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Effects of triangle-shape fiber on the transverse mechanical properties of unidirectional carbon fiber reinforced plastics
Composite Structures 152 (2016) 617–625
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Effects of triangle-shape fiber on the transverse mechanical properties of unidirectional carbon fiber reinforced plastics Lei Yang a, Xin Liu a,⇑, Zhanjun Wu a, Rongguo Wang b a b
State Key Laboratory of Structural Analysis for Industrial Equipment, School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, PR China National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 14 December 2015 Revised 5 May 2016 Accepted 23 May 2016 Available online 24 May 2016 Keywords: Carbon fiber reinforced plastics (CFRPs) Transverse mechanical properties Triangle-shape fiber Finite element method (FEM) Representative volume element (RVE)
a b s t r a c t In this paper, the transverse mechanical properties of triangle-shape carbon fiber reinforced plastic (TCFRP) were studied by numerical micromechanical approach, compared with the traditional roundshape carbon fiber reinforced plastic (RCFRP), to reveal the effect of triangle-shape fiber on the transverse mechanical properties of CFRP and its intrinsic mechanisms. The simulation results indicate the fiber shape can affect both the microscopic deformation and damage behavior of the composites. The triangle-shape fibers provide more restriction to the deformation of matrix than round-shape fibers do, resulting in higher stiffness of the composites; the triangle-shape fiber can improve both the transverse tension and compression strength of the composites with respect to round-shape fiber. These numerical findings are backed by experimental results. Thus, with both good wave-absorbing performance, and excellent mechanical properties that are as good as or even better than RCFRPs, the TCFRPs can serve as very promising structural and functional materials. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Over the past few decades carbon fiber reinforced plastics (CFRPs) have been increasingly used in aerospace [1] and other industries [2] for their high specific stiffness, high specific strength, and outstanding designability. The physical and mechanical properties of CFRP are dependent largely on the fiber content, the fiber orientation, and the cross-section shape of fibers [3]. Generally, the cross-section shape of the reinforcing fibers in composites is round-shape. However, from the viewpoint of structural mechanics, it is recognized that non-round-shape fiber will be better than round-shape fiber in mechanical properties of CFRPs, because the non-round-shape carbon fibers have higher specific surface area than the conventional round-shape carbon fibers (RCFs). The larger area in the surface contacting with the matrix can increase the interfacial bonding force, consequently improve the mechanical properties of the composites [4]. Besides, in some fields, nonround-shape carbon fiber reinforced composites may be better choice for their special properties. For example, for composites reinforced by triangle-shape carbon fibers (TCFs), the TCFs in the composites can reflect incidence microwave many times as its
⇑ Corresponding author. E-mail address: [email protected] (X. Liu). http://dx.doi.org/10.1016/j.compstruct.2016.05.065 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.
microstructure can work like anechoic chamber [5], contributing to higher absorbing ratio of electro-magnetic wave. However, up to date, there are only a limited number of studies on the non-round-shape carbon fibers and their composites. Park et al. [6–8] studied the mechanical properties of various shapes of carbon fibers reinforced cement composites. They found that C-shape carbon fiber reinforced cement composites showed higher tensile and flexural strength than round-shape and any other shape carbon fibers reinforced composites. Xu et al. [9,10] conducted a comprehensive experimental study to identify the differences of the kidney section carbon fibers and circular section carbon fibers in the surface characteristics of fibers and mechanical properties of composites. It was revealed that the kidney fibers with larger specific surface area have a better adsorption characteristic and higher impregnating performance compared with the circular fibers. Pakravan et al. [11] studied the influence of acrylic fibers shape on the flexural behavior of cement composite. It was found that by increasing the fibers’ shape factor, both flexural strength and toughness of the composite increased. In the previous work, the authors [4] manufactured the triangle-shape carbon fiber reinforced plastics (TCFRPs) and round-shape carbon fiber reinforced plastics (RCFRPs), as shown in Fig. 1, and their flexural properties were experimentally investigated. It was found that the TCFRPs showed higher flexural strength and flexural modulus than RCFRPs, and the tensile
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Fig. 1. SEM micrographs of triangle and round shape carbon fibers and their composites: (a) TCF, (b) RCF, (c) TCFRP, (d) RCFRP.
strength and tensile modulus did not reduce. To the best knowledge of the authors, there is no other study on the mechanical properties of TCFRPs. Besides, all the existing investigations as mentioned above about non-round-shape carbon fiber reinforced composites were based on experimental methods. In order to thoroughly understand the effect of fiber shape on the mechanical properties of CFRPs, a micromechanics approach based on numerical method is a good choice to reveal the intrinsic mechanisms of this effect. Many researchers have presented various micromechanics approaches [12–17]. The authors [18] previously developed a micromechanical model for fiber reinforced plastics, which can precisely simulate the mechanical and damage behavior of unidirectional fiber-reinforced polymer composites. In this study, this model is used to simulate the micromechanical and damage behavior of unidirectional TCFRPs and RCFRPs subjected to transverse tension and compression loads. The simulated elastic properties, damage behavior and strength of TCFRPs and RCFRPs are compared, and simulation results are also compared with experimental results, so as to determine the effects of triangleshape fiber on the transverse mechanical properties of CFRPs. 2. Modeling strategies 2.1. FEM model To perform micromechanical analysis of composites, a representative volume element (RVE) of the microstructure large enough to possess the same properties with the macroscopic material should be generated. From Fig. 1 it can be seen that the fibers are randomly embedded in the matrix, which should be taken into account in the RVE of composites. From the figure it can also be seen that the section of triangle fiber is not an exact triangle, but with fillet at each vertex. These fillets were naturally formed
during the manufacture process of the fibers, thus should be retained in the RVE. The random distribution of fibers in the RVE is generated by the random sequential expansion (RSE) algorithm [19] developed by the authors. Shown in Fig. 2 are the generated RVEs of TCFRP and RCFRP. Each RVE contains 30 fibers [20], with fiber volume fraction of 50%. For RCFRP, the radius of the round-shape fibers is 5 lm; for TCFRP, the side length of the triangle-shape fibers is 10 lm, and the fillet radius is 1.6 lm. Five separate RVEs with different fiber distributions are generated for both TCFRP (TCF-1–TCF-5) and RCFRP (RCF-1–RCF-5) to take into account the effect of microscopic configuration. The modeling and simulation platform of this study is the FEM package ABAQUS. The fibers and matrix are meshed with 4-node bilinear plane strain quadrilateral, reduced integration elements. As the interface between fibers and matrix can have significant influence on the properties of the composites, a layer of 4-node two-dimensional cohesive elements with very small thickness (0.01 lm) are introduced between each fiber and the surrounding matrix to simulate the interfacial debonding. Taking the RVE of TCF-1 as an example, shown in Fig. 3 is the finite element discretization of TCF-1. Periodic boundary conditions are applied to the RVEs to ensure a macroscopically uniform stress/displacement field, which are expressed as follows:
uR uRB ¼ uL uLB
ð1Þ
uT uLT ¼ uB uLB
ð2Þ
where u is the displacement vector of any node on the boundary, and subscripts L, R, B and T refer to the left, right, bottom and top edges, while subscripts with two letters correspond to the vertexes of the RVE. These relations between displacements are included in
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Fig. 2. RVEs of TCFRP and RCFRP: (a) TCF-1, (b) RCF-1.
Fig. 3. Finite element model of TCF-1.
the FE model by the ‘‘equation” constrains in ABAQUS, and horizontal displacement (positive for tension and negative for compression) is applied to the right-bottom vertex incrementally until final fracture happens. 2.2. Material models The triangle-shape and round-shape carbon fibers were supplied by Shanxi Institute of Coal Chemistry, China. The epoxy resin TDE-85 was supplied by Tianjin Resin Plant, China. Their transverse mechanical properties are listed in Table 1. As fiber fracture is unlikely to happen under transverse loading, the carbon fibers are modeled as linear elastic and isotropic solids without damage behavior. The material and damage models for the matrix and interface have been discussed in detail in [18], and they are briefly described as follows. The extended linear Drucker–Prager criterion is employed to predict the yielding of the matrix, including the effect of hydrostatic stress on yielding behavior: Table 1 Transverse mechanical properties of the carbon fiber and epoxy resin.
Young’s modulus/GPa Poisson’s ratio Tensile strength/MPa Compressive strength/MPa
Fig. 4. Stress–strain response of the matrix.
Table 2 Material parameters of the matrix and interface.
Carbon fiber
Epoxy resin
Matrix
23.34 0.25 – –
3.45 0.35 85.7 232.5
Interface
d (MPa)
b
104.8
37.7°
Kn = Ks (GPa/m) 108
k
epl0þ
0.8
0.025
tn0 = t0s (MPa) 85.7
epl 0
Gm (J/m2)
0.25
5
Gn = Gs (J/m2) 100
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" 3 # 1 1 1 r F ¼ t p tan b d ¼ 0; t ¼ q 1 þ 1 2 k k q
Table 3 The transverse elastic modulus of TCFRP and RCFRP. TCFRP
Modulus/GPa
RCFRP
Modulus/GPa
TCF-1 TCF-2 TCF-3 TCF-4 TCF-5 Average value Experiment result Relative error
8.28 8.32 8.21 8.31 8.15 8.25 7.86 4.96%
RCF-1 RCF-2 RCF-3 RCF-4 RCF-5 Average value Experiment result Relative error
8.12 8.36 8.12 8.11 7.98 8.14 7.74 5.17%
ð3Þ
where p is the hydrostatic stress, q is the Mises equivalent stress, r is the third invariant of deviatoric stress, b is the slope of the linear yield surface in the p t stress plane, d is the cohesion of the material, and k is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression and, thus, introduces different yield behaviors between tension and compression. The ductile criterion is used to predict the onset of damage for the matrix, which assumes the equivalent plastic strain at the onset of damage as a function of stress triaxiality g (g ¼ p=q),
Fig. 5. The maximum principal stress (kPa) contour plots of (a) TCF-1 and (b) RCF-1 at 0.1% equivalent strain.
Fig. 6. Damage initiation and evolution process of RCFRP under transverse tension: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experimental result.
L. Yang et al. / Composite Structures 152 (2016) 617–625
thus predicting the damage onset discriminating between different triaxial stress states. Here, the equivalent plastic strain at damage initiation for uniaxial tension epl 0þ (g ¼ 1=3) and uniaxial compression epl 0 (g ¼ 1=3) are used to achieve different behaviors between tension and compression. After the onset of failure, the damage evolution is controlled by a progressive failure procedure based on energy criterion, with the fracture energy of the matrix defined as Gm. The damage manifests itself in two forms: softening of the yield stress and degradation of the elasticity, both of which are related to the damage variable that increases with the evolution of damage. The stress–strain response of the matrix is illustrated in Fig. 4. The constitutive response of the cohesive element is defined in terms of a bi-linear traction-separation law which relates the separation displacement between the top and bottom faces of the element to the traction vector acting upon it. The initial response is linear in absence of damage with an elastic stiffness of K:
t n ¼ K n dn ; ts ¼ K s ds
ð4Þ
The initiation of damage is predicted by the maximum stress criterion:
max
( ) ht n i ts ¼1 ; t 0n t0s
ð5Þ
where h i is the Macaulay brackets, which return the argument if positive and zero otherwise, to impede the development of damage when the interface is under compression, and t 0n and t 0s are the normal and tangential interfacial strengths.
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After damage onset, the traction stress is reduced depending on the interface damage parameter, following a linear traction–separation law. The energy necessary to completely break the interface is equal to the interface fracture energy Gn (for Mode I damage) or Gs (for Mode II damage). The involved material parameters used in the simulation for the matrix and interface are listed in Table 2. As the interface strength is very difficult to obtain by experiment, it is assumed that the normal and tangential interface strengths are identical and equal to the tensile strength of the matrix. 3. Results and discussion In order to determine the effects of triangle-shape fiber on the transverse mechanical properties of unidirectional CFRP, the microscopic stress fields of TCFRP and RCFRP are illustrated and compared, and their equivalent elastic modulus are calculated by volumetric homogenization procedure; the damage initiation and evolution process of TCFRP and RCFRP are discussed and compared with experimental results, and their transverse tension and compression strength are also compared. 3.1. Stress field and stiffness Shown in Fig. 5 are the maximum principal stress contour plots of TCF-1 and RCF-1 respectively, at the loading level of 0.1% equivalent strain. As the loading level is very low, all the constituent materials are still in their elastic states, i.e., without plastic deformation and damage initiation. As can be seen, due to the significant
Fig. 7. Damage initiation and evolution process of TCFRP under transverse tension: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experimental result.
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Fig. 8. Transverse tension stress–strain curves of TCFRP and RCFRP.
distinction of elastic properties between fibers and matrix, stress concentration happens at the junction regions between fibers and matrix. For RCFRP, the maximum stress occurs at the locations where two round fibers are closely adjacent with the connecting line of the two fiber centers approximately parallel to the loading direction. While for TCFRP, owing to more diversity of the position relations between different triangle fibers, the microscopic stress field is more complicated. But generally the maximum stress also
occurs at the locations where two fibers are close. The maximum principal stress in TCF-1 (17.2 MPa) is higher than that in RCF-1 (14.8 MPa). However, the maximum stress is greatly influenced by the microscopic configuration of the RVE, especially the interfiber distance. In other RVEs, the maximum stress of RCFRP can be higher than that of TCFRP. Thus, it is the overall stress field, instead of the maximum stress, that reflects the stiffness characteristic of the composites.
Fig. 9. Damage initiation and evolution process of RCFRP under transverse compression: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experiment result.
L. Yang et al. / Composite Structures 152 (2016) 617–625
The volumetric homogenization procedure is implemented to calculate the effective elastic properties of the composites through the following formula [21]:
PN
rijj Ai i i i¼1 ejj A
i¼1
E j ¼ PN
ð6Þ
where N is the total number of elements in the model, rijj and eijj are the average j-component of stress and strain calculated in each element respectively, and Ai is the area of that element. The transverse elastic moduli for all five RVEs of TCFRP and RCFRP are calculated, with the results listed in Table 3. As can be seen, the results of different RVEs are very close, and the computed average transverse elastic modulus of TCFRP is a little higher than that of RCFRP. This is because the relative position relation of fibers in TCFRP is more complicated than that in RCFRP, and the triangleshape fibers can have more restriction to the deformation of matrix than round-shape fibers do. The experiment results also support this conclusion, although there is some error between the simulated and experiment results. 3.2. Damage behavior and strength The simulation results can help to reveal the microscopic damage mechanisms of the composites, and explain how the fiber shape affects the damage behavior of the composites. Taking RCF-1 and TCF-1 for example, shown in Figs. 6 and 7 are the
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damage initiation and evolution process of RCFRP and TCFRP under transverse tension. Experimental result of the fracture morphology observed by optical microscope is also provided as a comparison. For RCFRP, interfacial debonding first occurs at the equator of the fibers where the inter-fiber distance is small, as can be seen more clearly by the partial enlarged drawing. Then, matrix plastic damage begins to appear at the vicinity of the interfacial debonding, and more interfacial debonding occurs at other locations. Finally, interfacial debonding at different locations is linked by matrix cracks throughout the RVE, causing the ultimate fracture of the RVE. For TCFRP, the damage initiation and evolution process is similar, but interfacial debonding first occurs at the vertex of the triangle fibers. Besides, as the matrix crack tends to propagate along the edge of fibers, for RCFRP, it is more likely to form a relative smooth crack through the RVE; while for triangle-shape fibers, as the edges of the triangle fibers are not aligned, it is more likely to form a zigzag matrix crack through the RVE. For both RCFRP and TCFRP, there is good agreement on the failure morphology between the simulated and experimental results. The simulated stress–strain curves of all the five RVEs for TCFRP and RCFRP under transverse tension are shown in Fig. 8. As can be seen, the initial stress–strain relation of both TCFRP and RCFRP is linear, following with a short plastic stage, and then failure happens as a sudden drop in the stress–strain curves. For different RVEs, the elastic regimes of the curves are almost superposed, and divergence does not occur until the onset of damage. Compared with RCFRP, there seems a more obvious plastic stage for
Fig. 10. Damage initiation and evolution process of TCFRP under transverse compression: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experiment result.
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Fig. 11. Transverse compression stress–strain curves of TCFRP and RCFRP.
TCFRP. This is resulted from the difference in the matrix crack propagation mode as stated above. For transverse compression, the damage initiation and evolution process of TCF-1 and RCF-1 is illustrated in Figs. 9 and 10, respectively. As can be seen, the damage behavior is different from that of transverse tension. For both RCFRP and TCFRP, matrix plastic damage first happens at the location where two fibers are closely adjacent in the direction perpendicular to the load. Then, more matrix cracks occur at different locations, mostly adjoining the fibers. And finally the matrix cracks at different locations are linked to form a main crack, having a certain angle with the loading direction, which is called the plastic shear band. Thus it is concluded that the damage behavior of the composites under transverse compression is dominated by the matrix. But there is also some difference in the damage behavior between TCFRP and RCFRP. For RCFRP, there is often an obvious shear band throughout the RVE. While for TCFRP, due to the misaligning of different fiber edges, the through crack may fail to form in the RVE, replaced by some regional short cracks. Shown in Fig. 11 are the stress–strain curves of all the five RVEs for TCFRP and RCFRP under transverse compression. Compared with transverse tension, there is more dispersion in the stress– strain curves between different RVEs. On the other hand, the influence of fiber shape on the transverse compression stress–strain curves of composites is not obvious, with similar stress–strain curves for TCFRP and RCFRP. Listed in Table 4 are the simulated results of transverse tension and compression strength of all five RVEs for TCFRP and RCFRP together with experimental results. As can be seen, the simulated transverse tension strength of all five RVEs of TCFRP is higher than that of RCFRP, and the average tension strength of TCFRP is 3.5% higher than that of RCFRP. The experimental results have the same trend. Thus the triangle-shape fiber can have enhanced effect on the transverse tension strength of composites than round-shape fiber. On the other hand, the simulated transverse compression strength of some TCFRP RVEs is higher than that of RCFRP RVEs, but the others are opposite. And the average compression strength of TCFRP is only 0.9% higher than that of RCFRP, which can almost be neglected. This phenomenon is also backed by the experimental results, which can be explained by the micromechanical damage mechanisms of the composites: the transverse tension damage of the composites is mainly controlled by interfacial debonding, the zig-zag propagation mode of the crack in TCFRP delays the failure of the RVEs, resulting in a higher tension strength than RCFRP; but the transverse compression damage of the composites is dominated by matrix plastic damage, the effect of fiber shape on the damage behavior is relatively small, thus there is no obvious
Table 4 The transverse tension and compression strength of TCFRP and RCFRP. RVEs
TCF-1/RCF-1 TCF-2/RCF-2 TCF-3/RCF-3 TCF-4/RCF-4 TCF-5/RCF-5 Average value Experiment result Relative error
Tension strength/ MPa
Compression strength/MPa
TCFRP
RCFRP
TCFRP
RCFRP
86.3 87.7 85.7 87.0 85.6 86.5 80.6 7.3%
83.2 84.8 81.5 85.6 83.1 83.6 76.8 8.9%
211.7 211.8 229.6 205.6 211.3 214.0 192.9 10.9%
231.7 202.6 206.9 200.0 218.7 212.0 189.7 11.8%
difference in the transverse compression strength between TCFRP and RCFRP. 4. Conclusions The transverse mechanical properties and damage behavior of triangle-shape carbon fiber reinforced plastics (TCFRPs) were simulated by finite element analysis based on computational micromechanics, compared with round-shape carbon fiber reinforced plastics (RCFRPs) and experimental results, to investigate the effect of fiber shape on the transverse mechanical properties of CFRPs. The transverse elastic moduli of both TCFRP and RCFRP were calculated and compared with experimental results. It is found that the average transverse elastic modulus of TCFRP is a little higher than that of RCFRP, as the triangle fibers can have more restriction to the deformation of matrix than round fibers do. The damage initiation and evolution process of TCFRP and RCFRP under transverse tension and compression was also simulated and compared with experiment. For transverse tension, the damage of the composites is mainly controlled by interfacial debonding; the matrix crack propagates in zig–zag form for TCFRP, compared with a relatively smooth crack propagating path for RCFRP, which makes the average tension strength of TCFRP higher than RCFRP. For transverse compression, the damage behavior of the composites is dominated by matrix plastic damage; it is more likely to form a shear band through the RVE for RCFRP than TCFRP. In conclusion, the triangle-shape carbon fiber can improve both the transverse stiffness and strength of the composites with respect to round-shape fiber, though the enhanced effect is relatively small. Therefore, with both good wave-absorbing performance, and excellent mechanical properties that are as good as or even better than
L. Yang et al. / Composite Structures 152 (2016) 617–625
RCFRPs, the TCFRPs are very promising structural and functional materials. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11402045, 11302036), the China Postdoctoral Science Foundation Funded Project (No. 2014M560204), and the Fundamental Research Funds for the Central Universities (No. DUT16LK35). References [1] Soutis C. Fibre reinforced composites in aircraft construction. Prog Aerosp Sci 2005;41(2):143–51. [2] Hollaway LC. A review of the present and future utilisation of FRP composites in the civil infrastructure with reference to their important in-service properties. Constr Build Mater 2010;24(12):2419–45. [3] Squires CA, Netting KH, Chambers AR. Understanding the factors affecting the compressive testing of unidirectional carbon fibre composites. Composites Part B 2007;38:481–7. [4] Liu X, Wang RG, Wu ZJ, et al. The effect of triangle-shape carbon fiber on the flexural properties of the carbon fiber reinforced plastics. Mater Lett 2012;73:21–3. [5] Wang RG, Liu X, Liu WB. The B-basis value of the shearing strength of triangleshape carbon fibers reinforced plastics. Polym Polym Compos 2011;19(4– 5):327–32. [6] Park SJ, Seo MK, Shim HB. Effect of fiber shapes on physical characteristics of noncircular carbon fibers-reinforced composites. Mater Sci Eng A 2003;352(1– 2):34–9. [7] Kim TJ, Park CK. Flexural and tensile strength developments of various shape carbon fiber-reinforced lightweight cementitious composites. Cement Concr Res 1998;28(7):955–60. [8] Park SJ, Seo MK, Shim HB, et al. Effect of different cross-section types on mechanical properties of carbon fibers- reinforced cement composites. Mater Sci Eng A 2004;366(2):348–55.
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[9] Xu ZW, Huang YD, Liu L, et al. Surface characteristics of kidney and circular section carbon fibers and mechanical behavior of composites. Mater Chem Phys 2007;106(1):16–21. [10] Xu ZW, Li JL, Wu XQ, et al. Effect of kidney-type and circular cross sections on carbon fiber surface and composite interface. Composites Part A 2008;39:301–7. [11] Pakravan HR, Jamshidi M, Latif M, et al. Influence of acrylic fibers geometry on the mechanical performance of fiber-cement composites. J Appl Polym Sci 2012;125(4):3050–7. [12] Aghdam MM, Dezhsetan A. Micromechanics based analysis of randomly distributed fiber reinforced composites using simplified unit cell model. Compos Struct 2005;71(3–4):327–32. [13] Barbero EJ, Abdelal GF, Caceres A. A micromechanics approach for damage modeling of polymer matrix composites. Compos Struct 2005;67(4):427–36. [14] Segurado J, LLorca J. A computational micromechanics study of the effect of interface decohesion on the mechanical behavior of composites. Acta Mater 2005;53(18):4931–42. [15] Vaughan TJ, McCarthy CT. Micromechanical modelling of the transverse damage behaviour in fibre reinforced composites. Compos Sci Technol 2011;71(3):388–96. [16] Greco F, Leonetti L, Lonetti P. A two-scale failure analysis of composite materials in presence of fiber/matrix crack initiation and propagation. Compos Struct 2013;95:582–97. [17] Greco F, Leonetti L, Nevone Blasi P. Adaptive multiscale modeling of fiberreinforced composite materials subjected to transverse microcracking. Compos Struct 2014;113:249–63. [18] Yang L, Yan Y, Liu YJ, et al. Microscopic failure mechanisms of fibre-reinforced polymer composites under transverse tension and compression. Compos Sci Technol 2012;72(15):1818–25. [19] Yang L, Yan Y, Ran ZG, et al. A new method for generating random fibre distributions for fibre reinforced composites. Compos Sci Technol 2013;76 (4):14–20. [20] González C, LLorca J. Mechanical behavior of unidirectional fiber-reinforced polymers under transverse compression: microscopic mechanisms and modeling. Compos Sci Technol 2007;67(13):2795–806. [21] Melro AR, Camanho PP, Pinho ST. Generation of random distribution of fibres in long-fibre reinforced composites. Compos Sci Technol 2008;68 (9):2092–102.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Effects of triangle-shape fiber on the transverse mechanical properties of unidirectional carbon fiber reinforced plastics Lei Yang a, Xin Liu a,⇑, Zhanjun Wu a, Rongguo Wang b a b
State Key Laboratory of Structural Analysis for Industrial Equipment, School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, PR China National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 14 December 2015 Revised 5 May 2016 Accepted 23 May 2016 Available online 24 May 2016 Keywords: Carbon fiber reinforced plastics (CFRPs) Transverse mechanical properties Triangle-shape fiber Finite element method (FEM) Representative volume element (RVE)
a b s t r a c t In this paper, the transverse mechanical properties of triangle-shape carbon fiber reinforced plastic (TCFRP) were studied by numerical micromechanical approach, compared with the traditional roundshape carbon fiber reinforced plastic (RCFRP), to reveal the effect of triangle-shape fiber on the transverse mechanical properties of CFRP and its intrinsic mechanisms. The simulation results indicate the fiber shape can affect both the microscopic deformation and damage behavior of the composites. The triangle-shape fibers provide more restriction to the deformation of matrix than round-shape fibers do, resulting in higher stiffness of the composites; the triangle-shape fiber can improve both the transverse tension and compression strength of the composites with respect to round-shape fiber. These numerical findings are backed by experimental results. Thus, with both good wave-absorbing performance, and excellent mechanical properties that are as good as or even better than RCFRPs, the TCFRPs can serve as very promising structural and functional materials. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Over the past few decades carbon fiber reinforced plastics (CFRPs) have been increasingly used in aerospace [1] and other industries [2] for their high specific stiffness, high specific strength, and outstanding designability. The physical and mechanical properties of CFRP are dependent largely on the fiber content, the fiber orientation, and the cross-section shape of fibers [3]. Generally, the cross-section shape of the reinforcing fibers in composites is round-shape. However, from the viewpoint of structural mechanics, it is recognized that non-round-shape fiber will be better than round-shape fiber in mechanical properties of CFRPs, because the non-round-shape carbon fibers have higher specific surface area than the conventional round-shape carbon fibers (RCFs). The larger area in the surface contacting with the matrix can increase the interfacial bonding force, consequently improve the mechanical properties of the composites [4]. Besides, in some fields, nonround-shape carbon fiber reinforced composites may be better choice for their special properties. For example, for composites reinforced by triangle-shape carbon fibers (TCFs), the TCFs in the composites can reflect incidence microwave many times as its
⇑ Corresponding author. E-mail address: [email protected] (X. Liu). http://dx.doi.org/10.1016/j.compstruct.2016.05.065 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.
microstructure can work like anechoic chamber [5], contributing to higher absorbing ratio of electro-magnetic wave. However, up to date, there are only a limited number of studies on the non-round-shape carbon fibers and their composites. Park et al. [6–8] studied the mechanical properties of various shapes of carbon fibers reinforced cement composites. They found that C-shape carbon fiber reinforced cement composites showed higher tensile and flexural strength than round-shape and any other shape carbon fibers reinforced composites. Xu et al. [9,10] conducted a comprehensive experimental study to identify the differences of the kidney section carbon fibers and circular section carbon fibers in the surface characteristics of fibers and mechanical properties of composites. It was revealed that the kidney fibers with larger specific surface area have a better adsorption characteristic and higher impregnating performance compared with the circular fibers. Pakravan et al. [11] studied the influence of acrylic fibers shape on the flexural behavior of cement composite. It was found that by increasing the fibers’ shape factor, both flexural strength and toughness of the composite increased. In the previous work, the authors [4] manufactured the triangle-shape carbon fiber reinforced plastics (TCFRPs) and round-shape carbon fiber reinforced plastics (RCFRPs), as shown in Fig. 1, and their flexural properties were experimentally investigated. It was found that the TCFRPs showed higher flexural strength and flexural modulus than RCFRPs, and the tensile
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Fig. 1. SEM micrographs of triangle and round shape carbon fibers and their composites: (a) TCF, (b) RCF, (c) TCFRP, (d) RCFRP.
strength and tensile modulus did not reduce. To the best knowledge of the authors, there is no other study on the mechanical properties of TCFRPs. Besides, all the existing investigations as mentioned above about non-round-shape carbon fiber reinforced composites were based on experimental methods. In order to thoroughly understand the effect of fiber shape on the mechanical properties of CFRPs, a micromechanics approach based on numerical method is a good choice to reveal the intrinsic mechanisms of this effect. Many researchers have presented various micromechanics approaches [12–17]. The authors [18] previously developed a micromechanical model for fiber reinforced plastics, which can precisely simulate the mechanical and damage behavior of unidirectional fiber-reinforced polymer composites. In this study, this model is used to simulate the micromechanical and damage behavior of unidirectional TCFRPs and RCFRPs subjected to transverse tension and compression loads. The simulated elastic properties, damage behavior and strength of TCFRPs and RCFRPs are compared, and simulation results are also compared with experimental results, so as to determine the effects of triangleshape fiber on the transverse mechanical properties of CFRPs. 2. Modeling strategies 2.1. FEM model To perform micromechanical analysis of composites, a representative volume element (RVE) of the microstructure large enough to possess the same properties with the macroscopic material should be generated. From Fig. 1 it can be seen that the fibers are randomly embedded in the matrix, which should be taken into account in the RVE of composites. From the figure it can also be seen that the section of triangle fiber is not an exact triangle, but with fillet at each vertex. These fillets were naturally formed
during the manufacture process of the fibers, thus should be retained in the RVE. The random distribution of fibers in the RVE is generated by the random sequential expansion (RSE) algorithm [19] developed by the authors. Shown in Fig. 2 are the generated RVEs of TCFRP and RCFRP. Each RVE contains 30 fibers [20], with fiber volume fraction of 50%. For RCFRP, the radius of the round-shape fibers is 5 lm; for TCFRP, the side length of the triangle-shape fibers is 10 lm, and the fillet radius is 1.6 lm. Five separate RVEs with different fiber distributions are generated for both TCFRP (TCF-1–TCF-5) and RCFRP (RCF-1–RCF-5) to take into account the effect of microscopic configuration. The modeling and simulation platform of this study is the FEM package ABAQUS. The fibers and matrix are meshed with 4-node bilinear plane strain quadrilateral, reduced integration elements. As the interface between fibers and matrix can have significant influence on the properties of the composites, a layer of 4-node two-dimensional cohesive elements with very small thickness (0.01 lm) are introduced between each fiber and the surrounding matrix to simulate the interfacial debonding. Taking the RVE of TCF-1 as an example, shown in Fig. 3 is the finite element discretization of TCF-1. Periodic boundary conditions are applied to the RVEs to ensure a macroscopically uniform stress/displacement field, which are expressed as follows:
uR uRB ¼ uL uLB
ð1Þ
uT uLT ¼ uB uLB
ð2Þ
where u is the displacement vector of any node on the boundary, and subscripts L, R, B and T refer to the left, right, bottom and top edges, while subscripts with two letters correspond to the vertexes of the RVE. These relations between displacements are included in
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Fig. 2. RVEs of TCFRP and RCFRP: (a) TCF-1, (b) RCF-1.
Fig. 3. Finite element model of TCF-1.
the FE model by the ‘‘equation” constrains in ABAQUS, and horizontal displacement (positive for tension and negative for compression) is applied to the right-bottom vertex incrementally until final fracture happens. 2.2. Material models The triangle-shape and round-shape carbon fibers were supplied by Shanxi Institute of Coal Chemistry, China. The epoxy resin TDE-85 was supplied by Tianjin Resin Plant, China. Their transverse mechanical properties are listed in Table 1. As fiber fracture is unlikely to happen under transverse loading, the carbon fibers are modeled as linear elastic and isotropic solids without damage behavior. The material and damage models for the matrix and interface have been discussed in detail in [18], and they are briefly described as follows. The extended linear Drucker–Prager criterion is employed to predict the yielding of the matrix, including the effect of hydrostatic stress on yielding behavior: Table 1 Transverse mechanical properties of the carbon fiber and epoxy resin.
Young’s modulus/GPa Poisson’s ratio Tensile strength/MPa Compressive strength/MPa
Fig. 4. Stress–strain response of the matrix.
Table 2 Material parameters of the matrix and interface.
Carbon fiber
Epoxy resin
Matrix
23.34 0.25 – –
3.45 0.35 85.7 232.5
Interface
d (MPa)
b
104.8
37.7°
Kn = Ks (GPa/m) 108
k
epl0þ
0.8
0.025
tn0 = t0s (MPa) 85.7
epl 0
Gm (J/m2)
0.25
5
Gn = Gs (J/m2) 100
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" 3 # 1 1 1 r F ¼ t p tan b d ¼ 0; t ¼ q 1 þ 1 2 k k q
Table 3 The transverse elastic modulus of TCFRP and RCFRP. TCFRP
Modulus/GPa
RCFRP
Modulus/GPa
TCF-1 TCF-2 TCF-3 TCF-4 TCF-5 Average value Experiment result Relative error
8.28 8.32 8.21 8.31 8.15 8.25 7.86 4.96%
RCF-1 RCF-2 RCF-3 RCF-4 RCF-5 Average value Experiment result Relative error
8.12 8.36 8.12 8.11 7.98 8.14 7.74 5.17%
ð3Þ
where p is the hydrostatic stress, q is the Mises equivalent stress, r is the third invariant of deviatoric stress, b is the slope of the linear yield surface in the p t stress plane, d is the cohesion of the material, and k is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression and, thus, introduces different yield behaviors between tension and compression. The ductile criterion is used to predict the onset of damage for the matrix, which assumes the equivalent plastic strain at the onset of damage as a function of stress triaxiality g (g ¼ p=q),
Fig. 5. The maximum principal stress (kPa) contour plots of (a) TCF-1 and (b) RCF-1 at 0.1% equivalent strain.
Fig. 6. Damage initiation and evolution process of RCFRP under transverse tension: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experimental result.
L. Yang et al. / Composite Structures 152 (2016) 617–625
thus predicting the damage onset discriminating between different triaxial stress states. Here, the equivalent plastic strain at damage initiation for uniaxial tension epl 0þ (g ¼ 1=3) and uniaxial compression epl 0 (g ¼ 1=3) are used to achieve different behaviors between tension and compression. After the onset of failure, the damage evolution is controlled by a progressive failure procedure based on energy criterion, with the fracture energy of the matrix defined as Gm. The damage manifests itself in two forms: softening of the yield stress and degradation of the elasticity, both of which are related to the damage variable that increases with the evolution of damage. The stress–strain response of the matrix is illustrated in Fig. 4. The constitutive response of the cohesive element is defined in terms of a bi-linear traction-separation law which relates the separation displacement between the top and bottom faces of the element to the traction vector acting upon it. The initial response is linear in absence of damage with an elastic stiffness of K:
t n ¼ K n dn ; ts ¼ K s ds
ð4Þ
The initiation of damage is predicted by the maximum stress criterion:
max
( ) ht n i ts ¼1 ; t 0n t0s
ð5Þ
where h i is the Macaulay brackets, which return the argument if positive and zero otherwise, to impede the development of damage when the interface is under compression, and t 0n and t 0s are the normal and tangential interfacial strengths.
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After damage onset, the traction stress is reduced depending on the interface damage parameter, following a linear traction–separation law. The energy necessary to completely break the interface is equal to the interface fracture energy Gn (for Mode I damage) or Gs (for Mode II damage). The involved material parameters used in the simulation for the matrix and interface are listed in Table 2. As the interface strength is very difficult to obtain by experiment, it is assumed that the normal and tangential interface strengths are identical and equal to the tensile strength of the matrix. 3. Results and discussion In order to determine the effects of triangle-shape fiber on the transverse mechanical properties of unidirectional CFRP, the microscopic stress fields of TCFRP and RCFRP are illustrated and compared, and their equivalent elastic modulus are calculated by volumetric homogenization procedure; the damage initiation and evolution process of TCFRP and RCFRP are discussed and compared with experimental results, and their transverse tension and compression strength are also compared. 3.1. Stress field and stiffness Shown in Fig. 5 are the maximum principal stress contour plots of TCF-1 and RCF-1 respectively, at the loading level of 0.1% equivalent strain. As the loading level is very low, all the constituent materials are still in their elastic states, i.e., without plastic deformation and damage initiation. As can be seen, due to the significant
Fig. 7. Damage initiation and evolution process of TCFRP under transverse tension: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experimental result.
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Fig. 8. Transverse tension stress–strain curves of TCFRP and RCFRP.
distinction of elastic properties between fibers and matrix, stress concentration happens at the junction regions between fibers and matrix. For RCFRP, the maximum stress occurs at the locations where two round fibers are closely adjacent with the connecting line of the two fiber centers approximately parallel to the loading direction. While for TCFRP, owing to more diversity of the position relations between different triangle fibers, the microscopic stress field is more complicated. But generally the maximum stress also
occurs at the locations where two fibers are close. The maximum principal stress in TCF-1 (17.2 MPa) is higher than that in RCF-1 (14.8 MPa). However, the maximum stress is greatly influenced by the microscopic configuration of the RVE, especially the interfiber distance. In other RVEs, the maximum stress of RCFRP can be higher than that of TCFRP. Thus, it is the overall stress field, instead of the maximum stress, that reflects the stiffness characteristic of the composites.
Fig. 9. Damage initiation and evolution process of RCFRP under transverse compression: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experiment result.
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The volumetric homogenization procedure is implemented to calculate the effective elastic properties of the composites through the following formula [21]:
PN
rijj Ai i i i¼1 ejj A
i¼1
E j ¼ PN
ð6Þ
where N is the total number of elements in the model, rijj and eijj are the average j-component of stress and strain calculated in each element respectively, and Ai is the area of that element. The transverse elastic moduli for all five RVEs of TCFRP and RCFRP are calculated, with the results listed in Table 3. As can be seen, the results of different RVEs are very close, and the computed average transverse elastic modulus of TCFRP is a little higher than that of RCFRP. This is because the relative position relation of fibers in TCFRP is more complicated than that in RCFRP, and the triangleshape fibers can have more restriction to the deformation of matrix than round-shape fibers do. The experiment results also support this conclusion, although there is some error between the simulated and experiment results. 3.2. Damage behavior and strength The simulation results can help to reveal the microscopic damage mechanisms of the composites, and explain how the fiber shape affects the damage behavior of the composites. Taking RCF-1 and TCF-1 for example, shown in Figs. 6 and 7 are the
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damage initiation and evolution process of RCFRP and TCFRP under transverse tension. Experimental result of the fracture morphology observed by optical microscope is also provided as a comparison. For RCFRP, interfacial debonding first occurs at the equator of the fibers where the inter-fiber distance is small, as can be seen more clearly by the partial enlarged drawing. Then, matrix plastic damage begins to appear at the vicinity of the interfacial debonding, and more interfacial debonding occurs at other locations. Finally, interfacial debonding at different locations is linked by matrix cracks throughout the RVE, causing the ultimate fracture of the RVE. For TCFRP, the damage initiation and evolution process is similar, but interfacial debonding first occurs at the vertex of the triangle fibers. Besides, as the matrix crack tends to propagate along the edge of fibers, for RCFRP, it is more likely to form a relative smooth crack through the RVE; while for triangle-shape fibers, as the edges of the triangle fibers are not aligned, it is more likely to form a zigzag matrix crack through the RVE. For both RCFRP and TCFRP, there is good agreement on the failure morphology between the simulated and experimental results. The simulated stress–strain curves of all the five RVEs for TCFRP and RCFRP under transverse tension are shown in Fig. 8. As can be seen, the initial stress–strain relation of both TCFRP and RCFRP is linear, following with a short plastic stage, and then failure happens as a sudden drop in the stress–strain curves. For different RVEs, the elastic regimes of the curves are almost superposed, and divergence does not occur until the onset of damage. Compared with RCFRP, there seems a more obvious plastic stage for
Fig. 10. Damage initiation and evolution process of TCFRP under transverse compression: (a) Damage initiation; (b) Damage propagation; (c) Ultimate failure; (d) Experiment result.
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Fig. 11. Transverse compression stress–strain curves of TCFRP and RCFRP.
TCFRP. This is resulted from the difference in the matrix crack propagation mode as stated above. For transverse compression, the damage initiation and evolution process of TCF-1 and RCF-1 is illustrated in Figs. 9 and 10, respectively. As can be seen, the damage behavior is different from that of transverse tension. For both RCFRP and TCFRP, matrix plastic damage first happens at the location where two fibers are closely adjacent in the direction perpendicular to the load. Then, more matrix cracks occur at different locations, mostly adjoining the fibers. And finally the matrix cracks at different locations are linked to form a main crack, having a certain angle with the loading direction, which is called the plastic shear band. Thus it is concluded that the damage behavior of the composites under transverse compression is dominated by the matrix. But there is also some difference in the damage behavior between TCFRP and RCFRP. For RCFRP, there is often an obvious shear band throughout the RVE. While for TCFRP, due to the misaligning of different fiber edges, the through crack may fail to form in the RVE, replaced by some regional short cracks. Shown in Fig. 11 are the stress–strain curves of all the five RVEs for TCFRP and RCFRP under transverse compression. Compared with transverse tension, there is more dispersion in the stress– strain curves between different RVEs. On the other hand, the influence of fiber shape on the transverse compression stress–strain curves of composites is not obvious, with similar stress–strain curves for TCFRP and RCFRP. Listed in Table 4 are the simulated results of transverse tension and compression strength of all five RVEs for TCFRP and RCFRP together with experimental results. As can be seen, the simulated transverse tension strength of all five RVEs of TCFRP is higher than that of RCFRP, and the average tension strength of TCFRP is 3.5% higher than that of RCFRP. The experimental results have the same trend. Thus the triangle-shape fiber can have enhanced effect on the transverse tension strength of composites than round-shape fiber. On the other hand, the simulated transverse compression strength of some TCFRP RVEs is higher than that of RCFRP RVEs, but the others are opposite. And the average compression strength of TCFRP is only 0.9% higher than that of RCFRP, which can almost be neglected. This phenomenon is also backed by the experimental results, which can be explained by the micromechanical damage mechanisms of the composites: the transverse tension damage of the composites is mainly controlled by interfacial debonding, the zig-zag propagation mode of the crack in TCFRP delays the failure of the RVEs, resulting in a higher tension strength than RCFRP; but the transverse compression damage of the composites is dominated by matrix plastic damage, the effect of fiber shape on the damage behavior is relatively small, thus there is no obvious
Table 4 The transverse tension and compression strength of TCFRP and RCFRP. RVEs
TCF-1/RCF-1 TCF-2/RCF-2 TCF-3/RCF-3 TCF-4/RCF-4 TCF-5/RCF-5 Average value Experiment result Relative error
Tension strength/ MPa
Compression strength/MPa
TCFRP
RCFRP
TCFRP
RCFRP
86.3 87.7 85.7 87.0 85.6 86.5 80.6 7.3%
83.2 84.8 81.5 85.6 83.1 83.6 76.8 8.9%
211.7 211.8 229.6 205.6 211.3 214.0 192.9 10.9%
231.7 202.6 206.9 200.0 218.7 212.0 189.7 11.8%
difference in the transverse compression strength between TCFRP and RCFRP. 4. Conclusions The transverse mechanical properties and damage behavior of triangle-shape carbon fiber reinforced plastics (TCFRPs) were simulated by finite element analysis based on computational micromechanics, compared with round-shape carbon fiber reinforced plastics (RCFRPs) and experimental results, to investigate the effect of fiber shape on the transverse mechanical properties of CFRPs. The transverse elastic moduli of both TCFRP and RCFRP were calculated and compared with experimental results. It is found that the average transverse elastic modulus of TCFRP is a little higher than that of RCFRP, as the triangle fibers can have more restriction to the deformation of matrix than round fibers do. The damage initiation and evolution process of TCFRP and RCFRP under transverse tension and compression was also simulated and compared with experiment. For transverse tension, the damage of the composites is mainly controlled by interfacial debonding; the matrix crack propagates in zig–zag form for TCFRP, compared with a relatively smooth crack propagating path for RCFRP, which makes the average tension strength of TCFRP higher than RCFRP. For transverse compression, the damage behavior of the composites is dominated by matrix plastic damage; it is more likely to form a shear band through the RVE for RCFRP than TCFRP. In conclusion, the triangle-shape carbon fiber can improve both the transverse stiffness and strength of the composites with respect to round-shape fiber, though the enhanced effect is relatively small. Therefore, with both good wave-absorbing performance, and excellent mechanical properties that are as good as or even better than
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RCFRPs, the TCFRPs are very promising structural and functional materials. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11402045, 11302036), the China Postdoctoral Science Foundation Funded Project (No. 2014M560204), and the Fundamental Research Funds for the Central Universities (No. DUT16LK35). References [1] Soutis C. Fibre reinforced composites in aircraft construction. Prog Aerosp Sci 2005;41(2):143–51. [2] Hollaway LC. A review of the present and future utilisation of FRP composites in the civil infrastructure with reference to their important in-service properties. Constr Build Mater 2010;24(12):2419–45. [3] Squires CA, Netting KH, Chambers AR. Understanding the factors affecting the compressive testing of unidirectional carbon fibre composites. Composites Part B 2007;38:481–7. [4] Liu X, Wang RG, Wu ZJ, et al. The effect of triangle-shape carbon fiber on the flexural properties of the carbon fiber reinforced plastics. Mater Lett 2012;73:21–3. [5] Wang RG, Liu X, Liu WB. The B-basis value of the shearing strength of triangleshape carbon fibers reinforced plastics. Polym Polym Compos 2011;19(4– 5):327–32. [6] Park SJ, Seo MK, Shim HB. Effect of fiber shapes on physical characteristics of noncircular carbon fibers-reinforced composites. Mater Sci Eng A 2003;352(1– 2):34–9. [7] Kim TJ, Park CK. Flexural and tensile strength developments of various shape carbon fiber-reinforced lightweight cementitious composites. Cement Concr Res 1998;28(7):955–60. [8] Park SJ, Seo MK, Shim HB, et al. Effect of different cross-section types on mechanical properties of carbon fibers- reinforced cement composites. Mater Sci Eng A 2004;366(2):348–55.
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