þ 2w1 2w1

ð35Þ

/1 ð4  5m0 Þ 2w1   / x2 ð3v1 þ 2v2 Þ þ v1 n3 ¼ 2 3x2 þ 2w2 2w2

n2 ¼

n4 ¼

/2 v2 2w2

x1 ¼ 15ð1  m0 Þ

ð37Þ ð38Þ

l1 k0  l0 k1 þ 5m0  1 ðl1  l0 Þf3ðk1  k0 Þ þ 2ðl1  l0 Þg

x2 ¼ 1200að1  m0 Þ w1 ¼

ð36Þ

2

l2 k0  l0 k2 þ v1 ðl2  l0 Þf3ðk2  k0 Þ þ 2ðl2  l0 Þg

ð7  5m0 Þl0 þ ð8  10m0 Þl1 2ðl1  l0 Þ

ð39Þ ð40Þ ð41Þ

H.K. Lee, S.H. Pyo / International Journal of Solids and Structures 44 (2007) 8390–8406

w2 ¼ 600að1  m0 Þ2

l0 þ v2 l2  l0

8405

ð42Þ

with 2

v1 ¼ ð80 þ 480m0  400m20 Þa þ 500ð1  2m0 Þ k0 b þ ð98 þ 140m0  50m20 Þl0 a þ ð268  1240m0 þ 1300m20 Þl0 b

ð43Þ 2

v2 ¼ ð320  720m0 þ 400m20 Þa þ l0 ð3a þ 2bÞð7  5m0 Þ

ð44Þ

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