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Numerical characterization of compressive response and damage evolution in laminated plates containing a cutout
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 67 (2007) 2221–2230 www.elsevier.com/locate/compscitech
Numerical characterization of compressive response and damage evolution in laminated plates containing a cutout H.K. Lee *, B.R. Kim Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejon 305-701, Republic of Korea Received 4 August 2006; received in revised form 17 January 2007; accepted 25 January 2007 Available online 15 February 2007
Abstract A micromechanical constitutive model [Liang Z, Lee HK, Suaris W. Micromechanics-based constitutive modeling for unidirectional laminated composites. Int J Solid Struct 2006;43:5674–89], based on the concept of the ensemble-volume average for laminated composites, is implemented into a finite element program to numerically characterize the compressive response and damage evolution in laminated plates containing a cutout. Prior to the implementation of the model into the finite element program, the predicted moduli of laminated composites are compared with analytical bounds and experimental data for the validation and verification of the constitutive model. A series of numerical simulations for a uniaxial test of laminated plate specimens containing a cutout are conducted using the implemented constitutive model. The predictions are compared with experimental data [Lessard LB, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part II-Experiment. J Compos Mater 1991;25:44–64; Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I-Analysis. J Compos Mater 1991;25:2–43] to verify the accuracy of the implemented constitutive model. A parametric study is also carried out to illustrate the influence of the geometry of the specimens on the behavior of laminated plates. It is shown that the implemented constitutive model is suitable for the analysis of the constitutive behavior of laminated plates having a dilute or moderate fiber volume fraction. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Carbon fibres; Computational simulation; Damage mechanics; Laminates; Debonding
1. Introduction The performance prediction and estimation of structures made of fiber-reinforced, laminated composites has been increasingly important as they become primary load-carrying elements in many fields [12]. Many theoretical and numerical methods have been proposed to predict the effective properties of the fiber-reinforced, laminated composites. Hill [9] and Hashin [5,6] derived theoretical upper and lower bounds based on variational principles for the transverse moduli of the composites. The effective transverse shear modulus of unidirectional composites were *
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derived by Christensen and Lo [4] using a three-phase cylinder model. Ju and Zhang [12,13] proposed a two-dimensional, micromechanics-based model for the prediction of the effective elastoplastic moduli of two-phase, fiber-reinforced composites. Micromechanical approaches such as the self-consistent method, the differential scheme and the Mori-Tanaka method have been introduced by many researchers to estimate the effective moduli of the composites. A review of the micromechanical approaches for the prediction of the effective moduli of laminated composites can be found in Aboudi [1], Ju and Zhang [12], and NematNasser and Hori [24]. Based on the concept of the ensemble-volume average and the Eshelby’s inclusion approach, we recently proposed a three-dimensional, micromechanical constitutive
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H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
model for unidirectional laminated composites [22]. In our derivation, unidirectional fibers were assumed to be circular cylindrical inclusions that were embedded in a matrix. A newly developed Eshelby’s tensor for an infinite cylindrical inclusion [3] was adopted to model the unidirectional fibers and was incorporated into the micromechanical framework. The progressive loss of strength resulting from the partial fiber debonding and the nucleation of microcracks was incorporated into the constitutive model. The Weibull’s probabilistic function was used to model the varying probability of progressive fiber debonding [27,28] and the nucleation of microcracks was simulated by adopting a continuum damage model proposed by Karihaloo and Fu [14,15]. Damage mechanisms and evolutions of damage in laminated composites containing notches, holes or cutouts are not still well addressed [21]. In particular, the accurate evaluation of the compressive response and damage in laminated composites is vital to the complete understanding of the laminated composite structures [21]. The present study implements the previous work published in Liang et al. [22] into the finite element program ABQUS [29] to numerically characterize the response and damage evolution in laminated plates containing a centrally located cutout subjected to uniaxial compression. Prior to the implementation of the model into the finite element program, the predicted moduli of laminated composites are compared with Hashin’s theoretical bounds [7] and experimental data [26] for the validation and verification of the constitutive model. A series of numerical simulations on laminated plate specimens containing a cutout in uniaxial compression are conducted using the implemented constitutive model. The predictions are compared with experimental data [21,2] to verify the accuracy of the implemented constitutive model. Moreover, a parametric study is carried out to address the influence the geometry of the specimens on the behavior of laminated plates. 2. A constitutive model for unidirectional laminated composites The three-dimensional, micromechanical constitutive model for unidirectional laminated composites proposed by Liang et al. [22] is here summarized. Let us start by considering an initially perfectly bonded, unidirectional, threephase composite consisting of an elastic matrix (phase 0), aligned continuous fibers (phase 1), and (penny-shaped) microcracks (phase 2) of radius c. The 1-direction is chosen to be fiber direction and the plane 2–3 corresponds to the transversely isotropic plane in the unidirectional fiber reinforced composite as shown in Fig. 2 of Liang et al. [22]. It is assumed that penny-shaped microcracks are aligned spheroidal voids with the aspect ratio a2 ! 0 (see Fig. 2 of Lee [17]). The continuous fibers are assumed to be infinitely long, elastic cylindrical inclusions. As loadings or deformations proceed, some fibers in the composite are partially debonded (phase 3) and microcracks are nucleated. Fol-
lowing Zhao et al. [27,28], a partially debonded fiber is replaced by an equivalent, perfectly bonded fiber that possesses transversely isotropic moduli. The nucleation of microcracks is simulated by adopting a continuum damage model proposed by Karihaloo and Fu [14]. While the Eshelby’s tensor for a spheroidal inclusion derived by Sun [25] is employed to model penny-shaped microcracks, the Eshelby’s tensor for an infinite cylindrical inclusion previously proposed by Cheng and Batra [3] is considered to model the unidirectional fibers. Specifically, for the implementation of the Eshelby’s tensor proposed by Cheng and Batra [3] into the present micromechanical formulation, the Eshelby’s tensor is rephrased in a transe The transversely versely isotropic fourth-rank tensor F. e isotropic fourth-rank tensor F is defined by six parameters bm (m = 1–6) Fe ijkl ðbm Þ ¼ b1 ~ni ~nj ~nk ~nl þ b2 ðdik ~nj ~nl þ dil ~nj ~nk þ djk ~ni ~nl þ djl ~ni ~nk Þ þ b3 dij ~nk ~nl þ b4 dkl ~ni ~nj þ b5 dij dkl þ b6 ðdik djl þ dil djk Þ
ð1Þ
with the unit direction vector ~n and index m = 1–6. If the 1-direction is chosen to be symmetric, then we have ~n1 ¼ 1; ~n2 ¼ ~n3 ¼ 0. The details of the implementation of the Eshelby’s tensors into the micromechanical framework can be found in Lee and Simunovic [20] and Liang et al. [22]. Based on the governing field equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions [10,11], the effective stiffness tensor C* for the fourphase composite can be derived as h i 1 1 1 C ¼ C0 I þ R3r¼1 f/r ðAr þ Sr Þ ½I /r Sr ðAr þ Sr Þ g ð2Þ where Cr denotes the elasticity tensor of the r-th phase, ‘‘Æ’’ is the tensor multiplication, I is the fourth-rank identity tensor, and /r signifies the volume fraction of the r-th phase inclusion. While the Eshelby’s tensors S1 and S3 for perfectly bonded and partially debonded unidirectional fibers, respectively, were previously derived by Cheng and Batra [3] (see also Liang et al. [22]), the Eshelby’s tensor S2 for penny-shaped microcracks also was previously derived by Sun [25]. In addition, the fourth-rank tensor Ar is defined as 1
Ar ðCr C0 Þ
C0
ð3Þ
Accordingly, the effective elastic stiffness tensor C* for the four-phase composites can be derived as C ¼ Fe ijkl ðs1 ; s2 ; s3 ; s4 ; s5 ; s6 Þ
ð4Þ
where the six parameters in Eq. (4) are given in the Appendix A of Liang et al. [22]. Now we consider rotations through an angle h about the X3-axis to derive the stiffness of off-axis unidirectional fibrous composites. The angle h is measured positive coun-
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
terclockwise from the X1-axis to the x1-axis (see Fig. 2 of [22]). Following the stiffness transformation law by Herakovich [8] and Liang et al. [22], the stiffness matrix of offaxis unidirectional fibrous composites C, which is the transformed stiffness matrix through an arbitrary angle h about the 3-axis, is derived as
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By combining Eq. (48) of Lee and Simunovic [19] with Eqs. (4) and (1), the effective transverse Young’s modulus for unidirectional laminated composites can be derived as 2
ET ¼
4s6 ½ðs1 þ 4s2 þ s3 þ s4 þ s5 þ 2s6 Þðs5 þ s6 Þ ðs3 þ s5 Þ ðs1 þ 4s2 þ s3 þ s4 þ s5 þ 2s6 Þðs5 þ 2s6 Þ ðs3 þ s5 Þ
2
ð5Þ
ð7Þ
where the stiffness matrix of the unidirectional fibrous composites with the fibers oriented off-axis takes the form: 3 2 C 11 C 12 C 13 0 0 C 16 6C 0 C 26 7 7 6 12 C 22 C 23 0 7 6 7 6 C 13 C 23 C 33 0 0 C 36 7 6 ð6Þ C¼6 0 0 C 44 C 45 0 7 7 60 7 6 40 0 0 C 45 C 55 0 5
where the parameters s1, . . . , s6 are given by Liang et al. [22]. In addition, the transverse Young’s modulus given in Eq. (48) of Lee and Simunovic [19] are
r¼C:
C 36
0
0
7.0 6.0
C 66
Here, the components of C are given in Appendix C of Liang et al. [22]. A more detailed description on the micromechanical formulations for the effective elastic stiffness tensor for the off-axis unidirectional fibrous composites can be found in Liang et al. [22]. It is assumed that damage in laminate composites is controlled by the interfacial fiber debonding and nucleation of microcracks in the matrix. The Weibull probabilistic distribution function is used for modeling the interfacial fiber debonding [27,28], and a damage model proposed by Karihaloo and Fu [14,15] is utilized for modeling the nucleation of microcracks in the matrix. Details of the damage models can be found in Lee [17], Lee and Liang [18], and Liang et al. [22]. The constitutive model incorporating the interfacial fiber debonding and nucleation of microcracks in the matrix is then implemented into the finite element program ABAQUS using a usersupplied material subroutine to characterize numerically the damage evolutions and the corresponding load–deflection (p–u) behaviors of laminated composite structures. Computational algorithms used in the user-supplied material subroutine are based on the strain driven algorithm in which the stress history is to be determined by a given strain history.
5.0
*
C 26
Present prediction Experimental data (Uemura,1968) Hashin's lower bound Hashin's upper bound
8.0
E T /E 0
C 16
9.0
4.0 3.0 2.0 1.0 0.0 0.0
0.1
0.2
0.3
0.4
Fiber volume fraction
0.5
0.6
0.7
f
Fig. 1. The effective transverse Young’s modulus versus the fiber volume fraction.
l
3. Validation and verification of the constitutive model for unidirectional laminated composites For the validation and verification of the proposed constitutive model, the predictions on a glass fiber/epoxy matrix composite based on the proposed constitutive model are compared with Hashin’s theoretical bounds [7] and experimental data [26]. Similar to the analytical comparison conducted by Ju and Zhang [12], we adopt the same material properties for the glass fiber/epoxy matrix composite as those used in Kondo and Sato [16] as follows: Ef = 11,660 kg/mm2, mf = 0.22; and Em = 550 kg/mm2, mm = 0.35, where the subscripts f and m denote fibers and the matrix, respectively.
d
e
h
w
Fig. 2. The schematic description of laminated plates containing a cutout [21].
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Table 1 The ply orientations and gemetry of the laminated plate specimens (see also [21]) Ply orientations Cross-ply Angle-ply Quasi-isotropic *
h (mm) [(0/90)6]s [(±30)6]s [(±45)6]s [(0/±45/90)3]s [(0/±45)4]s
l (mm)
w (mm)
d (mm)
e (mm)
101.6
25.4
6.35
25.4
3.302
4^c6 ½ð^c1 þ 4^c2 þ ^c3 þ ^c4 þ ^c5 þ 2^c6 Þð^c5 þ ^c6 Þ ð^c3 þ ^c5 Þ 2
ð^c1 þ 4^c2 þ ^c3 þ ^c4 þ ^c5 þ 2^c6 Þð^c5 þ 2^c6 Þ ð^c3 þ ^c5 Þ ð8Þ
Hashin’s theoretical bounds for ET of the composites were given by [7] ¼
4jðÞ GTðÞ jðÞ þ mðÞ GTðÞ
ð9Þ
where mðÞ ¼ 1 þ
E1(GPa)
m0
m1
/1
4.62
238.56
0.36
0.20
0.66
jðÞ ¼ jm þ
/f 1=ðjf jm Þ þ /m =ðjm þ Gm Þ
4ðmLf mLm Þ2 /m /f /m =jf þ /f =jm þ 1=Gf ðm Lf mLm Þð1=jm 1=jf Þ/m /f mLðÞ ¼ mLm /m þ mLf /f þ /m =jf þ /f =jm þ 1=Gm ELðÞ ¼ ELm /m þ ELf /f þ
2
ETðÞ
E0(GPa)
3.429
All layups have 24 layers of plies.
ET ¼
Table 2 The material properties of the laminated plate specimens [2,8]
4jðÞ mLðÞ ELðÞ
in which the subscripts (+) and () indicate the upper and lower bounds of each of the properties, respectively. In addition, the lower bounds for the effective longitudinal Young’s modulus EL , transverse shear modulus GT , bulk modulus j*, longitudinal Poisson’s ratio mL can be expressed as (see Eqs. (31)–(33), (35a) of Hashin [7] for details)
ð10Þ ð11Þ
ð12Þ / f GTðÞ ¼ GTm þ 1=ðGTm Gm Þ þ /m ðjm þ 2Gm Þ=½2Gm ðjm þ Gm Þ ð13Þ
where /m and /f denote the volume fraction of the matrix and fibers, respectively. The corresponding upper bounds for those properties can be obtained by interchanging the position of each subscript [7]. The predicted effective (normalized) transverse Young’s modulus ET =E0 of the glass fiber/epoxy matrix composite according to Eq. (7) is shown in Fig. 1. The theoretical predictions of ET =E0 based on Hashin’s bounds given in Eq. (9) and experimental data reported by Uemura [26] for the composite are also plotted in the figure for comparison. Clearly, the present prediction is well within the Hashin’s theoretical bounds. It is also observed from the figure that the present prediction and the experimental data match well up to the fiber volume fraction / = 0.45. However, the predicted ET =E0 is lower than that based on the exper-
Fig. 3. The finite element model (a, b) and boundary conditions (c, d) used in the simulation (h = 3.43 mm, d = 6.35 mm, w/d = 4).
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230 40 Experimental data #1 Experimental data #2 Present prediction
35
Applied load, P (kN)
30 25 20 15 10 5 0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Displacement (mm)
Fig. 4. The comparison of load–displacement curves between the present prediction and the experiment [21] on cross-ply laminated plates.
2225
averaged perturbations in a fiber due to other surrounding fibers [10,11] whose effects on the mechanical behavior of composites become noticeable as the fiber volume fraction increases. These issues are beyond the scope of the present work; however, should be considered in future work. It is also worth mentioning that the interaction effect between fibers must be considered in modeling the mechanical behavior of composites having a high fiber volume fraction. Based on the comparison between the present analytical prediction and theoretical bounds as well as available experimental data, it is concluded that the proposed micromechanical constitutive model is capable of modeling the constitutive behavior of laminated composites having a dilute or moderate fiber volume fraction. 4. Numerical simulations for a uniaxial compression test of laminated plates containing a cutout 4.1. Overview
iment as the fiber volume fraction continues to increase. Variability between these results is partly due the present model not accounting for fiber interactions such as the
To gain insight into the constitutive behavior and damage evolution in laminated composites, a comprehensive
Fig. 5. The finite element discretization employed to simulate the compressive response and damage evolution in laminated plates containing a cutout during uniaxial compression.
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
4.2. Finite element implementation and modeling The constitutive model for laminated composites incorporating damage models for the fiber debonding and microcrack nucleation is implemented into the finite element program ABAQUS to simulate the constitutive behavior and damage evolution in structures made of laminated composites. Details of the implementation of a constitutive model into a finite element program can be founded in Lee and Liang [18] and Liang et al. [22]. The configuration of the laminated plate specimens containing a cutout is illustrated in Fig. 2. The plate specimens made of T300/976 graphite/epoxy resin prepreg tapes [21] are subjected to uniaxial compression as shown in Fig. 2. The details of the geometry of the plate specimens were listed in Table 1 of Lessard and Chang [21]. Following Lessard and Chang [21], various ply orientations listed in Table 1 are considered in this study. The three-dimensional, eight-node, linear brick solid element C3D8 in ABAQUS is used for modeling the laminated plate specimens. Only a quarter of the laminated plate specimens having a half of the specimen thickness is modeled in this study since the geometry and loading condition of the specimen are symmetry along the x-, y-, and z-directions as shown in Fig. 3. Here, we employ the same material properties for the laminated plate specimens according to Table 1 of Chang and Lessard [2] and Table 1.2 of Herakovich [8], and the employed material properties are summarized in Table 2. Following Lessard and Chang [21], we compute the compressive load and the corresponding displacement between two points located 12.7 mm above and below from the middle of the plate specimens. 4.3. Simulation results To implement the Weibull evolutionary debonding model [27,28] and the crack nucleation model [14,15], one needs to estimate the Weibull parameters S0, M and the crack nucleation parameters th, c1, c2. Based on the experimental data on cross-ply laminated plates under uniaxial compression [21], we estimate those parameters to be: S0 = 10,059.5 MPa, M = 4.0; th = 0.05%, c1 = 0.4, c2 = 1.8. Fig. 4 shows the predicted load–displacement curve
of the cross-ply laminated plates under uniaxial compression that best fits the experimental data [21]. Fig. 5 illustrates the finite element decretization along the x- and y-axes around the cutout. Fine meshes are used to model the vicinity of the cutout. The predicted evolution of volume fraction of debonded fibers versus displacement around the cutout is shown in Fig. 6(a). Fig. 6(b) exhibits the predicted evolution of microcrack density versus displacement around the cutout. As indicated in the experiment [21], the load–displacement curves for the present prediction and the experiment shown in Fig. 4 are linear in the early stages of loading. Despite the damage in the specimen continues to increase as shown in Fig. 6(a) and (b), a nearly linear load-displacement curve is obtained from the present prediction since the effect of the damage on the load-displacement curve is not substantial. Fig. 7 shows the sequence of deformed shape and von-Mises effective stress of the cross-ply laminated plates during uniaxial compression, illustrating the stress concentration around the cutout emanating from the edges of the cutout. On the basis of the estimated parameters, the present model is exercised against the experiment on two angleply laminated plates containing a cutout. The results are presented in Fig. 8(a). Fig. 8(a) shows the comparison of the load–displacement curves of the two angle-ply lami-
a
1.2E-03
1.0E-03
Segment A Segment B
8.0E-04
6.0E-04 4.0E-04
2.0E-04
0.0E+00 0.00
0.05
0.10
0.15
0.20
0.25
Displacement (mm)
b
5.0 4.5 4.0
Microcrack density
experimental and numerical study was performed by Lessard and Chang [21] and Chang and Lessard [2] for laminated plates containing a cutout. The laminated plates with a cutout subjected to uniaxial compression exhibited different types of response or damage, depending on the layup, material properties, and geometry of the plate specimens [21]. A series of numerical simulations for laminated plates containing a cutout are carried out in this study to examine whether the constitutive model implemented into a finite element program is able to predict the experimentally obtained response [21]. Modeling procedures and simulation results are summarized in this section.
Volume fraction of debonded fib
2226
3.5 3.0 2.5 2.0 1.5
Segment A
1.0
Segment B
0.5 0.0 0.00
0.05
0.10
0.15
0.20
0.25
Displacement (mm)
Fig. 6. The predicted damage evolution versus displacement around the cutout of cross-ply laminated plates under uniaxial compression: (a) volume fraction of debonded fibers and (b) microcrack density.
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
2227
Fig. 7. The sequence of the deformed shape and von-Mises effective stress of cross-ply laminated plates during uniaxial compression.
nated plates between the present prediction and the experiment [21]. As shown in Fig. 8(a), the load–displacement curves predicted by the present model are higher than those obtained by the experiment. The experimentally obtained load–displacement curve for the [(±45)6]s layup is highly
nonlinear, exhibiting the effect of material nonlinearities inherent to the composites, even in early stages of loading where there is no damage [21]. Since the present model does not consider material nonlinearities, the load–displacement curve for the [(±45)6]s layup predicted by the present model
2228
a
40
Applied load, P (kN)
35
Experimental data [(±30)6]s Present prediction [(±30)6]s Experimental data [(±45)6]s #1 Experimental data [(±45)6]s #2 Present prediction [(±45)6]s
30 25 20
Applied load, P (kN)
a
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
15
60 50 40
w/d=3 w/d=4 w/d=5 w/d=6
30 20 10 0 0
0.05
0.1 0.15 Displacement (mm)
10
b
5
50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Displacement (mm) 50 Experimental data [(0/±45/90)3]s #1 Experimental data [(0/±45/90)3]s #2 Present prediction [(0/±45/90)3]s Experimental data [(0/±45)4]s #1 Experimental data [(0/±45/90)3]s #2 Present prediction [(0/±45)4]s
45
Applied load, P (kN)
40 35
w/d=5 [(±30)6]s w/d=6 [(±30)6]s w/d=3 [(±45)6]s w/d=4 [(±45)6]s w/d=5 [(±45)6]s w/d=6 [(±45)6]s
40 35 30 25 20 15 10 5 0
0
30
20
c 70
15
60
10 5 0 0
0.05
0.1
0.15
0.2
Displacement (mm) Fig. 8. The comparison of load–displacement curves between the present prediction and the experiment [21]: (a) two angle-ply laminated plates and (b) two quasi-isotropic laminated plates.
0.1 Displacement (mm)
0.15
0.2
0.15
0.2
Table 3 The w/d ratios used in the parametric study (see also [21]) Ply orientations
h (mm)
Quasi-isotropic
[(0/90)6]s [(±30)6]s [(±45)6]s [(0/±45/90)3]s [(0/±45)4]s
w/d=3 [(0/±45)4]s w/d=4 [(0/±45)4]s
40
w/d=5 [(0/±45)4]s w/d=6 [(0/±45)4]s
30 20 10 0
0.05
0.1 Displacement (mm)
w = 25.4 mm
3.429 3, 4, 5, 6
w/d=5 [(0/±45/90)3]s w/d=6 [(0/±45/90)3]s
Fig. 9. The load–displacement curves of laminated plates with various w/d ratios (d = 6.35 mm): (a) cross-ply, (b) angle-ply, and (c) quasi-isotropic.
w/d ratios d = 6.35 mm
Cross-ply Angle-ply
w/d=3 [(0/±45/90)3]s w/d=4 [(0/±45/90)3]s
50
0
*
0.05
25
Applied load, P (kN)
b
Applied load, P (kN)
0
0.25
w/d=3 [(±30)6]s w/d=4 [(±30)6]s
45
0
0.2
2, 4
3.302
All layups have 24 layers of plies (l = 101.6 mm, e = 25.4 mm).
is exhibited to be higher than that obtained from the experiment. The predicted load–displacement curve for the [(±30)6]s layup is also shown to be higher than that obtained from the experiment. The present model is further exercised against the experiment on two quasi-isotropic laminated plates containing a cutout using the estimated parameters. Fig. 8(b) shows the comparison of the load–displacement curves of the two quasi-isotropic laminated plates between the present prediction and the experiment [21]. The load–displacement
curves of the two quasi-isotropic laminated composite plates between the present prediction and the experiment show slight deviations as shown in the figure. 5. Parametric study To illustrate the influence of the geometry of the plate specimens on the compressive response and damage evolution in laminated plates containing a cutout, a parametric study is conducted. In particular, the ratio of the width of the plate specimens to the diameter of the cutout (w/d ratio) is considered as the parameter of this parametric study. The w/d ratios used in this parametric study are summarized in Table 3. The load–displacement curves of laminated plates with different values of w (wd ¼ 3, 4, 5, 6; d = 6.35 mm) are presented in Fig. 9. It is observed from the figure that (1)
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
Applied load, P (kN)
b
c
1.0E-02
8.0E-03
[(0/90)6]s [(+30/-30)6]s [(+45/-45)6]s [(0/+45/-45/90)3]s [(0/+45/-45)4]s
6.0E-03
4.0E-03
2.0E-03
0.05
0.10
0.15
0.20
0.25
Displacement (mm)
b
5.0E+00 4.5E+00 4.0E+00
50 45 40 35 30 25 20 15 10 5 0
50 45 40 35 30 25 20 15 10 5 0
d=6.35 mm (w/d=4) d=12.7 mm (w/d=2)
3.0E+00 2.5E+00 [(0/90)6]s [(+30/-30)6]s [(+45/-45)6]s [(0/+45/-45/90)3]s [(0/+45/-45)4]s
2.0E+00 1.5E+00
5.0E-01 0.0E+00 0.00
0
0.05
0.1 0.15 Displacement (mm)
0.2
0.05
0.25
0.10
0.15 Displacement (mm)
0.20
0.25
Fig. 11. The predicted damage evolution versus displacement of laminated plates (w/d = 3 and d = 6.35 mm): (a) volume fraction of debonded fibers and (b) microcrack density.
d=12.7mm (w/d=2, [(±30)6]s) d=6.35mm (w/d=4, [(±45)6]s)
of their ply orientations. We observe from Fig. 11(b) that the cracks nucleate rapidly once they begin to nucleate. The ratio of the crack nucleation is also seen to be approximately the same between the plate specimens with different ply orientations.
d=12.7mm (w/d=2, [(±45)6]s)
6. Concluding remarks 0
0.05
0.1 0.15 Displacement (mm)
0.2
0.25
60 d=6.35mm (w/d=4, [(0/±45/90)3]s) d=12.7mm (w/d=2, [(0/±45/90)3]s) d=6.35mm (w/d=4, [(0/±45)4]s) d=12.7mm (w/d=2, [(0/±45)4]s)
40 30 20 10 0
3.5E+00
1.0E+00
d=6.35mm (w/d=4, [(±30)6]s)
50 Applied load, P (kN)
1.2E-02
0.0E+00 0.00
Microcrack density
Applied load, P (kN)
a
a Volume fraction of debonded fiber
the width of specimens significantly affects the load–deflection behavior; and (2) a stiffer fashion of load–deflection behavior is predicted with the use of a higher value of w. The load–deflection curves of laminated plates with different values of d (wd ¼ 2, 4; w = 25.4 mm) are also presented in Fig. 10 to illustrate the influence of the size of the cutout on the behavior of the laminated plates. Fig. 10 shows that the diameter of the cutout has a significant influence on the deflection behavior of the laminated plates. Typical plots for the predicted evolutions of volume fraction of debonded fibers and microcrack density of laminated plates are shown in Fig. 11(a) and (b). It is seen from Fig. 11(a), which exhibits the evolution of debonded fibers in laminated plates as functions of displacement, that the onset points of fiber debonding for cross-ply, angle-ply, and quasi-isotropic layups are almost identical regardless
2229
0
0.05
0.1 0.15 Displacement (mm)
0.2
0.25
Fig. 10. The load–displacement curves of laminated plates with various diameters (w = 25.4 mm): (a) cross-ply, (b) angle-ply, and (c) quasiisotropic.
A computational investigation of the behavior of laminated plates containing a cutout under uniaxial compression has been presented. A constitutive model for laminated composites [22] is implemented into the finite element program ABAQUS for an accurate characterization of the compression response and damage evolution in structures made of laminated composites. Prior to the implementation, the predicted moduli of laminated composites are compared with analytical bounds and experimental data to validate and verify the constitutive model. Numerical simulations for laminated plates containing a cutout are carried out to examine whether the constitutive model implemented into a finite element program is able to predict the experimentally obtained response [21]. Moreover, a parametric study is conducted to address the influence of the geometry of the specimens on the behavior of laminated plates.
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Based on the comparison between the present analytical prediction and theoretical bounds as well as available experimental data, it is concluded that the proposed micromechanical constitutive model is suitable for modeling the constitutive behavior of laminated composites having a dilute or moderate fiber volume fraction (up to 45%). It is shown from the numerical simulations and experimental comparison that the geometry of the plate specimens influence the load–displacement behavior and damage evolution in laminated plates. Specifically, 37% 58% increases in peak load are observed from the plates having an intermediate w/d ratio (w/d = 4) over the plates having a small w/d ratio (w/d = 3), while high w/d ratios (w/d = 5, 6) increase them by 15% 28% and 40% 55% over the intermediate w/d ratio, respectively. The comparison of load–displacement curves of laminated plates between the present prediction and the experiment [21] demonstrates the validity of the implemental constitutive model. In particular, the model is valid for modeling the constitutive behavior of laminated composites having a dilute or moderate fiber volume fraction (up to = 45%). In a forthcoming paper, the Eshelby’s tensor for unidirectional fibers employed in this study will be compared with Mura’s [23] to confirm uniqueness of the tensor and elastoplastic behavior of ductile laminated composites will be considered in detail. Specifically, an effective yield function based on the ensemble-volume average process will be incorporated into the proposed constitutive model to estimate the elastoplastic response of ductile laminated composites. Acknowledgements This research was sponsored by the Ministry of Science and Technology, Korea, for the financial support by a grant (NC33676, R11-2002-101-02004-0) from the Smart Infra-Structure Technology Center (SISTeC). References [1] Aboudi J. Mechanics of composite materials – a unified micromechanical approach. Amsterdam: Elsevier; 1991. [2] Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I-analysis. J Compos Mater 1991;25:2–43. [3] Cheng ZQ, Batra RC. Exact Eshelby’s tensor for a dynamic circular cylindrical inclusion. ASME J Appl Mech 1999;66:563–5. [4] Christensen RM, Lo KH. Solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solids 1979;27:315–30. [5] Hashin Z. On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J Mech Phys Solids 1965;13:119–34.
[6] Hashin Z. Theory of Fiber Reinforced Materials, NASA CR-1974, 1972. [7] Hashin Z. Analysis of properties of fiber composites with anisotropic constituents. ASME J Appl Mech 1979;46:543–50. [8] Herakovich CT. Mechanics of fibrous composites. New York: John Wiley and Sons; 1998. [9] Hill R. Theory of mechanical properties of fibre-strengthened materials-I. Elastic behaviour. J Mech Phys Solids 1964;12:199–212. [10] Ju JW, Chen TM. Micromechanics and effective moduli of elastic composites containing randomly dispersed llipsoidal inhomogeneities. Acta Mechanica 1994;103:103–21. [11] Ju JW, Chen TM. Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mechanica 1994;103:123–44. [12] Ju JW, Zhang XD. Micromechanics and effective transverse elastic moduli of composites with randomly located aligned circular fibers. Int J Solids Struct 1998;35:941–60. [13] Ju JW, Zhang XD. Effective elastoplastic behavior of ductile matrix composites containing randomly located aligned circular fibers. Int J Solids Struct 2001;38:4045–69. [14] Karihaloo BL, Fu D. A damage-based constitutive law for plain concrete in tension. Eur J Mech A Solids 1989;8:373–84. [15] Karihaloo BL, Fu D. Orthotropic damage model for plain concrete in tension. ACI Mater J 1990;87:62–7. [16] Kondo K, Sato N. The influence of random fiber packing on the elastic properties of unidirectional composites, In Composites’86: Recent Advances in Japan and the United States, Japan-US CCMIII, Tokyo, 1986. [17] Lee HK. Computational approach to the investigation of impact damage evolution in discontinuously reinforced fiber composites. Comput Mech 2001;27:504–12. [18] Lee HK, Liang Z. Computational modeling of the response and damage behavior of fiber-reinforced cellular concrete. Comput Struct 2004;82:581–92. [19] 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. [20] Lee HK, Simunovic S. A damage constitutive model of progressive debonding in aligned discontinuous fiber composites. Int J Solids Struct 2001;38:875–95. [21] Lessard LB, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part II-Experiment. J Compos Mater 1991;25:44–64. [22] Liang Z, Lee HK, Suaris W. Micromechanics-based constitutive modeling for unidirectional laminated composites. Int J Solids Struct 2006;43:5674–89. [23] Mura T. Micromechanics of defects in solids. 2nd ed. The Netherlands: Martinus Nijhoff; 1987. [24] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials. 2nd ed. Elsevier Science Publishers; 1999. [25] Sun LZ. Micromechanics and Overall Elastoplasticity of Discontinuously Reinforced Metal Matrix Composites, Ph.D. Dissertation, University of California, Los Angeles, 1998. [26] Uemura M et al. On the stiffness of filament wound materials (in Japanese). Rep Inst Space Aeronaut Sci 1968;4:448–63. [27] Zhao YH, Weng GJ. Plasticity of a two-phase composite with partially debonded inclusion. Int J Plast 1996;12:781–804. [28] Zhao YH, Weng GJ. Transversely isotropic moduli of two partially debonded composites. Int J Solids Struct 1997;34:493–507. [29] ABAQUS Example Problems Manual, Version 6.5, Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI, 2004.
Numerical characterization of compressive response and damage evolution in laminated plates containing a cutout H.K. Lee *, B.R. Kim Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejon 305-701, Republic of Korea Received 4 August 2006; received in revised form 17 January 2007; accepted 25 January 2007 Available online 15 February 2007
Abstract A micromechanical constitutive model [Liang Z, Lee HK, Suaris W. Micromechanics-based constitutive modeling for unidirectional laminated composites. Int J Solid Struct 2006;43:5674–89], based on the concept of the ensemble-volume average for laminated composites, is implemented into a finite element program to numerically characterize the compressive response and damage evolution in laminated plates containing a cutout. Prior to the implementation of the model into the finite element program, the predicted moduli of laminated composites are compared with analytical bounds and experimental data for the validation and verification of the constitutive model. A series of numerical simulations for a uniaxial test of laminated plate specimens containing a cutout are conducted using the implemented constitutive model. The predictions are compared with experimental data [Lessard LB, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part II-Experiment. J Compos Mater 1991;25:44–64; Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I-Analysis. J Compos Mater 1991;25:2–43] to verify the accuracy of the implemented constitutive model. A parametric study is also carried out to illustrate the influence of the geometry of the specimens on the behavior of laminated plates. It is shown that the implemented constitutive model is suitable for the analysis of the constitutive behavior of laminated plates having a dilute or moderate fiber volume fraction. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Carbon fibres; Computational simulation; Damage mechanics; Laminates; Debonding
1. Introduction The performance prediction and estimation of structures made of fiber-reinforced, laminated composites has been increasingly important as they become primary load-carrying elements in many fields [12]. Many theoretical and numerical methods have been proposed to predict the effective properties of the fiber-reinforced, laminated composites. Hill [9] and Hashin [5,6] derived theoretical upper and lower bounds based on variational principles for the transverse moduli of the composites. The effective transverse shear modulus of unidirectional composites were *
Corresponding author. Tel.: +82 42 869 3623; fax: +82 42 869 3610. E-mail address: [email protected] (H.K. Lee).
0266-3538/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.01.031
derived by Christensen and Lo [4] using a three-phase cylinder model. Ju and Zhang [12,13] proposed a two-dimensional, micromechanics-based model for the prediction of the effective elastoplastic moduli of two-phase, fiber-reinforced composites. Micromechanical approaches such as the self-consistent method, the differential scheme and the Mori-Tanaka method have been introduced by many researchers to estimate the effective moduli of the composites. A review of the micromechanical approaches for the prediction of the effective moduli of laminated composites can be found in Aboudi [1], Ju and Zhang [12], and NematNasser and Hori [24]. Based on the concept of the ensemble-volume average and the Eshelby’s inclusion approach, we recently proposed a three-dimensional, micromechanical constitutive
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model for unidirectional laminated composites [22]. In our derivation, unidirectional fibers were assumed to be circular cylindrical inclusions that were embedded in a matrix. A newly developed Eshelby’s tensor for an infinite cylindrical inclusion [3] was adopted to model the unidirectional fibers and was incorporated into the micromechanical framework. The progressive loss of strength resulting from the partial fiber debonding and the nucleation of microcracks was incorporated into the constitutive model. The Weibull’s probabilistic function was used to model the varying probability of progressive fiber debonding [27,28] and the nucleation of microcracks was simulated by adopting a continuum damage model proposed by Karihaloo and Fu [14,15]. Damage mechanisms and evolutions of damage in laminated composites containing notches, holes or cutouts are not still well addressed [21]. In particular, the accurate evaluation of the compressive response and damage in laminated composites is vital to the complete understanding of the laminated composite structures [21]. The present study implements the previous work published in Liang et al. [22] into the finite element program ABQUS [29] to numerically characterize the response and damage evolution in laminated plates containing a centrally located cutout subjected to uniaxial compression. Prior to the implementation of the model into the finite element program, the predicted moduli of laminated composites are compared with Hashin’s theoretical bounds [7] and experimental data [26] for the validation and verification of the constitutive model. A series of numerical simulations on laminated plate specimens containing a cutout in uniaxial compression are conducted using the implemented constitutive model. The predictions are compared with experimental data [21,2] to verify the accuracy of the implemented constitutive model. Moreover, a parametric study is carried out to address the influence the geometry of the specimens on the behavior of laminated plates. 2. A constitutive model for unidirectional laminated composites The three-dimensional, micromechanical constitutive model for unidirectional laminated composites proposed by Liang et al. [22] is here summarized. Let us start by considering an initially perfectly bonded, unidirectional, threephase composite consisting of an elastic matrix (phase 0), aligned continuous fibers (phase 1), and (penny-shaped) microcracks (phase 2) of radius c. The 1-direction is chosen to be fiber direction and the plane 2–3 corresponds to the transversely isotropic plane in the unidirectional fiber reinforced composite as shown in Fig. 2 of Liang et al. [22]. It is assumed that penny-shaped microcracks are aligned spheroidal voids with the aspect ratio a2 ! 0 (see Fig. 2 of Lee [17]). The continuous fibers are assumed to be infinitely long, elastic cylindrical inclusions. As loadings or deformations proceed, some fibers in the composite are partially debonded (phase 3) and microcracks are nucleated. Fol-
lowing Zhao et al. [27,28], a partially debonded fiber is replaced by an equivalent, perfectly bonded fiber that possesses transversely isotropic moduli. The nucleation of microcracks is simulated by adopting a continuum damage model proposed by Karihaloo and Fu [14]. While the Eshelby’s tensor for a spheroidal inclusion derived by Sun [25] is employed to model penny-shaped microcracks, the Eshelby’s tensor for an infinite cylindrical inclusion previously proposed by Cheng and Batra [3] is considered to model the unidirectional fibers. Specifically, for the implementation of the Eshelby’s tensor proposed by Cheng and Batra [3] into the present micromechanical formulation, the Eshelby’s tensor is rephrased in a transe The transversely versely isotropic fourth-rank tensor F. e isotropic fourth-rank tensor F is defined by six parameters bm (m = 1–6) Fe ijkl ðbm Þ ¼ b1 ~ni ~nj ~nk ~nl þ b2 ðdik ~nj ~nl þ dil ~nj ~nk þ djk ~ni ~nl þ djl ~ni ~nk Þ þ b3 dij ~nk ~nl þ b4 dkl ~ni ~nj þ b5 dij dkl þ b6 ðdik djl þ dil djk Þ
ð1Þ
with the unit direction vector ~n and index m = 1–6. If the 1-direction is chosen to be symmetric, then we have ~n1 ¼ 1; ~n2 ¼ ~n3 ¼ 0. The details of the implementation of the Eshelby’s tensors into the micromechanical framework can be found in Lee and Simunovic [20] and Liang et al. [22]. Based on the governing field equations for linear elastic composites containing arbitrarily non-aligned and/or dissimilar inclusions [10,11], the effective stiffness tensor C* for the fourphase composite can be derived as h i 1 1 1 C ¼ C0 I þ R3r¼1 f/r ðAr þ Sr Þ ½I /r Sr ðAr þ Sr Þ g ð2Þ where Cr denotes the elasticity tensor of the r-th phase, ‘‘Æ’’ is the tensor multiplication, I is the fourth-rank identity tensor, and /r signifies the volume fraction of the r-th phase inclusion. While the Eshelby’s tensors S1 and S3 for perfectly bonded and partially debonded unidirectional fibers, respectively, were previously derived by Cheng and Batra [3] (see also Liang et al. [22]), the Eshelby’s tensor S2 for penny-shaped microcracks also was previously derived by Sun [25]. In addition, the fourth-rank tensor Ar is defined as 1
Ar ðCr C0 Þ
C0
ð3Þ
Accordingly, the effective elastic stiffness tensor C* for the four-phase composites can be derived as C ¼ Fe ijkl ðs1 ; s2 ; s3 ; s4 ; s5 ; s6 Þ
ð4Þ
where the six parameters in Eq. (4) are given in the Appendix A of Liang et al. [22]. Now we consider rotations through an angle h about the X3-axis to derive the stiffness of off-axis unidirectional fibrous composites. The angle h is measured positive coun-
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
terclockwise from the X1-axis to the x1-axis (see Fig. 2 of [22]). Following the stiffness transformation law by Herakovich [8] and Liang et al. [22], the stiffness matrix of offaxis unidirectional fibrous composites C, which is the transformed stiffness matrix through an arbitrary angle h about the 3-axis, is derived as
2223
By combining Eq. (48) of Lee and Simunovic [19] with Eqs. (4) and (1), the effective transverse Young’s modulus for unidirectional laminated composites can be derived as 2
ET ¼
4s6 ½ðs1 þ 4s2 þ s3 þ s4 þ s5 þ 2s6 Þðs5 þ s6 Þ ðs3 þ s5 Þ ðs1 þ 4s2 þ s3 þ s4 þ s5 þ 2s6 Þðs5 þ 2s6 Þ ðs3 þ s5 Þ
2
ð5Þ
ð7Þ
where the stiffness matrix of the unidirectional fibrous composites with the fibers oriented off-axis takes the form: 3 2 C 11 C 12 C 13 0 0 C 16 6C 0 C 26 7 7 6 12 C 22 C 23 0 7 6 7 6 C 13 C 23 C 33 0 0 C 36 7 6 ð6Þ C¼6 0 0 C 44 C 45 0 7 7 60 7 6 40 0 0 C 45 C 55 0 5
where the parameters s1, . . . , s6 are given by Liang et al. [22]. In addition, the transverse Young’s modulus given in Eq. (48) of Lee and Simunovic [19] are
r¼C:
C 36
0
0
7.0 6.0
C 66
Here, the components of C are given in Appendix C of Liang et al. [22]. A more detailed description on the micromechanical formulations for the effective elastic stiffness tensor for the off-axis unidirectional fibrous composites can be found in Liang et al. [22]. It is assumed that damage in laminate composites is controlled by the interfacial fiber debonding and nucleation of microcracks in the matrix. The Weibull probabilistic distribution function is used for modeling the interfacial fiber debonding [27,28], and a damage model proposed by Karihaloo and Fu [14,15] is utilized for modeling the nucleation of microcracks in the matrix. Details of the damage models can be found in Lee [17], Lee and Liang [18], and Liang et al. [22]. The constitutive model incorporating the interfacial fiber debonding and nucleation of microcracks in the matrix is then implemented into the finite element program ABAQUS using a usersupplied material subroutine to characterize numerically the damage evolutions and the corresponding load–deflection (p–u) behaviors of laminated composite structures. Computational algorithms used in the user-supplied material subroutine are based on the strain driven algorithm in which the stress history is to be determined by a given strain history.
5.0
*
C 26
Present prediction Experimental data (Uemura,1968) Hashin's lower bound Hashin's upper bound
8.0
E T /E 0
C 16
9.0
4.0 3.0 2.0 1.0 0.0 0.0
0.1
0.2
0.3
0.4
Fiber volume fraction
0.5
0.6
0.7
f
Fig. 1. The effective transverse Young’s modulus versus the fiber volume fraction.
l
3. Validation and verification of the constitutive model for unidirectional laminated composites For the validation and verification of the proposed constitutive model, the predictions on a glass fiber/epoxy matrix composite based on the proposed constitutive model are compared with Hashin’s theoretical bounds [7] and experimental data [26]. Similar to the analytical comparison conducted by Ju and Zhang [12], we adopt the same material properties for the glass fiber/epoxy matrix composite as those used in Kondo and Sato [16] as follows: Ef = 11,660 kg/mm2, mf = 0.22; and Em = 550 kg/mm2, mm = 0.35, where the subscripts f and m denote fibers and the matrix, respectively.
d
e
h
w
Fig. 2. The schematic description of laminated plates containing a cutout [21].
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Table 1 The ply orientations and gemetry of the laminated plate specimens (see also [21]) Ply orientations Cross-ply Angle-ply Quasi-isotropic *
h (mm) [(0/90)6]s [(±30)6]s [(±45)6]s [(0/±45/90)3]s [(0/±45)4]s
l (mm)
w (mm)
d (mm)
e (mm)
101.6
25.4
6.35
25.4
3.302
4^c6 ½ð^c1 þ 4^c2 þ ^c3 þ ^c4 þ ^c5 þ 2^c6 Þð^c5 þ ^c6 Þ ð^c3 þ ^c5 Þ 2
ð^c1 þ 4^c2 þ ^c3 þ ^c4 þ ^c5 þ 2^c6 Þð^c5 þ 2^c6 Þ ð^c3 þ ^c5 Þ ð8Þ
Hashin’s theoretical bounds for ET of the composites were given by [7] ¼
4jðÞ GTðÞ jðÞ þ mðÞ GTðÞ
ð9Þ
where mðÞ ¼ 1 þ
E1(GPa)
m0
m1
/1
4.62
238.56
0.36
0.20
0.66
jðÞ ¼ jm þ
/f 1=ðjf jm Þ þ /m =ðjm þ Gm Þ
4ðmLf mLm Þ2 /m /f /m =jf þ /f =jm þ 1=Gf ðm Lf mLm Þð1=jm 1=jf Þ/m /f mLðÞ ¼ mLm /m þ mLf /f þ /m =jf þ /f =jm þ 1=Gm ELðÞ ¼ ELm /m þ ELf /f þ
2
ETðÞ
E0(GPa)
3.429
All layups have 24 layers of plies.
ET ¼
Table 2 The material properties of the laminated plate specimens [2,8]
4jðÞ mLðÞ ELðÞ
in which the subscripts (+) and () indicate the upper and lower bounds of each of the properties, respectively. In addition, the lower bounds for the effective longitudinal Young’s modulus EL , transverse shear modulus GT , bulk modulus j*, longitudinal Poisson’s ratio mL can be expressed as (see Eqs. (31)–(33), (35a) of Hashin [7] for details)
ð10Þ ð11Þ
ð12Þ / f GTðÞ ¼ GTm þ 1=ðGTm Gm Þ þ /m ðjm þ 2Gm Þ=½2Gm ðjm þ Gm Þ ð13Þ
where /m and /f denote the volume fraction of the matrix and fibers, respectively. The corresponding upper bounds for those properties can be obtained by interchanging the position of each subscript [7]. The predicted effective (normalized) transverse Young’s modulus ET =E0 of the glass fiber/epoxy matrix composite according to Eq. (7) is shown in Fig. 1. The theoretical predictions of ET =E0 based on Hashin’s bounds given in Eq. (9) and experimental data reported by Uemura [26] for the composite are also plotted in the figure for comparison. Clearly, the present prediction is well within the Hashin’s theoretical bounds. It is also observed from the figure that the present prediction and the experimental data match well up to the fiber volume fraction / = 0.45. However, the predicted ET =E0 is lower than that based on the exper-
Fig. 3. The finite element model (a, b) and boundary conditions (c, d) used in the simulation (h = 3.43 mm, d = 6.35 mm, w/d = 4).
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230 40 Experimental data #1 Experimental data #2 Present prediction
35
Applied load, P (kN)
30 25 20 15 10 5 0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Displacement (mm)
Fig. 4. The comparison of load–displacement curves between the present prediction and the experiment [21] on cross-ply laminated plates.
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averaged perturbations in a fiber due to other surrounding fibers [10,11] whose effects on the mechanical behavior of composites become noticeable as the fiber volume fraction increases. These issues are beyond the scope of the present work; however, should be considered in future work. It is also worth mentioning that the interaction effect between fibers must be considered in modeling the mechanical behavior of composites having a high fiber volume fraction. Based on the comparison between the present analytical prediction and theoretical bounds as well as available experimental data, it is concluded that the proposed micromechanical constitutive model is capable of modeling the constitutive behavior of laminated composites having a dilute or moderate fiber volume fraction. 4. Numerical simulations for a uniaxial compression test of laminated plates containing a cutout 4.1. Overview
iment as the fiber volume fraction continues to increase. Variability between these results is partly due the present model not accounting for fiber interactions such as the
To gain insight into the constitutive behavior and damage evolution in laminated composites, a comprehensive
Fig. 5. The finite element discretization employed to simulate the compressive response and damage evolution in laminated plates containing a cutout during uniaxial compression.
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
4.2. Finite element implementation and modeling The constitutive model for laminated composites incorporating damage models for the fiber debonding and microcrack nucleation is implemented into the finite element program ABAQUS to simulate the constitutive behavior and damage evolution in structures made of laminated composites. Details of the implementation of a constitutive model into a finite element program can be founded in Lee and Liang [18] and Liang et al. [22]. The configuration of the laminated plate specimens containing a cutout is illustrated in Fig. 2. The plate specimens made of T300/976 graphite/epoxy resin prepreg tapes [21] are subjected to uniaxial compression as shown in Fig. 2. The details of the geometry of the plate specimens were listed in Table 1 of Lessard and Chang [21]. Following Lessard and Chang [21], various ply orientations listed in Table 1 are considered in this study. The three-dimensional, eight-node, linear brick solid element C3D8 in ABAQUS is used for modeling the laminated plate specimens. Only a quarter of the laminated plate specimens having a half of the specimen thickness is modeled in this study since the geometry and loading condition of the specimen are symmetry along the x-, y-, and z-directions as shown in Fig. 3. Here, we employ the same material properties for the laminated plate specimens according to Table 1 of Chang and Lessard [2] and Table 1.2 of Herakovich [8], and the employed material properties are summarized in Table 2. Following Lessard and Chang [21], we compute the compressive load and the corresponding displacement between two points located 12.7 mm above and below from the middle of the plate specimens. 4.3. Simulation results To implement the Weibull evolutionary debonding model [27,28] and the crack nucleation model [14,15], one needs to estimate the Weibull parameters S0, M and the crack nucleation parameters th, c1, c2. Based on the experimental data on cross-ply laminated plates under uniaxial compression [21], we estimate those parameters to be: S0 = 10,059.5 MPa, M = 4.0; th = 0.05%, c1 = 0.4, c2 = 1.8. Fig. 4 shows the predicted load–displacement curve
of the cross-ply laminated plates under uniaxial compression that best fits the experimental data [21]. Fig. 5 illustrates the finite element decretization along the x- and y-axes around the cutout. Fine meshes are used to model the vicinity of the cutout. The predicted evolution of volume fraction of debonded fibers versus displacement around the cutout is shown in Fig. 6(a). Fig. 6(b) exhibits the predicted evolution of microcrack density versus displacement around the cutout. As indicated in the experiment [21], the load–displacement curves for the present prediction and the experiment shown in Fig. 4 are linear in the early stages of loading. Despite the damage in the specimen continues to increase as shown in Fig. 6(a) and (b), a nearly linear load-displacement curve is obtained from the present prediction since the effect of the damage on the load-displacement curve is not substantial. Fig. 7 shows the sequence of deformed shape and von-Mises effective stress of the cross-ply laminated plates during uniaxial compression, illustrating the stress concentration around the cutout emanating from the edges of the cutout. On the basis of the estimated parameters, the present model is exercised against the experiment on two angleply laminated plates containing a cutout. The results are presented in Fig. 8(a). Fig. 8(a) shows the comparison of the load–displacement curves of the two angle-ply lami-
a
1.2E-03
1.0E-03
Segment A Segment B
8.0E-04
6.0E-04 4.0E-04
2.0E-04
0.0E+00 0.00
0.05
0.10
0.15
0.20
0.25
Displacement (mm)
b
5.0 4.5 4.0
Microcrack density
experimental and numerical study was performed by Lessard and Chang [21] and Chang and Lessard [2] for laminated plates containing a cutout. The laminated plates with a cutout subjected to uniaxial compression exhibited different types of response or damage, depending on the layup, material properties, and geometry of the plate specimens [21]. A series of numerical simulations for laminated plates containing a cutout are carried out in this study to examine whether the constitutive model implemented into a finite element program is able to predict the experimentally obtained response [21]. Modeling procedures and simulation results are summarized in this section.
Volume fraction of debonded fib
2226
3.5 3.0 2.5 2.0 1.5
Segment A
1.0
Segment B
0.5 0.0 0.00
0.05
0.10
0.15
0.20
0.25
Displacement (mm)
Fig. 6. The predicted damage evolution versus displacement around the cutout of cross-ply laminated plates under uniaxial compression: (a) volume fraction of debonded fibers and (b) microcrack density.
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
2227
Fig. 7. The sequence of the deformed shape and von-Mises effective stress of cross-ply laminated plates during uniaxial compression.
nated plates between the present prediction and the experiment [21]. As shown in Fig. 8(a), the load–displacement curves predicted by the present model are higher than those obtained by the experiment. The experimentally obtained load–displacement curve for the [(±45)6]s layup is highly
nonlinear, exhibiting the effect of material nonlinearities inherent to the composites, even in early stages of loading where there is no damage [21]. Since the present model does not consider material nonlinearities, the load–displacement curve for the [(±45)6]s layup predicted by the present model
2228
a
40
Applied load, P (kN)
35
Experimental data [(±30)6]s Present prediction [(±30)6]s Experimental data [(±45)6]s #1 Experimental data [(±45)6]s #2 Present prediction [(±45)6]s
30 25 20
Applied load, P (kN)
a
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
15
60 50 40
w/d=3 w/d=4 w/d=5 w/d=6
30 20 10 0 0
0.05
0.1 0.15 Displacement (mm)
10
b
5
50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Displacement (mm) 50 Experimental data [(0/±45/90)3]s #1 Experimental data [(0/±45/90)3]s #2 Present prediction [(0/±45/90)3]s Experimental data [(0/±45)4]s #1 Experimental data [(0/±45/90)3]s #2 Present prediction [(0/±45)4]s
45
Applied load, P (kN)
40 35
w/d=5 [(±30)6]s w/d=6 [(±30)6]s w/d=3 [(±45)6]s w/d=4 [(±45)6]s w/d=5 [(±45)6]s w/d=6 [(±45)6]s
40 35 30 25 20 15 10 5 0
0
30
20
c 70
15
60
10 5 0 0
0.05
0.1
0.15
0.2
Displacement (mm) Fig. 8. The comparison of load–displacement curves between the present prediction and the experiment [21]: (a) two angle-ply laminated plates and (b) two quasi-isotropic laminated plates.
0.1 Displacement (mm)
0.15
0.2
0.15
0.2
Table 3 The w/d ratios used in the parametric study (see also [21]) Ply orientations
h (mm)
Quasi-isotropic
[(0/90)6]s [(±30)6]s [(±45)6]s [(0/±45/90)3]s [(0/±45)4]s
w/d=3 [(0/±45)4]s w/d=4 [(0/±45)4]s
40
w/d=5 [(0/±45)4]s w/d=6 [(0/±45)4]s
30 20 10 0
0.05
0.1 Displacement (mm)
w = 25.4 mm
3.429 3, 4, 5, 6
w/d=5 [(0/±45/90)3]s w/d=6 [(0/±45/90)3]s
Fig. 9. The load–displacement curves of laminated plates with various w/d ratios (d = 6.35 mm): (a) cross-ply, (b) angle-ply, and (c) quasi-isotropic.
w/d ratios d = 6.35 mm
Cross-ply Angle-ply
w/d=3 [(0/±45/90)3]s w/d=4 [(0/±45/90)3]s
50
0
*
0.05
25
Applied load, P (kN)
b
Applied load, P (kN)
0
0.25
w/d=3 [(±30)6]s w/d=4 [(±30)6]s
45
0
0.2
2, 4
3.302
All layups have 24 layers of plies (l = 101.6 mm, e = 25.4 mm).
is exhibited to be higher than that obtained from the experiment. The predicted load–displacement curve for the [(±30)6]s layup is also shown to be higher than that obtained from the experiment. The present model is further exercised against the experiment on two quasi-isotropic laminated plates containing a cutout using the estimated parameters. Fig. 8(b) shows the comparison of the load–displacement curves of the two quasi-isotropic laminated plates between the present prediction and the experiment [21]. The load–displacement
curves of the two quasi-isotropic laminated composite plates between the present prediction and the experiment show slight deviations as shown in the figure. 5. Parametric study To illustrate the influence of the geometry of the plate specimens on the compressive response and damage evolution in laminated plates containing a cutout, a parametric study is conducted. In particular, the ratio of the width of the plate specimens to the diameter of the cutout (w/d ratio) is considered as the parameter of this parametric study. The w/d ratios used in this parametric study are summarized in Table 3. The load–displacement curves of laminated plates with different values of w (wd ¼ 3, 4, 5, 6; d = 6.35 mm) are presented in Fig. 9. It is observed from the figure that (1)
H.K. Lee, B.R. Kim / Composites Science and Technology 67 (2007) 2221–2230
Applied load, P (kN)
b
c
1.0E-02
8.0E-03
[(0/90)6]s [(+30/-30)6]s [(+45/-45)6]s [(0/+45/-45/90)3]s [(0/+45/-45)4]s
6.0E-03
4.0E-03
2.0E-03
0.05
0.10
0.15
0.20
0.25
Displacement (mm)
b
5.0E+00 4.5E+00 4.0E+00
50 45 40 35 30 25 20 15 10 5 0
50 45 40 35 30 25 20 15 10 5 0
d=6.35 mm (w/d=4) d=12.7 mm (w/d=2)
3.0E+00 2.5E+00 [(0/90)6]s [(+30/-30)6]s [(+45/-45)6]s [(0/+45/-45/90)3]s [(0/+45/-45)4]s
2.0E+00 1.5E+00
5.0E-01 0.0E+00 0.00
0
0.05
0.1 0.15 Displacement (mm)
0.2
0.05
0.25
0.10
0.15 Displacement (mm)
0.20
0.25
Fig. 11. The predicted damage evolution versus displacement of laminated plates (w/d = 3 and d = 6.35 mm): (a) volume fraction of debonded fibers and (b) microcrack density.
d=12.7mm (w/d=2, [(±30)6]s) d=6.35mm (w/d=4, [(±45)6]s)
of their ply orientations. We observe from Fig. 11(b) that the cracks nucleate rapidly once they begin to nucleate. The ratio of the crack nucleation is also seen to be approximately the same between the plate specimens with different ply orientations.
d=12.7mm (w/d=2, [(±45)6]s)
6. Concluding remarks 0
0.05
0.1 0.15 Displacement (mm)
0.2
0.25
60 d=6.35mm (w/d=4, [(0/±45/90)3]s) d=12.7mm (w/d=2, [(0/±45/90)3]s) d=6.35mm (w/d=4, [(0/±45)4]s) d=12.7mm (w/d=2, [(0/±45)4]s)
40 30 20 10 0
3.5E+00
1.0E+00
d=6.35mm (w/d=4, [(±30)6]s)
50 Applied load, P (kN)
1.2E-02
0.0E+00 0.00
Microcrack density
Applied load, P (kN)
a
a Volume fraction of debonded fiber
the width of specimens significantly affects the load–deflection behavior; and (2) a stiffer fashion of load–deflection behavior is predicted with the use of a higher value of w. The load–deflection curves of laminated plates with different values of d (wd ¼ 2, 4; w = 25.4 mm) are also presented in Fig. 10 to illustrate the influence of the size of the cutout on the behavior of the laminated plates. Fig. 10 shows that the diameter of the cutout has a significant influence on the deflection behavior of the laminated plates. Typical plots for the predicted evolutions of volume fraction of debonded fibers and microcrack density of laminated plates are shown in Fig. 11(a) and (b). It is seen from Fig. 11(a), which exhibits the evolution of debonded fibers in laminated plates as functions of displacement, that the onset points of fiber debonding for cross-ply, angle-ply, and quasi-isotropic layups are almost identical regardless
2229
0
0.05
0.1 0.15 Displacement (mm)
0.2
0.25
Fig. 10. The load–displacement curves of laminated plates with various diameters (w = 25.4 mm): (a) cross-ply, (b) angle-ply, and (c) quasiisotropic.
A computational investigation of the behavior of laminated plates containing a cutout under uniaxial compression has been presented. A constitutive model for laminated composites [22] is implemented into the finite element program ABAQUS for an accurate characterization of the compression response and damage evolution in structures made of laminated composites. Prior to the implementation, the predicted moduli of laminated composites are compared with analytical bounds and experimental data to validate and verify the constitutive model. Numerical simulations for laminated plates containing a cutout are carried out to examine whether the constitutive model implemented into a finite element program is able to predict the experimentally obtained response [21]. Moreover, a parametric study is conducted to address the influence of the geometry of the specimens on the behavior of laminated plates.
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Based on the comparison between the present analytical prediction and theoretical bounds as well as available experimental data, it is concluded that the proposed micromechanical constitutive model is suitable for modeling the constitutive behavior of laminated composites having a dilute or moderate fiber volume fraction (up to 45%). It is shown from the numerical simulations and experimental comparison that the geometry of the plate specimens influence the load–displacement behavior and damage evolution in laminated plates. Specifically, 37% 58% increases in peak load are observed from the plates having an intermediate w/d ratio (w/d = 4) over the plates having a small w/d ratio (w/d = 3), while high w/d ratios (w/d = 5, 6) increase them by 15% 28% and 40% 55% over the intermediate w/d ratio, respectively. The comparison of load–displacement curves of laminated plates between the present prediction and the experiment [21] demonstrates the validity of the implemental constitutive model. In particular, the model is valid for modeling the constitutive behavior of laminated composites having a dilute or moderate fiber volume fraction (up to = 45%). In a forthcoming paper, the Eshelby’s tensor for unidirectional fibers employed in this study will be compared with Mura’s [23] to confirm uniqueness of the tensor and elastoplastic behavior of ductile laminated composites will be considered in detail. Specifically, an effective yield function based on the ensemble-volume average process will be incorporated into the proposed constitutive model to estimate the elastoplastic response of ductile laminated composites. Acknowledgements This research was sponsored by the Ministry of Science and Technology, Korea, for the financial support by a grant (NC33676, R11-2002-101-02004-0) from the Smart Infra-Structure Technology Center (SISTeC). References [1] Aboudi J. Mechanics of composite materials – a unified micromechanical approach. Amsterdam: Elsevier; 1991. [2] Chang FK, Lessard LB. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part I-analysis. J Compos Mater 1991;25:2–43. [3] Cheng ZQ, Batra RC. Exact Eshelby’s tensor for a dynamic circular cylindrical inclusion. ASME J Appl Mech 1999;66:563–5. [4] Christensen RM, Lo KH. Solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solids 1979;27:315–30. [5] Hashin Z. On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J Mech Phys Solids 1965;13:119–34.
[6] Hashin Z. Theory of Fiber Reinforced Materials, NASA CR-1974, 1972. [7] Hashin Z. Analysis of properties of fiber composites with anisotropic constituents. ASME J Appl Mech 1979;46:543–50. [8] Herakovich CT. Mechanics of fibrous composites. New York: John Wiley and Sons; 1998. [9] Hill R. Theory of mechanical properties of fibre-strengthened materials-I. Elastic behaviour. J Mech Phys Solids 1964;12:199–212. [10] Ju JW, Chen TM. Micromechanics and effective moduli of elastic composites containing randomly dispersed llipsoidal inhomogeneities. Acta Mechanica 1994;103:103–21. [11] Ju JW, Chen TM. Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mechanica 1994;103:123–44. [12] Ju JW, Zhang XD. Micromechanics and effective transverse elastic moduli of composites with randomly located aligned circular fibers. Int J Solids Struct 1998;35:941–60. [13] Ju JW, Zhang XD. Effective elastoplastic behavior of ductile matrix composites containing randomly located aligned circular fibers. Int J Solids Struct 2001;38:4045–69. [14] Karihaloo BL, Fu D. A damage-based constitutive law for plain concrete in tension. Eur J Mech A Solids 1989;8:373–84. [15] Karihaloo BL, Fu D. Orthotropic damage model for plain concrete in tension. ACI Mater J 1990;87:62–7. [16] Kondo K, Sato N. The influence of random fiber packing on the elastic properties of unidirectional composites, In Composites’86: Recent Advances in Japan and the United States, Japan-US CCMIII, Tokyo, 1986. [17] Lee HK. Computational approach to the investigation of impact damage evolution in discontinuously reinforced fiber composites. Comput Mech 2001;27:504–12. [18] Lee HK, Liang Z. Computational modeling of the response and damage behavior of fiber-reinforced cellular concrete. Comput Struct 2004;82:581–92. [19] 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. [20] Lee HK, Simunovic S. A damage constitutive model of progressive debonding in aligned discontinuous fiber composites. Int J Solids Struct 2001;38:875–95. [21] Lessard LB, Chang FK. Damage tolerance of laminated composites containing an open hole and subjected to compressive loadings: Part II-Experiment. J Compos Mater 1991;25:44–64. [22] Liang Z, Lee HK, Suaris W. Micromechanics-based constitutive modeling for unidirectional laminated composites. Int J Solids Struct 2006;43:5674–89. [23] Mura T. Micromechanics of defects in solids. 2nd ed. The Netherlands: Martinus Nijhoff; 1987. [24] Nemat-Nasser S, Hori M. Micromechanics: overall properties of heterogeneous materials. 2nd ed. Elsevier Science Publishers; 1999. [25] Sun LZ. Micromechanics and Overall Elastoplasticity of Discontinuously Reinforced Metal Matrix Composites, Ph.D. Dissertation, University of California, Los Angeles, 1998. [26] Uemura M et al. On the stiffness of filament wound materials (in Japanese). Rep Inst Space Aeronaut Sci 1968;4:448–63. [27] Zhao YH, Weng GJ. Plasticity of a two-phase composite with partially debonded inclusion. Int J Plast 1996;12:781–804. [28] Zhao YH, Weng GJ. Transversely isotropic moduli of two partially debonded composites. Int J Solids Struct 1997;34:493–507. [29] ABAQUS Example Problems Manual, Version 6.5, Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI, 2004.