- Email: [email protected]
Simple constitutive model for nonlinear response of fiber-reinforced composites with loading-directional dependence
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 67 (2007) 111–118 www.elsevier.com/locate/compscitech
Simple constitutive model for nonlinear response of fiber-reinforced composites with loading-directional dependence Tomohiro Yokozeki a
a,*
, Shinji Ogihara b, Shunsuke Yoshida b, Toshio Ogasawara
a
Advanced Composite Technology Center, Institute of Aerospace Technology (IAT), Japan Aerospace Exploration Agency (JAXA), 6-13-1 Osawa, Mitaka, Tokyo, 181-0015, Japan b Department of Mechanical Engineering, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan Received 5 January 2006; received in revised form 23 March 2006; accepted 27 March 2006 Available online 24 May 2006
Abstract Polymeric composites generally exhibit different nonlinear response in tension and compression tests. In this study, Sun and Chen’s one-parameter plasticity model [Sun C.T. and Chen J.L., 1989. A simple flow rule for characterizing nonlinear behavior of fiber composites, J Compos Mater 29, 457–463] is extended to incorporate the loading-directional dependence on the plastic flow in composite laminates. Off-axis specimens of unidirectional carbon/epoxy composites were tested in compression and tension. Based on experimental results of compression tests, a simple two-parameter constitutive model for characterizing nonlinear mechanical behavior of fiber-reinforced composites was established. Finally, comparison of model predictions with experimental results of off-axis tension tests indicated a good correlation between the two. The developed simple plasticity model can explain the difference of nonlinear response in tension and compression tests. 2006 Elsevier Ltd. All rights reserved. Keywords: A. Polymer-matrix composites (PMCs); B. Plastic deformation; B. Non-linear behaviour; Loading-directional dependence
1. Introduction Interest in advanced composite materials is growing for use in structures for aerospace, maritime and civil applications. Predicting the mechanical response and failure behavior of fiber-reinforced composite is considerably important for efficient design of composite structures. Detailed description of macroscopic behavior of lamina and laminate is necessary to enhance the applicability of such composite structures. Fiber-reinforced composites are widely recognized as exhibiting nonlinear mechanical response under off-axis loading. Nonlinear behaviors of polymeric composites include plastic deformation, microscopic failure (e.g., matrix cracking, fiber–matrix debonding), geometric nonlinearity (e.g., fiber rotation), and others. Therefore, micro*
Corresponding author. Tel.: +81 422 40 3384; fax: +81 422 40 3549. E-mail address: [email protected] (T. Yokozeki).
0266-3538/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.03.024
scopic modeling of fiber-reinforced composites using fiber and matrix properties has been studied extensively [2–5]. These micromechanical models are then used to evaluate the effective mechanical response of a lamina and laminate. In addition to the request for avoidance of time-consuming micromechanical calculations, the constituent fiber and matrix properties of recent high-performance composites produced from prepregs are not often disclosed. Therefore, macroscopic approaches using composite properties are also necessary to describe the nonlinear behavior of composite laminates. Macroscopic characterization is preferable for incorporation of the constitutive relations into structural analytical tools (e.g., finite element analysis). Many macroscopic approaches have been developed for characterizing nonlinear behaviors of fiber-reinforced composites. Hahn and Tsai [6] predicted nonlinear stress–strain responses of unidirectional laminates under off-axis tensile loading using nonlinear elastic constitutive equations based on higher order energy functions. Spencer et al. [7,8]
112
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
developed a macro-mechanical plasticity model for a transversely isotropic material in which the yield surface is described by two stress invariants using two parameters. Robinson and Duffy [9] expanded the previous yield surface [7] to include a third invariant. Hansen et al. [10,11] proposed an invariant-based anisotropic plasticity model using a scalar hardening parameter instead of determining effective stress–strain relations. It is difficult to obtain a unique effective stress–strain relation for fiber-reinforced composites that can fit experimental results obtained under various loading conditions, as noted by Hansen et al. [10]. Sun and Chen [1] proposed a one-parameter plasticity model using a Hill-type [12] quadratic plastic potential by assuming a plane stress state and no plastic deformation in the fiber direction. They showed that effective stress-effective plastic strain curves were obtainable by fitting the experimental results of offaxis tension tests. They also showed that nonlinear responses of unidirectional fiber-reinforced composites under various loading conditions were well characterized. Chen and Sun [13] proposed a 3-D plastic potential that is suitable for modeling plastic flow in anisotropic fiber composites. Xie and Adams [14] developed a 3-D plasticity model similar to that of Sun and Chen [1]. Ogi and Takeda [15] proposed a modified one-parameter model in which an anisotropy parameter changes concomitant with plastic deformation. Ogihara et al. [16] applied a oneparameter plasticity model to characterize the nonlinear behavior of angle-ply laminates. The family of one-parameter plasticity models was extended to describe the ratedependent nonlinear behavior of polymeric composites [17–20]. One-parameter models are attractive for characterizing nonlinear behaviors in various polymeric composites because of their simplicity and consistency with experimental results. In general, polymeric composites exhibit different nonlinear behavior in compression and tension tests. Tsai and Sun [19] indicated that tension and compression stress–strain curves of off-axis specimens appear to be different, specifically in nonlinear portions of the curves. This phenomenon is attributed to: (1) the residual stress in the matrix because the coefficients of thermal expansion are different between the fiber and matrix; and (2) the matrix itself exhibits pressure-dependent behavior. Quadratic plastic potentials used in the above-mentioned models (e.g., Sun and Chen [1]) cannot explain the different nonlinear behaviors in compression and tension tests. Therefore, this loading-directional dependence on the plastic deformation should be incorporated for characterizing the nonlinear behavior of polymeric composites under various loading conditions. In the case of isotropic materials, hydrostatic pressure terms are incorporated in the stress potential (e.g., Drucker–Prager model [21]) to represent pressuredependent yielding behavior. Some anisotropic yield criteria can account for the different strengths that are exhibited by composites in compression and tension, such as those presented in Hoffman [22], and Tsai and Wu [23]. Voyiadjis
and Thiagarajan [24,25] proposed a six-parameter anisotropic yield surface to formulate plasticity in anisotropic materials in which pressure-dependent behavior is included. A simple but accurate model for nonlinear behaviors of fiber-reinforced composites under general loading conditions must be established. This study develops a simple constitutive model that incorporates the above-mentioned loading-directional dependence on plastic flow in composite laminates under monotonic loading. It is developed by extending Sun and Chen’s one-parameter plasticity model [1]. Off-axis specimens of unidirectional carbon/epoxy composites are tested in compression and tension. Based on the compressive experimental results, a simple constitutive model for characterizing nonlinear mechanical behavior of fiber-reinforced composites was established using two parameters: one is the anisotropic parameter and the other reflects the loading-directional effect. Finally, the predictive capability of the present model is verified through comparison with experimental results of off-axis tension tests. 2. Constitutive model Mechanical behavior of an orthotropic lamina of fiberreinforced composites is considered. It is assumed that total strain can be decomposed into elastic and plastic strains within infinitesimal strain conditions, as deij ¼ deeij þ depij ;
ð1Þ
where superscripts e and p, respectively, denote elastic and plastic. The incremental form of the elastic constitutive equation under a plane stress state is expressed as 9 8 e 9 2 38 1=E1 m12 =E1 0 > > = = < de11 > < dr11 > 6 7 dee22 ¼ 4 ð2Þ 1=E2 0 5 dr22 : > > > ; ; : e > : dc12 sym 1=G12 ds12 Note that the 1, 2, 3-directions are defined, respectively, as the fiber, in-plane transverse to the fiber, out-of-plane transverse to the fiber direction. Although the linear stress–strain relation is assumed in Eq. (2), one can include nonlinear elastic properties (e.g., nonlinear shear response [6], nonlinear response in fiber direction [26]). To model the plastic constitutive relation, Sun and Chen [1] defined the effective stress as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ~¼ r ðr þ 2a66 s212 Þ; ð3Þ 2 22 where a66 is an anisotropic parameter. This is a plane stress case of the following 3-D effective stress of a transversely isotropic material rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 3n 2 3D ~ ¼ ðr22 r33 Þ þ 2a66 ðs212 þ s213 Þ þ 2a44 s223 : ð4Þ r 2 In Weeks and Sun [17], a44 is set as 2 by assuming the descriptive equivalency between plastic potential and strain energy.
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
In this study, effective stress including the loadingdirectional parameter (a1) is assumed to be in the following form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 3n ¼ ðr22 r33 Þ2 þ 2a66 ðs212 þ s213 Þ þ 2a44 s223 þ a21 r211 r 2 þ a1 ðr11 þ r22 þ r33 Þ: ð5Þ In the case of plane stress state, the effective stress can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ¼ r ðr þ 2a66 s212 Þ þ a21 r211 þ a1 ðr11 þ r22 Þ 2 22 ~eff þ a1 ðr11 þ r22 Þ: r ð6Þ The r11 term in the square root is introduced to satisfy the positive-definite condition of effective stress. It is noteworthy that the following condition is assumed in this model in view of the result that plastic deformation in the fiber direction is negligible within the realistic strain ranges of fiber composites pffiffiffiffiffiffi a1 1; a66 : ð7Þ
depij ¼
o r p de ; orij
ð8Þ
where ep is the effective plastic strain. Using Eqs. (5), (6) and (8), 9 8 8 p 9 > a21 r11 þ a > 1 de > > 11 > > > r~eff > > > > > > > > p > > > > = < de = < 3r22 þ a > 1 22 2~ reff dep ¼ ð9Þ p > > > > 3r22 de > > > > 33 > > > > þ a 1 > > > > 2~reff > > ; : > > ; : dcp12 12 3a66 r~seff is obtained. The effective stress–strain relation is approximated by the power law in reference to Sun and Chen [1] ep ¼ A rn :
ð10Þ
Using Eqs. (6), (9) and (10), the following incremental constitutive equation is obtainable under the plane stress condition
2 2 2 a1 r11 8 p 9 þ a 1 6 r~eff > = < de11 > 6 p n1 6 r 6 de22 ¼ An > 6 ; : p > 4 dc12 sym
a21 r11 ~eff r
þ a1 3r22 2~ reff
3r22 2~ reff
þ a1
and (11). It is necessary to determine two parameters (a1 and a66) in this model. 3. Application to off-axis tension/compression test The effective stress-effective plastic strain relation is determined using tension/compression tests of off-axis specimens in reference to Sun and Chen [1]. Let the (x–y) coordinate and the (1–2) coordinate, respectively, represent the global and the local (material principal) axes. Under uniaxial loading, the stresses in the local coordinate are expressed as: r11 ¼ rx cos2 h r22 ¼ rx sin2 h
þ a1
2
ð12Þ
s12 ¼ rx sin h cos h; where h is an off-axis angle. The plastic strain in the global coordinate is expressible as: depx ¼ dep11 cos2 h þ dep22 sin2 h dcp12 sin h cos h depy ¼ dep11 sin2 h þ dep22 cos2 h þ dcp12 sin h cos h
The incremental plastic strain is expressed as
113
depz
¼
ð13Þ
dep33
Substituting Eq. (9) in Eq. (13) with Eqs. (6) and (12) yields depx ¼ ðhðhÞ þ a1 Þdep ! 3 3a66 þ a21 sin2 h cos2 h p 2 þ a1 dep dey ¼ hðhÞ 3 sin2 h þ a1 dep depz ¼ 2 hðhÞ
ð14Þ
where
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 32 sin4 h þ 3a66 sin2 h cos2 h þ a21 cos4 h ðrx P 0Þ; hðhÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 3 sin4 h þ 3a66 sin2 h cos2 h þ a2 cos4 h ðrx 6 0Þ: 1 2 ð15Þ
From Eqs. (6), (12) and (14), the effective stress and effective plastic strain under uniaxial loading are
12 3a66 r~seff 12 3a66 r~seff
a21 r11 ~eff r
3r22 2~ reff s2
9a266 r~212
3 9 þ a1 78 dr11 > > = < 7 7 dr 7 þ a1 7> 22 >: 5: ds12 ;
ð11Þ
eff
Once this effective stress-effective plastic strain relation is obtained, one can predict nonlinear mechanical behavior under various monotonic loading conditions using Eqs. (2)
¼ ðhðhÞ þ a1 Þrx r epx ep ¼ hðhÞ þ a1
ð16Þ
114
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
Therefore, the effective stress and effective plastic strain can be expressed simply in terms of experimental values in off-axis tests. Experimental results of off-axis tension/compression tests are convertible to effective stress-effective plastic strain relation using Eqs. (10), (16) and epx ¼ ex
rx : Ex
ð17Þ
The effective stress-effective plastic strain relation depends on the choice of a66 and a1. The values of two parameters should be selected so that all curves derived from different off-axis specimens collapse into a single curve. Using Eq. (14), the in-plane and out-of-plane plastic Poisson’s ratio are 3 depy 3a66 þ a21 sin2 h cos2 h þ a1 hðhÞ p 2 mxy ¼ p ¼ hðhÞðhðhÞ þ a1 Þ dex ð18Þ 2 p de 3 sin h þ a1 hðhÞ : mpxz ¼ zp ¼ dex 2 hðhÞ2 þ a1 hðhÞ Eq. (18) predicts different plastic Poisson’s ratios resulting from compression and tension. The present model is reducible to the expression in Sun and Chen [1] by setting a1 = 0. 4. Experimental procedure This study used a T800H/3633 carbon/epoxy system supplied by Toray Co. Ltd. Off-axis tension and compression tests were performed, respectively, using [0]20 and [0]40 unidirectional laminates (Fiber volume fractions are almost identical between the two, Vf = 57%). The respective resultant average thicknesses were 2.8 mm and 5.7 mm. All tests were conducted using a mechanical testing machine (4482; Instron Corp.). For compression tests, 50 mm long and 6 mm wide offaxis specimens were prepared by cutting the specimens with
φ:oblique angle
40
70 120
50
6
Fig. 2. Apparatus of off-axis compression test.
off-axis angles of 15, 30, 45, 60, and 90 (one specimen for each angle). Off-axis specimens were bonded to potting-type fixtures using epoxy adhesives. The specimen configuration is summarized in Fig. 1. Fig. 2 shows that compression loading was applied to the specimens using a supporting guide. Back-to-back strain gauges were attached to the specimens in longitudinal, transverse, and out-of-plane directions; the applied crosshead speed was 0.5 mm/min. Off-axis tensile specimens with 120 mm length and 10 mm width were prepared by cutting the specimens with off-axis angles of 15, 30, 45, 60, and 90(one specimen for each angle). Oblique end tabs were adopted for off-axis tension specimens in reference to Sun and Chung [27] to measure accurate nonlinear stress–strain curves using small specimens. The specimen configuration of the tension test is summarized in Fig. 1. The oblique end tabs were bonded to the specimens using epoxy adhesives at the oblique angles (see Fig. 1) to the loading axis. The oblique angles were calculated in accordance with Ref. [27]. Back-to-back strain gauges were attached to specimens in longitudinal and transverse directions; the applied crosshead speed was 1.0 mm/min. 5. Results and discussion
unit:mm 10 Fig. 1. Specimen configuration: (left) compression test, (right) tension test.
Experimental longitudinal stress–strain curves of offaxis compression tests are presented in Fig. 3. Plotted strains are average values of back-to-back gauges. Table 1 shows the initial longitudinal Young’s moduli, in-plane Poisson’s ratios, and out-of-plane Poisson’s ratios, which were evaluated using the experimental results in the strain
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
500
T800H/3633 15oo-experiment 15o-analysis 30o-experiment 30 -analysis
400
effective stress [MPa]
compressive stress [MPa]
500
300 200 100 0
ε p = 3.14E-13σ 4.19
400 300 T800H/3633 15o 30o o 45o 60 90o fitted curve
200 100 0
0
0.01 0.02 compressive strain
0.03
0
0.005 0.01 0.015 effective plastic strain
0.02
Fig. 4. Effective stress-effective plastic strain curve obtained from off-axis compression tests.
250 compressive stress [MPa]
115
200 Table 2 Plastic parameters obtained from compression test results
150 T800H/3633 45oo-experiment 45o-analysis 60o-experiment 60o-analysis 90o-experiment 90 -analysis
100 50
a66
a1
A (MPan)
n
2.7
0.09
3.14 · 1013
4.19
experimental plastic strain curves (e.g., epx vs. epy ) exhibited almost linear relationship, the experimental plastic Poisson’s ratios were determined by linear fitting of the plastic
0 0.03
Fig. 3. Longitudinal stress–strain curves of off-axis compression tests.
Table 1 Evaluated elastic properties based on compression and tension tests Angle
15 30 45 60 90
Compression
Tension
Ex (GPa)
mxy
mxz
Ex (GPa)
mxy
mxz
55.1 24.6 14.0 11.6 9.26
0.364 0.343 0.286 0.139 0.016
0.285 0.360 0.393 0.454 0.536
50.9 21.1 13.2 10.5 9.33
0.381 0.352 0.273 0.175 0.024
N/A N/A N/A N/A N/A
range between 0.1% and 0.2%. Using the compression test results, the effective stress-effective plastic strain relation was obtained as shown in Fig. 4. The parameters (a66 and a1) were parametrically changed, and then, the optimum values were determined by minimizing the leastsquares error of the fitted curve in this study. It is noteworthy that some sophisticated techniques may be necessary to find the global optimum combination in the cases other than this experimental result. The obtained parameters are summarized in Table 2. The analytical stress–strain curves using the obtained parameters are compared with experimental results (see Fig. 3). Good correlation between experimental and analytical curves is apparent for all offaxis compression tests. The predicted in-plane and outof-plane plastic Poisson’s ratios are plotted in Fig. 5, which shows a comparison to experimental results. Because the
in-plane plastic Poisson's ratio
0.01 0.02 compressive strain
1
0.5 compression experiment analysis tension experiment analysis
0 0
out-of-plane plastic Poisson's ratio
0
30 60 off-axis angle [degree]
90
1 compression experiment analysis tension analysis
0.8 0.6 0.4 0.2 0 0
30 60 off-axis angle [degree]
90
Fig. 5. Comparison of measured and predicted plastic Poisson’s ratio: (upper) in-plane, (lower) out-of-plane.
116
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
250 T800H/3633 15[deg]
longitudinal stress [MPa]
longitudinal stress [MPa]
500 400 300 compression experiment analysis tension experiment analysis
200 100 0
200 150 compression experiment analysis tension experiment analysis
100 50 0
0
0.005 0.01 longitudinal strain (a) θ = 15˚
0.015
0
300
0.01 0.02 longitudinal strain (c) θ = 45˚
0.03
200 T800H/3633 30[deg]
longitudinal stress [MPa]
longitudinal stress [MPa]
T800H/3633 45[deg]
200
compression experiment analysis tension experiment analysis
100
0
T800H/3633 60[deg]
150
100 compression experiment analysis tension experiment analysis
50
0 0
0.01 0.02 longitudinal strain (b) θ = 30˚
0.03
0
0.01 0.02 longitudinal strain (d) θ = 60˚
0.03
longitudinal stress [MPa]
200 T800H/3633 90[deg]
150
100 compression experiment analysis tension experiment analysis
50
0 0
0.01 0.02 longitudinal strain (e) θ = 90˚
0.03
Fig. 6. Comparison of experimental and predicted stress–strain curves under compression and tension loading.
strain curves. Fig. 5 also indicates the validity of the present model for characterizing nonlinear behavior in carbon/ epoxy unidirectional laminates under various loading conditions. A comparison of experimental longitudinal stress–strain curves between off-axis tension and compression tests is presented in Fig. 6. The initial longitudinal Young’s moduli and in-plane Poisson’s ratios obtained in the off-axis tension test are presented in Table 1 in comparison with
compression test results. Apparently, the initial Young’s moduli are comparable in tension and compression tests, but stress–strain curves are different in nonlinear portions; tensile loading induces considerable plastic deformation. In addition, specimens with smaller off-axis angle exhibit larger differences between tension and compression curves. Longitudinal stress–strain curves of off-axis tension specimens are predicted using the obtained parameters based on compression test results, as shown in Fig. 6. The
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
predicted in-plane plastic Poisson’s ratios are also compared with experimental results of tension tests (see Fig. 5). The predicted results agree well with experiments. The proposed model can predict differences of nonlinear behavior depending on loading conditions (i.e., tension or compression). The present model demonstrates the potential to predict nonlinear behavior of fiber-reinforced polymeric composites with loading-directional dependence. It is noteworthy that the residual stress in the matrix is one reason for different nonlinear behavior in tension and compression; this might be valid because smaller off-axis specimens exhibit greater difference in nonlinear behaviors during tension and compression tests. Therefore, one might infer that a plastic potential including residual thermal stress is suitable for description of the loading-directional dependence. For example, a plastic potential of 2
2 2 2f ¼ a11 ðr11 þ rR 11 Þ þ a22 r22 þ 2a66 r12 ;
ð19Þ
is definable under a plane stress condition, where the rR 11 parameter expresses the residual thermal stress effect. However, it is rather difficult to obtain the unique effective stress-effective plastic strain relation from a simple experimental program in this case, as opposed to the case presented in this study. In addition, the number of parameters to be determined is three even when a22 is set as 1. Although this plastic potential might be investigated further, the present two-parameter model is rather simple and useful for description of nonlinear behaviors including loading-directional dependence. In this article, off-axis tension/compression tests were performed to establish a directional dependent constitutive model for polymeric composites. Within the cases of this study, nonlinear stress–strain responses are well characterized using the present model. Although further verifications under various loading conditions are necessary for the present simple model (e.g., different material systems, multi-directionally reinforced laminates under various loading), applicability of the present simple model to macroscopic nonlinear behavior of polymeric composites was verified using the off-axis tension and compression tests. The present model can be extended to a rate-dependent constitutive model in the same manner as Sun and coauthors [19,20]. For example, in Eq. (10), it is assumed that A is a power law function of the effective plastic strain rate (e_ p ) as m A ¼ vðe_ p Þ ; ð20Þ where v and m are fitted parameters. It might be also interesting to investigate the validity of this direction-dependent and rate-dependent constitutive model for polymeric composites. 6. Conclusions A simple constitutive model incorporating loadingdirectional dependence on plastic flow of fiber-reinforced composites is presented. Sun and Chen’s one-parameter
117
plasticity model was extended to a two-parameter model by incorporating normal stress terms into the stress potential. Off-axis specimens of unidirectional carbon/epoxy composites were tested in compression and tension. Using compression test results, a constitutive relation for nonlinear mechanical behavior of fiber-reinforced composites was established. The obtained constitutive model can predict different nonlinear behaviors for compression and tension; good correlation between the predictions and experiments was observed. The present simple constitutive model turned out to have the potential to predict nonlinear behavior of fiber-reinforced composites with loading-directional dependence. References [1] Sun CT, Chen JL. A simple flow rule for characterizing nonlinear behavior of fiber composites. J Compos Mater 1989;23:1009–20. [2] Aboudi J. Closed form constitutive equations for metal matrix composites. Int J Eng Sci 1987;25:1229–40. [3] Sun CT, Chen JL. A micromechanical model for plastic behavior of fibrous composites. Compos Sci Technol 1991;40:115–29. [4] Aboudi J. Micromechanical analysis of composites by the method of cells-update. Appl Mech Rev 1996;49:S83–91. [5] Pindera M-J, Bednarcyk BA. An efficient implementation of the generalized method of cells for unidirectional, multi-phased composites with complex microstructures. Compos part B 1999;30:87–105. [6] Hahn HT, Tsai SW. Nonlinear elastic behavior of unidirectional composite laminae. J Compos Mater 1973;7:102–18. [7] Mulhern JR, Rogers TG, Spencer AJM. A continuum model for a fiber reinforced plastic material. Proc Royal Soc Lond 1967;A301:473–92. [8] Spencer AJM. Plasticity theory for fibre-reinforced composites. J Eng Math 1992;26:107–18. [9] Robinson DN, Duffy SF. Continuum deformation theory for hightemperature metallic composites. J Eng Mech 1990;116:832–44. [10] Hansen AC, Blackketter DM, Walrath DE. An invariant-based flow rule for anisotropic plasticity applied to composite materials. J Appl Mech 1991;58:881–8. [11] Schmidt RJ, Wang D-Q, Hansen AC. Plasticity model for transversely isotropic materials. J Eng Mech 1993;119:748–66. [12] Hill R. The mathematical theory of plasticity. London: Oxford University Press; 1950. [13] Chen JL, Sun CT. A plastic potential function suitable for anisotropic fiber composites. J Compos Mater 1993;27:1379–90. [14] Xie M, Adams DF. A plasticity model for unidirectional composite materials and its applications in modeling composites testing. Compos Sci Technol 1995;54:11–21. [15] Ogi K, Takeda N. Effects of moisture content on nonlinear deformation behavior of CF/epoxy composites. J Compos Mater 1997;31:530–51. [16] Ogihara S, Kobayashi S, Reifsnider K. Characterization of nonlinear behavior of carbon/epoxy unidirectional and angle-ply laminates. Adv Compos Mater 2003;11:239–54. [17] Gates TS, Sun CT. Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites. AIAA J 1991;29:457–63. [18] Weeks CA, Sun CT. Modeling non-linear rate-dependent behavior in fiber-reinforced composites. Compos Sci Technol 1998;58:603–11. [19] Tsai J, Sun CT. Constitutive model for high strain rate response of polymeric composites. Compos Sci Technol 2002;62:1289–97. [20] Bing Q, Sun CT. Modeling and testing strain rate-dependent compressive strength of carbon/epoxy composites. Compos Sci Technol 2005;65:2481–91. [21] Drucker DC, Prager W. Soil mechanics and plastic analysis or limit design. Quart Appl Math 1952;10:157–65.
118
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
[22] Hoffman O. The brittle strength of composite materials. J Compos Mater 1967;1:200–6. [23] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Compos Mater 1971;5:58–80. [24] Voyiadjis GZ, Thiagarajan G. An anisotropic yield surface model for directionally reinforced metal-matrix composites. Int J Plastic 1995;11:867–94.
[25] Voyiadjis GZ, Thiagarajan G. A cyclic anisotropic-plasticity model for metal matrix composites. Int J Plastic 1996;12:69–91. [26] Ishikawa T, Matsushima M, Hayashi Y. Hardening non-linear behaviour in longitudinal tension of unidirectional carbon composites. J Mater Sci 1985;20:4075–83. [27] Sun CT, Chung I. An oblique end-tab design for testing off-axis composite specimens. Composites 1993;24:619–23.
Simple constitutive model for nonlinear response of fiber-reinforced composites with loading-directional dependence Tomohiro Yokozeki a
a,*
, Shinji Ogihara b, Shunsuke Yoshida b, Toshio Ogasawara
a
Advanced Composite Technology Center, Institute of Aerospace Technology (IAT), Japan Aerospace Exploration Agency (JAXA), 6-13-1 Osawa, Mitaka, Tokyo, 181-0015, Japan b Department of Mechanical Engineering, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan Received 5 January 2006; received in revised form 23 March 2006; accepted 27 March 2006 Available online 24 May 2006
Abstract Polymeric composites generally exhibit different nonlinear response in tension and compression tests. In this study, Sun and Chen’s one-parameter plasticity model [Sun C.T. and Chen J.L., 1989. A simple flow rule for characterizing nonlinear behavior of fiber composites, J Compos Mater 29, 457–463] is extended to incorporate the loading-directional dependence on the plastic flow in composite laminates. Off-axis specimens of unidirectional carbon/epoxy composites were tested in compression and tension. Based on experimental results of compression tests, a simple two-parameter constitutive model for characterizing nonlinear mechanical behavior of fiber-reinforced composites was established. Finally, comparison of model predictions with experimental results of off-axis tension tests indicated a good correlation between the two. The developed simple plasticity model can explain the difference of nonlinear response in tension and compression tests. 2006 Elsevier Ltd. All rights reserved. Keywords: A. Polymer-matrix composites (PMCs); B. Plastic deformation; B. Non-linear behaviour; Loading-directional dependence
1. Introduction Interest in advanced composite materials is growing for use in structures for aerospace, maritime and civil applications. Predicting the mechanical response and failure behavior of fiber-reinforced composite is considerably important for efficient design of composite structures. Detailed description of macroscopic behavior of lamina and laminate is necessary to enhance the applicability of such composite structures. Fiber-reinforced composites are widely recognized as exhibiting nonlinear mechanical response under off-axis loading. Nonlinear behaviors of polymeric composites include plastic deformation, microscopic failure (e.g., matrix cracking, fiber–matrix debonding), geometric nonlinearity (e.g., fiber rotation), and others. Therefore, micro*
Corresponding author. Tel.: +81 422 40 3384; fax: +81 422 40 3549. E-mail address: [email protected] (T. Yokozeki).
0266-3538/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.03.024
scopic modeling of fiber-reinforced composites using fiber and matrix properties has been studied extensively [2–5]. These micromechanical models are then used to evaluate the effective mechanical response of a lamina and laminate. In addition to the request for avoidance of time-consuming micromechanical calculations, the constituent fiber and matrix properties of recent high-performance composites produced from prepregs are not often disclosed. Therefore, macroscopic approaches using composite properties are also necessary to describe the nonlinear behavior of composite laminates. Macroscopic characterization is preferable for incorporation of the constitutive relations into structural analytical tools (e.g., finite element analysis). Many macroscopic approaches have been developed for characterizing nonlinear behaviors of fiber-reinforced composites. Hahn and Tsai [6] predicted nonlinear stress–strain responses of unidirectional laminates under off-axis tensile loading using nonlinear elastic constitutive equations based on higher order energy functions. Spencer et al. [7,8]
112
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
developed a macro-mechanical plasticity model for a transversely isotropic material in which the yield surface is described by two stress invariants using two parameters. Robinson and Duffy [9] expanded the previous yield surface [7] to include a third invariant. Hansen et al. [10,11] proposed an invariant-based anisotropic plasticity model using a scalar hardening parameter instead of determining effective stress–strain relations. It is difficult to obtain a unique effective stress–strain relation for fiber-reinforced composites that can fit experimental results obtained under various loading conditions, as noted by Hansen et al. [10]. Sun and Chen [1] proposed a one-parameter plasticity model using a Hill-type [12] quadratic plastic potential by assuming a plane stress state and no plastic deformation in the fiber direction. They showed that effective stress-effective plastic strain curves were obtainable by fitting the experimental results of offaxis tension tests. They also showed that nonlinear responses of unidirectional fiber-reinforced composites under various loading conditions were well characterized. Chen and Sun [13] proposed a 3-D plastic potential that is suitable for modeling plastic flow in anisotropic fiber composites. Xie and Adams [14] developed a 3-D plasticity model similar to that of Sun and Chen [1]. Ogi and Takeda [15] proposed a modified one-parameter model in which an anisotropy parameter changes concomitant with plastic deformation. Ogihara et al. [16] applied a oneparameter plasticity model to characterize the nonlinear behavior of angle-ply laminates. The family of one-parameter plasticity models was extended to describe the ratedependent nonlinear behavior of polymeric composites [17–20]. One-parameter models are attractive for characterizing nonlinear behaviors in various polymeric composites because of their simplicity and consistency with experimental results. In general, polymeric composites exhibit different nonlinear behavior in compression and tension tests. Tsai and Sun [19] indicated that tension and compression stress–strain curves of off-axis specimens appear to be different, specifically in nonlinear portions of the curves. This phenomenon is attributed to: (1) the residual stress in the matrix because the coefficients of thermal expansion are different between the fiber and matrix; and (2) the matrix itself exhibits pressure-dependent behavior. Quadratic plastic potentials used in the above-mentioned models (e.g., Sun and Chen [1]) cannot explain the different nonlinear behaviors in compression and tension tests. Therefore, this loading-directional dependence on the plastic deformation should be incorporated for characterizing the nonlinear behavior of polymeric composites under various loading conditions. In the case of isotropic materials, hydrostatic pressure terms are incorporated in the stress potential (e.g., Drucker–Prager model [21]) to represent pressuredependent yielding behavior. Some anisotropic yield criteria can account for the different strengths that are exhibited by composites in compression and tension, such as those presented in Hoffman [22], and Tsai and Wu [23]. Voyiadjis
and Thiagarajan [24,25] proposed a six-parameter anisotropic yield surface to formulate plasticity in anisotropic materials in which pressure-dependent behavior is included. A simple but accurate model for nonlinear behaviors of fiber-reinforced composites under general loading conditions must be established. This study develops a simple constitutive model that incorporates the above-mentioned loading-directional dependence on plastic flow in composite laminates under monotonic loading. It is developed by extending Sun and Chen’s one-parameter plasticity model [1]. Off-axis specimens of unidirectional carbon/epoxy composites are tested in compression and tension. Based on the compressive experimental results, a simple constitutive model for characterizing nonlinear mechanical behavior of fiber-reinforced composites was established using two parameters: one is the anisotropic parameter and the other reflects the loading-directional effect. Finally, the predictive capability of the present model is verified through comparison with experimental results of off-axis tension tests. 2. Constitutive model Mechanical behavior of an orthotropic lamina of fiberreinforced composites is considered. It is assumed that total strain can be decomposed into elastic and plastic strains within infinitesimal strain conditions, as deij ¼ deeij þ depij ;
ð1Þ
where superscripts e and p, respectively, denote elastic and plastic. The incremental form of the elastic constitutive equation under a plane stress state is expressed as 9 8 e 9 2 38 1=E1 m12 =E1 0 > > = = < de11 > < dr11 > 6 7 dee22 ¼ 4 ð2Þ 1=E2 0 5 dr22 : > > > ; ; : e > : dc12 sym 1=G12 ds12 Note that the 1, 2, 3-directions are defined, respectively, as the fiber, in-plane transverse to the fiber, out-of-plane transverse to the fiber direction. Although the linear stress–strain relation is assumed in Eq. (2), one can include nonlinear elastic properties (e.g., nonlinear shear response [6], nonlinear response in fiber direction [26]). To model the plastic constitutive relation, Sun and Chen [1] defined the effective stress as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ~¼ r ðr þ 2a66 s212 Þ; ð3Þ 2 22 where a66 is an anisotropic parameter. This is a plane stress case of the following 3-D effective stress of a transversely isotropic material rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 3n 2 3D ~ ¼ ðr22 r33 Þ þ 2a66 ðs212 þ s213 Þ þ 2a44 s223 : ð4Þ r 2 In Weeks and Sun [17], a44 is set as 2 by assuming the descriptive equivalency between plastic potential and strain energy.
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
In this study, effective stress including the loadingdirectional parameter (a1) is assumed to be in the following form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o 3n ¼ ðr22 r33 Þ2 þ 2a66 ðs212 þ s213 Þ þ 2a44 s223 þ a21 r211 r 2 þ a1 ðr11 þ r22 þ r33 Þ: ð5Þ In the case of plane stress state, the effective stress can be expressed as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 ¼ r ðr þ 2a66 s212 Þ þ a21 r211 þ a1 ðr11 þ r22 Þ 2 22 ~eff þ a1 ðr11 þ r22 Þ: r ð6Þ The r11 term in the square root is introduced to satisfy the positive-definite condition of effective stress. It is noteworthy that the following condition is assumed in this model in view of the result that plastic deformation in the fiber direction is negligible within the realistic strain ranges of fiber composites pffiffiffiffiffiffi a1 1; a66 : ð7Þ
depij ¼
o r p de ; orij
ð8Þ
where ep is the effective plastic strain. Using Eqs. (5), (6) and (8), 9 8 8 p 9 > a21 r11 þ a > 1 de > > 11 > > > r~eff > > > > > > > > p > > > > = < de = < 3r22 þ a > 1 22 2~ reff dep ¼ ð9Þ p > > > > 3r22 de > > > > 33 > > > > þ a 1 > > > > 2~reff > > ; : > > ; : dcp12 12 3a66 r~seff is obtained. The effective stress–strain relation is approximated by the power law in reference to Sun and Chen [1] ep ¼ A rn :
ð10Þ
Using Eqs. (6), (9) and (10), the following incremental constitutive equation is obtainable under the plane stress condition
2 2 2 a1 r11 8 p 9 þ a 1 6 r~eff > = < de11 > 6 p n1 6 r 6 de22 ¼ An > 6 ; : p > 4 dc12 sym
a21 r11 ~eff r
þ a1 3r22 2~ reff
3r22 2~ reff
þ a1
and (11). It is necessary to determine two parameters (a1 and a66) in this model. 3. Application to off-axis tension/compression test The effective stress-effective plastic strain relation is determined using tension/compression tests of off-axis specimens in reference to Sun and Chen [1]. Let the (x–y) coordinate and the (1–2) coordinate, respectively, represent the global and the local (material principal) axes. Under uniaxial loading, the stresses in the local coordinate are expressed as: r11 ¼ rx cos2 h r22 ¼ rx sin2 h
þ a1
2
ð12Þ
s12 ¼ rx sin h cos h; where h is an off-axis angle. The plastic strain in the global coordinate is expressible as: depx ¼ dep11 cos2 h þ dep22 sin2 h dcp12 sin h cos h depy ¼ dep11 sin2 h þ dep22 cos2 h þ dcp12 sin h cos h
The incremental plastic strain is expressed as
113
depz
¼
ð13Þ
dep33
Substituting Eq. (9) in Eq. (13) with Eqs. (6) and (12) yields depx ¼ ðhðhÞ þ a1 Þdep ! 3 3a66 þ a21 sin2 h cos2 h p 2 þ a1 dep dey ¼ hðhÞ 3 sin2 h þ a1 dep depz ¼ 2 hðhÞ
ð14Þ
where
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 32 sin4 h þ 3a66 sin2 h cos2 h þ a21 cos4 h ðrx P 0Þ; hðhÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 3 sin4 h þ 3a66 sin2 h cos2 h þ a2 cos4 h ðrx 6 0Þ: 1 2 ð15Þ
From Eqs. (6), (12) and (14), the effective stress and effective plastic strain under uniaxial loading are
12 3a66 r~seff 12 3a66 r~seff
a21 r11 ~eff r
3r22 2~ reff s2
9a266 r~212
3 9 þ a1 78 dr11 > > = < 7 7 dr 7 þ a1 7> 22 >: 5: ds12 ;
ð11Þ
eff
Once this effective stress-effective plastic strain relation is obtained, one can predict nonlinear mechanical behavior under various monotonic loading conditions using Eqs. (2)
¼ ðhðhÞ þ a1 Þrx r epx ep ¼ hðhÞ þ a1
ð16Þ
114
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
Therefore, the effective stress and effective plastic strain can be expressed simply in terms of experimental values in off-axis tests. Experimental results of off-axis tension/compression tests are convertible to effective stress-effective plastic strain relation using Eqs. (10), (16) and epx ¼ ex
rx : Ex
ð17Þ
The effective stress-effective plastic strain relation depends on the choice of a66 and a1. The values of two parameters should be selected so that all curves derived from different off-axis specimens collapse into a single curve. Using Eq. (14), the in-plane and out-of-plane plastic Poisson’s ratio are 3 depy 3a66 þ a21 sin2 h cos2 h þ a1 hðhÞ p 2 mxy ¼ p ¼ hðhÞðhðhÞ þ a1 Þ dex ð18Þ 2 p de 3 sin h þ a1 hðhÞ : mpxz ¼ zp ¼ dex 2 hðhÞ2 þ a1 hðhÞ Eq. (18) predicts different plastic Poisson’s ratios resulting from compression and tension. The present model is reducible to the expression in Sun and Chen [1] by setting a1 = 0. 4. Experimental procedure This study used a T800H/3633 carbon/epoxy system supplied by Toray Co. Ltd. Off-axis tension and compression tests were performed, respectively, using [0]20 and [0]40 unidirectional laminates (Fiber volume fractions are almost identical between the two, Vf = 57%). The respective resultant average thicknesses were 2.8 mm and 5.7 mm. All tests were conducted using a mechanical testing machine (4482; Instron Corp.). For compression tests, 50 mm long and 6 mm wide offaxis specimens were prepared by cutting the specimens with
φ:oblique angle
40
70 120
50
6
Fig. 2. Apparatus of off-axis compression test.
off-axis angles of 15, 30, 45, 60, and 90 (one specimen for each angle). Off-axis specimens were bonded to potting-type fixtures using epoxy adhesives. The specimen configuration is summarized in Fig. 1. Fig. 2 shows that compression loading was applied to the specimens using a supporting guide. Back-to-back strain gauges were attached to the specimens in longitudinal, transverse, and out-of-plane directions; the applied crosshead speed was 0.5 mm/min. Off-axis tensile specimens with 120 mm length and 10 mm width were prepared by cutting the specimens with off-axis angles of 15, 30, 45, 60, and 90(one specimen for each angle). Oblique end tabs were adopted for off-axis tension specimens in reference to Sun and Chung [27] to measure accurate nonlinear stress–strain curves using small specimens. The specimen configuration of the tension test is summarized in Fig. 1. The oblique end tabs were bonded to the specimens using epoxy adhesives at the oblique angles (see Fig. 1) to the loading axis. The oblique angles were calculated in accordance with Ref. [27]. Back-to-back strain gauges were attached to specimens in longitudinal and transverse directions; the applied crosshead speed was 1.0 mm/min. 5. Results and discussion
unit:mm 10 Fig. 1. Specimen configuration: (left) compression test, (right) tension test.
Experimental longitudinal stress–strain curves of offaxis compression tests are presented in Fig. 3. Plotted strains are average values of back-to-back gauges. Table 1 shows the initial longitudinal Young’s moduli, in-plane Poisson’s ratios, and out-of-plane Poisson’s ratios, which were evaluated using the experimental results in the strain
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
500
T800H/3633 15oo-experiment 15o-analysis 30o-experiment 30 -analysis
400
effective stress [MPa]
compressive stress [MPa]
500
300 200 100 0
ε p = 3.14E-13σ 4.19
400 300 T800H/3633 15o 30o o 45o 60 90o fitted curve
200 100 0
0
0.01 0.02 compressive strain
0.03
0
0.005 0.01 0.015 effective plastic strain
0.02
Fig. 4. Effective stress-effective plastic strain curve obtained from off-axis compression tests.
250 compressive stress [MPa]
115
200 Table 2 Plastic parameters obtained from compression test results
150 T800H/3633 45oo-experiment 45o-analysis 60o-experiment 60o-analysis 90o-experiment 90 -analysis
100 50
a66
a1
A (MPan)
n
2.7
0.09
3.14 · 1013
4.19
experimental plastic strain curves (e.g., epx vs. epy ) exhibited almost linear relationship, the experimental plastic Poisson’s ratios were determined by linear fitting of the plastic
0 0.03
Fig. 3. Longitudinal stress–strain curves of off-axis compression tests.
Table 1 Evaluated elastic properties based on compression and tension tests Angle
15 30 45 60 90
Compression
Tension
Ex (GPa)
mxy
mxz
Ex (GPa)
mxy
mxz
55.1 24.6 14.0 11.6 9.26
0.364 0.343 0.286 0.139 0.016
0.285 0.360 0.393 0.454 0.536
50.9 21.1 13.2 10.5 9.33
0.381 0.352 0.273 0.175 0.024
N/A N/A N/A N/A N/A
range between 0.1% and 0.2%. Using the compression test results, the effective stress-effective plastic strain relation was obtained as shown in Fig. 4. The parameters (a66 and a1) were parametrically changed, and then, the optimum values were determined by minimizing the leastsquares error of the fitted curve in this study. It is noteworthy that some sophisticated techniques may be necessary to find the global optimum combination in the cases other than this experimental result. The obtained parameters are summarized in Table 2. The analytical stress–strain curves using the obtained parameters are compared with experimental results (see Fig. 3). Good correlation between experimental and analytical curves is apparent for all offaxis compression tests. The predicted in-plane and outof-plane plastic Poisson’s ratios are plotted in Fig. 5, which shows a comparison to experimental results. Because the
in-plane plastic Poisson's ratio
0.01 0.02 compressive strain
1
0.5 compression experiment analysis tension experiment analysis
0 0
out-of-plane plastic Poisson's ratio
0
30 60 off-axis angle [degree]
90
1 compression experiment analysis tension analysis
0.8 0.6 0.4 0.2 0 0
30 60 off-axis angle [degree]
90
Fig. 5. Comparison of measured and predicted plastic Poisson’s ratio: (upper) in-plane, (lower) out-of-plane.
116
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
250 T800H/3633 15[deg]
longitudinal stress [MPa]
longitudinal stress [MPa]
500 400 300 compression experiment analysis tension experiment analysis
200 100 0
200 150 compression experiment analysis tension experiment analysis
100 50 0
0
0.005 0.01 longitudinal strain (a) θ = 15˚
0.015
0
300
0.01 0.02 longitudinal strain (c) θ = 45˚
0.03
200 T800H/3633 30[deg]
longitudinal stress [MPa]
longitudinal stress [MPa]
T800H/3633 45[deg]
200
compression experiment analysis tension experiment analysis
100
0
T800H/3633 60[deg]
150
100 compression experiment analysis tension experiment analysis
50
0 0
0.01 0.02 longitudinal strain (b) θ = 30˚
0.03
0
0.01 0.02 longitudinal strain (d) θ = 60˚
0.03
longitudinal stress [MPa]
200 T800H/3633 90[deg]
150
100 compression experiment analysis tension experiment analysis
50
0 0
0.01 0.02 longitudinal strain (e) θ = 90˚
0.03
Fig. 6. Comparison of experimental and predicted stress–strain curves under compression and tension loading.
strain curves. Fig. 5 also indicates the validity of the present model for characterizing nonlinear behavior in carbon/ epoxy unidirectional laminates under various loading conditions. A comparison of experimental longitudinal stress–strain curves between off-axis tension and compression tests is presented in Fig. 6. The initial longitudinal Young’s moduli and in-plane Poisson’s ratios obtained in the off-axis tension test are presented in Table 1 in comparison with
compression test results. Apparently, the initial Young’s moduli are comparable in tension and compression tests, but stress–strain curves are different in nonlinear portions; tensile loading induces considerable plastic deformation. In addition, specimens with smaller off-axis angle exhibit larger differences between tension and compression curves. Longitudinal stress–strain curves of off-axis tension specimens are predicted using the obtained parameters based on compression test results, as shown in Fig. 6. The
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
predicted in-plane plastic Poisson’s ratios are also compared with experimental results of tension tests (see Fig. 5). The predicted results agree well with experiments. The proposed model can predict differences of nonlinear behavior depending on loading conditions (i.e., tension or compression). The present model demonstrates the potential to predict nonlinear behavior of fiber-reinforced polymeric composites with loading-directional dependence. It is noteworthy that the residual stress in the matrix is one reason for different nonlinear behavior in tension and compression; this might be valid because smaller off-axis specimens exhibit greater difference in nonlinear behaviors during tension and compression tests. Therefore, one might infer that a plastic potential including residual thermal stress is suitable for description of the loading-directional dependence. For example, a plastic potential of 2
2 2 2f ¼ a11 ðr11 þ rR 11 Þ þ a22 r22 þ 2a66 r12 ;
ð19Þ
is definable under a plane stress condition, where the rR 11 parameter expresses the residual thermal stress effect. However, it is rather difficult to obtain the unique effective stress-effective plastic strain relation from a simple experimental program in this case, as opposed to the case presented in this study. In addition, the number of parameters to be determined is three even when a22 is set as 1. Although this plastic potential might be investigated further, the present two-parameter model is rather simple and useful for description of nonlinear behaviors including loading-directional dependence. In this article, off-axis tension/compression tests were performed to establish a directional dependent constitutive model for polymeric composites. Within the cases of this study, nonlinear stress–strain responses are well characterized using the present model. Although further verifications under various loading conditions are necessary for the present simple model (e.g., different material systems, multi-directionally reinforced laminates under various loading), applicability of the present simple model to macroscopic nonlinear behavior of polymeric composites was verified using the off-axis tension and compression tests. The present model can be extended to a rate-dependent constitutive model in the same manner as Sun and coauthors [19,20]. For example, in Eq. (10), it is assumed that A is a power law function of the effective plastic strain rate (e_ p ) as m A ¼ vðe_ p Þ ; ð20Þ where v and m are fitted parameters. It might be also interesting to investigate the validity of this direction-dependent and rate-dependent constitutive model for polymeric composites. 6. Conclusions A simple constitutive model incorporating loadingdirectional dependence on plastic flow of fiber-reinforced composites is presented. Sun and Chen’s one-parameter
117
plasticity model was extended to a two-parameter model by incorporating normal stress terms into the stress potential. Off-axis specimens of unidirectional carbon/epoxy composites were tested in compression and tension. Using compression test results, a constitutive relation for nonlinear mechanical behavior of fiber-reinforced composites was established. The obtained constitutive model can predict different nonlinear behaviors for compression and tension; good correlation between the predictions and experiments was observed. The present simple constitutive model turned out to have the potential to predict nonlinear behavior of fiber-reinforced composites with loading-directional dependence. References [1] Sun CT, Chen JL. A simple flow rule for characterizing nonlinear behavior of fiber composites. J Compos Mater 1989;23:1009–20. [2] Aboudi J. Closed form constitutive equations for metal matrix composites. Int J Eng Sci 1987;25:1229–40. [3] Sun CT, Chen JL. A micromechanical model for plastic behavior of fibrous composites. Compos Sci Technol 1991;40:115–29. [4] Aboudi J. Micromechanical analysis of composites by the method of cells-update. Appl Mech Rev 1996;49:S83–91. [5] Pindera M-J, Bednarcyk BA. An efficient implementation of the generalized method of cells for unidirectional, multi-phased composites with complex microstructures. Compos part B 1999;30:87–105. [6] Hahn HT, Tsai SW. Nonlinear elastic behavior of unidirectional composite laminae. J Compos Mater 1973;7:102–18. [7] Mulhern JR, Rogers TG, Spencer AJM. A continuum model for a fiber reinforced plastic material. Proc Royal Soc Lond 1967;A301:473–92. [8] Spencer AJM. Plasticity theory for fibre-reinforced composites. J Eng Math 1992;26:107–18. [9] Robinson DN, Duffy SF. Continuum deformation theory for hightemperature metallic composites. J Eng Mech 1990;116:832–44. [10] Hansen AC, Blackketter DM, Walrath DE. An invariant-based flow rule for anisotropic plasticity applied to composite materials. J Appl Mech 1991;58:881–8. [11] Schmidt RJ, Wang D-Q, Hansen AC. Plasticity model for transversely isotropic materials. J Eng Mech 1993;119:748–66. [12] Hill R. The mathematical theory of plasticity. London: Oxford University Press; 1950. [13] Chen JL, Sun CT. A plastic potential function suitable for anisotropic fiber composites. J Compos Mater 1993;27:1379–90. [14] Xie M, Adams DF. A plasticity model for unidirectional composite materials and its applications in modeling composites testing. Compos Sci Technol 1995;54:11–21. [15] Ogi K, Takeda N. Effects of moisture content on nonlinear deformation behavior of CF/epoxy composites. J Compos Mater 1997;31:530–51. [16] Ogihara S, Kobayashi S, Reifsnider K. Characterization of nonlinear behavior of carbon/epoxy unidirectional and angle-ply laminates. Adv Compos Mater 2003;11:239–54. [17] Gates TS, Sun CT. Elastic/viscoplastic constitutive model for fiber reinforced thermoplastic composites. AIAA J 1991;29:457–63. [18] Weeks CA, Sun CT. Modeling non-linear rate-dependent behavior in fiber-reinforced composites. Compos Sci Technol 1998;58:603–11. [19] Tsai J, Sun CT. Constitutive model for high strain rate response of polymeric composites. Compos Sci Technol 2002;62:1289–97. [20] Bing Q, Sun CT. Modeling and testing strain rate-dependent compressive strength of carbon/epoxy composites. Compos Sci Technol 2005;65:2481–91. [21] Drucker DC, Prager W. Soil mechanics and plastic analysis or limit design. Quart Appl Math 1952;10:157–65.
118
T. Yokozeki et al. / Composites Science and Technology 67 (2007) 111–118
[22] Hoffman O. The brittle strength of composite materials. J Compos Mater 1967;1:200–6. [23] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Compos Mater 1971;5:58–80. [24] Voyiadjis GZ, Thiagarajan G. An anisotropic yield surface model for directionally reinforced metal-matrix composites. Int J Plastic 1995;11:867–94.
[25] Voyiadjis GZ, Thiagarajan G. A cyclic anisotropic-plasticity model for metal matrix composites. Int J Plastic 1996;12:69–91. [26] Ishikawa T, Matsushima M, Hayashi Y. Hardening non-linear behaviour in longitudinal tension of unidirectional carbon composites. J Mater Sci 1985;20:4075–83. [27] Sun CT, Chung I. An oblique end-tab design for testing off-axis composite specimens. Composites 1993;24:619–23.