Modeling of tensile and bending properties of biaxial weft knitted composites

1 Introduction

Due to the potential benefits of cost-effective manufacturing of knitting with advanced fibers, such as glass and aramid, to produce near-net-shape preforms, it has, in recent years, received increasing interest [1].

Knitted composites are generally considered to have inferior mechanical properties due to their highly looped structure and low fiber volume fraction. However, attractive properties, such as high energy absorption or good impact resistance or in cases where the component is complex in shape and demands exceptional formability, can be achieved by using knitted composites [2]. At a constant fiber volume fraction, introduction of in-lay yarns can significantly improve the properties of knitting composite, as was reported by Leong et al. [3].

Knitted fabrics are basically categorized into two types, namely, warp knitted fabrics and weft knitted fabrics. Warp knitted fabrics are produced by knitting in the lengthwise direction (wale direction) of fabrics. Weft knitted fabrics are produced by knitting in the horizontal direction (course direction) of fabrics [4]. Biaxial weft knitted (BWK) fabrics include weft and warp yarn layers, which are held together by a stitching yarn system. Reinforcing yarns, such as glass or aramid fibers, can be used within all yarn systems. The strength and the stiffness of composite can be improved by reinforcing yarns [5].

Knitted fabrics have an orientation of yarns not only in thickness but also in fabric plane. In the case of woven fabrics, they have crimp on the reinforcement yarns in the thickness direction. Therefore, the orientation of yarns in knitted fabric is more complex in comparison with that in woven fabrics. Finite-element analysis is a well-known method for the analysis of complex construction with the complex material properties [6]. Therefore, we used finite-element modeling (FEM) to express the possibility of predicting the mechanical properties of complex BWK composite structures. This model consists of several steps: (a) building the model of geometry of the BWK structure, (b) building the material properties of the yarns and resin, (c) building the boundary conditions of the BWK composites, and (d) modeling the mechanical properties of the structure [7].

Contributions on the modeling of textile composites have been reported in the literature [8–11]. Sheng and Hoa [12] studied the three-dimensional (3D) micromechanical modeling of woven fabric composites. They predicted the elastic constants of the woven fabric composites by using the potential energy method [12]. Sheng and Hoa [13] also proposed a 3D angle interlock modeling for woven fabric composites. They predicted the elastic constants of 3D angle interlock woven fabric composites. Their results obtained from modeling agreed well with experimental values [13].

Verpoest and Lomov [14] studied the virtual textile composite software WiseTex: Integration with micromechanical, permeability, and structural analysis. They developed a model and software to describe the internal geometry and properties of textile fabrics and composites on a unit cell level [14]. Lomov et al. [15, 16] studied full-field strain measurements for validation of meso-finite element (FE) analysis of textile composites. They developed a mesoscale (unit cell of an impregnated textile reinforcement) FE modeling of the textile composites to determine the mechanical properties of the composites [15, 16].

In the literature, contributions on the modeling of knitted composites have been reported [17–19]. Ramakrishna et al. [20] developed an analytical method for the prediction of the elastic properties of plain knitted composites. They found good agreement between the experimental and analytical results. Araújo et al. [21] studied modeling of the mechanical behavior of weft knitted composite materials. In this research, tensile modulus was calculated both numerically and experimentally. However, the modeling and experimental results yielded differences of up to 40%.

There were a few studies in the literature that predicted the bending properties of the composites using modeling. Nishiwaki et al. [22–26] studied the quasi 3D model to predict the bending properties of fiber-reinforced composites. They found that the quasi 3D model was effective in predicting the mechanical behavior (bending properties) of heterogeneous composites [22–26]. However, no research about the numerical analysis of tensile and bending properties of the BWK composites was found in the literature.

There were only a few contributions on the modeling of BWK composites. Li et al. [27] developed a FEM modeling for the calculation of the impact properties of the 3D BWK composites. They achieved good agreement between the experimental results and the FEM calculation.

The bending modulus of a composite could be predicted by using the laminate beam theory. However, in the literature, no research on the calculation of the bending modulus of the BWK composites by using the laminate beam theory was found. Baba et al. [28] investigated the prediction of modulus of injection molded carbon fiber (CF)/liquid crystal polymer (LCP) thin plates. They used the laminate beam theory to calculate the bending modulus in the molded plate. They found that the calculated values were in good agreement with the experimental values.

Recent studies have shown that the tensile strength of the BWK composite with a fiber combination of glass-glass-glass (315.3 MPa) [29] in the course direction was around four times higher than that with the glass weft knitted fabric composites (75.8 MPa) [30]. In addition, the total absorbed impact energy of the BWK composites with the glass-glass-glass fibers (46.0 J) based on the plate bending impact test [31] was also more than three times higher than that with the glass weft knitted fabric composites (13.9 J) [32].

Because the fabrication method of BWK fabrics is comparatively very new compared with that of traditional knitting fabrics, it was necessary to characterize the mechanical properties of composites with the BWK fabric both numerically and experimentally. The present work investigated the tensile and bending properties of four types of BWK composite both experimentally and numerically: (1) BWK composite with the glass stitch yarn, (2) BWK composite with the aramid stitch yarn, (3) BWK composite with the glass biaxial yarn, and (4) BWK composite with the aramid biaxial yarn. The obtained results of the tensile and three-point bending tests can be used to design new textile preforms during the development of different composite materials. The validity of our model, with results and discussions, is presented below.

2 Materials and methods

2.1 Composite constituents

Three types of BWK fabrics were produced on a flat-bed knitting machine (SHIMA SEIKI MFG., Ltd., Japan). Figure 1 depicts the fabricated BWK reinforcement fabric. 520 tex E-glass yarn (Nippon Electric Glass Co. Ltd., Japan) and 110 tex aramid yarn (Kevlar-29; Dupont-Toray Co. Ltd., Japan) were used as a biaxial material. 68 tex E-Glass (ECG 75 1/0 1.0 OZ; Nittobo Co. Ltd., Japan) and 28/2 Ne (21 tex) aramid (Kevlar-29, Dupont-Toray Co. Ltd., Japan) were used as stitch yarns (Table 1). Vinyl ester resin (Ripoxy R-806; Showa High Polymer Co. Ltd., Japan) was used as matrix.

Table 1

Parameters of the BWK fabric.

Sample name	Biaxial yarn		Stitch yarn	Density of warp yarn in fabric (end/cm)	Density of weft yarn in fabric (end/cm)	Loop density in the wale direction (loop/cm)
	Warp yarn	Weft yarn
GF-GF-GF	GF	GF	GF	2.0	4.6	6.0
(one layer)	520 T	520 T	68 T
GF-GF-AR	GF	GF	AR	2.0	6.3	6.5
(one layer)	520 T	520 T	28/2 Ne (21 T)
GF-GF-AR	GF	GF	AR	2.0	6.3	6.5
(six layers)	520 T	520 T	28/2 Ne (21 T)
AR-AR-AR	AR	AR	AR	2.1	6.3	6.0
(six layers)	110 T	110 T	28/2 Ne (21 T)

T, Tex.

Figure 1

Photographs of BWK reinforcement fabrics.

(A) GF-GF-GF. (B) GF-GF-AR. (C) AR-AR-AR.

2.3 Mechanical characterization

Tensile and three-point bending tests were conducted on the specimens according to ASTM-D303 and ASTM-D790 standards; the measurements were performed using universal testing machine type 55R4206 (Instron) under displacement control with a speed of 1 mm/min. Figure 2A and B shows the geometry of the specimen from the tensile and three-point bending tests. In this figure, the lamina and the aluminum thicknesses are shown with t_c and t_Al, respectively. The thickness of the aluminum tabs was 0.5 mm. The composite coupons have a nominal dimension: (a) 200 mm×20 mm for the tensile test and (b) 90 mm×15 mm for the bending test. The composite coupons were cut from produced panels parallel to the wale and course directions. Strain gages with 10-mm gage length were used to measure the tensile strain. There was no variation within a single specimen. All specimens were gripped with the same squeezing condition. A 10.000-N load cell was used for the tensile test, and 500 N for the three-point bending test. The tensile and bending measurements were performed at ambient conditions of 23±2°C and 50%±5% relative humidity. The test span length was 70 mm in the three-point bending test. Between two and three specimens for the tensile test and two specimens for the three-point bending test were tested.

Figure 2

Geometry of the specimen.

(A) Tensile test. (B) Three-point bending test.

2.4 Results of experiments

2.4.1 Results of the tensile test

Due to the higher loop density of aramid stitch yarn compared with GF stitch yarn in the wale direction, the final composite thickness of the GF-GF-AR specimen (thickness=0.9 mm) was higher than that of the GF-GF-GF specimen (thickness=0.7 mm) (Table 1). Table 2 demonstrates the mean and SD tensile strength and modulus of one-layer composites in the course and wale directions. The tensile modulus of the specimens in the course direction was higher than that in the wale direction. This was caused by the different warp and weft yarn densities in the BWK (Table 1). In the GF-GF-GF specimen, the weft yarn density was 2.3 times higher than the warp yarn density. In the GF-GF-AR specimen, the weft yarn density was 3.2 times higher than the warp yarn density. The tensile modulus and strength of the GF-GF-GF specimen with the glass stitch yarn were higher than those of the GF-GF-AR specimen with the aramid stitch yarn in the wale and course directions.

Table 2

Experimental results from the tensile test.

Samples	Tensile modulus (GPa)	Tensile strength (MPa)
GF-GF-GF (C)	14.3±0.2	239±18
GF-GF-AR (C)	13.3±0.1	185±24
GF-GF-GF (W)	11.1±0.5	119±3.9
GF-GF-AR (W)	8.40±0.1	94.9±13

Data are presented as mean±SD. C, course; W, wale.

The calculated fiber volume fraction of the AR stitch fiber (8.1%) was lower than that of the GF stitch fiber (18%). This is the reason for the lower tensile modulus results from the experiment of composites with the AR stitch fiber (8.4 and 13.3 GPa, respectively) compared with the GF stitch fiber (11.1 and 14.3 GPa, respectively) in the wale and course directions. The volume fraction of each stich fiber bundle was calculated as follows.

Resin was involved in the fiber element. The area of fiber bundles from the catalog value was divided by the area of fiber bundles from the cross-sectional observation. Then, the volume fraction of the stitch fiber bundles was obtained. The volume fraction of the GF stitch fiber bundle (18%) was calculated: The area of the 68 tex GF fiber bundle (0.027 mm²) from the catalog value was divided by the area of the GF fiber bundle (0.15 mm²) from the cross-sectional observation.

2.4.2 Results of three-point bending test

Table 3 demonstrates the mean and SD bending strength and modulus of the six-layer composites in the course direction. Both the modulus and the strength of the composites were higher in the GF-GF-AR specimen compared with the AR-AR-AR specimen. The bending modulus results from the experiments of the composites with the AR biaxial fiber (4.4 GPa) was lower compared with the GF biaxial fiber (9.8 GPa). The volume fraction of the composites with the AR biaxial fiber (16.7%) was lower than that with the GF biaxial fiber (27.9%). This is the reason for the higher bending modulus obtained with the GF biaxial fiber compared with the AR biaxial fiber. The average bending strength of the GF-GF-AR composite was found to be 213.3 MPa, which was 35.5% higher than the bending strength of the AR-AR-AR composite. Hence, bending strength and stiffness were controlled by the strength of the reinforcement fibers. It can be expected that more fiber breakages occurred in the specimen with the GF biaxial yarn compared with the AR biaxial yarn.

Table 3

Experimental results from the three-point bending test.

Samples	Bending modulus (GPa)	Bending strength (MPa)
GF-GF-AR (C)	9.80±0.6	213.3±4.5
AR-AR-AR (C)	4.40±0.1	137.5±2.9

Data are presented as mean±SD. C, course.

Figure 3A shows the load-displacement curves of the BWK composites during the three-point bending test. The area under the load-displacement curves gives the absorbed energy during three-point bending test. The initiation energy was determined to calculate the area under the load-displacement curve until the maximum load and the propagation energy after the maximum load (Figure 3A). The energy results from the three-point bending test are shown in Figure 3B. It can be seen that the area under the stress-strain curve of the GF biaxial yarn was about 1.7 times higher than that with the AR biaxial yarns. This means that the specimen with the GF biaxial yarn absorbed more than 1.7 times higher total energy than did the specimen with the AR biaxial yarn during the three-point bending test. The maximum load with the GF biaxial yarn (1078 N) was also about 1.7 times higher than that with the AR biaxial yarn (619.7 N).

Figure 3

(A) Load-displacement curves of BWK composites during the three-point bending test and (B) energy results from the three-point bending test.

3 Finite-element modeling

3.1 FEM of tensile properties

Finite-element methods for the tensile tests were performed using the MARC software on a personal computer with Windows 7 operating system. The tensile modulus of the one-layer composites of the GF-GF-GF and GF-GF-AR composites was calculated by the FEM (E_FEM). Two different modeling procedures with two different boundary conditions were performed by applying the tensile loads in the wale and course directions of specimens: (a) tensile load was applied in the wale direction in the GF-GF-AR specimen, (b) tensile load was applied in the course direction in the GF-GF-AR specimen, (c) tensile load was applied in the wale direction in the GF-GF-GF specimen, and (d) tensile load was applied in the course direction in the GF-GF-GF specimen. Figure 4A (left) and B (left) demonstrates the actual structure of the composite after consolidation. The FEM includes repeating unit cell structure of the BWK composite. The FEM consists of 3D beam elements. Each FEM modeling consists of 6886 elements and 2520 nodes. The straight yarn in the warp (vertical) and the weft (horizontal) directions, the loop yarns, and the resin (surface) were modeled as depicted in Figure 4A (right) and B (right). Another resin element, which was called the resin (cross of fiber bundle), was also used between the fiber bundles. Under consideration of the geometrical differences of the modeling of the specimens, tensile modulus was calculated. The length of the sinker loop in the GF-GF-AR specimen (L1=2.2 mm) (Figure 4A) was longer than that in the GF-GF-GF specimen (L2=1.5 mm) (Figure 4B). In the wale direction, the length of the repetitive unit cell (4.3 mm) in the GF-GF-GF specimen was longer than the length of the repetitive unit cell in the GF-GF-AR specimen (3.26 mm). In the course direction, the length of the repetitive unit cell (8.89 mm) in the GF-GF-GF specimen was shorter than the length of the repetitive unit cell in the GF-GF-AR specimen (9.84 mm). According to the observations of laminas, warp yarns in the GF-GF-AR specimen were placed in the middle of the sinker loops, which are shown in Figure 4A (right). However, warp yarns were placed in the middle of the needle loops in the GF-GF-GF specimen, which are shown in Figure 4B (right).

Figure 4

(A) Actual structure of the composite in the wale direction (left) and the FEM of the GF-GF-AR composite in the wale direction (right); (B) actual structure of the composite in the wale direction (left) and the FEM of the GF-GF-GF composite in the wale direction (right).

The internal structures of the warp, the weft, and the stitch fiber bundle elements are presented in Table 4. In both laminas, the cross-sectional area of the warp fiber bundle elements was about one and half times higher than the cross-sectional area of the weft fiber bundle elements (Table 4). The cross-sections in the warp direction of the GF-GF-AR and GF-GF-GF specimens are shown in Figure 5A and B. Based on the cross-sectional observations, the shapes of the warp, the weft, and the stitch fiber bundles were expressed as a rectangular shape. The geometric properties of the fiber bundle elements were calculated with the help of the internal structure. The width and the thickness of all fiber bundle elements of modeling were determined using the aspect ratio. The aspect ratio can be calculated by dividing the major axis by the minor axis of the cross-section of fiber bundles, which is shown in Figure 5A. The moments of inertia of fiber bundle elements were calculated with the obtained results of the width and the thickness.

Table 4

Internal structure of warp, weft, and stitch yarns.

GF-GF-GF and GF-GF-AR	Major axis (mm)	Minor axis (mm)	Aspect ratio (major axis/minor axis)	Area (mm2)
Warp fiber bundle (GF)	2.43	0.31	7.82	0.48 (1.5 times higher than weft fiber bundles)
Weft fiber bundle (GF)	1.67	0.26	6.34	0.33
Stitch fiber bundle (GF)	0.55	0.35	1.57	0.15
Stitch fiber bundle (AR)	0.71	0.29	2.45	0.18

Figure 5

Cross-section of the (A) GF-GF-AR composite and (B) GF-GF-GF composite in the warp direction.

Table 5 exhibits the input values for the calculation of Young’s modulus of the fiber bundle elements [33]. The material constants of the each fiber bundle element are shown in Table 6. The resin was involved in the fiber element. The area of the fiber bundles from the catalog value was divided by the area of the fiber bundles from the cross-sectional observation. Then, the volume fraction of the fiber bundle elements was obtained. Young’s modulus of the fiber bundle elements was calculated using the equation for rule of mixture.

Table 5

Input values for the calculation of Young’s modulus of fiber bundle elements [33].

	Young’s modulus (longitudinal) (GPa)	Young’s modulus (transverse) (GPa)	Poisson’s ratio (vL)
Glass fiber bundle (warp, weft, and stitch)	90	90	0.22
Aramid stitch yarn	85	2.6	0.3
Matrix	3.3	3.3	0.3

Table 6

Material constants of FEM of tensile modulus.

GF-GF-GF and GF-GF-AR	Fiber volume fraction (%)	Modulus (GPa)	Poisson’s ratio
Warp fiber bundle (GF)	42	39.7	0.25
Weft fiber bundle (GF)	61	56.2	0.23
Stitch fiber bundle (GF)	18	19.2	0.28
Stitch fiber bundle (AR)	8.1	9.84	0.35

(1)EL = EfL ×Vf+Em×(1−Vf) (1)

where E is the Young’s modulus and V_f is the fiber volume fraction.

The boundary conditions were given to achieve the calculation under the tensile load. In the wale direction of the modeling of both composites, the longitudinal displacement in the longitudinal (y) and thickness (z) directions of the nodes on the CD surface of the geometry of the FEM [Figure 4A (right) and B (right)] was fixed. Also, the longitudinal displacement of the warp fiber bundles in one of the nodes on the CD was given in the longitudinal (y), thickness (z), and width (x) directions. The constant displacement was given to the nodes on the AB, which is shown in Figure 4A (right) and B (right).

The axial forces on the x coordinate were obtained from the numerical results. The total of these forces was the tensile load. The stress of the BWK composite could be calculated from these forces. For the stress calculation, the forces in the load direction for each element were calculated from the axial forces occurring in the fiber and the surface resin elements. Tensile stress was obtained by dividing the tensile load by the section area. The thickness of the model was determined by adding the distance of the cross part between the fiber bundles and the thickness of the surface elements in both faces. For the calculation of strain, the load-displacement value was divided by the length of the model. Thus, the tensile modulus was attained from the tensile stress and the strain [6].

3.2 FEM of bending properties

The finite-element methods of the three-point bending tests were performed on a personal computer with Windows 7 operating system using the MARC software.

The bending modulus of the GF-GF-AR and the AR-AR-AR composites was calculated by FEM (E_FEM). Two different modeling procedures were performed by applying the bending loads in the course direction of the specimens: (a) bending load was applied in the course direction in the GF-GF-AR specimen and (b) bending load was applied in the course direction in the AR-AR-AR specimen.

In the numerical simulation of the composite materials, the heterogeneity of the material was considered because the mechanical behavior involves the fiber and the matrix failures and these failures affect the overall mechanical properties of composites. Figure 6A–C shows our concept of the proposed modeling. Figure 6A shows the cross-section of each layer of the laminated composite. The heterogeneous cross-section can be found in Figure 6B. Each layer was divided into the three layers, namely, a fiber concentrated layer and two resin layers. Fiber was assumed to be a hexagonal closet packed arrangement. The theoretical maximum fiber volume fraction was 0.907 [26]. Figure 6C shows the quasi 3D model. The finite-element model, the quasi 3D model, consists of the shell and the beam elements. The shell elements correspond to the fiber plate with V_f=90.7% and the isotropic beam elements correspond to the resin layers with V_f=0. In order to express the lamina, the shell elements were connected with the beam elements in the thickness direction. The thickness of the fiber-plate layer of the shell elements (h₁) can be expressed as

Figure 6

Basic concept of the quasi 3D model.

(A) Composite laminate (six layers). (B) Intermediate. (c) Quasi 3D model.

(2)h1=h0×(Vf/0.907), (2)

where h₀ is the thickness of the ply and V_f is the fiber volume fraction of the composite.

Figure 7 shows the geometry of the finite-element model. Because of the symmetry, half of the specimen was modeled. The overall length of the model was 45 mm, the span length was 35 mm, the width was 15 mm, and the thickness was 5.3–5.6 mm. The loads were applied on the right side of the top three points by incremental method in 20 steps. In this model, the failure of the shell elements by the compressive stress was not considered. Table 7 shows the material constant of the quasi 3D model. The material properties used in the FEM for the shell elements were obtained by performing tensile tests on one layer of the GF-GF-AR and the AR-AR-AR composites. Those for the beam elements were obtained from the catalog value. The FEM analysis was repeated with an increased number of nodes to verify convergence. The number of the shell elements, beam elements, and nodes was 108, 150, and 180, respectively.

Figure 7

Quasi 3D finite-element model of the composite.

Table 7

Material properties of the quasi 3D model.

Material	Element	Material properties
Matrix	Beam	E=3 GPa, vL=0.3
GF-GF-AR	Shell	E=13.6 GPa, vL=0.33
AR-AR-AR	Shell	E=6.6 GPa, vL=0.33

4 Laminate beam theory

The laminate beam theory was utilized as another modeling method for the calculation of the bending modulus. The flowchart of the estimation procedure of the bending modulus by the calculation method is shown in Figure 8. First, the modulus of each layer of the composite was calculated by the laminate theory. After, the modulus of the composite was calculated by using the modulus of each layer of the composite. In the laminate theory, biaxial reinforcement inside the knitted structure was assumed as 0°/90° laminate of two uniaxial composite layers. The effect of the stitch yarns was ignored.

Figure 8

Flowchart of the estimation procedure of bending modulus by the calculation method.

A schematic drawing of the six-layer composite with the neutral axis for the calculation of bending modulus with the laminate beam theory is shown in Figure 9. The GF-GF-AR and AR-AR-AR specimens were symmetrical in the thickness direction. The formula for the laminate beam theory is as follows [28]:

Figure 9

Schematic of the six-layer composite with neutral axis.

where EI is the bending rigidity of the laminated plate, E_i is the modulus of the ith layer, A_i is the cross-section area of the ith layer, z_0i is the distance between the outer surface of the laminate and the neutral axis of the ith layer, I_oi is the second moment of area against the neutral axis of the ith layer, and z₀ is the distance between the outer surface of the laminate and the neutral axis of the laminate plate.

5 Results and discussion

The results of the tensile modulus from the FEM (E_FEM) and from the experiment (E_Exp) are demonstrated in Table 8. The tensile modulus yields good agreement between the FEM analysis (E_FEM) and the experimental results (E_Exp) (from -4.7% to +19.5%). This result shows that the FEM can be utilized for the prediction of tensile modulus of composites with the BWK preforms. The difference between E_FEM and E_Exp with the AR stitch fiber (-4.7% and +19.5%, respectively) was higher than with the GF stitch fiber (-1.8% and +3.5%, respectively) in the wale and course directions.

In the experiments, the AR spun fibers were used instead of the AR filament fibers. In the FEM modeling, the mechanical properties of the AR stitch fiber were obtained according to the input values for the calculation of Young’s modulus of the fiber bundle elements in Table 5. When we wrote the properties of the AR stitch yarn (Young’s modulus of longitudinal and transverse directions and Poisson’s ratio) in Table 5 and in the FEM, it was considered that the AR was filament yarn. However, in the experiments, the AR spun fibers were used. The mechanical properties of the AR spun fiber would be lower than those of the filament fiber. This was not taken into account in the modeling process, so the modeling results were higher than the experimental results with the AR stitch fiber in the course direction. Moreover, the difference between E_FEM and E_Exp was +19.5% in such condition in the course direction with the AR stitch fiber (GF-GF-AR). The effect of the AR spun yarns in the course direction would be about three times higher than that in the wale direction. This was because of the higher number of the AR stitch fiber bundles in the course direction (the proportion of the length of the knitting yarn in the course direction by the wale direction was about three) compared with the wale direction in the GF-GF-AR preform (Figure 4A).

In the biaxial weft knitting fabrics, in the front surface of the fabric, the rows of the legs of the loops were orientated in the wale direction. Because of this, residual stresses occurred in the fabric in the wale and compression directions. These residual stresses would increase the total tensile apparent stress and the tensile modulus in the wale direction. On the other hand, the residual stress in the compression and wale directions with the AR stitch fiber was higher compared with that with the GF stitch fiber. The exact amount of residual stresses could not be determined enough and these could not be reflected in the FEM. Therefore, the FEM results were lower than the results from the experiments in the wale direction. In addition, the difference between E_FEM and E_Exp in the wale direction in the GF-GF-GF and the GF-GF-AR composites was negative, such as -1.8% and -4.7%.

Table 9 shows the comparison of the bending modulus between the experiment and the FEM in the course direction. The bending modulus yields good agreement between the FEM analysis (E_FEM) and the experimental results (E_Exp) (from +2% to +10.2%). This result showed that the FEM can be utilized for the prediction of the bending modulus of composites with the BWK preforms.

The difference between E_FEM and E_Exp with the GF biaxial fiber (+10.2%) in the GF-GF-AR composite was higher than that with the AR biaxial fiber (+2%) in the AR-AR-AR composite in the course direction. During knitting of the BWK preforms, the GF weft fibers should be bended to carry and to insert the fibers on the machine with the carrier. Moreover, the fibers could be rubbed against not only the parts of the knitting machine but also the other glass warp fibers during knitting. Due to the brittleness of the GF fibers, the bending and the rubbing caused breaking of the GF weft fibers, which could change the tensile and bending strengths of the composites with the GF biaxial fibers (Figure 10A). In our FEM, we could not reflect the broken GF and the AR fibers in the modeling. If the number of the broken GF fibers were reflected in the modeling, the calculated modulus of composite with the GF fiber would be lower. This effect would be ignored in the composites with the AR biaxial fiber because the bending and the rubbing of the AR weft fibers against the machine parts and against the warp yarns could not break them much (Figure 10B). Therefore, the difference between E_FEM and E_Exp with the GF biaxial fiber was higher than that with the AR biaxial fiber in the course direction.

Figure 10

Photographs of BWK-reinforcement fabrics.

(A) GF-GF-AR (some of GF weft fibers were broken). (B) AR-AR-AR (AR weft fibers were not broken).

The comparison of bending modulus between the GF-GF-AR and AR-AR-AR composites with the same volume fraction is shown in Table 10. The bending modulus of the composites with the AR biaxial fiber was recalculated with the same volume fraction as the composites with the GF biaxial fiber (26.7%). As shown in Table 10, the bending modulus of the composites with the AR biaxial fiber (8.79 GPa) was lower than that with the GF biaxial fiber (10.8 GPa). This result can be also used to compare the experimental results between the GF biaxial fiber and the AR. This means that we can expect higher bending properties with the GF biaxial fiber compared with the AR biaxial fiber with the same volume fraction.

Table 11 shows the bending modulus results from the experiment, the FEM, and the laminate beam theory in the course (C) direction. The calculated results from the laminate beam theory (GF-GF-AR=17.9 GPa, AR-AR-AR=9.4 GPa) and from the experiments (GF-GF-AR=9.8 GPa, AR-AR-AR=4.4 GPa) were not in good agreement for both the GF-GF-AR and AR-AR-AR specimens. The effect of the stitch yarns on the calculated modulus could not be considered in the laminate and the laminate beam theory. Therefore, the bending modulus results with the laminate beam theory were higher compared with the experimental results. In addition, there was a big difference between E_Exp and E_Lam in the GF-GF-AR and AR-AR-AR specimens (GF-GF-AR=+82.6%, AR-AR-AR=+114%).

The result of the laminate theory with the GF biaxial yarn delivers less deviation from the experimental results (+82.6%) than that with the AR biaxial yarn (+114%) in the course direction. The laminate theory assumed that the layers in the composites were perfectly bonded. Moreover, the interfacial properties between the layers were considered perfect. However, the interfacial properties of the AR/epoxy composites were lower compared with those of the GF/epoxy [34–38]. In our calculation of the bending modulus with the lamination theory in both of the composites, the various interfacial properties could not be reflected in the calculation. If this were reflected in the calculation of the bending modulus in the AR-AR-AR composite, then the bending modulus from the theory and the difference between E_Lam and E_Exp would be lower than the obtained results (9.4 GPa and +114%, respectively). This effect could be ignored in the GF biaxial fibers because of the good interfacial properties of the GF fibers.

The difference between E_FEM and E_Exp was lower than the difference between E_Lam and E_Exp. This means that the calculated bending modulus from the FEM (E_FEM) showed very good agreement compared with the calculated bending modulus from the laminate beam theory (E_Lam). The effect of the stitch yarn on the calculated bending modulus could not be considered in the laminate theory. However, the effect of stitch yarn on the calculated bending modulus could be considered in the FEM. This is the reason for the obtained better agreement of the results between E_FEM and E_Exp compared with E_Lam and E_Exp.

6 Conclusions

This study shows that the GF-GF-GF composite with the BWK preform with the GF stitch yarns has higher tensile properties (modulus and strength) in the course and wale directions than the GF-GF-AR composite with the BWK preform with the AR stitch yarns. FEM can be used to estimate the tensile modulus of the composites with good agreement. In all specimens, E_FEM was close to the experimentally determined tensile modulus (E_Exp).

The quasi 3D model for the three-point bending properties of the BWK composites was established successfully. FEM can be used to estimate the bending modulus of the composites with good agreement. In the first step of the modeling, the experimental results were validated with the numerical results. In the second step of the modeling, the bending modulus of both specimens was compared with the same volume fraction. This study shows that the GF-GF-AR composites with the GF biaxial yarns has higher bending properties (modulus and strength) than the AR-AR-AR composites with the AR biaxial yarns in the course direction. The laminate beam theory was utilized as another modeling method for the calculation of the bending modulus.