Investigation of tensile and flexural behavior of biaxial and rib 1 × 1 weft-knitted composite using experimental tests and multi-scale finite element modeling

Abstract

In this study, tensile and flexural behavior of biaxial and rib weft-knitted composite is obtained numerically and experimentally. Multi-scale finite element modeling is employed to simulate the tensile and flexural behavior of composite samples. In the finite element modeling, the geometry of a unit cell of each fabric is initially modeled in ABAQUS software, and then periodic boundary conditions were applied to a unit cell. The stiffness matrix for each structure was obtained by a python code via meso scale modeling and used as input data for the macro modeling. To validate the numerical model, two types of weft-knitted fabrics (rib 1 × 1 and biaxial fabrics) are produced by a flat weft knitting machine. Epoxy resin is used to construct composite by the vacuum injection process (VIP). After that, the tensile and three-point bending tests were applied to composite samples. The experimental results showed that tensile strength and tensile modulus of biaxial composites are greater than rib composites, in both wale and course directions. Moreover, in three-point bending test, biaxial composite showed more strength and more stiffness in comparison to rib composite. Finite element results were compared to experimental results in tensile and bending tests. The results showed that good agreement with experimental results in the linear section of tensile and flexural behavior of composites. Consequently, the current multi-scale modeling can be used to predict the stiffness matrix and mechanical behavior of complex composite structures such as knitted composites.

Keywords

Multi-scale modeling biaxial weft-knitted fabric composite finite element tensile and flexural

Introduction

Recently, the use of composites has been growing in different industries such as construction, civil engineering, automobile industry, aerospace and etc.^1–3 Textile composites due to their lightweight structure are suitable for using in composite materials as the reinforcement part.² Textile structures could be fabricated by weaving, knitting, nonwoven manufacturing, and braiding methods.⁴ In knitting technology, straight yarns are deformed during some processes to a loop, so that they can interact with each other. Extensive studies have been conducted to investigate the mechanical properties of composites reinforced by knitted fabrics. According to some researches,^5,6 composites reinforced by knitted fabrics show higher failure deformation as well as energy absorbency under different loadings in comparison other 3D textile composites. On the other hand, knitted composite shows good energy absorbing capability under impact load. According to Leong et al.⁷ and Semnani,⁸ the limited fibers stiffness and reduction strength of the yarns during the loop formation can diminish the mechanical properties of the knitted composite. In another study, Verpoest et al.⁹ stated that the stiffness and strength of knitted fabric are relatively lower in comparison to those of other textile composites.

Mechanical properties of knitted fabrics strongly depend on the loop curved architecture of the yarns. The maximum strength of knitted preforms is generally influenced by the fiber strength, modulus, and type of yarn, knitting pattern, stitch density, number of knitted fabric plies, amount of pre-tension applied to the fed yarns, number of reinforcing in-laid yarns and knitting process parameters. Therefore, the mechanical properties of composite mostly depend on fabric properties.¹⁰ Limitations in the section of the yarn because of curving structure of loops is one of the weaknesses in the knitting process (such as carbon yarn).¹¹ Considering the previous studies, using biaxial knitted fabric is a solution to improve the mechanical properties of knitted composites. In biaxial knitted fabric, yarns insert to knitted fabric thorough warp (wale-wise) and weft (course-wise) directions as a reinforcement part.¹² Many studies have been conducted to investigate the mechanical properties of biaxial weft-knitted fabric for reinforcing thermoplastic and thermoset composites.^13–16 In studies presented by Qi, Liu, and Geng,^17–19 the bending properties of three, four and five-layer biaxial weft-knitted fabric were investigated. The results showed that increasing the number of weft-knitted fabrics layer could significantly enhance the bending strength of the fabricated composite.

On the other hand, the use of analytical and numerical models to predict the mechanical behavior of the knitted fabrics under different load conditions has been considered by researchers to achieve a better performance structure.²⁰ Various studies have been focused on modeling and simulating the mechanical performance of the textile-reinforced composite.^21–24 Multi-scale finite element (FE) modeling is one of the methods to model and predict the mechanical behavior of textile composite as considered by some researches.^25,26 In this method, a unit cell of a model can be analyzed and obtained properties are assigned to the macro model for different load conditions. Some researchers employed multi-scale FE modeling to the numerical simulation of knitted fabric materials recently.²⁷ Shekarchizadeh et al.²⁸ used FE modeling in meso scale to predict the mechanical behavior of plain weft-knitted composite. They used Vassiliadis model²⁰ to model the geometry of the fabric loop. FE model outputs include stiffness constants have good agreements with experimental results. Hamedi et al.²⁹ simulated bending behavior of 3D weft-knitted fabric composite which reinforced with single jersey knitted fabric. Vassiliadis model was used to create the geometry of knitted loop and applied periodic boundary conditions to unit cell for calculating mechanical constants of the composite. In another study, Demircan et al.³⁰ used FE modeling to study the tensile properties of biaxial weft-knitted composite. They used a simple model based on 3D beam element to simulate unit cell of model. In other paper, multi-scale modeling was used to model the tensile and shear behavior of plain knitted weft fabric by Dinh et al.³¹ Vassiliadis model was considered for modeling knitted loop geometry. Weeger et al.³² applied multiscale modeling in yarn and fabric level to simulate nonlinear mechanical behavior of single jersey plain fabric. For modeling the geometry of knitted loop, Vassiliadis model was used. They modeled the knitted loop by 3D beam formulation and predicted nonlinear orthotropic mechanical behavior of fabric. In addition, Pham et al.³³ used meso scale FE modeling to simulate biaxial weft-knitted fabric for forming process. The knitting yarn system in the biaxial reinforced weft-knitted fabric was a single jersey structure. They used Choi and Lo³⁴ equations to model loop geometry of knitted structure. As can be seen, Vassiliadis model is a famous model for 3D geometry of weft-knitted loop that is used in many types of research recently. As can be seen, most studies have been focused on the mechanical behavior of biaxial warp-knitted composites or single jersey fabric and there is limited works on meso–macro mechanical behavior of the double jersey and biaxial weft-knitted composites. Therefore, in the present study, multi-scale FE modeling is presented for simulating mechanical properties of the biaxial weft-knitted composite under tensile and flexural load. The tensile load was applied in wale and course directions. On the other hand, a knitted composite with rib 1 × 1 pattern was fabricated and mechanical properties under tensile and flexural load were compared with biaxial weft-knitted composite.

Experimental procedure

In this study, high-tenacity (HT) polyester yarn (with the linear density of 1000 denier) was used to fabricate the biaxial weft-knitted fabrics. The biaxial weft-knitted fabric based on rib 1 × 1 structure was knitted on a flat weft knitting machine (Stoll CMS 400, E5). Epoxy resin (EPL 370) was used as a matrix part. The tensile test was carried out on HT polyester yarn according to ASTM D2256 to specify the mechanical properties of the yarn. The mechanical properties of HT polyester yarn and epoxy resin are shown in Table 1.

Table 1.

Mechanical properties of materials.

Material type	Poisson ratio	Young's modulus
Polyester (HT)	0.3	19 GPa
Epoxy resin	0.35	3.1 GPa

To produce the biaxial weft-knitted fabric, a rib 1 × 1 structure was chosen as the base of fabric and straight yarns, as weft and warp were inserted to the fabric in the wale and course directions. The rib 1 × 1 knit pattern was chosen to produce simple weft-knitted fabrics. Figure 1 shows the knitted biaxial weft-knitted fabric (BWKF) graphically. The number of wale and course per centimeter for the weft-knitted fabric and BWKF was 4 and 6, respectively.

Figure 1.

BWKF structure.

All composites samples were manufactured by using the vacuum injection process (VIP) which is schematically shown in Figure 2(a). The epoxy resin was mixed thoroughly with its corresponded hardener with the ratio of 100:30 weight fraction. Then, prior to be used in vacuum bag molding process, the deaeration process was performed by placing the prepared mixture in the vacuum oven under a pressure 65 bar, at 30℃ temperature for 15 min. The molded samples were kept at room temperature for 24 h to be cured. For improving their mechanical properties, the samples were then post-cured during further three different heating process in an oven: for 2 h at 45℃ temperature, for the next 2 h at 60℃ temperature, and for the final 8 h at 80℃ temperature. Figure 2(b) shows the final bagging and vacuum applied.

Figure 2.

(a) VIP molding configuration,³⁵ and (b) vacuum injection process.

Tensile test

Tensile tests were carried out on fabricated composite samples, according to ASTM D3039 instructions, in wale and course directions by Zwick tensile tester machine. Five samples of each composite type were tested, and average results were reported. The dimensions of samples for the tensile test were 1.5 cm × 25 cm. The speed of test was chosen 2.5 mm/min. Figure 3 shows the tensile test conducted on the composite sample.

Figure 3.

Tensile load is applied to composite sample, (a) tensile test and (b) schematic view of tensile test.

Three-point bending test

Specimens (1.5 cm×10 cm) were cut off from the prepared composite sample. Three-point bending test under speed of 2 mm/min was applied to composite samples according to the ASTM C393 standard method. Span length for bending test was chosen to be 5 cm.

Multi-scale FE modeling

The implementation of FE modeling required the geometry of unit cell to be created. In this regard, a proper geometrical model for designing the loop geometry based on rib 1 × 1 pattern had to be selected. Figure 4 shows rib 1 × 1 structure and unit-cell of fabric.

Figure 4.

Rib 1 × 1 weft-knitted fabric and unit cell.

In this research, the geometrical model developed by Vassiliadis²⁰ was used to model the loop geometry in Abaqus software. Thus, due to the presence of symmetry in a single loop, equations were only used for simulating a quarter of the loop. Figure 5 schematically illustrates the proposed geometrical structure of a knit loop.

Figure 5.

A schematic of the geometrical model of the knit loop, (a) top view, (b) front view and (c) side view.²⁰

Based on Figure 5, a quarter of a loop consisted of three sections; ΣM (from the side view), MK (from the front view) and KA (from the top view). Therefore, it needs to define curve equation for these three sections. The section ΣM is considered as an elliptic arc in 3D space that can be represented as a circular with radius ( $r + \frac{D}{2}$ ) in the YZ plane and a straight line in the XY plane. D is diameter of yarn and r can be calculated as follows

r = {(c - \frac{D}{2} - \frac{t}{2}) 2 - (\frac{D}{2} + \frac{t}{2}) 2} / (2 D)

(1)

where c is courses spacing and t is a parameter that related to loop curvature (

0 \leq t \leq c - 2 D

). The next section (MK) is also considered as an elliptic arc in 3D. It is allocated as an arc of the mentioned circle of in the YZ plane and a quarter of an ellipse of minor and major radius

a = \frac{D}{2} + h

and

b = R

in the XY plane. R can be calculated by following equation

R = \frac{c}{2} - \frac{t}{2} - \frac{D}{2}

(2)

Thus, the coordinates of the sections ΣM and MK are derived from below equations

Part \sum M (0 < y < c / 2)

x (y) = - \frac{D}{c} y

(3)

z (y) = \sqrt{(r + D / 2) 2 - y^{2}} - (r + \frac{D}{2})

(4)

Part MK ( $\frac{c}{2} < y < \frac{c}{2} + R$ )

x (y) = h - a \sqrt{1 - (\frac{y - c / 2}{b}) 2}

(5)

z (y) = \sqrt{(r + D / 2) 2 - y^{2}} - (r + \frac{D}{2})

(6)

Considering Figure 5(a), the section KA is calculated as circular equations could be allocated to this section as follows

Part KA ( $c / 2 + R < x < w / 4$ )

z (x) = OZ - \sqrt{A^{2} - {(x - OX)}^{2}}

(7)

y (x) = \sqrt{(r + D / 2) 2 - {(z + r + D / 2)}^{2}}

(8)

where

OX = w / 4

(9)

OZ = \frac{{(x_{2} - OX)}^{2} - (x_{1} - OX) 2 + z_{2}^{2} - z_{1}^{2}}{2 z_{2} - 2 z_{1}}

(10)

A = \sqrt{{(x_{1} - OX)}^{2} + {(z_{1} + OZ)}^{2}}

(11)

In equations (7) to (9), w is wales spacing.

As illustrated in Figure 4, the geometry of rib 1 × 1 is divided into two segments. First segment consists of face and reverse loops as modeled by Vassiliadis et al.²⁰ equations. Another segment is the linking portion between the face and the reverse loops that is created by Abghary et al.²⁷ model.

For modeling the geometry of a unit cell of fabrics, a python code based on the mentioned geometry was written. Some of the knitted fabrics' structural parameters including wale and course distance and yarn diameter were used as inputs in python code to design a three-dimensional fabric unit cell in Abaqus software. The geometrical inputs used for the modeling are given in Table 2.

Table 2.

Geometrical parameters used for the modeling.

c (mm)	w (mm)	d (mm)	t (mm)	Thickness (mm)
3.1	4.5	1.2	0.05	2

A unit cell for rib structure and biaxial fabric is illustrated in Figure 6.

Figure 6.

A unit cell of structure, (a) rib 1 × 1 and (b) biaxial weft-knitted.

In the fiber level, the fabric yarn considered as a transversely isotropic material. In these materials, mechanical properties are the same in the directions that are perpendicular to the yarn axis but different along the fiber axis. Therefore, the elastic parameters of the yarn are assumed as transversely isotropic as shown in Table 3.

Table 3.

Transverse isotropic constants for mechanical properties of yarns.

E1 (MPa)	E2 (MPa)	E3 (MPa)	ν12	ν13	ν13	G12 (MPa)	G13 (MPa)	G23 (MPa)
19,000	250	250	0.3	0.3	0.3	175	175	125

During composite production process, the yarn is impregnated with resin. Impregnated yarns are assumed as a unidirectional composite with fiber volume fraction equal to the packing density of the yarn

V_{f} = p_{d}

(12)

where P_d is the packing density of the yarn. The resin volume fraction in impregnated yarn is

V_{r} = 1 - p_{d}

(13)

The Chamisa³⁶ micromechanical model is used to calculate the elastic constants of a transversely isotropic unidirectional composite.

E_{11} = V_{f} E_{11 f} + V_{r} E_{r}

(14)

E_{11} = V_{f} E_{11 f} + V_{r} E_{r}

(15)

E_{22} = \frac{E_{r}}{1 - \sqrt{V_{f}} (1 - \frac{E_{r}}{E_{22 f}})}

(16)

G_{12} = \frac{G_{r}}{1 - \sqrt{V_{f}} (1 - \frac{G_{r}}{G_{12 f}})}

(17)

G_{23} = \frac{G_{r}}{1 - \sqrt{V_{f}} (1 - \frac{G_{r}}{G_{23 f}})}

(18)

υ_{12} = V_{f} ν_{12 f} + V_{r} ν_{r}

(19)

ν_{23} = \frac{E_{22}}{2 G_{23}} - 1

(20)

The elastic constants of the yarn impregnated with resin are given in Table 4.

Table 4.

Elastic constants of the yarn impregnated with resin.

E11 (MPa)	E22 (MPa)	E33 (MPa)	ν12	ν13	ν13	G12 (MPa)	G13 (MPa)	G23 (MPa)
13,435	3109	3109	0.31	0.31	0.34	1152.6	1152.6	1154.7

To implement transversely isotropic properties of the yarn impregnated with resin, material orientation in Abaqus software was used. The direction 1 is along the yarn axis, while the direction 2 is the direction normal to the yarn axis and the direction 3 is defined so that it can comply with the other directions to create a right-handed, orthogonal coordinate system. The material orientation of fabric yarns is shown in Figure 7.

Figure 7.

Material direction in the fabric yarn; (a) direction 1 is defined along yarn axis, (b) direction 2 is defined perpendicularly to yarn axis and (c) direction 3, it can form a right-handed coordinate system.

The knitted fabrics were embedded in the epoxy resin. After that, for determining the elastic constants of the composite, periodic boundary conditions were applied to unit cell. The composite unit cell includes matrix and yarns, and the applied load is shown in Figure 8.

Figure 8.

Some steps of FE simulation, (a) matrix, (b) fibers embedded into the matrix and (c) applying periodic boundary condition.

C3D4 element type in ABAQUS/Standard element library, which is a 4-node linear, was applied for meshing the model and a total number of 126504 elements were used for meshing. Figure 9 shows the unit cell after meshing process. Mesh dependency study was performed on the model to make sure that the results are independent of the mesh size. Table 5 shows the mesh sizes of three different models, and the predicted elastic constants are compared in Figure 10. As it can be seen in this figure, the results do not face any significant variation by decreasing the mesh size.

Figure 9.

Mesh generation of unit cell; (a) matrix and (b) fabric yarns.

Figure 10.

Elastic moduli calculated for different mesh sizes.

Table 5.

Mesh sizes in different models.

Model no.	1	2	3
Mesh size	0.45	0.4	0.3
Elements number	83,773	93,332	126,504

To evaluate the multi-scale modeling, at first, mechanical constants for macroscopic stiffness matrix of composite should be obtained from the meso scale model. According to Hooke's law, for an elastic and isotropic material, stress–strain relation can be defined as following³⁷

{σ} = [\bar{C_{ij}}] {ɛ}

(21)

{ɛ} = [\bar{S_{ij}}] {σ}

(22)

where

{σ}

is the macroscopic stress vector,

{ɛ}

is the macroscopic strain vector,

[\bar{C_{ij}}]

is the macroscopic stiffness matrix and

[\bar{S_{ij}}] = [\bar{C_{ij}}]^{- 1}

is the macroscopic compliance matrix. The representative unit cell of composite samples is assumed to be homogeneous and orthotropic. Thus, the stiffness matrix for an orthotropic material are given in equations (23)³⁷

\begin{matrix} [\bar{C_{ij}}] = [\begin{matrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{21} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{31} & C_{32} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{matrix}] \end{matrix}

(23)

For the homogeneous composite material, the relationship between average stress and strain is

{\bar{σ}}_{α} = C_{α β} {\bar{ɛ}}_{β}

(24)

where

{\bar{σ}}_{α}

and

{\bar{ɛ}}_{β}

are the average stress and strain and

C_{α β}

is the stiffness matrix of the composite. It is possible to compute the stress field

σ_{α}

, whose average gives the required components of the elastic matrix, one column at a time, as³⁸

C_{α β} = {\bar{σ}}_{α} = \frac{1}{V} V σ_{α} (x 1, x 2, x 3) d V with ɛ_{β} = 1

(25)

where V is the volume of unit cell. Therefore, a python code was employed in order to calculate the average volume stress as well as obtain stiffness matrix components. The mechanical constants based on the stiffness matrix and compliance matrix are defined as equations (26) to (28)

\begin{matrix} [{\bar{C}}_{ij}] = [\begin{matrix} \frac{1 - υ_{23} υ_{32}}{E_{2} E_{3} Δ} & \frac{υ_{21} + υ_{23} υ_{31}}{E_{2} E_{3} Δ} & \frac{υ_{21} + υ_{23} υ_{31}}{E_{2} E_{3} Δ} & 0 & 0 & 0 \\ \frac{υ_{21} + υ_{23} υ_{31}}{E_{2} E_{3} Δ} & \frac{1 - υ_{13} υ_{31}}{E_{1} E_{3} Δ} & \frac{υ_{32} + υ_{12} υ_{13}}{E_{1} E_{3} Δ} & 0 & 0 & 0 \\ \frac{υ_{31} + υ_{21} υ_{32}}{E_{2} E_{3} Δ} & \frac{υ_{32} + υ_{12} υ_{13}}{E_{1} E_{3} Δ} & \frac{1 - υ_{13} υ_{31}}{E_{1} E_{2} Δ} & 0 & 0 & 0 \\ 0 & 0 & 0 & G_{23} & 0 & 0 \\ 0 & 0 & 0 & 0 & G_{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & G_{12} \end{matrix}] \end{matrix}

(26)

Δ = (1 - υ_{12} υ_{21} - υ_{23} υ_{32} - υ_{13} υ_{31} - 2 υ_{21} υ_{32} υ_{13}) / (E_{1} E_{2} E_{3})

(27)

\begin{matrix} [\bar{S_{ij}}] = [\begin{matrix} \frac{1}{E_{1}} & - \frac{υ_{12}}{E_{1}} & - \frac{υ_{13}}{E_{1}} & 0 & 0 & 0 \\ - \frac{υ_{21}}{E_{2}} & \frac{1}{E_{2}} & - \frac{υ_{23}}{E_{2}} & 0 & 0 & 0 \\ - \frac{υ_{31}}{E_{3}} & - \frac{υ_{32}}{E_{3}} & \frac{1}{E_{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{matrix}] \end{matrix}

(28)

In macro scale modeling, a shell geometry, according to test dimensions, was modeled in FE software. Lamina property because of insignificant the thickness of composite in compared the other dimensions were assigned to macro. The mechanical constants were obtained from meso scale by python code. Tensile load according to experiment test was applied to macro model in wale and course directions. For simulation bending test in FE software, three rigid parts were modeled as bases and bar load. S4R elements which are suitable for the shell geometry were used in mesh generation of macro model. Bases and bar load were meshed by R3D4 elements. In three-point bending simulation, for defining contact between components, surface-to-surface contact was used with 0.18 friction confession. Figure 11 shows the mesh generation for tensile and bending tests on macro scale. Static and Explicit solve method were used for tensile and bending tests simulations, respectively.

Figure 11.

Mesh generation in macro model; (a) tensile test and (b) bending test.

Results and FE validation

Tensile test results

According to the experimental method, tensile tests were performed in wale and course directions for rib knitted and BWK composites. The obtained results for composite samples are illustrated in Figures 12 and 13, respectively, for wale and course directions.

Figure 12.

Results of tensile test in wale direction.

Figure 13.

Results of tensile test in course direction.

As can be observed, the strength of biaxial composite in both directions is more than rib composite. The presence of warp and weft yarns in BWKF composites results in increasing of composite's strength in wale and course directions. Warp and weft yarns in biaxial fabric structure are straight, so they show their maximum resistance in biaxial fabric structure under tensile loading. Due to the loop shape of yarns in rib weft-knitted fabric, yarns do not participate effectively in bearing the tensile load which results in a reduction of composite strength. On the other hand, mechanical behavior of rib-knitted composite is more non-linear in comparison to biaxial weft-knitted composite in both wale and course directions, which is due to the fact that warp and weft yarns are being kept straight in biaxial fabric structure. When tensile load is applied to biaxial composite, weft or warp yarn shows more resistance in comparison to other yarns. As tensile load increases, straight yarns start to fracture and then looped yarns play the role as a reinforcement part in composite. In this point, composite's response under tensile loading changes from linear to nonlinear behavior.

Three-point bending test results

The results for three-point bending test are shown in Figure 14.

Figure 14.

Three-point bending test diagrams.

According to Figure 14, bending strength of the biaxial composite is more than rib composite. The presence of straight yarns in biaxial composite structures leads to improved mechanical properties and strength of the composite. On the other hand, as shown in the linear section of diagrams, the stiffness of biaxial composite, because of weft and warp yarns, is more than the rib composite. As shown in Figure 14, a precipitate force drop occurred by increasing extension. Fracture in the matrix part leads to a sudden increase in the bending force and reinforcement part bear bending load. The failure point in the biaxial composite is more than the rib composite because of the effective role of reinforcement part and straight yarns in the composite structure.

Numerical results

In this study, a unit cell of rib and biaxial weft-knitted composite was modeled in ABAQUS software. Periodic boundary conditions were applied to each unit cell. Figure 15 shows the stress contours of matrix and reinforcement parts.

Figure 15.

Stress contours for unit cell of composite structures, (a) matrix, (b) biaxial fabric and (c) rib fabric.

A Python code was used to calculate stiffness matrix components. Stiffness matrices for each composite are given in Table 6.

Table 6.

Predicted stiffness matrices for all composites.

Composite sample	Stiffness matrix	E11 (GPa)	E22 (GPa)	G12 (GPa)
Rib composite	$\begin{matrix} C_{ij} = [\begin{matrix} 5.9293.0303.0230.0000.0000.000 \\ 3.0305.9833.0550.0000.0000.000 \\ 3.0233.0555.9110.0000.0000.000 \\ 0.0000.0000.0001.9710.0000.000 \\ 0.0000.0000.0000.0001.0600.000 \\ 0.0000.0000.0000.0000.0001.233 \end{matrix}] \end{matrix}$ GPa	3.889	3.912	1.971
Biaxial composite	$\begin{matrix} C_{ij} = [\begin{matrix} 6.3533.0993.1020.0000.0000.000 \\ 3.0996.5273.1300.0000.0000.000 \\ 3.1023.1306.1200.0000.0000.000 \\ 0.0000.0000.0002.0560.0000.000 \\ 0.0000.0000.0000.0001.1180.000 \\ 0.0000.0000.0000.0000.0001.273 \end{matrix}] \end{matrix}$ GPa	4.315	4.448	2.056

Stiffness matrix, which was obtained from meso scale, was used for macro model. Figure 16 shows the stress–strain curve comparison between experimental and FE modeling results in the linear section.

Figure 16.

Stress–strain curves for FE and experimental results, (a) rib composite in the course direction, (b) biaxial composite in the course direction, (c) rib composite in the wale direction and (d) biaxial composite in the wale direction.

Based on Figure 16, multi-scale modeling results show good agreement with experimental results in predicting the elastic behavior of composites. The stress contours and deformation of macro model under bending load are shown in Figure 17.

Figure 17.

Three-point bending deformation, (a) FE modeling and (b) experimental.

The compression between FE and experimental results in three-point bending is shown in Figure 18.

Figure 18.

Force–extension curve for FE and experimental results, (a) rib composite and (b) biaxial composite.

The predicted diagrams of FE modeling have good agreement with experimental results as shown in Figure 18. On other hand, the elastic modulus of the biaxial composite is more than the rib composite which is similar to the experimental results. According to the stiffness matrices of composites, the elastic modulus of the biaxial composite in both wale and course directions is more than the rib composite but in the thickness direction the difference is low. The difference of shearing modulus between biaxial and rib composites is low. The warp and weft yarns stayed on the wale and course direction and do not play a reinforced role in composite structures. Therefore, the shear modulus of composites is close together. In this study, just one yarn was inserted to the fabric structure in the wale and course directions to produce a biaxial fabric that improves mechanical properties in wale and course directions. Hence, using more than one straight yarn in fabric structure leads to increase in the volume fraction of reinforcement parts in composite and increase in the stiffness and the strength of composite in different directions. For comparing between the experimental and numerical results, a table containing error values is shown in Table 7.

Table 7.

Experimental and numerical results data.

Composite sample	Experimental modulus in course direction (MPa)	Numerical modulus in course direction (MPa)	Error value in course direction (%)	Experimental modulus in wale direction (MPa)	Numerical modulus in wale direction (MPa)	Error value in wale direction (%)
Rib	3703	3889	5.02	3805	3912	2.81
Biaxial	4232	4315	1.96	4306	4448	3.29

To calculate the stiffness matrix of the composite by theoretical method, a transformation matrix should be defined according to the geometry of the reinforcement part. This method is too difficult for composites reinforced by complex fabric structures (such as knitted fabric). Therefore, using multi-scale modeling to calculate the stiffness matrix and the mechanical behavior of composite is a reasonable method that improves the precision and reduction-solving time.

Conclusions

In this study, tensile and flexural behavior of rib weft-knitted composite and biaxial weft-knitted composite was investigated by experimental and multi-scale FE modeling. According to the experimental results, tensile strength of biaxial composite in both wale and course directions is more than the rib composite. On the other hand, the presence of warp and weft yarn in biaxial weft-knitted fabric structure leads to increase in the tensile modulus of the composite in wale and course directions. The results of three-point bending test show that the stiffness of biaxial composite is more than the rib composite, and failure point in matrix was accrue in more extension and force. In multi-scale modeling, a unit cell of fabrics structure was modeled in ABAQUS FE software, and periodic boundary conditions were applied to unit cell. Stiffness matrix for each composite structure was obtained by a python code. According to the stiffness matrix, the stiffness of biaxial composite, because of the presence of warp and weft yarns, is more than the rib composite in wale and course directions, but they do not have a significant effect on the shear modulus. The stiffness matrices, which obtained from meso scale, were assigned to the macro model for tensile and three-point bending tests. The results of FE modeling showed good agreements with the experimental results, which demonstrates that the predicted outputs are suitable for experimental results. Using multi-scale modeling leads to reduction in solution time in meso scale and improvement in the precision of modeling.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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