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Analysis of effect of fiber orientation on Young’s modulus for unidirectional fiber reinforced composites
Composites: Part B 56 (2014) 733–739
Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Analysis of effect of fiber orientation on Young’s modulus for unidirectional fiber reinforced composites H.W. Wang a,⇑, H.W. Zhou b, L.L. Gui b, H.W. Ji a, X.C. Zhang a a b
Tianjin Key Laboratory of Refrigeration Technology, Tianjin University of Commerce, Tianjin 300134, China State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 17 April 2013 Received in revised form 4 September 2013 Accepted 7 September 2013 Available online 17 September 2013 Keywords: A. Glass fibers B. Mechanical properties C. Analytical modeling C. Numerical analysis
a b s t r a c t Young’s modulus of unidirectional glass fiber reinforced polymer (GFRP) composites for wind energy applications were studied using analytical, numerical and experimental methods. In order to explore the effect of fiber orientation angle on the Young’s modulus of composites, from the basic theory of elastic mechanics, a procedure which can be applied to evaluate the elastic stiffness matrix of GFRP composite as an analytical function of fiber orientation angle (from 0° to 90°), was developed. At the same time, different finite element models with inclined glass fiber were developed via the ABAQUS Scripting Interface. Results indicate that Young’s modulus of the composites strongly depends on the fiber orientation angles. A U-shaped dependency of the Young’s modulus of composites on the inclined angle of fiber is found, which agree well with the experimental results. The shear modulus is found to have significant effect on the composites’ Young’s modulus, too. The effect of volume content of glass fiber on the Young’s modulus of composites was investigated. Results indicate the relation between them is nearly linear. The results of the investigation are expected to provide some design guideline for the microstructural optimization of the glass fiber reinforced composites. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduction Polymer materials reinforced with glass fiber have received tremendous attention in both scientific and industrial communities due to their extraordinary enhanced properties, such as lower weight, higher toughness and higher strength characteristics. In response to these requirements, research on composites has attracted much attention, which results in numerous publications [1–11]. In order to evaluate the mechanical behaviors of composites materials, different approaches, including experimental investigation, numerical simulations and theoretical modeling, were employed [12–16]. For example, the fiber bundle model has been used to study the damage behaviors of fiber while loading along the fibers [17]. The numerical continuum mechanical models, such as finite element models, allow the incorporation of many different features of the nonlinear material behaviors and the analysis of the interaction of available and evolving microstructural elements [18]. Many computational experiments of damage and failure in composites have been done by employing numerical continuum mechanical models [19–22]. The shear lag and other analytical models based on simplifying assumptions are applicable mainly ⇑ Corresponding author. Tel.: +86 13512806141.
to the linear elastic material behaviors and relatively simple, periodic microgeometrics. They are often used to analyze the load transfer and multiple cracking in composites [23]. The fracture mechanics-based models are often used to the cases of fiber bridging analysis of elastic or homogeneous material [24]. Additional, experimental investigation about mechanical behaviors of GFRP also made great advance. For example, SEM (scanning electron microscopy) in situ experiments of damage growth in GFRP composite under three-point bending loads were carried out. The dependence of mechanical parameters on the orientation angles of fibers was analyzed [25]. The tensile strength and fracture surface characterization of sized and unsized glass fibers were examined by single fiber tensile tests. The experimental tests clearly indicated that the unsized fibers were weaker in the low strength range, but had similar strength in the high strength range [26]. The interfacial shear strength between the fiber and the matrix of the fiber embedded matrix specimen was calculated by single fiber fragmentation test and fiber strengths of both sized and unsized fiber were found [27]. As a typical transverse isotropic material, the elastic properties of GFRP are characterized by five elastic constants. Fiber orientation with respect to loading direction is one of the most important parameters affecting mechanical properties of fiber reinforced composites. However, in case of Young’s modulus, there is less reported on this aspect from theory, experiment or numerical
E-mail address: [email protected] (H.W. Wang). 1359-8368/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.09.020
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simulation. In the present work, theoretical analysis, finite element models as well as experimental investigations were used to study the stiffness, i.e. Young’s modulus, of glass fiber reinforced composites. Composites cells with different glass fiber volume content were also simulated and the effect of shear modulus G on the Young’s modulus was also analyzed.
Thus, it is possible to evaluate the stiffness matrix of any fiber orientation angle, which is the angle between fiber direction and loading direction, according to transverse stiffness matrix for transverse isotropic material.
2. Theoretical analysis
A special subclass of orthotropy is transverse isotropy, which is characterized by a plane of isotropy at every point in the material. GFRP composites could be considered as transverse isotropy material macroscopically. Assuming the 2–3 plane to be the plane of isotropy at every point, transverse isotropy requires E2 = E3 = Ep, m12 = m13 = mtp, m21 = m31 = mpt and G21 = G31 = Gt, where p and t stand for ‘‘in-plane’’ and ‘‘transverse,’’ respectively. Thus, while mtp has the physical interpretation of the Poisson’s ratio that characterizes the strain in the plane of isotropy resulting from stress normal to it, mpt characterizes the transverse strain in the direction normal to the plane of isotropy resulting from stress in the plane of isotropy. In general, the quantities mtp is not equal to mpt and they are related by mtp/Et = mpt/Ep. The stress–strain laws reduce to:
2.1. Transformation of strain and stress Define the stress and strain vectors, in the coordinate system xoy, as shown in Fig. 1(a), as:
r ¼ ½r11 r22 r33 s12 s13 s23 T e ¼ ½e11 e22 e33 c12 c13 c23 T
ð1Þ ð2Þ
The stiffness matrix in the coordinate system xoy is D. Then the stress vector, strain vector and the stiffness matrix satisfy:
r ¼ De
ð3Þ
After a transformation, as shown in Fig. 1(c), we can obtain: 0
r ¼ T1 ðaÞr e0 ¼ T2 ðaÞe
ð4Þ ð5Þ
where,
2
cos2 a 6 6 sin2 a 6 6 0 6 T1 ðaÞ ¼ 6 6 1 sin 2a 6 2 6 0 4 0
2
sin a
0
cos2 a
0 sin 2a
0 1 2
sin 2a
1
sin 2a 0
0 cos 2a
0
0
0
0
0
0
0
0
0
0
3
7 7 7 0 0 7 7 7 0 0 7 7 7 cos a sin a 5 sin a cos a
2
0
0
0
0
sin a
ð7Þ
cos a
0 0 0
Then in the coordinate system x o y ,
r0 ¼ D0 ðaÞe0
8 e11 9 > > > > > > > > > e22 > > > > > > > < =
2
mpt =Ep 1=Ep mp =Ep 0 0 0
1=Et 6 mtp =Et 6 6 6 mtp =Et e33 ¼6 6 > > 0 c12 > > 6 > > > > 6 > > > > 4 0 c > > > : 13 > ; 0 c23
mpt =Et mp =Ep 1=Ep 0 0 0
ð8Þ
where,
2
0:067 0:02 0:02 0 0 6 0:02 0:1 0:025 0 0 6 6 6 0:02 0:025 0:1 0 0 C¼6 6 0 0 0 0:5 0 6 6 4 0 0 0 0 0:5 0
0
0
ð9Þ
(a)
0 0 0 0 1=Gt 0
38 0 r11 9 > > > > > > 7 > > 0 7> r22 > > > > > > 7> < 0 7 r33 = 7 > 0 7 r12 > > 7> > > > 7> > > 0 5> r13 > > > > > : ; 1=Gp r23
where Gp = Ep/2(1 + mp) and the total number of independent constants is five. The stability criterion requires that E > 0, G > 0 and 1 < m < 0.5. The value of Et can be taken from experiment data. Et took 15GPa in this study. Transverse isotropy materials have Et = E11 > E22 = E33 = Ep, Ep was assumed to be 10 GPa. According to typical mechanical parameters for transverse isotropy material, we took mp = 0.25, mtp = 0.3, and Gt = 2 GPa. Substituting these values into the flexibility matrix, then
The stiffness matrix D will be:
D0 ¼ T1 ðaÞDT1 2 ðaÞ
0 0 0 1=Gt 0 0
ð10Þ
ð6Þ 3 2 cos2 a sin a 0 12 sin 2a 0 0 7 6 6 sin2 a cos2 a 0 12 sin 2a 0 0 7 7 6 7 6 6 0 0 1 0 0 0 7 T2 ðaÞ ¼ 6 7 6 sin 2a sin 2a 0 cos 2a 0 0 7 7 6 7 6 4 0 0 0 0 cos a sin a 5
2.2. Calculation of transformed stiffness matrix
(b) Fig. 1. Definition of two coordinate systems.
(c)
0
0
0
3
0 7 7 7 0 7 7 0 7 7 7 0 5 0:25
ð11Þ
H.W. Wang et al. / Composites: Part B 56 (2014) 733–739
D ¼ C1
17:86 4:76 4:76 0 0 6 4:76 11:94 3:94 0 0 6 6 6 4:76 3:94 11:94 0 0 ¼6 6 0 0 0 2 0 6 6 4 0 0 0 0 2 0
0
0
0
735
3
2
0 07 7 7 07 7 07 7 7 05
ð12Þ
0 4
And the inversion of matrix T2(a) is,
2
2
cos2 a sin a 6 2 6 sin2 a cos a 6 6 6 0 0 T1 2 ðaÞ ¼ 6 6 sin 2a sin 2a 6 6 4 0 0 0
0 12 sin 2a
0
0
0
3
0
0
7 7 7 7 0 0 7 7 0 0 7 7 7 cos a sin a 5
0
0
sin a
0
0
0
1 2
sin 2a
1
0
0
cos 2a
Fig. 3. Young’s modulus vs. fiber orientation angle curves with different Gp values.
3. Numerical simulations
cos a ð13Þ
Substituting Eq. (13) into Eq. (9), we can get the global stiffness matrix of transverse isotropy GFRP. The first element is the Young’s modulus of loading direction, that is:
4
2
E0t ¼ 17:86 cos4 a þ 11:94 sin a þ 4:38 sin 2a
ð14Þ
Thus, the relationship between the composites’ Young’s modulus and fiber orientation angle can be shown in Fig. 2. From Fig. 2, it can be seen that the Young’s modulus of the composites strongly depends on the fiber orientation angle. When the angle is 0° (loading along fiber direction), the stiffness of the composites reaches its highest value. The lowest stiffness is when this angle around 60°. After this angle, the composites Young’s modulus increase slightly with the rise of fiber orientation angle. Moreover, it should be noted that the shear modulus Gp was found to have significant effect on the Young’s modulus while no similar effect was found for other parameters. Fig. 3 shows the relationship between the Young’s modulus and fiber orientation angle when Gp takes the values of 1 GPa, 2 GPa and 5 GPa. It can be seen from Fig. 3 that: (1) the shear modulus Gp almost have no difference to the Young’s modulus when fiber orientation angle equal 0° and 90°; (2) lower value of Gp led to lower Young’s modulus at the same fiber orientation angle; (3) the angle for the least Young’s modulus decreases with the smaller Gp.
Young's Modulus/GPa
18
3.1. Effect of fiber obliquity on composites’ Young’s modulus In order to explore the relationship between the Young’s modulus and fiber orientation in GFRP, models with different fiber orientation were developed. As we know, the representative volume element (RVE) plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties [18,28–32]. Here, the dimensions of the cell models, i.e. RVE, were 10 10 1 mm3. Single-ply fiber was placed into the cell uniformly. The fiber volume content is 20%. The unit cells were subject to a uniaxial tensile displacement along vertical direction (applied strain was 10%). The boundary conditions for all simulations were as follows: the bottom side of the cell model, all degree of freedoms of point (0, 0, 0) were fixed, the 3rd degree of freedom of bottom surface with coordinate z = 0 was restricted, and the 1st degree of freedom of line with x = 0 and z = 0 was restricted [33,34]. The following properties of the phases were used in the simulations. The glass fibers behaved as elastic isotropic solids, with Young modulus E = 72 GPa and Poisson’s ratio 0.26 [35]. The elastic properties of the epoxy matrix were as follows: Young modulus 3790 MPa, Poisson’s ratio 0.37 [36]. The cells were automatically produced in ABAQUS CAE by Scripting Interface. Three models with different fiber orientations, such as 15°, 45° and 75°, are shown in Fig. 4. The curve of Young’s modulus of composites vs. fiber obliquity is shown in Fig. 5(a). From Fig. 5, a similar trend as theoretical analysis is found, i.e. a U-shaped dependency of the Young modulus of composites on the fiber orientation angle is obtained. When the angle is 0° (loading along fiber direction), the stiffness of the composites reaches its highest value. The lowest stiffness is when this angle around 60°.
16 Calculated Young's modulus
3.2. Effect of fiber volume content on composites Young’s modulus
14
12
10 0
15
30
45
60
Fiber Orientation Angle/
75
90
o
Fig. 2. Relationship between the composites’ Young’s modulus and fiber orientation angle (0° means the along the fiber direction and 90° means perpendicular to the fiber direction).
As we know, the volume content of glass fiber has great effect on the properties of GFRP. The quantitative relationship between them is the foundation for design the composites. In order to get this relationship, cell models with different glass fiber volume content were developed. The dimensions of the cell models were 10 10 3 mm3. Three-ply fibers were placed into the cell uniformly. The boundary condition and material’s properties were the same as Section 3.1. Unit cells with 10% and 40% fiber volume content are shown in Fig. 6. The curve of Young’s modulus of composites vs. fiber volume content is shown in Fig. 7. The Mises stress distribution for one case with fiber volume content 40% is shown in Fig. 8
H.W. Wang et al. / Composites: Part B 56 (2014) 733–739
Composites' Young's Modulus/GPa
736
18
15
12
9
6
3 0
20
40
60
Fiber Orientation Angle/
80
100
o
Fig. 5. Simulated results of relationship between the composite Young’s modulus and fiber orientation angle.
Fig. 4. Unit cells with different fiber orientation angles: (a) 15°, (b) 45° and (c) 75°.
It can be seen from Fig. 7 that the relationship between Young’s modulus and fiber volume content is nearly linear. The fitted line results in 3.7947 GPa when fiber content equal 0, which agree with the Young’s modulus of matrix 3790 MPa. And the fitted line results in 72.03 GPa when the fiber content equal 100%, which agree with the Young’s modulus of fiber 72 GPa. That is to say the ruleof-mixture can be used when loading direction alone the fiber axes for fiber reinforced composites if the fiber/matrix interface is supposed as strong interface. From Fig. 8, it can be seen that the main load is carried by the glass fiber. The more glass fiber in the composites, the higher Young’s modulus of the composites.
Fig. 6. Unit cells with different fiber volume content: (a)10% and (b) 40%.
H.W. Wang et al. / Composites: Part B 56 (2014) 733–739
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provided by the Knowledge Centre of Wind Turbine Materials and Constructions (WMC), Netherlands. The specifications of these materials are as follows: identification number 595/c, system resin RIM 135, system hardener RIM/H 134/137, system mix ratio 100/30, vacuum curing 1000, post-cure 10 h at 70 °C, material density 194079, saertex weaver, curing tabs glue 2 h at 65 °C. All the specimens are prepared by water jet technology. In order to investigate the influence of fiber orientation angles on the mechanical behaviour of composite materials, the fibers were arranged at the orientation angles 0°, 15°, 30°, 45°, 60°, 75° and 90° with respect to horizontal direction. Three specimens were prepared for each orientation angle. There were 21 specimens in total, divided into 3 groups. The nominal dimensions of the specimens were 10 10 3 mm3, as shown in Fig. 9. Fig. 7. Composites Young’s modulus vs. fiber volume content.
4.2. Experiments and results In this experiment, the device employed was JSM-5410LV scanning electron microscope with electrohydraulic servo experimental system, which is shown in Fig. 10. The specimens were gild, and fixed onto the platform of the loading cell. The displacement-control mode was selected with the loading speed of 1.0 103 mm/s. Interval sampling and peak value sampling were carried out. Displacement and load data were recorded automatically by the test system. According the obtained displacement and load data, stress– strain curves can be achieved. Thus, the Young’s modulus can be calculated by:
E¼
Fig. 8. Mises stress distribution of the cell model with fiber volume content 40%.
Dr De
ð15Þ
The Young’s modulus of the specimens with different fiber orientation angles were calculated, and plotted in Fig. 11. It is of interest that similar results (i.e. U-shaped dependency of the Young modulus of composites on the orientation angles of fiber) were obtained in the FEM simulations carried out on polymer composites reinforced by cylindrical short fibers [37]. The difference between analytical, simulated and experimental results is that the fiber orientation angle is around 45° when Young’s modulus reaches its lowest value for experimental case. This is caused by: (1) the material properties for analytical, simulations and experiments are not completely the same; (2) for the experimental case, there are many facts, such as material fault and experimental conditions, affect the results. But for simulation and analysis, the model is simplified and some conditions are ideal. 5. Conclusion
Fig. 9. Dimensions of specimens (Loading direction is horizontal in these experiments. a is the fiber orientation angle. Unit: mm).
4. Experimental investigation 4.1. Materials and specimen The specimens in the paper named laminates GF/UD (4 layers, lay-up (90/0–0/90–90/0)S, fibers PPG 2002) were
The effect of fiber orientation on Young’s modulus for unidirectional GFRP was studied by theoretical analysis, finite element numerical simulations as well as experimental investigations. All results indicate that Young’s modulus of the composites strongly depends on the fiber orientation angles. A U-shaped dependency of the Young’s modulus of composites on the inclined angle of fiber is found. At the same time, there is slight difference among analytical, simulated and experimental results. The difference is that the fiber orientation angle is around 45° when Young’s modulus reaches its lowest value for experimental case and the fiber orientation angle is around 60° when Young’s modulus reaches its lowest value for numerical and analytical cases. This is caused by: (1) the material properties for analytical, simulations and experiments are not completely the same; (2) for the experimental case, there are many facts affect the results. But for simulation and analysis, the model is simplified and some conditions are ideal.
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SEM specimen room Electrohydraulic servo experimental system Specimen
Composites Young's modulus/Gpa
Fig. 10. JSM-5410LV SEM experimental system.
18 Group 1
16
Group 2 Group 3
14
Average
12 10 8 6 4 0
20
40
60
80
100
Fiber Orientation Angle/ o Fig. 11. Experimental results of relationship between the composite Young’s modulus and fiber orientation angle.
In addition, analytical results also indicate that the shear modulus Gp has significant effect on the composites Young’s modulus. Lower value of Gp led to lower Young’s modulus at the same fiber orientation angle and the angle for the least Young’s modulus decreases with the smaller Gp. simulated results indicate the relationship between Young’s modulus and fiber volume content is nearly linear. The more glass fiber in the composites, the higher Young’s modulus of the composites. The rule-of-mixture can be used when loading direction alone the fiber axes for fiber reinforced composites if the fiber/matrix interface is supposed as strong interface.
Acknowledgments This work is supported by Program of International S&T Cooperation, MOST (Grant No. 2010DFA64560). Danish side partner of this international S&T cooperation program is Technical University of Denmark (DTU). The authors would like to express their special thanks to Dr. Leon Mishnaevsky Jr. for the discussions and communications. The support by the National Natural Science Foundation of China (Grant Nos. 11072176, 11002100) and Tianjin Natural Science Foundation (No. 11JCYBJC26800) are greatly acknowledged.
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Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Analysis of effect of fiber orientation on Young’s modulus for unidirectional fiber reinforced composites H.W. Wang a,⇑, H.W. Zhou b, L.L. Gui b, H.W. Ji a, X.C. Zhang a a b
Tianjin Key Laboratory of Refrigeration Technology, Tianjin University of Commerce, Tianjin 300134, China State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 17 April 2013 Received in revised form 4 September 2013 Accepted 7 September 2013 Available online 17 September 2013 Keywords: A. Glass fibers B. Mechanical properties C. Analytical modeling C. Numerical analysis
a b s t r a c t Young’s modulus of unidirectional glass fiber reinforced polymer (GFRP) composites for wind energy applications were studied using analytical, numerical and experimental methods. In order to explore the effect of fiber orientation angle on the Young’s modulus of composites, from the basic theory of elastic mechanics, a procedure which can be applied to evaluate the elastic stiffness matrix of GFRP composite as an analytical function of fiber orientation angle (from 0° to 90°), was developed. At the same time, different finite element models with inclined glass fiber were developed via the ABAQUS Scripting Interface. Results indicate that Young’s modulus of the composites strongly depends on the fiber orientation angles. A U-shaped dependency of the Young’s modulus of composites on the inclined angle of fiber is found, which agree well with the experimental results. The shear modulus is found to have significant effect on the composites’ Young’s modulus, too. The effect of volume content of glass fiber on the Young’s modulus of composites was investigated. Results indicate the relation between them is nearly linear. The results of the investigation are expected to provide some design guideline for the microstructural optimization of the glass fiber reinforced composites. Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduction Polymer materials reinforced with glass fiber have received tremendous attention in both scientific and industrial communities due to their extraordinary enhanced properties, such as lower weight, higher toughness and higher strength characteristics. In response to these requirements, research on composites has attracted much attention, which results in numerous publications [1–11]. In order to evaluate the mechanical behaviors of composites materials, different approaches, including experimental investigation, numerical simulations and theoretical modeling, were employed [12–16]. For example, the fiber bundle model has been used to study the damage behaviors of fiber while loading along the fibers [17]. The numerical continuum mechanical models, such as finite element models, allow the incorporation of many different features of the nonlinear material behaviors and the analysis of the interaction of available and evolving microstructural elements [18]. Many computational experiments of damage and failure in composites have been done by employing numerical continuum mechanical models [19–22]. The shear lag and other analytical models based on simplifying assumptions are applicable mainly ⇑ Corresponding author. Tel.: +86 13512806141.
to the linear elastic material behaviors and relatively simple, periodic microgeometrics. They are often used to analyze the load transfer and multiple cracking in composites [23]. The fracture mechanics-based models are often used to the cases of fiber bridging analysis of elastic or homogeneous material [24]. Additional, experimental investigation about mechanical behaviors of GFRP also made great advance. For example, SEM (scanning electron microscopy) in situ experiments of damage growth in GFRP composite under three-point bending loads were carried out. The dependence of mechanical parameters on the orientation angles of fibers was analyzed [25]. The tensile strength and fracture surface characterization of sized and unsized glass fibers were examined by single fiber tensile tests. The experimental tests clearly indicated that the unsized fibers were weaker in the low strength range, but had similar strength in the high strength range [26]. The interfacial shear strength between the fiber and the matrix of the fiber embedded matrix specimen was calculated by single fiber fragmentation test and fiber strengths of both sized and unsized fiber were found [27]. As a typical transverse isotropic material, the elastic properties of GFRP are characterized by five elastic constants. Fiber orientation with respect to loading direction is one of the most important parameters affecting mechanical properties of fiber reinforced composites. However, in case of Young’s modulus, there is less reported on this aspect from theory, experiment or numerical
E-mail address: [email protected] (H.W. Wang). 1359-8368/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.09.020
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simulation. In the present work, theoretical analysis, finite element models as well as experimental investigations were used to study the stiffness, i.e. Young’s modulus, of glass fiber reinforced composites. Composites cells with different glass fiber volume content were also simulated and the effect of shear modulus G on the Young’s modulus was also analyzed.
Thus, it is possible to evaluate the stiffness matrix of any fiber orientation angle, which is the angle between fiber direction and loading direction, according to transverse stiffness matrix for transverse isotropic material.
2. Theoretical analysis
A special subclass of orthotropy is transverse isotropy, which is characterized by a plane of isotropy at every point in the material. GFRP composites could be considered as transverse isotropy material macroscopically. Assuming the 2–3 plane to be the plane of isotropy at every point, transverse isotropy requires E2 = E3 = Ep, m12 = m13 = mtp, m21 = m31 = mpt and G21 = G31 = Gt, where p and t stand for ‘‘in-plane’’ and ‘‘transverse,’’ respectively. Thus, while mtp has the physical interpretation of the Poisson’s ratio that characterizes the strain in the plane of isotropy resulting from stress normal to it, mpt characterizes the transverse strain in the direction normal to the plane of isotropy resulting from stress in the plane of isotropy. In general, the quantities mtp is not equal to mpt and they are related by mtp/Et = mpt/Ep. The stress–strain laws reduce to:
2.1. Transformation of strain and stress Define the stress and strain vectors, in the coordinate system xoy, as shown in Fig. 1(a), as:
r ¼ ½r11 r22 r33 s12 s13 s23 T e ¼ ½e11 e22 e33 c12 c13 c23 T
ð1Þ ð2Þ
The stiffness matrix in the coordinate system xoy is D. Then the stress vector, strain vector and the stiffness matrix satisfy:
r ¼ De
ð3Þ
After a transformation, as shown in Fig. 1(c), we can obtain: 0
r ¼ T1 ðaÞr e0 ¼ T2 ðaÞe
ð4Þ ð5Þ
where,
2
cos2 a 6 6 sin2 a 6 6 0 6 T1 ðaÞ ¼ 6 6 1 sin 2a 6 2 6 0 4 0
2
sin a
0
cos2 a
0 sin 2a
0 1 2
sin 2a
1
sin 2a 0
0 cos 2a
0
0
0
0
0
0
0
0
0
0
3
7 7 7 0 0 7 7 7 0 0 7 7 7 cos a sin a 5 sin a cos a
2
0
0
0
0
sin a
ð7Þ
cos a
0 0 0
Then in the coordinate system x o y ,
r0 ¼ D0 ðaÞe0
8 e11 9 > > > > > > > > > e22 > > > > > > > < =
2
mpt =Ep 1=Ep mp =Ep 0 0 0
1=Et 6 mtp =Et 6 6 6 mtp =Et e33 ¼6 6 > > 0 c12 > > 6 > > > > 6 > > > > 4 0 c > > > : 13 > ; 0 c23
mpt =Et mp =Ep 1=Ep 0 0 0
ð8Þ
where,
2
0:067 0:02 0:02 0 0 6 0:02 0:1 0:025 0 0 6 6 6 0:02 0:025 0:1 0 0 C¼6 6 0 0 0 0:5 0 6 6 4 0 0 0 0 0:5 0
0
0
ð9Þ
(a)
0 0 0 0 1=Gt 0
38 0 r11 9 > > > > > > 7 > > 0 7> r22 > > > > > > 7> < 0 7 r33 = 7 > 0 7 r12 > > 7> > > > 7> > > 0 5> r13 > > > > > : ; 1=Gp r23
where Gp = Ep/2(1 + mp) and the total number of independent constants is five. The stability criterion requires that E > 0, G > 0 and 1 < m < 0.5. The value of Et can be taken from experiment data. Et took 15GPa in this study. Transverse isotropy materials have Et = E11 > E22 = E33 = Ep, Ep was assumed to be 10 GPa. According to typical mechanical parameters for transverse isotropy material, we took mp = 0.25, mtp = 0.3, and Gt = 2 GPa. Substituting these values into the flexibility matrix, then
The stiffness matrix D will be:
D0 ¼ T1 ðaÞDT1 2 ðaÞ
0 0 0 1=Gt 0 0
ð10Þ
ð6Þ 3 2 cos2 a sin a 0 12 sin 2a 0 0 7 6 6 sin2 a cos2 a 0 12 sin 2a 0 0 7 7 6 7 6 6 0 0 1 0 0 0 7 T2 ðaÞ ¼ 6 7 6 sin 2a sin 2a 0 cos 2a 0 0 7 7 6 7 6 4 0 0 0 0 cos a sin a 5
2.2. Calculation of transformed stiffness matrix
(b) Fig. 1. Definition of two coordinate systems.
(c)
0
0
0
3
0 7 7 7 0 7 7 0 7 7 7 0 5 0:25
ð11Þ
H.W. Wang et al. / Composites: Part B 56 (2014) 733–739
D ¼ C1
17:86 4:76 4:76 0 0 6 4:76 11:94 3:94 0 0 6 6 6 4:76 3:94 11:94 0 0 ¼6 6 0 0 0 2 0 6 6 4 0 0 0 0 2 0
0
0
0
735
3
2
0 07 7 7 07 7 07 7 7 05
ð12Þ
0 4
And the inversion of matrix T2(a) is,
2
2
cos2 a sin a 6 2 6 sin2 a cos a 6 6 6 0 0 T1 2 ðaÞ ¼ 6 6 sin 2a sin 2a 6 6 4 0 0 0
0 12 sin 2a
0
0
0
3
0
0
7 7 7 7 0 0 7 7 0 0 7 7 7 cos a sin a 5
0
0
sin a
0
0
0
1 2
sin 2a
1
0
0
cos 2a
Fig. 3. Young’s modulus vs. fiber orientation angle curves with different Gp values.
3. Numerical simulations
cos a ð13Þ
Substituting Eq. (13) into Eq. (9), we can get the global stiffness matrix of transverse isotropy GFRP. The first element is the Young’s modulus of loading direction, that is:
4
2
E0t ¼ 17:86 cos4 a þ 11:94 sin a þ 4:38 sin 2a
ð14Þ
Thus, the relationship between the composites’ Young’s modulus and fiber orientation angle can be shown in Fig. 2. From Fig. 2, it can be seen that the Young’s modulus of the composites strongly depends on the fiber orientation angle. When the angle is 0° (loading along fiber direction), the stiffness of the composites reaches its highest value. The lowest stiffness is when this angle around 60°. After this angle, the composites Young’s modulus increase slightly with the rise of fiber orientation angle. Moreover, it should be noted that the shear modulus Gp was found to have significant effect on the Young’s modulus while no similar effect was found for other parameters. Fig. 3 shows the relationship between the Young’s modulus and fiber orientation angle when Gp takes the values of 1 GPa, 2 GPa and 5 GPa. It can be seen from Fig. 3 that: (1) the shear modulus Gp almost have no difference to the Young’s modulus when fiber orientation angle equal 0° and 90°; (2) lower value of Gp led to lower Young’s modulus at the same fiber orientation angle; (3) the angle for the least Young’s modulus decreases with the smaller Gp.
Young's Modulus/GPa
18
3.1. Effect of fiber obliquity on composites’ Young’s modulus In order to explore the relationship between the Young’s modulus and fiber orientation in GFRP, models with different fiber orientation were developed. As we know, the representative volume element (RVE) plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties [18,28–32]. Here, the dimensions of the cell models, i.e. RVE, were 10 10 1 mm3. Single-ply fiber was placed into the cell uniformly. The fiber volume content is 20%. The unit cells were subject to a uniaxial tensile displacement along vertical direction (applied strain was 10%). The boundary conditions for all simulations were as follows: the bottom side of the cell model, all degree of freedoms of point (0, 0, 0) were fixed, the 3rd degree of freedom of bottom surface with coordinate z = 0 was restricted, and the 1st degree of freedom of line with x = 0 and z = 0 was restricted [33,34]. The following properties of the phases were used in the simulations. The glass fibers behaved as elastic isotropic solids, with Young modulus E = 72 GPa and Poisson’s ratio 0.26 [35]. The elastic properties of the epoxy matrix were as follows: Young modulus 3790 MPa, Poisson’s ratio 0.37 [36]. The cells were automatically produced in ABAQUS CAE by Scripting Interface. Three models with different fiber orientations, such as 15°, 45° and 75°, are shown in Fig. 4. The curve of Young’s modulus of composites vs. fiber obliquity is shown in Fig. 5(a). From Fig. 5, a similar trend as theoretical analysis is found, i.e. a U-shaped dependency of the Young modulus of composites on the fiber orientation angle is obtained. When the angle is 0° (loading along fiber direction), the stiffness of the composites reaches its highest value. The lowest stiffness is when this angle around 60°.
16 Calculated Young's modulus
3.2. Effect of fiber volume content on composites Young’s modulus
14
12
10 0
15
30
45
60
Fiber Orientation Angle/
75
90
o
Fig. 2. Relationship between the composites’ Young’s modulus and fiber orientation angle (0° means the along the fiber direction and 90° means perpendicular to the fiber direction).
As we know, the volume content of glass fiber has great effect on the properties of GFRP. The quantitative relationship between them is the foundation for design the composites. In order to get this relationship, cell models with different glass fiber volume content were developed. The dimensions of the cell models were 10 10 3 mm3. Three-ply fibers were placed into the cell uniformly. The boundary condition and material’s properties were the same as Section 3.1. Unit cells with 10% and 40% fiber volume content are shown in Fig. 6. The curve of Young’s modulus of composites vs. fiber volume content is shown in Fig. 7. The Mises stress distribution for one case with fiber volume content 40% is shown in Fig. 8
H.W. Wang et al. / Composites: Part B 56 (2014) 733–739
Composites' Young's Modulus/GPa
736
18
15
12
9
6
3 0
20
40
60
Fiber Orientation Angle/
80
100
o
Fig. 5. Simulated results of relationship between the composite Young’s modulus and fiber orientation angle.
Fig. 4. Unit cells with different fiber orientation angles: (a) 15°, (b) 45° and (c) 75°.
It can be seen from Fig. 7 that the relationship between Young’s modulus and fiber volume content is nearly linear. The fitted line results in 3.7947 GPa when fiber content equal 0, which agree with the Young’s modulus of matrix 3790 MPa. And the fitted line results in 72.03 GPa when the fiber content equal 100%, which agree with the Young’s modulus of fiber 72 GPa. That is to say the ruleof-mixture can be used when loading direction alone the fiber axes for fiber reinforced composites if the fiber/matrix interface is supposed as strong interface. From Fig. 8, it can be seen that the main load is carried by the glass fiber. The more glass fiber in the composites, the higher Young’s modulus of the composites.
Fig. 6. Unit cells with different fiber volume content: (a)10% and (b) 40%.
H.W. Wang et al. / Composites: Part B 56 (2014) 733–739
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provided by the Knowledge Centre of Wind Turbine Materials and Constructions (WMC), Netherlands. The specifications of these materials are as follows: identification number 595/c, system resin RIM 135, system hardener RIM/H 134/137, system mix ratio 100/30, vacuum curing 1000, post-cure 10 h at 70 °C, material density 194079, saertex weaver, curing tabs glue 2 h at 65 °C. All the specimens are prepared by water jet technology. In order to investigate the influence of fiber orientation angles on the mechanical behaviour of composite materials, the fibers were arranged at the orientation angles 0°, 15°, 30°, 45°, 60°, 75° and 90° with respect to horizontal direction. Three specimens were prepared for each orientation angle. There were 21 specimens in total, divided into 3 groups. The nominal dimensions of the specimens were 10 10 3 mm3, as shown in Fig. 9. Fig. 7. Composites Young’s modulus vs. fiber volume content.
4.2. Experiments and results In this experiment, the device employed was JSM-5410LV scanning electron microscope with electrohydraulic servo experimental system, which is shown in Fig. 10. The specimens were gild, and fixed onto the platform of the loading cell. The displacement-control mode was selected with the loading speed of 1.0 103 mm/s. Interval sampling and peak value sampling were carried out. Displacement and load data were recorded automatically by the test system. According the obtained displacement and load data, stress– strain curves can be achieved. Thus, the Young’s modulus can be calculated by:
E¼
Fig. 8. Mises stress distribution of the cell model with fiber volume content 40%.
Dr De
ð15Þ
The Young’s modulus of the specimens with different fiber orientation angles were calculated, and plotted in Fig. 11. It is of interest that similar results (i.e. U-shaped dependency of the Young modulus of composites on the orientation angles of fiber) were obtained in the FEM simulations carried out on polymer composites reinforced by cylindrical short fibers [37]. The difference between analytical, simulated and experimental results is that the fiber orientation angle is around 45° when Young’s modulus reaches its lowest value for experimental case. This is caused by: (1) the material properties for analytical, simulations and experiments are not completely the same; (2) for the experimental case, there are many facts, such as material fault and experimental conditions, affect the results. But for simulation and analysis, the model is simplified and some conditions are ideal. 5. Conclusion
Fig. 9. Dimensions of specimens (Loading direction is horizontal in these experiments. a is the fiber orientation angle. Unit: mm).
4. Experimental investigation 4.1. Materials and specimen The specimens in the paper named laminates GF/UD (4 layers, lay-up (90/0–0/90–90/0)S, fibers PPG 2002) were
The effect of fiber orientation on Young’s modulus for unidirectional GFRP was studied by theoretical analysis, finite element numerical simulations as well as experimental investigations. All results indicate that Young’s modulus of the composites strongly depends on the fiber orientation angles. A U-shaped dependency of the Young’s modulus of composites on the inclined angle of fiber is found. At the same time, there is slight difference among analytical, simulated and experimental results. The difference is that the fiber orientation angle is around 45° when Young’s modulus reaches its lowest value for experimental case and the fiber orientation angle is around 60° when Young’s modulus reaches its lowest value for numerical and analytical cases. This is caused by: (1) the material properties for analytical, simulations and experiments are not completely the same; (2) for the experimental case, there are many facts affect the results. But for simulation and analysis, the model is simplified and some conditions are ideal.
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SEM specimen room Electrohydraulic servo experimental system Specimen
Composites Young's modulus/Gpa
Fig. 10. JSM-5410LV SEM experimental system.
18 Group 1
16
Group 2 Group 3
14
Average
12 10 8 6 4 0
20
40
60
80
100
Fiber Orientation Angle/ o Fig. 11. Experimental results of relationship between the composite Young’s modulus and fiber orientation angle.
In addition, analytical results also indicate that the shear modulus Gp has significant effect on the composites Young’s modulus. Lower value of Gp led to lower Young’s modulus at the same fiber orientation angle and the angle for the least Young’s modulus decreases with the smaller Gp. simulated results indicate the relationship between Young’s modulus and fiber volume content is nearly linear. The more glass fiber in the composites, the higher Young’s modulus of the composites. The rule-of-mixture can be used when loading direction alone the fiber axes for fiber reinforced composites if the fiber/matrix interface is supposed as strong interface.
Acknowledgments This work is supported by Program of International S&T Cooperation, MOST (Grant No. 2010DFA64560). Danish side partner of this international S&T cooperation program is Technical University of Denmark (DTU). The authors would like to express their special thanks to Dr. Leon Mishnaevsky Jr. for the discussions and communications. The support by the National Natural Science Foundation of China (Grant Nos. 11072176, 11002100) and Tianjin Natural Science Foundation (No. 11JCYBJC26800) are greatly acknowledged.
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