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Multi-level modeling of effective elastic behavior and progressive weakened interface in particulate composites
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COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 387–397 www.elsevier.com/locate/compscitech
Multi-level modeling of effective elastic behavior and progressive weakened interface in particulate composites H.K. Lee *, S.H. Pyo Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea Received 29 March 2007; received in revised form 29 May 2007; accepted 28 June 2007 Available online 5 July 2007
Abstract This paper proposes a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model to predict the effective elastic behavior and progressive weakened interface in particulate composites. The progression of weakened interface is assumed to be gradual and sequential. A multi-level damage model in accordance with the Weibull’s probabilistic function is developed to describe a sequential, progressive weakened interface in the composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu J. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. J Appl Mech 1993;60:1048–50; Qu J. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech Mater 1993;14:269–81] is adopted to model particles having mildly or severely weakened interface and is incorporated into the micromechanical formulation by Ju and Chen [Ju JW, Chen TM. Micromechanics and effective elastoplastic behavior of two-phase metal matrix composites, J Eng Mater 1994;116:310–18]. A numerical example corresponding to uniaxial tension loading is solved to illustrate the potential of the proposed multi-level elastic damage model. A parametric analysis is also carried out to address the influence of model parameters on the progressive weakened interface in the composites. Furthermore, the present predictions are compared with available experimental data in the literature to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. 2007 Elsevier Ltd. All rights reserved. Keywords: A. Particle-reinforced composites; B. Interface; B. Modelling; C. Probabilistic methods
1. Introduction Debonding phenomenon existed in between particles and matrix is one of well-known damage mechanisms in particulate composites and its effect on the mechanical behavior of the composites has to be well addressed for an accurate analysis of the composites. The progressive interfacial debonding between particles and matrix may occur under increasing deformation or stresses from the viewpoint of failure analysis. One-step Weibull distribution has been used by many researchers to describe the interfacial debonding between particles (fibers) and matrix, e.g. *
Corresponding author. Tel.: +82 42 869 3623; fax: +82 442 869 3610. E-mail address: [email protected] (H.K. Lee).
0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.06.026
[25,28–30,8,9,11–13,17,14,10]. However, a sequential probabilistic debonding analysis is necessary to realistically reflect the effect of loading history on the interfacial debonding. Gurvich and Pipes [3,4], Tabiei and Sun [23,24] and Hwang et al. [5] introduced sequential multi-step failure models to model the sequential probabilistic failure between fibers and matrix in laminated composites. In their sequential multi-step failure models, a structure was modeled as a parallel arrangement of elements based on the fiber bundle theory [5]. When an element fails, the load is redistributed among the remaining elements and final failure occurs when all of the elements fail [5]. Another approach to model the sequential probabilistic failure is proposed by Liu et al. [15,16] and Ju and Ko [7]. In their
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formulations, a progressive interfacial debonding model was developed based on Weibull distribution function [15,7] and one damage mode was transformed to another as applied external loads increased. That is, another damage mode is initiated when the previous local principal radial normal stress reaches the critical interfacial debonding strength [15,7]. Recently, a multi-level model for damage evolution in microstructurally debonded composites was developed by Ghosh et al. [2] for multi-scale analysis of damaged composite structures due to microstructural damage induced by interfacial debonding. In this study, a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model is proposed to predict the effective elastic behavior and progressive weakened interface in particulate composites. Since the progression of weakened interface is assumed to be gradual and sequential, as loads
or deformations continue to increase, some initially perfectly bonded particles are transformed to particles with mildly weakened interface. Some particles with mildly weakened interface are then transformed to particles with severely weakened interface, and all particles are transformed to completely debonded particles asymptotically within the proposed framework. A multi-level damage model in accordance with the Weibull’s probabilistic function is developed to describe the aforementioned sequential, progressive weakened interface in the composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [18,19] is adopted to model particles having mildly or severely weakened interface and is incorporated into the micromechanical formulation by Ju and Chen [6]. The schematic of a particulate composite under uniaxial tension, illustrating a sequential progression of weakened interface in the composite, is shown in Fig. 1.
Perfectly bonded particle (phase 1)
Perfectly bonded particle (phase 1) Matrix (phase 0)
Matrix (phase 0)
Particle with mildly weakened interface (phase 2)
Particle with mildly weakened
Particle with severely weakened
interface (phase 2)
interface (phase 3) Perfectly bonded particle (phase 1) Matrix (phase 0)
Matrix (phase 0) Particle with mildly weakened interface (phase 2) Void (phase 4)
Perfectly bonded particle (phase 1) Particle with severely weakened interface (phase 3)
Fig. 1. Schematics of a particulate composite subjected to uniaxial tension: (a) Level 0 of two-phase composite state (initial state). (b) Level 1 of threephase composite state. (c) Level 2 of four-phase composite state. (d) Level 3 of five-phase composite state.
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In particular, the following four-level elastic damage model is considered for a complete description of the sequential progression of weakened interface in the composite: (1) Level 0 of two-phase composite state consisting of matrix and perfectly bonded particles; (2) Level 1 of three-phase composite state consisting of matrix, perfectly bonded particles and particles having mildly weakened interface; (3) Level 2 of four-phase composite state consisting of matrix, perfectly bonded particles, particles having mildly weakened interface and particles having severely weakened interface; and (4) Level 3 of five-phase composite state consisting of matrix, perfectly bonded particles, particles having mildly weakened interface, particles having severely weakened interface and completely debonded particles. We assume that the level of severity of weakened interface can be classified simplistically as either mildly (moderately) weakened interface or severely weakened interface. It is obvious that the severity of weakened interface may vary as loads or deformations continue to increase. However, our premise is that its cumulative effect on the effective stress–strain behavior can be quantitatively described by using this simple approximation. The two weakened interface states, mildly weakened interface and severely weakened interface, are also assumed to occur sequentially. Further, particles are assumed to be randomly dispersed, non-interacting elastic spheres that are initially embedded firmly in the matrix with perfect interfaces. It is also assumed that the progression of weakened interface is governed by the average internal stresses of the particles as well as the Weibull parameters [27]. A numerical example corresponding to uniaxial tension loading is solved to illustrate the potential of the proposed multi-level elastic damage model. A parametric analysis is also carried out to address the influence of model parameters on the progressive weakened interface in the composites. Comparisons between the present prediction with available experimental data are conducted to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. 2. Multi-level damage modeling Weakened interfaces between particles and matrix in particulate composites may occur as deformations or loadings continue to increase, and may affect the load-carrying capacity and overall stress–strain behavior of the composites. As an effort to realistically reflect the effect of loading history on the progression of weakened interface, a multilevel damage model in accordance with the Weibull’s probabilistic function is developed. Specifically, a four-level damage model is considered in the present study, in the order of sequence of progressive weakened interface, for a complete description of the sequential, progressive weakened interface in the composite as illustrated in Fig. 1 and is discussed in detail in this section.
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2.1. Level 0 of two-phase composite state As illustrated in Fig. 1a, in the initial state, an initially perfectly bonded two-phase composite state consists of an elastic matrix (Phase 0), and randomly dispersed elastic spherical particles (Phase 1) with the volume fraction /. 2.2. Level 1 of three-phase composite state Following Tohgo and Weng [25] and Zhao and Weng [28–30], the probability of mildly weakened interface is modeled as a two-parameter Weibull process. Assuming that the Weibull [27] statistics governs and the average internal stresses of perfect bonded particles (Phase 1) are the controlling factor of the Weibull function, the current volume fractions of particles having mildly weakened interface /2 and perfectly bonded particles /1 in the three-phase composite at a given level of ð r11 Þ1 are (see also [9]) ( " M #) ð r11 Þ1 /2 ¼ /P d ½ð r11 Þ1 ¼ / 1 exp ð1Þ S0 /1 ¼ / /2
ð2Þ
where / is the original particle volume fraction, Pd signifies the cumulative probability distribution function of mildly weakened interface for the uniaxial tension loading case (in the 1-direction), ð r11 Þ1 is the internal stress of particles (Phase 1) in the 1-direction, the subscript (Æ)1 denotes the particle phase, and S0 and M are the Weibull parameters. For the biaxial and hydrostatic loading cases, ð r11 Þ1 in Eq. (1) is replaced by ½ð r22 Þ1 þ ð r33 Þ1 =2 and ½ð r11 Þ1 þ ð r22 Þ1 þ ð r33 Þ1 =3, respectively (see also [8]). The internal stresses of particles required for the initiation of the weakened interface were explicitly derived by Ju and Lee [8] as 1 ¼ C1 ½I S1 ðA1 þ S1 Þ1 r " #1 n X 1 I /r Sr ðAr þ Sr Þ : U :
ð3Þ
r¼1
where C1 is the elasticity tensor of the perfectly bonded particles, ‘‘Æ’’ is the tensor multiplication, I is the fourthrank identity tensor, and /r denotes the volume fraction of the r-phase inclusion. The fourth-rank tensor Ar is defined as Ar ðCr C0 Þ
1
C0
ð4Þ
The components of Eshelby’s tensor S1 for a (perfectly bonded) spherical particle and S4 for a (completely debonded) spherical particle embedded in an isotropic linear elastic and infinite matrix can be derived as ðS1;4 Þijkl ¼
1 fð5m0 1Þdij dkl þ ð4 5m0 Þ 15ð1 m0 Þ ðdik djl þ dil djk Þg
ð5Þ
where m0 denotes the Poisson’s ratio of the matrix and dij signifies the Kronecker delta.
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Following Lee and Pyo [10], the Eshelby’s tensors Sm+1 for a spherical particle with weakened interface embedded in an isotropic linear elastic and infinite matrix can be expressed as ðSmþ1 Þijkl ¼
1 1200að1 m0 Þ
2
½fð80 þ 480m0 400m20 Þa
þ 500ð1 2m0 Þ2 k0 bm þ ð98 þ 140m0 50m20 Þl0 am þ ð268 1240m0 þ 1300m2o Þl0 bm gdij dkl þ fð320 720m0 þ 400m20 Þa þ l0 ð3am þ 2bm Þð7 5m0 Þ2 g ðdik djl þ dil djk Þ
ð6Þ
where m = 1, 2 in which 1 indicates the mildly weakened interface stage, while 2 denotes the severely weakened interface stage. In addition, a denotes the radius of particles, k0 and l0 denote the Lame constants of the matrix, and a and b represent the compliance parameters in the tangential and normal directions of the interface. Solutions to the case of small a with b = 0 yield approximations to the free-sliding interfaces, whereas the free-sliding can be achieved by setting a ! 1 with b = 0 [19]. Following Qu’s [19] definitions on a and b, the severity of weakened
2.3. Level 2 of four-phase composite state Similar to the Level 1 of three-phase composite state, assuming that the Weibull [27] statistics governs and some particles with mildly weakened interface are transformed to particles with severely weakened interface as deformations or loads increase, the current volume fractions of particles having severely weakened interface /3, particles having mildly weakened interface /2 and perfectly bonded particles /1 in the four-phase composite state at a given level of ð r11 Þ1 can be derived through the following two-step Weibull approach ( " M #) ð r11 Þ1 /2 ¼ / 1 exp ð10Þ S0 ( " M #) ð r Þ 11 1 2 1 exp ð11Þ /3 ¼ / S0 2 / /2 ¼ / 3 / ¼ / /2 1
ð1 2n2 2n4 2n6 Þfn1 ð3k1 þ 2l1 Þ þ k1 ð/1 2n2 Þg þ 2l1 ðn1 þ n3 þ n5 Þð/1 2n2 Þ /1 ð1 2n2 2n4 2n6 Þð1 3n1 2n2 3n3 2n4 3n5 2n6 Þ l1 ð/1 2n2 Þ U2 ¼ /1 ð1 2n2 2n4 2n6 Þ
Uijkl ¼ U 1 dij dkl þ U 2 ðdik djl þ dil djk Þ
ð7Þ
where U1 and U2 for the three-phase composite can be expressed as see Lee and Pyo [10]
where k1 and l1 are the Lame constants of particles and the components n1, . . ., n4 are listed in the Appendix.
ð14Þ ð15Þ
where the components n1, . . ., n6 are listed in the Appendix. 2.4. Level 3 of five-phase composite state Similar to the Level 2 of four-phase composite state, assuming that the Weibull [27] statistics governs and some particles with severely weakened interface are transformed to completely debonded particles as deformations or loads increase, the current volume fractions of completely debonded particles /4, particles having severely weakened interface /3, particles having mildly weakened interface /2 and perfectly bonded particles /1 in the five-phase com-
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
ð13Þ
The components of the positive definite fourth-rank tensor U for the four-phase composite state in Eq. (7), denoted by U1 and U2, can be expressed as
U1 ¼
interface is assumed to be controlled by the values of a and b in the present study. It should also be noted that those parameters are not position dependent, indicating that no interfacial partial debonding is considered in this framework. By carrying out the lengthy algebra, the components of the positive definite fourth-rank tensor U for the multiphase composite state are explicitly given by
ð12Þ
ð8Þ ð9Þ
posite at a given level of ð r11 Þ1 can be derived through the following three-step Weibull approach
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
( " M #) ð r11 Þ1 /2 ¼ / 1 exp S0 ( " M #) ð r Þ 11 1 3 ¼ / 2 1 exp / S0 ( " M #) ð r11 Þ1 /4 ¼ /3 1 exp S0
ð16Þ ð17Þ ð18Þ
3 / /3 ¼ / 4 2 / 3 / ¼/
ð19Þ
2 /1 ¼ / /
ð21Þ
ð20Þ
2
The components of the positive definite fourth-rank tensor U for the five-phase composite state in Eq. (7), denoted by U1 and U2, can be expressed as
3.1. Level 0 of two-phase composite state The effective Lame constants k* and l* in Eq. (25) for the two-phase composite state can be expressed as (see Lee and Pyo [10]) 1 þ K2 k ¼ ð3k0 þ 2l0 ÞK1 þ 2k0 ð26Þ 2 1 þ K2 ð27Þ l ¼ 2l0 2 where the components K1 and K2 are listed in the Appendix. 3.2. Level 1 of three-phase composite state The effective Lame constants k* and l* in Eq. (25) for the three-phase composite state can be expressed as (see Lee and Pyo [10])
ð1 2n2 2n4 2n6 2n8 Þfn1 ð3k1 þ 2l1 Þ þ k1 ð/1 2n2 Þg þ 2l1 ðn1 þ n3 þ n5 þ n7 Þð/1 2n2 Þ /1 ð1 2n2 2n4 2n6 2n8 Þð1 3n1 2n2 3n3 2n4 3n5 2n6 3n7 2n8 Þ l1 ð/1 2n2 Þ U2 ¼ /1 ð1 2n2 2n4 2n6 2n8 Þ 1 where the components n1, . . ., n8 are listed in the Appendix. k ¼ ð3k0 þ 2l0 ÞðK1 þ K3 Þ þ 2k0 þ K2 þ K4 2 1 3. A micromechanics-based, multi-level elastic model for þ K2 þ K4 l ¼ 2l0 2 particulate composites with weakened interfaces
U1 ¼
We start with considering 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 loads or deformations continue to increase, some initially perfectly bonded particles are transformed to particles with mildly weakened interface, some particles with mildly weakened interface are then transformed to particles with severely weakened interface, and all particles are transformed to completely debonded particles asymptotically within the proposed framework. The effective stiffness tensor C* for the multi-phase particulate composite can be derived based on the governing equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions [6] as C ¼ C0 ½I þ Rnr¼1 f/r ðAr þ Sr Þ
1
1 1
½I /r Sr ðAr þ Sr Þ g
ð24Þ
where Cr is the elasticity tensor of the r-phase. Substituting Eqs. (5) and (6) into Eq. (24) yields the effective stiffness tensor C* for the multi-phase particulate composite as C ¼ k dij dkl þ l ðdik djl þ dil djk Þ:
ð25Þ
391
ð22Þ ð23Þ
ð28Þ ð29Þ
where the components K1, . . ., K4 are listed in the Appendix. 3.3. Level 2 of four-phase composite state The effective Lame constants k* and l* in Eq. (25) for the four-phase composite state can be expressed as 1 k ¼ ð3k0 þ2l0 ÞðK1 þK3 þK5 Þþ2k0 þK2 þK4 þK6 ð30Þ 2 1 ð31Þ l ¼ 2l0 þK2 þK4 þK6 2 where the components K1, . . ., K6 are listed in the Appendix. 3.4. Level 3 of five-phase composite state The effective Lame constants k* and l* in Eq. (25) for the five-phase composite state can be expressed as k ¼ ð3k0 þ 2l0 ÞðK1 þ K3 þ K5 þ K7 Þ 1 þ K2 þ K4 þ K6 þ K8 þ 2k0 2 1 þ K2 þ K4 þ K6 þ K8 l ¼ 2l0 2
ð32Þ ð33Þ
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where the components K1, . . ., K8 are listed in the Appendix. 4. Numerical example To illustrate the elastic damage behavior of the present framework, we consider the uniaxial tension loading case ij ¼ 0 for all other stress components). For ( r11 6¼ 0 and r brevity, we adopt the material properties for the 6061-T6 aluminum alloy matrix/silicon carbide particle composites according to Zhao and Weng [28], Ju and Lee [9] and Lee and Pyo [10]: E0 = 68.3 GPa, m0 = 0.33, E1 = 490 GPa, m1 = 0.17, ry = 250 MPa, / = 0.2. As a result of the parametric analysis on compliance parameters a and b reported in a preceding paper Lee and Pyo [10], the compliance parameters of the mildly weakened interface, denoted by a1 and b1, and the severely weakened interface, denoted by a2 and b2, in Eq. (6) are chosen to be a1 = 2.0 · 105, b1 = 3.0 · 105; a2 = 2.0, b2 = 3.0. Moreover, the Weibull parameters in Eq. (1) are assumed to be S0 = 1.09*ry and M = 5. To investigate the influence of the level of damage on the stress–strain response of the composites, four different phase composite states illustrated in Fig. 1 are considered in this simulation. Fig. 2a shows the present predicted stress–strain responses of particulate composites under uniaxial tension. The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles (e.g. particles with mildly weakened interface,
particles with severely weakened interface, completely debonded particles) at the three-phase, four-phase and fivephase composite states are exhibited in Fig. 2b. At the early stage of loading, the stress–strain curves are shown to be linear since the volume fractions of the damaged particles are low (Fig. 2a). As strains or stresses continue to increase, the volume fractions of the damaged particles increase gradually and the composites show nonlinear stress–strain responses as shown in Fig. 2. Clearly, the high-level composite states give rise to substantial weaker stress–strain response in comparison with the low-level composite states. It is demonstrated through this simulation that the current multi-level elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged particles exist simultaneously within the proposed framework. 5. Parametric analysis The Weibull parameter S0 in Eq. (1) is closely related to the strength at the particle–matrix interface in particulate composites. A parametric analysis is conducted to examine the influence of the Weibull parameter S0 on the progression of weakened interface in the composites. As primary parameters governing the progressive weakened interface, three representative sets of Weibull parameters are used: S0 = 1.09*ry, M = 5; S0 = 2.18*ry, M = 5; S0 = 3.27*ry, M = 5. For comparison, a uniaxial tensile simulation is
500
0.2
450 2-phase composite state
400
3-phase composite state 4-phase composite state
0.15
5-phase composite state
Particle Volume Fraction
Stress [σ11] (MPa)
350
300
250
200
0.1 perfectly bonded (3-phase) mildly weakened (3-phase) perfectly bonded (4-phase) mildly weakened (4-phase) severely weakened (4-phase) perfectly bonded (5-phase)
150
mildly weakened (5-phase) severely weakened (5-phase)
0.05
completely debonded (5-phase)
100
50
0 0.000
0.001
0.002
0.003
Strain [ε11]
0.004
0.005
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε11]
Fig. 2. (a) The present predicted stress–strain responses of particulate composites under uniaxial tension. (b) The predicted evolution of volume fractions of perfectly bonded particles various types of damaged particles corresponding to (a).
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
conducted using the same material properties and model parameters as used in Section 4. The present predicted stress–strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S0 values are shown in
393
Fig. 3a. Fig. 3b exhibits the predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface, and particles with severely weakened interface, respectively, as a function of the uniaxial strain corresponding to Fig. 3a. It is clearly shown from Fig. 3 0.3 perfectly bonded (S0 = 1.09*σy ) mildly weakened (S 0 = 1.09*σy ) severely weakened (S0 = 1.09*σy ) perfectly bonded (S0 = 2.18*σy ) mildly weakened (S0 = 2.18*σy ) severely weakened (S0 = 2.18*σy ) perfectly bonded ( S0 = 3.27*σy ) mildly weakened (S0 = 3.27*σy ) severely weakened (S0 = 3.27*σy )
500 0.25
400
Particle Volume Fraction
Stress [σ11] (MPa)
0.2
300
200
0.15
0.1
S0 = 3.27*σy S0 = 2.18*σy S0 = 1.09*σy
100
0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.05
0
0.008
0.000
0.001
0.002
0.003
Strain [ε11]
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
350
300
Stress [σ11] (MPa)
250
200
150
100
α2 =2.0, β 2=3.0 α2=2.E-4, β 2=3.E-4 α 2=6.E-5, β 2=7.E-5 α2=2.E-5, β 2=3.E-5
50
0
0
0.005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
Strain [ε11]
Fig. 3. (a) The present predicted stress–strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S0 values. (b) The predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface and particles with severely weakened interface corresponding to (a). (c) The present predicted stress–strain responses of particulate composites under uniaxial tension at four-phase composite state with various a2 and b2 values.
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that if the strength at the particle–matrix interface is low (lower S0), most particles are mildly weakened or severely weakened in their interfaces in early stage and the composites show a nonlinear stress–strain behavior even in early stage of loading. It can be concluded from this parametric analysis that the influence of the Weibull parameter S0 on the progression of weakened interface and stress–strain behavior of the composites is quite significant; thus, an accurate characterization of S0 would be essential to realistically predict the progressive weakened interface in particulate composites. Another parametric analysis is conducted to evaluate the proposed multi-level elastic damage model sensitivity to the compliance parameters a and b and to address the influence of varying a and b values on the stress–strain response of the model. As parameters controlling the severity of weakened interface, four sets of the compliance parameters for the severely weakened interface are used: a2 = 2.0, b2 = 3.0; a2 = 2.0 · 104, b2 = 3.0 · 104; a2 = 6.0 · 105, b2 = 7.0 · 105; a2 = 2.0 · 105, b2 = 3.0 · 105. For simplicity, fixed values of the compliance parameters for the mildly weakened interface are used as follows: a1 = 2.0 · 105, b1 = 3.0 · 105. The present predicted stress–strain responses of particulate composites with severely weakened interfaces under uniaxial tension with various values of a2 and b2 are shown in Fig. 3c. As the values of the compliance parameters a2 and b2 become higher, the effect of the severely weakened interface on the stress–strain behavior of the composites is more pronounced.
6. Experimental comparisons The proposed multi-level elastic damage model is further exercised to predict the behavior of brittle ceramic composites. The uniaxial stress–strain behavior of unidirectional SiC fiber-reinforced CAS glass–ceramic matrix composites tested by Sørensen [20] is numerically predicted using the multi-level elastic damage model. The unidirectional fibers in the composites are modeled as spherical particles in this simulation. The elastic properties of the composites used in this simulation are as follows [20– 22,26]: E0 = 98 GPa, m0 = 0.3, E1 = 200 GPa, m1 = 0.15, / = 0.35. According to the experimentally obtained stress–strain curve in the longitudinal direction [20], the compliance parameters a1, b1, a2 and b2 given in Eq. (6) as well as the Weibull parameters S0 and M in Eq. (1) are estimated to be: a1 = 2.0 · 105 and b1 = 3.0 · 105, a2 = 2.0, b2 = 3.0; S0 = 450 MPa, M = 2. The present predicted uniaxial stress–strain curve of the composites at the five-phase composite state is compared with the experimentally obtained stress–strain curve in the longitudinal (fiber) direction [20] in Fig. 4a. It is noted that the present prediction and the experimental data match well within the small strain region (the corresponding threshold strain th = 0.005), but the predicted stress– strain curve starts to deviate from the experimentally obtained one after the threshold strain. The experiment in Sørensen [20] was designed to show the brittle behavior of ceramic composites reinforced with unidirectional fibers. It was shown from Fig. 4a that the experimentally obtained
500
0.4
450
perfectly bonded mildly weakened severely weakened completely debonded
0.35 400 0.3
Particle Volume Fraction
Stress [σ11] (MPa)
350
300
250
200
0.25
0.2
0.15
150 0.1
experiment
100 presentprediction (5-phase composite)
0.05
50
0
0
0.001
0.002
0.003
0.004 0.005 Strain [ε11]
0.006
0.007
0.008
0
0
0.001
0.002
0.003
0.004 0.005 Strain [ε11]
0.006
0.007
0.008
Fig. 4. (a) The comparison between the present prediction and experimental data [20] for overall uniaxial tensile responses of SiC fiber-reinforced CAS glass–ceramic matrix composites. (b) The predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface, severely weakened interface and completely debonded particles corresponding to the present prediction in (a).
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397 0.3
600
500
0.25
400
0.2 Particle Volume Fraction
Stress [σ11] (MPa)
395
300
perfectly bonded mildly weakened severely weakened completely debonded
0.15
0.1
200
experiment
100
0.05
present prediction (5-phase composite state)
0
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0
0.002
0.004
0.006
0.008
0.01
0.012
Strain [ε11]
Strain [ε11]
Fig. 5. (a) The comparison between the present prediction and experimental data [1] for overall uniaxial tensile responses of silicon carbide particlereinforced 2009 aluminum matrix composites. (b) The predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface, severely weakened interface and completely debonded particles corresponding to the present prediction in (a).
stress of the composites was increased substantially after the threshold strain due the unidirectional fibers may play the primary role in the increase of stress. On the other hand, inclusions are limited to be spherical particles in our framework and therefore our predictions cannot be compared with his experimental data after the threshold strain. The predicted evolution of volume fractions of particles corresponding to Fig. 4a is shown in Fig. 4b. To assess and demonstrate the accuracy of the proposed multi-level elastic damage model, the present predictions are compared with experimental data reported by Geiger and Welch [1] for the uniaxial stress–strain behavior of silicon carbide particulate-reinforced 2009 aluminum matrix composites. The matrix is assumed to behave elastically. We adopt the same material properties for the composites as those in Geiger and Welch [1] as follows: E0 = 72.4 GPa, m0 = 0.33, E1 = 450 GPa, m1 = 0.19, / = 0.294. Since model parameters of the proposed model were not reported by Geiger and Welch [1], one needs to estimate the compliance parameters a1, b1, a2 and b2 given in Eq. (6) as well as the Weibull parameters S0 and M in Eq. (1). Those compliance and Weibull parameters are fitted according to the experimentally obtained stress–strain curve [1]. The fitted model parameters are: a1 = 2.0 · 105, b1 = 3.0 · 105, a2 = 2.0, b2 = 3.0; S0 = 825 MPa, M = 3. The predicted uniaxial stress–strain curve of the composites at the five-phase composite state based on the above material properties and parameters is shown in Fig. 5a. The experimentally obtained stress–strain curve
is also plotted in the figure for comparison. As a whole, the present prediction and the experimental data match well. The predicted evolution of volume fractions of particles corresponding to Fig. 5a is shown in Fig. 5b. 7. Concluding remarks A multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model has been presented to predict the effective elastic behavior and progressive weakened interface in particulate composites. A four-level damage model based on the Weibull’s probabilistic function is developed to describe the progressive weakened interface in the composites. The Eshelby’s tensor for a slightly weakened ellipsoidal inclusion [18,19] to model particles having mildly or severely weakened interface is incorporated into a micromechanics-based multi-level elastic model. The proposed micromechanical elastic damage model is applied to the uniaxial tension loading to predict the corresponding stress–strain responses. A parametric analysis is also carried out to address the influence of the compliance and Weibull parameters on the progressive weakened interface in the composites. Finally, the present predictions are compared with experimental data [20,1] to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. The observations and findings of this numerical study can be summarized as:
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(1) The stress–strain curves of the composites are linear at the early stage of loading since the volume fractions of damaged particles are low. As strains or stresses continue to increase, the volume fractions of damaged particles increase gradually, resulting in a nonlinear stress–strain response. (2) The proposed multi-level elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged particles exist simultaneously within the proposed framework. (3) If the strength at the particle–matrix interface is low (lower S0), most particles are mildly weakened, severely weakened or completely debonded in their interfaces even in early stage of loading. (4) The influence of the Weibull parameter S0 on the progression of weakened interface and stress–strain behavior of the composites is quite significant. (5) As the values of the compliance parameters a2 and b2 become higher, the effect of the severely weakened interface on the stress–strain behavior of the composites is more pronounced. (6) The predicted stress–strain behavior of particulate composites featuring the multi-level damage progression of weakened interface is observed to be in good qualitative agreement with experimental data [1].
Appendix. Parameters K1, . . ., K8 in Eqs. (8)–(23) and (26)– (33) K1 ¼ 15ð1m0 Þ/1
2
K3 ¼ 600að1m0 Þ /2
ð35Þ
x2 ð2n4 1Þþn3 ð3x2 þ2w2 Þ w2 ð3x2 þ2w2 Þð12n4 Þð13n3 2n4 Þ
ð36Þ 2
K4 ¼
300að1m0 Þ /2 w2 ð12n4 Þ 2
K5 ¼ 600að1m0 Þ /3
ð37Þ
x3 ð2n6 1Þþn5 ð3x3 þ2w3 Þ w3 ð3x3 þ2w3 Þð12n6 Þð13n5 2n6 Þ
ð38Þ 2
300að1m0 Þ /3 ð39Þ w3 ð12n6 Þ x4 ð2n8 1Þþn7 ð3x4 þ2w4 Þ K7 ¼ 15ð1m0 Þ/4 2w4 ð3x4 þ2w4 Þð12n8 Þð13n7 2n8 Þ
K6 ¼
ð40Þ 15ð1m0 Þ/4 4w4 ð12n8 Þ
ð41Þ
in which
/1 5x1 ð1 þ m0 Þ n1 ¼ þ 5m0 1 3x1 þ 2w1 2w1 /1 ð4 5m0 Þ 2w1 / x2 ð3v1 þ 2v2 Þ þ v1 n3 ¼ 2 3x2 þ 2w2 2w2
n2 ¼
/ 2 v2 2w2 /3 x3 ð3v3 þ 2v4 Þ þ v3 n5 ¼ 3x3 þ 2w3 2w3
n4 ¼
/ 3 v4 2w3 /4 5x4 ð1 þ m0 Þ þ 5m0 1 n7 ¼ 3x4 þ 2w4 2w4
n6 ¼
n8 ¼
/4 ð4 5m0 Þ 2w4
x1 ¼ 15ð1 m0 Þ Acknowledgement
ð34Þ 15ð1m0 Þ/1 K2 ¼ 4w1 ð12n2 Þ
K8 ¼ As an extension of the present study, a series of characterization tests on a particulate composite need to be conducted in the future for an accurate calibration of the model parameters (S0, M, a1, b1, a2 and b2). Data points (e.g. stress–strain curves, load-deflection curves, etc.) will be sampled from the tests and will be used to obtain an optimal set of the parameters by minimizing the sum of the squares of differences between the predicted and measured loads at all data points. Since the lower and upper limits of the parameters can be determined from a series of parametric analysis, this would be a nonlinear leastsquare constrained minimization problem. Alternatively, these parameters can be determined simultaneously with a trial-and-error procedure. The authors are currently working on the extension of the proposed multi-level elastic damage model for the prediction of the damage evolution and elastoplastic behavior of particulate ductile matrix composites. Specifically, the classical rate-independent plasticity will be incorporated into the present framework and the corresponding computational algorithm will be systematically developed.
x1 ð2n2 1Þþn1 ð3x1 þ2w1 Þ 2w1 ð3x1 þ2w1 Þð12n2 Þð13n1 2n2 Þ
ð42Þ ð43Þ ð44Þ ð45Þ ð46Þ ð47Þ ð48Þ ð49Þ
l1 k0 l0 k1 þ 5m0 1 ðl1 l0 Þf3ðk1 k0 Þ þ 2ðl1 l0 Þg ð50Þ
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.
x2 ¼ 1200að1 m0 Þ
2
l2 k0 l0 k2 þ v1 ðl2 l0 Þf3ðk2 k0 Þ þ 2ðl2 l0 Þg ð51Þ
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
l3 k0 l0 k3 þ v3 ðl3 l0 Þf3ðk3 k0 Þ þ 2ðl3 l0 Þg ð52Þ x4 ¼ 5m0 1 ð53Þ ð7 5m0 Þl0 þ ð8 10m0 Þl1 ð54Þ w1 ¼ 2ðl1 l0 Þ l0 w2 ¼ 600að1 m0 Þ2 þ v2 ð55Þ l2 l0 l0 2 þ v4 ð56Þ w3 ¼ 600að1 m0 Þ l3 l0 5m0 7 ð57Þ w4 ¼ 2 with x3 ¼ 1200að1 m0 Þ2
2
v1 ¼ ð80 þ 480m0 400m20 Þa þ 500ð1 2m0 Þ k0 b1 þ ð98 þ 140m0 50m20 Þl0 a1 þ ð268 1240m0 þ 1300m20 Þl0 b1 v2 ¼ ð320 720m0 þ
400m20 Þa
ð58Þ
þ l0 ð3a1 þ 2b1 Þð7 5m0 Þ
2
ð59Þ v3 ¼
ð80 þ 480m0 400m20 Þa þ 500ð1 þ ð98 þ 140m0 50m20 Þl0 a2 þ ð268 1240m0 þ 1300m20 Þl0 b2
v4 ¼ ð320 720m0 þ
400m20 Þa
2
2m0 Þ k0 b2 ð60Þ 2
þ l0 ð3a2 þ 2b2 Þð7 5m0 Þ : ð61Þ
References [1] Geiger AL, Welch P. Tensile properties and thermal expansion of discontinuously reinforced aluminium composites at subambient temperatures. J Mater Sci 1997;32:2611–6. [2] Ghosh S, Bai J, Raghavan P. Concurrent multi-level model for damage evolution in microstructurally debonding composites. Mech Mater 2007;39:241–66. [3] Gurvich MR, Pipes RB. Strength size effect of laminated composites. Compos Sci Technol 1995;55:93–105. [4] Gurvich MR, Pipes RB. Probabilistic strength analysis of fourdirectional laminated composites. Compos Sci Technol 1996;56:649–56. [5] Hwang TK, Hong CS, Kim CG. Size effect on the fiber strength of composite pressure vessels. Compos Struct 2003;59:489–98. [6] Ju JW, Chen TM. Micromechanics and effective elastoplastic behavior of two-phase metal matrix composites. J Eng Mater 1994;116:310–8. [7] Ju JW, Ko YF. Micromechanical elastoplastic damage modeling for progressive interfacial arc debonding for fiber reinforced composites. Int J Damage Mech, [submitted for publication]. [8] Ju JW, Lee HK. A micromechanical damage model for effective elastoplastic behavior of ductile matrix composites considering evolutionary complete particle debonding. Comput Method Appl M 2000;183:201–22.
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[9] Ju JW, Lee HK. A micromechanical damage model for effective elastoplastic behavior of partially debonded ductile matrix composites. Int J Solids Struct 2001;38:6307–32. [10] Lee HK, Pyo SH. Micromechanics-based elastic damage modeling of particulate composites with weakened interfaces. Int J Solids Struct, [in press]. [11] Lee HK, Simunovic S. Modeling of progressive damage in aligned and randomly oriented discontinuous fiber polymer matrix composites. Compos Part B – Eng 2000;31:77–86. [12] Lee HK, Simunovic S. A damage constitutive model of progressive debonding in aligned discontinuous fiber composites. Int J Solids Struct 2001;38:875–95. [13] Lee HK, Simunovic S. A damage mechanics model of crackweakened, chopped fiber composites under impact loading. Compos Part B – Eng 2002;33:25–34. [14] Liang Z, Lee HK, Suaris W. Micromechanics-based constitutive modeling for unidirectional laminated composites. Int J Solids Struct 2006;43:5674–89. [15] Liu HT, Sun LZ, Ju JW. An interfacial debonding model for particlereinforced composites. Int J Damage Mech 2004;13:163–85. [16] Liu HT, Sun LZ, Ju JW. Elastoplastic modeling of progressive interfacial debonding for particle-reinforced metal matrix composites. Acta Mech 2006;181:1–17. [17] Liu HT, Sun LZ, Wu HC. Monte Carlo simulation of particlecracking damage evolution in metal matrix composites. J Eng Mater – T ASME 2005;127:318–24. [18] Qu J. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. J Appl Mech 1993;60:1048–50. [19] Qu J. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech Mater 1993;14: 269–81. [20] Sørensen BF. Effect of fibre roughness on the overall stress-transverse strain response of ceramic composites. Scripta Metall Mater 1993;28:435–9. [21] Sørensen BF, Talreja R. Analysis of damage in a ceramic matrix composite. Int J Damage Mech 1993;2:246–71. [22] Sørensen BF, Talreja R, Sørensen OT. Micromechanical analysis of damage mechanisms in ceramic matrix composites during mechanical and thermal loading. Composites 1993;24:129–40. [23] Tabiei A, Sun J. Statistical aspects of strength size effect of laminated composite materials. Compos Struct 1999;46:209–16. [24] Tabiei A, Sun J. Analytical simulation of strength size effect in composite materials. Compos Part B – Eng 2000;31:133–9. [25] Tohgo K, Weng GJ. A progress damage mechanics in particlereinforced metal–matrix composites under high hydrostatic tension. J Eng Mater 1994;116:414–20. [26] Vanswijgenhoven E, Van Der Biest O. The relationship between longitudinal stress and transverse strain during tensile testing of unidirectional fibre toughened ceramic matrix composites. Acta Mater 1997;45:3349–62. [27] Weibull W. A statistical distribution function of wide applicability. J Appl Mech 1951;18:293–7. [28] Zhao YH, Weng GJ. A theory of inclusion debonding and its influence on the stress–strain relations of a ductile matrix. Int J Damage Mech 1995;4:196–211. [29] Zhao YH, Weng GJ. Plasticity of a two-phase composite with partially debonded inclusions. Int J Plasticity 1996;12: 781–804. [30] Zhao YH, Weng GJ. Transversely isotropic moduli of two partially debonded composites. Int J Solids Struct 1997;34:493–507.
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 387–397 www.elsevier.com/locate/compscitech
Multi-level modeling of effective elastic behavior and progressive weakened interface in particulate composites H.K. Lee *, S.H. Pyo Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea Received 29 March 2007; received in revised form 29 May 2007; accepted 28 June 2007 Available online 5 July 2007
Abstract This paper proposes a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model to predict the effective elastic behavior and progressive weakened interface in particulate composites. The progression of weakened interface is assumed to be gradual and sequential. A multi-level damage model in accordance with the Weibull’s probabilistic function is developed to describe a sequential, progressive weakened interface in the composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [Qu J. Eshelby tensor for an elastic inclusion with slightly weakened interfaces. J Appl Mech 1993;60:1048–50; Qu J. The effect of slightly weakened interfaces on the overall elastic properties of composite materials. Mech Mater 1993;14:269–81] is adopted to model particles having mildly or severely weakened interface and is incorporated into the micromechanical formulation by Ju and Chen [Ju JW, Chen TM. Micromechanics and effective elastoplastic behavior of two-phase metal matrix composites, J Eng Mater 1994;116:310–18]. A numerical example corresponding to uniaxial tension loading is solved to illustrate the potential of the proposed multi-level elastic damage model. A parametric analysis is also carried out to address the influence of model parameters on the progressive weakened interface in the composites. Furthermore, the present predictions are compared with available experimental data in the literature to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. 2007 Elsevier Ltd. All rights reserved. Keywords: A. Particle-reinforced composites; B. Interface; B. Modelling; C. Probabilistic methods
1. Introduction Debonding phenomenon existed in between particles and matrix is one of well-known damage mechanisms in particulate composites and its effect on the mechanical behavior of the composites has to be well addressed for an accurate analysis of the composites. The progressive interfacial debonding between particles and matrix may occur under increasing deformation or stresses from the viewpoint of failure analysis. One-step Weibull distribution has been used by many researchers to describe the interfacial debonding between particles (fibers) and matrix, e.g. *
Corresponding author. Tel.: +82 42 869 3623; fax: +82 442 869 3610. E-mail address: [email protected] (H.K. Lee).
0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.06.026
[25,28–30,8,9,11–13,17,14,10]. However, a sequential probabilistic debonding analysis is necessary to realistically reflect the effect of loading history on the interfacial debonding. Gurvich and Pipes [3,4], Tabiei and Sun [23,24] and Hwang et al. [5] introduced sequential multi-step failure models to model the sequential probabilistic failure between fibers and matrix in laminated composites. In their sequential multi-step failure models, a structure was modeled as a parallel arrangement of elements based on the fiber bundle theory [5]. When an element fails, the load is redistributed among the remaining elements and final failure occurs when all of the elements fail [5]. Another approach to model the sequential probabilistic failure is proposed by Liu et al. [15,16] and Ju and Ko [7]. In their
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formulations, a progressive interfacial debonding model was developed based on Weibull distribution function [15,7] and one damage mode was transformed to another as applied external loads increased. That is, another damage mode is initiated when the previous local principal radial normal stress reaches the critical interfacial debonding strength [15,7]. Recently, a multi-level model for damage evolution in microstructurally debonded composites was developed by Ghosh et al. [2] for multi-scale analysis of damaged composite structures due to microstructural damage induced by interfacial debonding. In this study, a multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model is proposed to predict the effective elastic behavior and progressive weakened interface in particulate composites. Since the progression of weakened interface is assumed to be gradual and sequential, as loads
or deformations continue to increase, some initially perfectly bonded particles are transformed to particles with mildly weakened interface. Some particles with mildly weakened interface are then transformed to particles with severely weakened interface, and all particles are transformed to completely debonded particles asymptotically within the proposed framework. A multi-level damage model in accordance with the Weibull’s probabilistic function is developed to describe the aforementioned sequential, progressive weakened interface in the composites. The Eshelby’s tensor for an ellipsoidal inclusion with slightly weakened interface [18,19] is adopted to model particles having mildly or severely weakened interface and is incorporated into the micromechanical formulation by Ju and Chen [6]. The schematic of a particulate composite under uniaxial tension, illustrating a sequential progression of weakened interface in the composite, is shown in Fig. 1.
Perfectly bonded particle (phase 1)
Perfectly bonded particle (phase 1) Matrix (phase 0)
Matrix (phase 0)
Particle with mildly weakened interface (phase 2)
Particle with mildly weakened
Particle with severely weakened
interface (phase 2)
interface (phase 3) Perfectly bonded particle (phase 1) Matrix (phase 0)
Matrix (phase 0) Particle with mildly weakened interface (phase 2) Void (phase 4)
Perfectly bonded particle (phase 1) Particle with severely weakened interface (phase 3)
Fig. 1. Schematics of a particulate composite subjected to uniaxial tension: (a) Level 0 of two-phase composite state (initial state). (b) Level 1 of threephase composite state. (c) Level 2 of four-phase composite state. (d) Level 3 of five-phase composite state.
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
In particular, the following four-level elastic damage model is considered for a complete description of the sequential progression of weakened interface in the composite: (1) Level 0 of two-phase composite state consisting of matrix and perfectly bonded particles; (2) Level 1 of three-phase composite state consisting of matrix, perfectly bonded particles and particles having mildly weakened interface; (3) Level 2 of four-phase composite state consisting of matrix, perfectly bonded particles, particles having mildly weakened interface and particles having severely weakened interface; and (4) Level 3 of five-phase composite state consisting of matrix, perfectly bonded particles, particles having mildly weakened interface, particles having severely weakened interface and completely debonded particles. We assume that the level of severity of weakened interface can be classified simplistically as either mildly (moderately) weakened interface or severely weakened interface. It is obvious that the severity of weakened interface may vary as loads or deformations continue to increase. However, our premise is that its cumulative effect on the effective stress–strain behavior can be quantitatively described by using this simple approximation. The two weakened interface states, mildly weakened interface and severely weakened interface, are also assumed to occur sequentially. Further, particles are assumed to be randomly dispersed, non-interacting elastic spheres that are initially embedded firmly in the matrix with perfect interfaces. It is also assumed that the progression of weakened interface is governed by the average internal stresses of the particles as well as the Weibull parameters [27]. A numerical example corresponding to uniaxial tension loading is solved to illustrate the potential of the proposed multi-level elastic damage model. A parametric analysis is also carried out to address the influence of model parameters on the progressive weakened interface in the composites. Comparisons between the present prediction with available experimental data are conducted to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. 2. Multi-level damage modeling Weakened interfaces between particles and matrix in particulate composites may occur as deformations or loadings continue to increase, and may affect the load-carrying capacity and overall stress–strain behavior of the composites. As an effort to realistically reflect the effect of loading history on the progression of weakened interface, a multilevel damage model in accordance with the Weibull’s probabilistic function is developed. Specifically, a four-level damage model is considered in the present study, in the order of sequence of progressive weakened interface, for a complete description of the sequential, progressive weakened interface in the composite as illustrated in Fig. 1 and is discussed in detail in this section.
389
2.1. Level 0 of two-phase composite state As illustrated in Fig. 1a, in the initial state, an initially perfectly bonded two-phase composite state consists of an elastic matrix (Phase 0), and randomly dispersed elastic spherical particles (Phase 1) with the volume fraction /. 2.2. Level 1 of three-phase composite state Following Tohgo and Weng [25] and Zhao and Weng [28–30], the probability of mildly weakened interface is modeled as a two-parameter Weibull process. Assuming that the Weibull [27] statistics governs and the average internal stresses of perfect bonded particles (Phase 1) are the controlling factor of the Weibull function, the current volume fractions of particles having mildly weakened interface /2 and perfectly bonded particles /1 in the three-phase composite at a given level of ð r11 Þ1 are (see also [9]) ( " M #) ð r11 Þ1 /2 ¼ /P d ½ð r11 Þ1 ¼ / 1 exp ð1Þ S0 /1 ¼ / /2
ð2Þ
where / is the original particle volume fraction, Pd signifies the cumulative probability distribution function of mildly weakened interface for the uniaxial tension loading case (in the 1-direction), ð r11 Þ1 is the internal stress of particles (Phase 1) in the 1-direction, the subscript (Æ)1 denotes the particle phase, and S0 and M are the Weibull parameters. For the biaxial and hydrostatic loading cases, ð r11 Þ1 in Eq. (1) is replaced by ½ð r22 Þ1 þ ð r33 Þ1 =2 and ½ð r11 Þ1 þ ð r22 Þ1 þ ð r33 Þ1 =3, respectively (see also [8]). The internal stresses of particles required for the initiation of the weakened interface were explicitly derived by Ju and Lee [8] as 1 ¼ C1 ½I S1 ðA1 þ S1 Þ1 r " #1 n X 1 I /r Sr ðAr þ Sr Þ : U :
ð3Þ
r¼1
where C1 is the elasticity tensor of the perfectly bonded particles, ‘‘Æ’’ is the tensor multiplication, I is the fourthrank identity tensor, and /r denotes the volume fraction of the r-phase inclusion. The fourth-rank tensor Ar is defined as Ar ðCr C0 Þ
1
C0
ð4Þ
The components of Eshelby’s tensor S1 for a (perfectly bonded) spherical particle and S4 for a (completely debonded) spherical particle embedded in an isotropic linear elastic and infinite matrix can be derived as ðS1;4 Þijkl ¼
1 fð5m0 1Þdij dkl þ ð4 5m0 Þ 15ð1 m0 Þ ðdik djl þ dil djk Þg
ð5Þ
where m0 denotes the Poisson’s ratio of the matrix and dij signifies the Kronecker delta.
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Following Lee and Pyo [10], the Eshelby’s tensors Sm+1 for a spherical particle with weakened interface embedded in an isotropic linear elastic and infinite matrix can be expressed as ðSmþ1 Þijkl ¼
1 1200að1 m0 Þ
2
½fð80 þ 480m0 400m20 Þa
þ 500ð1 2m0 Þ2 k0 bm þ ð98 þ 140m0 50m20 Þl0 am þ ð268 1240m0 þ 1300m2o Þl0 bm gdij dkl þ fð320 720m0 þ 400m20 Þa þ l0 ð3am þ 2bm Þð7 5m0 Þ2 g ðdik djl þ dil djk Þ
ð6Þ
where m = 1, 2 in which 1 indicates the mildly weakened interface stage, while 2 denotes the severely weakened interface stage. In addition, a denotes the radius of particles, k0 and l0 denote the Lame constants of the matrix, and a and b represent the compliance parameters in the tangential and normal directions of the interface. Solutions to the case of small a with b = 0 yield approximations to the free-sliding interfaces, whereas the free-sliding can be achieved by setting a ! 1 with b = 0 [19]. Following Qu’s [19] definitions on a and b, the severity of weakened
2.3. Level 2 of four-phase composite state Similar to the Level 1 of three-phase composite state, assuming that the Weibull [27] statistics governs and some particles with mildly weakened interface are transformed to particles with severely weakened interface as deformations or loads increase, the current volume fractions of particles having severely weakened interface /3, particles having mildly weakened interface /2 and perfectly bonded particles /1 in the four-phase composite state at a given level of ð r11 Þ1 can be derived through the following two-step Weibull approach ( " M #) ð r11 Þ1 /2 ¼ / 1 exp ð10Þ S0 ( " M #) ð r Þ 11 1 2 1 exp ð11Þ /3 ¼ / S0 2 / /2 ¼ / 3 / ¼ / /2 1
ð1 2n2 2n4 2n6 Þfn1 ð3k1 þ 2l1 Þ þ k1 ð/1 2n2 Þg þ 2l1 ðn1 þ n3 þ n5 Þð/1 2n2 Þ /1 ð1 2n2 2n4 2n6 Þð1 3n1 2n2 3n3 2n4 3n5 2n6 Þ l1 ð/1 2n2 Þ U2 ¼ /1 ð1 2n2 2n4 2n6 Þ
Uijkl ¼ U 1 dij dkl þ U 2 ðdik djl þ dil djk Þ
ð7Þ
where U1 and U2 for the three-phase composite can be expressed as see Lee and Pyo [10]
where k1 and l1 are the Lame constants of particles and the components n1, . . ., n4 are listed in the Appendix.
ð14Þ ð15Þ
where the components n1, . . ., n6 are listed in the Appendix. 2.4. Level 3 of five-phase composite state Similar to the Level 2 of four-phase composite state, assuming that the Weibull [27] statistics governs and some particles with severely weakened interface are transformed to completely debonded particles as deformations or loads increase, the current volume fractions of completely debonded particles /4, particles having severely weakened interface /3, particles having mildly weakened interface /2 and perfectly bonded particles /1 in the five-phase com-
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
ð13Þ
The components of the positive definite fourth-rank tensor U for the four-phase composite state in Eq. (7), denoted by U1 and U2, can be expressed as
U1 ¼
interface is assumed to be controlled by the values of a and b in the present study. It should also be noted that those parameters are not position dependent, indicating that no interfacial partial debonding is considered in this framework. By carrying out the lengthy algebra, the components of the positive definite fourth-rank tensor U for the multiphase composite state are explicitly given by
ð12Þ
ð8Þ ð9Þ
posite at a given level of ð r11 Þ1 can be derived through the following three-step Weibull approach
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
( " M #) ð r11 Þ1 /2 ¼ / 1 exp S0 ( " M #) ð r Þ 11 1 3 ¼ / 2 1 exp / S0 ( " M #) ð r11 Þ1 /4 ¼ /3 1 exp S0
ð16Þ ð17Þ ð18Þ
3 / /3 ¼ / 4 2 / 3 / ¼/
ð19Þ
2 /1 ¼ / /
ð21Þ
ð20Þ
2
The components of the positive definite fourth-rank tensor U for the five-phase composite state in Eq. (7), denoted by U1 and U2, can be expressed as
3.1. Level 0 of two-phase composite state The effective Lame constants k* and l* in Eq. (25) for the two-phase composite state can be expressed as (see Lee and Pyo [10]) 1 þ K2 k ¼ ð3k0 þ 2l0 ÞK1 þ 2k0 ð26Þ 2 1 þ K2 ð27Þ l ¼ 2l0 2 where the components K1 and K2 are listed in the Appendix. 3.2. Level 1 of three-phase composite state The effective Lame constants k* and l* in Eq. (25) for the three-phase composite state can be expressed as (see Lee and Pyo [10])
ð1 2n2 2n4 2n6 2n8 Þfn1 ð3k1 þ 2l1 Þ þ k1 ð/1 2n2 Þg þ 2l1 ðn1 þ n3 þ n5 þ n7 Þð/1 2n2 Þ /1 ð1 2n2 2n4 2n6 2n8 Þð1 3n1 2n2 3n3 2n4 3n5 2n6 3n7 2n8 Þ l1 ð/1 2n2 Þ U2 ¼ /1 ð1 2n2 2n4 2n6 2n8 Þ 1 where the components n1, . . ., n8 are listed in the Appendix. k ¼ ð3k0 þ 2l0 ÞðK1 þ K3 Þ þ 2k0 þ K2 þ K4 2 1 3. A micromechanics-based, multi-level elastic model for þ K2 þ K4 l ¼ 2l0 2 particulate composites with weakened interfaces
U1 ¼
We start with considering 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 loads or deformations continue to increase, some initially perfectly bonded particles are transformed to particles with mildly weakened interface, some particles with mildly weakened interface are then transformed to particles with severely weakened interface, and all particles are transformed to completely debonded particles asymptotically within the proposed framework. The effective stiffness tensor C* for the multi-phase particulate composite can be derived based on the governing equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions [6] as C ¼ C0 ½I þ Rnr¼1 f/r ðAr þ Sr Þ
1
1 1
½I /r Sr ðAr þ Sr Þ g
ð24Þ
where Cr is the elasticity tensor of the r-phase. Substituting Eqs. (5) and (6) into Eq. (24) yields the effective stiffness tensor C* for the multi-phase particulate composite as C ¼ k dij dkl þ l ðdik djl þ dil djk Þ:
ð25Þ
391
ð22Þ ð23Þ
ð28Þ ð29Þ
where the components K1, . . ., K4 are listed in the Appendix. 3.3. Level 2 of four-phase composite state The effective Lame constants k* and l* in Eq. (25) for the four-phase composite state can be expressed as 1 k ¼ ð3k0 þ2l0 ÞðK1 þK3 þK5 Þþ2k0 þK2 þK4 þK6 ð30Þ 2 1 ð31Þ l ¼ 2l0 þK2 þK4 þK6 2 where the components K1, . . ., K6 are listed in the Appendix. 3.4. Level 3 of five-phase composite state The effective Lame constants k* and l* in Eq. (25) for the five-phase composite state can be expressed as k ¼ ð3k0 þ 2l0 ÞðK1 þ K3 þ K5 þ K7 Þ 1 þ K2 þ K4 þ K6 þ K8 þ 2k0 2 1 þ K2 þ K4 þ K6 þ K8 l ¼ 2l0 2
ð32Þ ð33Þ
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H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
where the components K1, . . ., K8 are listed in the Appendix. 4. Numerical example To illustrate the elastic damage behavior of the present framework, we consider the uniaxial tension loading case ij ¼ 0 for all other stress components). For ( r11 6¼ 0 and r brevity, we adopt the material properties for the 6061-T6 aluminum alloy matrix/silicon carbide particle composites according to Zhao and Weng [28], Ju and Lee [9] and Lee and Pyo [10]: E0 = 68.3 GPa, m0 = 0.33, E1 = 490 GPa, m1 = 0.17, ry = 250 MPa, / = 0.2. As a result of the parametric analysis on compliance parameters a and b reported in a preceding paper Lee and Pyo [10], the compliance parameters of the mildly weakened interface, denoted by a1 and b1, and the severely weakened interface, denoted by a2 and b2, in Eq. (6) are chosen to be a1 = 2.0 · 105, b1 = 3.0 · 105; a2 = 2.0, b2 = 3.0. Moreover, the Weibull parameters in Eq. (1) are assumed to be S0 = 1.09*ry and M = 5. To investigate the influence of the level of damage on the stress–strain response of the composites, four different phase composite states illustrated in Fig. 1 are considered in this simulation. Fig. 2a shows the present predicted stress–strain responses of particulate composites under uniaxial tension. The predicted evolution of volume fractions of perfectly bonded particles and various types of damaged particles (e.g. particles with mildly weakened interface,
particles with severely weakened interface, completely debonded particles) at the three-phase, four-phase and fivephase composite states are exhibited in Fig. 2b. At the early stage of loading, the stress–strain curves are shown to be linear since the volume fractions of the damaged particles are low (Fig. 2a). As strains or stresses continue to increase, the volume fractions of the damaged particles increase gradually and the composites show nonlinear stress–strain responses as shown in Fig. 2. Clearly, the high-level composite states give rise to substantial weaker stress–strain response in comparison with the low-level composite states. It is demonstrated through this simulation that the current multi-level elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged particles exist simultaneously within the proposed framework. 5. Parametric analysis The Weibull parameter S0 in Eq. (1) is closely related to the strength at the particle–matrix interface in particulate composites. A parametric analysis is conducted to examine the influence of the Weibull parameter S0 on the progression of weakened interface in the composites. As primary parameters governing the progressive weakened interface, three representative sets of Weibull parameters are used: S0 = 1.09*ry, M = 5; S0 = 2.18*ry, M = 5; S0 = 3.27*ry, M = 5. For comparison, a uniaxial tensile simulation is
500
0.2
450 2-phase composite state
400
3-phase composite state 4-phase composite state
0.15
5-phase composite state
Particle Volume Fraction
Stress [σ11] (MPa)
350
300
250
200
0.1 perfectly bonded (3-phase) mildly weakened (3-phase) perfectly bonded (4-phase) mildly weakened (4-phase) severely weakened (4-phase) perfectly bonded (5-phase)
150
mildly weakened (5-phase) severely weakened (5-phase)
0.05
completely debonded (5-phase)
100
50
0 0.000
0.001
0.002
0.003
Strain [ε11]
0.004
0.005
0 0.000
0.001
0.002
0.003
0.004
0.005
Strain [ε11]
Fig. 2. (a) The present predicted stress–strain responses of particulate composites under uniaxial tension. (b) The predicted evolution of volume fractions of perfectly bonded particles various types of damaged particles corresponding to (a).
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
conducted using the same material properties and model parameters as used in Section 4. The present predicted stress–strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S0 values are shown in
393
Fig. 3a. Fig. 3b exhibits the predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface, and particles with severely weakened interface, respectively, as a function of the uniaxial strain corresponding to Fig. 3a. It is clearly shown from Fig. 3 0.3 perfectly bonded (S0 = 1.09*σy ) mildly weakened (S 0 = 1.09*σy ) severely weakened (S0 = 1.09*σy ) perfectly bonded (S0 = 2.18*σy ) mildly weakened (S0 = 2.18*σy ) severely weakened (S0 = 2.18*σy ) perfectly bonded ( S0 = 3.27*σy ) mildly weakened (S0 = 3.27*σy ) severely weakened (S0 = 3.27*σy )
500 0.25
400
Particle Volume Fraction
Stress [σ11] (MPa)
0.2
300
200
0.15
0.1
S0 = 3.27*σy S0 = 2.18*σy S0 = 1.09*σy
100
0 0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.05
0
0.008
0.000
0.001
0.002
0.003
Strain [ε11]
0.004
0.005
0.006
0.007
0.008
Strain [ε11]
350
300
Stress [σ11] (MPa)
250
200
150
100
α2 =2.0, β 2=3.0 α2=2.E-4, β 2=3.E-4 α 2=6.E-5, β 2=7.E-5 α2=2.E-5, β 2=3.E-5
50
0
0
0.005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0.004
Strain [ε11]
Fig. 3. (a) The present predicted stress–strain responses of particulate composites under uniaxial tension at the four-phase composite state with various S0 values. (b) The predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface and particles with severely weakened interface corresponding to (a). (c) The present predicted stress–strain responses of particulate composites under uniaxial tension at four-phase composite state with various a2 and b2 values.
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that if the strength at the particle–matrix interface is low (lower S0), most particles are mildly weakened or severely weakened in their interfaces in early stage and the composites show a nonlinear stress–strain behavior even in early stage of loading. It can be concluded from this parametric analysis that the influence of the Weibull parameter S0 on the progression of weakened interface and stress–strain behavior of the composites is quite significant; thus, an accurate characterization of S0 would be essential to realistically predict the progressive weakened interface in particulate composites. Another parametric analysis is conducted to evaluate the proposed multi-level elastic damage model sensitivity to the compliance parameters a and b and to address the influence of varying a and b values on the stress–strain response of the model. As parameters controlling the severity of weakened interface, four sets of the compliance parameters for the severely weakened interface are used: a2 = 2.0, b2 = 3.0; a2 = 2.0 · 104, b2 = 3.0 · 104; a2 = 6.0 · 105, b2 = 7.0 · 105; a2 = 2.0 · 105, b2 = 3.0 · 105. For simplicity, fixed values of the compliance parameters for the mildly weakened interface are used as follows: a1 = 2.0 · 105, b1 = 3.0 · 105. The present predicted stress–strain responses of particulate composites with severely weakened interfaces under uniaxial tension with various values of a2 and b2 are shown in Fig. 3c. As the values of the compliance parameters a2 and b2 become higher, the effect of the severely weakened interface on the stress–strain behavior of the composites is more pronounced.
6. Experimental comparisons The proposed multi-level elastic damage model is further exercised to predict the behavior of brittle ceramic composites. The uniaxial stress–strain behavior of unidirectional SiC fiber-reinforced CAS glass–ceramic matrix composites tested by Sørensen [20] is numerically predicted using the multi-level elastic damage model. The unidirectional fibers in the composites are modeled as spherical particles in this simulation. The elastic properties of the composites used in this simulation are as follows [20– 22,26]: E0 = 98 GPa, m0 = 0.3, E1 = 200 GPa, m1 = 0.15, / = 0.35. According to the experimentally obtained stress–strain curve in the longitudinal direction [20], the compliance parameters a1, b1, a2 and b2 given in Eq. (6) as well as the Weibull parameters S0 and M in Eq. (1) are estimated to be: a1 = 2.0 · 105 and b1 = 3.0 · 105, a2 = 2.0, b2 = 3.0; S0 = 450 MPa, M = 2. The present predicted uniaxial stress–strain curve of the composites at the five-phase composite state is compared with the experimentally obtained stress–strain curve in the longitudinal (fiber) direction [20] in Fig. 4a. It is noted that the present prediction and the experimental data match well within the small strain region (the corresponding threshold strain th = 0.005), but the predicted stress– strain curve starts to deviate from the experimentally obtained one after the threshold strain. The experiment in Sørensen [20] was designed to show the brittle behavior of ceramic composites reinforced with unidirectional fibers. It was shown from Fig. 4a that the experimentally obtained
500
0.4
450
perfectly bonded mildly weakened severely weakened completely debonded
0.35 400 0.3
Particle Volume Fraction
Stress [σ11] (MPa)
350
300
250
200
0.25
0.2
0.15
150 0.1
experiment
100 presentprediction (5-phase composite)
0.05
50
0
0
0.001
0.002
0.003
0.004 0.005 Strain [ε11]
0.006
0.007
0.008
0
0
0.001
0.002
0.003
0.004 0.005 Strain [ε11]
0.006
0.007
0.008
Fig. 4. (a) The comparison between the present prediction and experimental data [20] for overall uniaxial tensile responses of SiC fiber-reinforced CAS glass–ceramic matrix composites. (b) The predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface, severely weakened interface and completely debonded particles corresponding to the present prediction in (a).
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397 0.3
600
500
0.25
400
0.2 Particle Volume Fraction
Stress [σ11] (MPa)
395
300
perfectly bonded mildly weakened severely weakened completely debonded
0.15
0.1
200
experiment
100
0.05
present prediction (5-phase composite state)
0
0 0
0.002
0.004
0.006
0.008
0.01
0.012
0
0.002
0.004
0.006
0.008
0.01
0.012
Strain [ε11]
Strain [ε11]
Fig. 5. (a) The comparison between the present prediction and experimental data [1] for overall uniaxial tensile responses of silicon carbide particlereinforced 2009 aluminum matrix composites. (b) The predicted evolution of volume fractions of perfectly bonded particles, particles with mildly weakened interface, severely weakened interface and completely debonded particles corresponding to the present prediction in (a).
stress of the composites was increased substantially after the threshold strain due the unidirectional fibers may play the primary role in the increase of stress. On the other hand, inclusions are limited to be spherical particles in our framework and therefore our predictions cannot be compared with his experimental data after the threshold strain. The predicted evolution of volume fractions of particles corresponding to Fig. 4a is shown in Fig. 4b. To assess and demonstrate the accuracy of the proposed multi-level elastic damage model, the present predictions are compared with experimental data reported by Geiger and Welch [1] for the uniaxial stress–strain behavior of silicon carbide particulate-reinforced 2009 aluminum matrix composites. The matrix is assumed to behave elastically. We adopt the same material properties for the composites as those in Geiger and Welch [1] as follows: E0 = 72.4 GPa, m0 = 0.33, E1 = 450 GPa, m1 = 0.19, / = 0.294. Since model parameters of the proposed model were not reported by Geiger and Welch [1], one needs to estimate the compliance parameters a1, b1, a2 and b2 given in Eq. (6) as well as the Weibull parameters S0 and M in Eq. (1). Those compliance and Weibull parameters are fitted according to the experimentally obtained stress–strain curve [1]. The fitted model parameters are: a1 = 2.0 · 105, b1 = 3.0 · 105, a2 = 2.0, b2 = 3.0; S0 = 825 MPa, M = 3. The predicted uniaxial stress–strain curve of the composites at the five-phase composite state based on the above material properties and parameters is shown in Fig. 5a. The experimentally obtained stress–strain curve
is also plotted in the figure for comparison. As a whole, the present prediction and the experimental data match well. The predicted evolution of volume fractions of particles corresponding to Fig. 5a is shown in Fig. 5b. 7. Concluding remarks A multi-level elastic damage model based on a combination of a micromechanical formulation and a multi-level damage model has been presented to predict the effective elastic behavior and progressive weakened interface in particulate composites. A four-level damage model based on the Weibull’s probabilistic function is developed to describe the progressive weakened interface in the composites. The Eshelby’s tensor for a slightly weakened ellipsoidal inclusion [18,19] to model particles having mildly or severely weakened interface is incorporated into a micromechanics-based multi-level elastic model. The proposed micromechanical elastic damage model is applied to the uniaxial tension loading to predict the corresponding stress–strain responses. A parametric analysis is also carried out to address the influence of the compliance and Weibull parameters on the progressive weakened interface in the composites. Finally, the present predictions are compared with experimental data [20,1] to further illustrate the elastic damage behavior of the present framework and to verify the validity of the proposed multi-level elastic damage model. The observations and findings of this numerical study can be summarized as:
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(1) The stress–strain curves of the composites are linear at the early stage of loading since the volume fractions of damaged particles are low. As strains or stresses continue to increase, the volume fractions of damaged particles increase gradually, resulting in a nonlinear stress–strain response. (2) The proposed multi-level elastic damage model naturally captures the gradual transition from previous damage level to next damage level and various types of damaged particles exist simultaneously within the proposed framework. (3) If the strength at the particle–matrix interface is low (lower S0), most particles are mildly weakened, severely weakened or completely debonded in their interfaces even in early stage of loading. (4) The influence of the Weibull parameter S0 on the progression of weakened interface and stress–strain behavior of the composites is quite significant. (5) As the values of the compliance parameters a2 and b2 become higher, the effect of the severely weakened interface on the stress–strain behavior of the composites is more pronounced. (6) The predicted stress–strain behavior of particulate composites featuring the multi-level damage progression of weakened interface is observed to be in good qualitative agreement with experimental data [1].
Appendix. Parameters K1, . . ., K8 in Eqs. (8)–(23) and (26)– (33) K1 ¼ 15ð1m0 Þ/1
2
K3 ¼ 600að1m0 Þ /2
ð35Þ
x2 ð2n4 1Þþn3 ð3x2 þ2w2 Þ w2 ð3x2 þ2w2 Þð12n4 Þð13n3 2n4 Þ
ð36Þ 2
K4 ¼
300að1m0 Þ /2 w2 ð12n4 Þ 2
K5 ¼ 600að1m0 Þ /3
ð37Þ
x3 ð2n6 1Þþn5 ð3x3 þ2w3 Þ w3 ð3x3 þ2w3 Þð12n6 Þð13n5 2n6 Þ
ð38Þ 2
300að1m0 Þ /3 ð39Þ w3 ð12n6 Þ x4 ð2n8 1Þþn7 ð3x4 þ2w4 Þ K7 ¼ 15ð1m0 Þ/4 2w4 ð3x4 þ2w4 Þð12n8 Þð13n7 2n8 Þ
K6 ¼
ð40Þ 15ð1m0 Þ/4 4w4 ð12n8 Þ
ð41Þ
in which
/1 5x1 ð1 þ m0 Þ n1 ¼ þ 5m0 1 3x1 þ 2w1 2w1 /1 ð4 5m0 Þ 2w1 / x2 ð3v1 þ 2v2 Þ þ v1 n3 ¼ 2 3x2 þ 2w2 2w2
n2 ¼
/ 2 v2 2w2 /3 x3 ð3v3 þ 2v4 Þ þ v3 n5 ¼ 3x3 þ 2w3 2w3
n4 ¼
/ 3 v4 2w3 /4 5x4 ð1 þ m0 Þ þ 5m0 1 n7 ¼ 3x4 þ 2w4 2w4
n6 ¼
n8 ¼
/4 ð4 5m0 Þ 2w4
x1 ¼ 15ð1 m0 Þ Acknowledgement
ð34Þ 15ð1m0 Þ/1 K2 ¼ 4w1 ð12n2 Þ
K8 ¼ As an extension of the present study, a series of characterization tests on a particulate composite need to be conducted in the future for an accurate calibration of the model parameters (S0, M, a1, b1, a2 and b2). Data points (e.g. stress–strain curves, load-deflection curves, etc.) will be sampled from the tests and will be used to obtain an optimal set of the parameters by minimizing the sum of the squares of differences between the predicted and measured loads at all data points. Since the lower and upper limits of the parameters can be determined from a series of parametric analysis, this would be a nonlinear leastsquare constrained minimization problem. Alternatively, these parameters can be determined simultaneously with a trial-and-error procedure. The authors are currently working on the extension of the proposed multi-level elastic damage model for the prediction of the damage evolution and elastoplastic behavior of particulate ductile matrix composites. Specifically, the classical rate-independent plasticity will be incorporated into the present framework and the corresponding computational algorithm will be systematically developed.
x1 ð2n2 1Þþn1 ð3x1 þ2w1 Þ 2w1 ð3x1 þ2w1 Þð12n2 Þð13n1 2n2 Þ
ð42Þ ð43Þ ð44Þ ð45Þ ð46Þ ð47Þ ð48Þ ð49Þ
l1 k0 l0 k1 þ 5m0 1 ðl1 l0 Þf3ðk1 k0 Þ þ 2ðl1 l0 Þg ð50Þ
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.
x2 ¼ 1200að1 m0 Þ
2
l2 k0 l0 k2 þ v1 ðl2 l0 Þf3ðk2 k0 Þ þ 2ðl2 l0 Þg ð51Þ
H.K. Lee, S.H. Pyo / Composites Science and Technology 68 (2008) 387–397
l3 k0 l0 k3 þ v3 ðl3 l0 Þf3ðk3 k0 Þ þ 2ðl3 l0 Þg ð52Þ x4 ¼ 5m0 1 ð53Þ ð7 5m0 Þl0 þ ð8 10m0 Þl1 ð54Þ w1 ¼ 2ðl1 l0 Þ l0 w2 ¼ 600að1 m0 Þ2 þ v2 ð55Þ l2 l0 l0 2 þ v4 ð56Þ w3 ¼ 600að1 m0 Þ l3 l0 5m0 7 ð57Þ w4 ¼ 2 with x3 ¼ 1200að1 m0 Þ2
2
v1 ¼ ð80 þ 480m0 400m20 Þa þ 500ð1 2m0 Þ k0 b1 þ ð98 þ 140m0 50m20 Þl0 a1 þ ð268 1240m0 þ 1300m20 Þl0 b1 v2 ¼ ð320 720m0 þ
400m20 Þa
ð58Þ
þ l0 ð3a1 þ 2b1 Þð7 5m0 Þ
2
ð59Þ v3 ¼
ð80 þ 480m0 400m20 Þa þ 500ð1 þ ð98 þ 140m0 50m20 Þl0 a2 þ ð268 1240m0 þ 1300m20 Þl0 b2
v4 ¼ ð320 720m0 þ
400m20 Þa
2
2m0 Þ k0 b2 ð60Þ 2
þ l0 ð3a2 þ 2b2 Þð7 5m0 Þ : ð61Þ
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