Multi-scale finite element modeling of bending behavior of 3D multi-cell spacer weft-knitted composites with different structures

Abstract

In the present study, a multi-scale finite element model is proposed to predict the linear and nonlinear behavior of the 3D multi-cell spacer weft-knitted composite under bending load. In this study, a unit-cell of the composite which includes plain and biaxial weft-knitted structures was modeled at the meso scale. Periodic boundary conditions were applied to the meso model to calculate the elastic constants of each composite structure. In order to obtain failure parameters of the composites, the Puck failure criterion model was utilized by a VUMAT code for the meso model. Afterward, the elastic constants of the composites based on a Python code were extracted from the meso model. Moreover, failure parameters that include tensile and compressive strength through the fiber and transverse directions were obtained from the meso model. All elastic and failure parameters were used for the macro model which is created with different profiles under bending load. The numerical results at the meso scale showed that the presence of the weft and warp yarns inside the biaxial weft-knitted composite increases the strength of the composite through the course and wale directions. Moreover, the stiffness of the composite would be improved. So, the samples that contained a biaxial composite had more stiffness and bending strength in comparison with plain composite samples because the top and bottom layers were manufactured by the biaxial weft-knitted structure. Besides, the comparison between numerical and experimental force–deflection curves showed that the proposed model could predict the linear and nonlinear behavior of the composites with high accuracy. So, this model can be used for other textile composites with complex shapes to predict the mechanical behavior of them.

Keywords

Multi-scale finite element model 3D weft-knitted composite failure bending Puck criterion

Introduction

In the recent era, utilizing 3D composites in various industries has increased.¹ These types of composite can be fabricated by different methods such as weaving, knitting, braiding, and nonwoven manufacturing.² Recently, some researches have been conducted to produce a 3D weft-knitted composite with complex shapes.^3–5 The curving structure of the loop in the weft-knitted structure is the weakest point of the knitted fabric which influences mechanical properties of the composite.⁶ In order to overcome this problem, straight yarns such as weft and warp yarn can be inserted into the fabric structure and make a biaxial fabric. The presence of straight yarns inside the fabric structure leads to the straight yarns playing a significant role during the loading process to reinforce the composite and reduce the stress concentration in other spots.⁷

Numerous experimental and numerical studies were carried out on weft-knitted composites.^8–12 Shekarchizadeh et al.¹³ analyzed the tensile behavior of the weft-knitted composite by micromechanics equations and the finite element model. In another study, Dinh et al.¹⁴ predicted the tensile and shear properties of the weft-knitted fabric at the meso scale. They applied a finite element (FE) model on the meso model and modeled a unit-cell of the fabric by Vassiliadis et al.’s¹⁵ equations. Hessami et al.⁹ compared tensile and bending behavior of rib and biaxial weft-knitted composites. They also utilized a multi-scale finite element model to predict the elastic behavior of the composites. Some researchers have focused on 3D weft-knitted composites and simulated mechanical properties of these types of composites. Hamedi et al.¹⁶ investigated the bending behavior of a 3D weft-knitted spacer composite experimentally and numerically. They just predict the linear behavior of the composite by a multi-scale finite element model. Omrani et al.¹⁷ applied a multi-scale finite element model to predict the mechanical behavior of tubular 3D weft-knitted composite. Hosseini et al.¹⁸ manufactured multi-cell 3D weft-knitted composites and analyzed the bending properties of these structures experimentally and numerically. Two structures, plain and biaxial weft-knitted, were utilized to fabricate 3D composites.

According to researches, the multi-scale finite element method is one of the effective methods to predict the mechanical behavior of composite structures with complex geometries. In this method, a unit-cell of the composite is modeled at the meso scale, and the essential parameters are extracted from the meso model.^19,20 Recently, researchers have used Vassiliadis et al.’s¹⁵ model to create weft-knitted loop geometry at the meso scale.^7,9,21–23 In this model, the geometry of the weft-knitted loop based on the coordinate points of the loop geometry is created in a three-dimensional space. Despite all efforts regarding prediction of the mechanical properties of the 3D spacer weft-knitted composites, only some limited researches have been focused on the failure behavior of these structures. So, in the present study, a multi-scale finite element model is proposed to predict the linear and nonlinear behavior of the 3D weft-knitted composite. The numerical model is developed based on Hosseini et al.’s¹⁸ experimental parameters, and eventually, the numerical results are compared with experimental results which are given in Ref [18]. It should be mentioned that elastic and failure parameters of each structure are extracted from a meso model and used for the macro model under bending load. The 3D composites are composed of plain and biaxial weft-knitted structures.

Multi-scale finite element model

Meso model

At the meso scale, a unit-cell of the composite preforms should be created in finite element software. The unit-cell of the plain structure includes two loop legs and one loop head which interlace each other. For the purpose of knitted loop modeling, Vassiliadis’s¹⁵ equations were applied by using a Python code, and the unit-cells of each structure were created in ABAQUS software. The yarn diameters, number of courses per centimeter, and number of wales per centimeter were the input data of the Python code. The biaxial architecture has two straight yarns (warp and weft) that were laid in the matrix vertically and horizontally according to Ref [18]. Figure 1 shows the unit-cells of structures.

Figure 1.

The unit-cells of the composite preforms, (a) plain knitted fabric and (b) biaxial knitted fabric.

The structural parameters of the weft-knitted fabrics are given in Table 1.

Table 1.

Structural parameters of weft-knitted fabrics¹⁸.

Fabric	Course per centimeter (CPC)	Wales per centimeter (WPC)	Yarn diameter (mm)
Plain	3.45	2.2	0.9
Biaxial	3.35	2.2	0.3

The embedded yarns within the simulated unit-cell are assumed to be transversely isotropic.²⁴ The matrix part’s mechanical behavior was considered isotropic. For impregnated yarns, Chamis’s²⁵ micromechanical relations for elastic constants and strengths were utilized to obtain the required properties to define the material in the ABAQUS environment

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

(1)

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

(2)

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

(3)

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

(4)

υ_{12} = V_{f} ν_{12 f} + V_{r} ν_{r}

(5)

ν_{23} = \frac{E_{22}}{2 G_{23}} - 1

(6)

{(σ_{1}^{T})}_{u l t} \approx V_{f} S_{f T}

(7)

{(σ_{1}^{C})}_{u l t} \approx V_{f} S_{f C}

(8)

{(σ_{1}^{C})}_{u l t} \approx 10 {(τ_{12})}_{u l t} + 2.5 S_{r T}

(9)

{(σ_{1}^{C})}_{u l t} \approx \frac{G_{r}}{1 - V_{f} (1 - \frac{G_{r}}{G_{f 12}})}

(10)

{(σ_{2}^{T})}_{u l t} \approx (1 - (\sqrt{V_{f}} - V_{f}) (1 - \frac{E_{r}}{E_{22 f}})) S_{r T}

(11)

{(σ_{2}^{C})}_{u l t} \approx (1 - (\sqrt{V_{f}} - V_{f}) (1 - \frac{E_{r}}{E_{22 f}})) S_{r C}

(12)

{(τ_{12})}_{u l t} \approx (1 - (\sqrt{V_{f}} - V_{f}) (1 - \frac{G_{r}}{G_{12 f}})) S_{r S}

(13)

In equations 1–13,

{(σ_{1}^{T})}_{u l t}

{(σ_{1}^{C})}_{u l t}

{(σ_{2}^{T})}_{u l t}

{(σ_{2}^{C})}_{u l t}

, and

{(τ_{12})}_{u l t}

are the tensile, and compressive, and shear strength in the first and second directions, respectively.

S_{r T}

S_{r C}

, and

S_{r S}

are the tensile, compressive, and shear strength of the matrix. V_f is the volume fraction of fibers. E_r, E_22f, G_r, and G_12f are the elastic modulus of resin, transverse modulus of fiber, shear modulus of resin, and fiber, respectively. Tables 2 and 3 illustrate elastic constants and strength values for the impregnated yarn.

Table 2.

Elastic constants of the yarn impregnated with resin.

E11 (MPa)	E22 (MPa)	E33 (MPa)	ν12	ν13	ν23	G12 (MPa)	G13 (MPa)	G23 (MPa)
18,802	3969	3969	0.33	0.33	0.34	1471	1471	1471

Table 3.

Strength values of weft-knitted composite.

Epoxy resin	Glass yarn	Impregnated yarn
$S_{rT}$ =50 MPa	$S_{f T}$ = 1150 MPa	${(σ_{1}^{T})}_{u l t}$ = 257.6 MPa
$S_{rC}$ =70 MPa		${(σ_{1}^{C})}_{u l t}$ = 257.6 MPa
$S_{rS}$ =29 MPa		${(σ_{2}^{T})}_{u l t}$ = 42.7 MPa
		${(σ_{2}^{C})}_{u l t}$ = 55.7 MPa
		${(τ_{12})}_{u l t}$ = 22.1 MPa

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

In Hosseini et al.’s¹⁸ study, the linear behavior of the 3D composite was considered, and there was no attempt to simulate the nonlinear behavior of the composite. So, in this study, Puck’s ²⁶ failure criterion was utilized to obtain failure parameters of composites for the macro model. To determine the constants of the composite’s stiffness matrix, periodic boundary conditions were applied to the meso model. The macroscopic strain increment was applied to the unit-cells by decomposing the displacement increment on the boundary into the macroscopic averaged displacement field and a period part repeated from one unit-cell to another one, such as

U_{i} = {\bar{ε}}_{i k} X_{k} + U_{i}^{*}

(14)

where

{\bar{ε}}_{i k} = {\dot{ε}}_{i k} Δ t

is the macroscopic strain increment. As the periodic part of

U_{i}^{*}

is equal on the corresponding nodes of the opposite parallel surface of the unit-cell, the total displacement at this node pair is expressed by

{(U_{i})}_{n_{1}^{p}} - {(U_{i})}_{n_{2}^{p}} = {\bar{ε}}_{i k} [{(X_{k})}_{n_{1}^{p}} - {(X_{k})}_{n_{2}^{p}}] = {\bar{ε}}_{i k} Δ X_{k}

(15)

where

Δ X_{k}

is the relative coordinates of the node pair, and

n_{1}^{p}

and

n_{2}^{p}

are the first and second nodes, respectively. For this aim, the corresponding nodes from parallel faces were chosen and a constraint was created between them. The composite’s unit-cell includes matrix and yarns. Next, the embedded element technique was applied, considering the yarn parts as the embedded region and the matrix part as the host. However, to eliminate the volume redundancy caused by the embedded element technique,²⁷ the stiffness of the epoxy resin was subtracted from the stiffness matrix of the unidirectional composite yarns. An 8-node linear brick with reduced integration and hourglass control (C3D8R) elements were employed to discretize the composite yarns with the element size of 0.15

m m

and also the matrix part with the element size of 0.2

m m

, as shown in Figure 2. Prior to performing the final finite element modeling, mesh dependency was checked. Thus, further refining the mesh size merely increases the computational cost without improving the accuracy significantly. Table 4 shows the numbers of elements and nodes that were used for the meshing of the meso models.

Figure 2.

Mesh generation on the unit-cells, (a) plain structure and (b) biaxial structure.

Table 4.

Numbers of the elements and nodes for the meso models.

Model	Number of elements	Number of nodes
Plain composite	6594	8385
Biaxial composite	8962	11,501

It should be mentioned that to calculate elastic constants of the composite, an implicit solver was applied. The elastic constants of the stiffness matrix could be calculated according to the following equations²⁸

{\bar{σ}}_{α} = \frac{1}{V} \int_{V} σ_{α} (x 1, x 2, x 3) d V

(16)

{\bar{ε}}_{β} = \frac{1}{V} \int_{V} ε_{β} (x 1, x 2, x 3) d V

(17)

C_{α β} = \frac{{\bar{σ}}_{α}}{{\bar{ε}}_{β}}

(18)

As mentioned before, in this study, the 3D Puck damage criterion was used to analyze the damage and failure mechanism of the composite. The Puck failure criterion can be written as follows²⁶:

(1) Fiber failure in tension

\frac{1}{ε_{1 T}} (ε_{1} + \frac{υ_{f 12}}{E_{f 1}} m_{σ f} σ_{22}) = 1

(19)

(2) Fiber failure in compression

\frac{1}{ε_{1 C}} | (ε_{1} + \frac{υ_{f 12}}{E_{f 1}} m_{σ f} σ_{22}) | + {(10 γ_{21})}^{2} = 1

(20)

(3) Matrix failure in transverse tension

\sqrt{{(\frac{τ_{21}}{S_{21}})}^{2} + {(1 - P_{v p}^{+} \frac{Y_{T}}{S_{21}})}^{2} {(\frac{σ_{22}}{Y_{T}})}^{2}} + P_{v p}^{+} \frac{σ_{22}}{S_{21}} + \frac{σ_{11}}{σ_{11 D}} = 1

(21)

(4) Matrix failure in moderate transverse compression

\frac{1}{S_{21}} (\sqrt{τ_{21}^{2} + {(P_{v p}^{-} σ_{22})}^{2}} + P_{v p}^{-} σ_{22}) + \frac{σ_{11}}{σ_{11 D}} = 1

(22)

(5) Matrix failure in large transverse compression

[{(\frac{τ_{21}}{2 (1 + P_{v v}^{-}) S_{21}})}^{2} + {(\frac{σ_{22}}{Y_{C}})}^{2}] \frac{Y_{C}}{- σ_{22}} + \frac{σ_{11}}{σ_{11 D}} = 1

(23)

where

ε_{1 T}

and

ε_{1 C}

are the tensile and compressive failure strains of a unidirectional layer in the x1 direction, respectively;

ε_{1}

is the normal strain of a unidirectional layer;

υ_{f 12}

and

E_{f 1}

are Poisson’s ratio and Young’s modulus for the fiber in the x1 direction, respectively;

m_{σ f}

is the mean stress magnification factor for the fibers in the ×2 direction;

σ_{11}

and

σ_{22}

are the normal stresses in a unidirectional layer;

γ_{21}

and

τ_{21}

are the shear strain and stress of a unidirectional layer in the elastic symmetry direction, respectively;

S_{21}

is the shear strength of a unidirectional layer transverse and parallel to the fiber direction;

P_{v p}^{+}

P_{v p}^{-}

, and

P_{v v}^{-}

are the fracture plane angle-dependent parameters; and

σ_{11 D}

is the stress value for linear degradation. The value for

P_{v p}^{+}

P_{v p}^{-}

, and

P_{v v}^{-}

were considered according to Puck et al.’s²⁶ recommendations(Table 5).

Table 5.

Recommended values for inclination parameters²⁶.

Parameter	Value
$P_{vp}^{+}$	0.3
$P_{vp}^{-}$	0.25
$P_{vv}^{-}$	0.2–0.25

The elastic stress–strain constitutive relation, fiber and matrix damage variable with respect to the tensile and compressive stress states, and material constants for the initial and damage-coupled material stiffness for orthotropic composite laminates can be written as follows

d_{f} = 1 - (1 - d_{f t}) (1 - d_{f c})

(24)

d_{m} = 1 - (1 - d_{m t}) (1 - d_{m c})

(25)

C_{11} = (1 - d_{f}) C_{11}^{0}

(26)

C_{22} = (1 - d_{f}) (1 - d_{m}) C_{22}^{0}

(27)

C_{33} = (1 - d_{f}) (1 - d_{m}) C_{33}^{0}

(28)

C_{12} = (1 - d_{f}) (1 - d_{m}) C_{12}^{0}

(29)

C_{23} = (1 - d_{f}) (1 - d_{m}) C_{23}^{0}

(30)

C_{13} = (1 - d_{f}) (1 - d_{m}) C_{13}^{0}

(31)

G_{12} = (1 - d_{f}) (1 - s_{m t} d_{m t}) (1 - s_{m t} d_{m c}) G_{12}^{0}

(32)

G_{23} = (1 - d_{f}) (1 - s_{m t} d_{m t}) (1 - s_{m t} d_{m c}) G_{23}^{0}

(33)

G_{31} = (1 - d_{f}) (1 - s_{m t} d_{m t}) (1 - s_{m t} d_{m c}) G_{31}^{0}

(34)

\begin{array}{l} C_{11}^{0} = \frac{E_{1} (1 - υ_{23} υ_{32})}{Δ} \\ C_{22}^{0} = \frac{E_{2} (1 - υ_{13} υ_{31})}{Δ} \\ C_{33}^{0} = \frac{E_{3} (1 - υ_{12} υ_{21})}{Δ} \\ C_{12}^{0} = \frac{E_{1} (υ_{21} - υ_{31} υ_{23})}{Δ} \\ C_{13}^{0} = \frac{E_{1} (υ_{31} - υ_{12} υ_{31})}{Δ} \\ C_{23}^{0} = \frac{E_{2} (υ_{32} - υ_{12} υ_{31})}{Δ} \\ Δ = 1 - υ_{12} υ_{21} - υ_{23} υ_{32} - υ_{13} υ_{31} - 2 υ_{21} υ_{32} υ_{13} \end{array}

(35)

C_{d} = [\begin{matrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{22} & C_{23} & 0 & 0 & 0 \\ C_{33} & 0 & 0 & 0 \\ C_{44} & 0 & 0 \\ C_{55} & 0 \\ s y m & C_{66} \end{matrix}]

(36)

where

d_{f}

and

d_{m}

are the accumulated damage variables for the fiber and matrix, respectively;

d_{f t}

and

d_{f c}

are the damage variables that corresponded to fiber failure under tension and compression, in that order;

d_{m t}

and

d_{m c}

are the damage variables corresponding to matrix failure under tension and compression, respectively; and

s_{m t}

and

s_{m c}

are shear modulus loss control factors resulted from the matrix failure under tension and compression. Based on the studies,^29,30 loss control factors were considered as

s_{m t} = 0.9

and

s_{m c} = 0.5

. Once the failure was disclosed, the material stiffness of the initially failed element was replaced by zero in compliance with equations 24–34 This method is the so-called “element weakening method (EWM)” in the finite element procedure.^31,32 Damage variables corresponding to each failure mode are updated: For the fiber tension mode,

d_{f t} = 1

; for fiber compression mode,

d_{f c} = 1

; for matrix tension mode,

d_{m t} = 1

; and for matrix compression mode,

d_{m c} = 1

. To apply the Puck failure criterion and element deletion in the model, a VUMAT code was written in ABAQUS software.

To determine damage parameters of the 3D weft-knitted spacer fabric-reinforced composite, tensile and compressive loads through the course and wale directions were applied to the unit-cells. Moreover, two shear loads were applied to the meso model to calculate the shear strengths of the composite.

Macro model

At the macro scale, the 3D composites according to their dimensions were created by a shell geometry, and the elastic and damage parameters which were extracted from the meso model were assigned to the macro model. In this study, the Hashin failure criterion³³ was applied for the damage and failure mechanisms at the macro model. Four failure modes are considered in Hashin’s failure criterion, named fiber tension, fiber compression, matrix tension, and matrix compression. These modes can be respectively written as:

Fiber tension $(σ_{11} \geq 0)$

{(\frac{σ_{11}}{X_{T}})}^{2} + α {(\frac{τ_{12}}{S_{L}})}^{2} \geq 1

(37)

Fiber compression

(σ_{11} < 0)

{(\frac{σ_{11}}{X_{C}})}^{2} \geq 1

Matrix tension

(σ_{22} \geq 0)

\frac{{(σ_{22})}^{2}}{Y_{T}^{2}} + {(\frac{τ_{12}}{S_{L}})}^{2} \geq 1

(39)

Matrix compression

(σ_{22} < 0)

{(\frac{σ_{22}}{2 S_{T}})}^{2} + [{(\frac{Y_{C}}{2 S_{T}})}^{2} - 1] \frac{σ_{22}}{Y_{C}} + {(\frac{τ_{12}}{S_{L}})}^{2} \geq 1

where

X_{T} / Y_{T}

and

X_{C} / Y_{C}

are the tensile and compressive strengths, respectively, and

S_{i j}

is the shear strength. The stiffness reduction for the macro scale model is the same as meso scale model based on equations (24–34). This algorithm is available in the ABAQUS interface. Moreover, for each element that d_f is equaled to one, that element will be deleted.

In Hosseini et al.’s¹⁸ article, four types of 3D composites as shown in Table 6 with the different structures were considered for the bending test.

Table 6.

Structural characteristics of the 3D composites.

		Layer’s structure
Sample code	Profile	Surface layer	Connecting layer	Middle layer
SU	Single decker	Plain-knitted fabric	Plain-knitted fabric	—
RSU	Single decker	Biaxial-knitted fabric	Plain-knitted fabric	—
DU	Double decker	Plain-knitted fabric	Plain-knitted fabric	Plain-knitted fabric
RDU	Double decker	Biaxial-knitted fabric	Plain-knitted fabric	Plain-knitted fabric

For the simulation bending test in finite element software, three rigid parts were modeled as bases and bar load. S4R elements that are suitable for the shell geometry were used in the mesh generation of the 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 a 0.18 friction coefficient. Figure 3 shows mesh generation for the bending test at the macro scale.

Figure 3.

Meshing on macro model, (a) single decker and (b) double decker.

Results and discussion

The procedure of this study could be divided into two sections: (1) calculation of elastic constants of plain and biaxial weft-knitted composites from the meso model and (2) obtaining failure parameters from the meso model. Both sections were used for modeling the macro model under bending load. Figure 4 shows the stress contours of the reinforcement part and matrix for both structures at the meso scale under tensile load through the wale direction.

Figure 4.

Stress contours for meso models under tensile load through the wale direction. (a) plain weft-knitted composite and (b) biaxial weft-knitted composite.

Once a tensile load is applied to the composite, straight yarns which aligned through the external load resist more stress in comparison with other yarns. So, for biaxial weft-knitted composite, weft or warp yarns play a significant role in the composite’s strength (Figure 4(b)). In the plain weft-knitted composite, loop legs are aligned through the load direction and show more stress when a tensile load in the wale direction is applied to the composite. However, the curved geometry of the loop leads to the yarns being unable to use all their abilities against the external load. As mentioned before, the stiffness matrix for each model was extracted by a Python code. Table 7 shows the stiffness matrices of each composite structure.

Table 7.

Stiffness matrices for each composite structure.

Composite sample	Stiffness matrix
Plain weft-knitted	$C_{i j} = [\begin{matrix} \begin{matrix} 9.589 & 3.193 & 1.822 \\ 3.123 & 10.117 & 3.282 \\ 2.710 & 2.906 & 8.501 \end{matrix} \begin{matrix} 3.008 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \\ \begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \begin{matrix} 1.135 & 0.000 & 0.000 \\ 0.000 & 1.582 & 0.000 \\ 0.000 & 0.000 & 1.482 \end{matrix} \end{matrix}]$ GPa
Biaxial weft-knitted	$C_{i j} = [\begin{matrix} \begin{matrix} 10.646 & 3.361 & 2.363 \\ 3.401 & 11.370 & 3.124 \\ 3.511 & 3.372 & 8.653 \end{matrix} \begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \\ \begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \begin{matrix} 1.245 & 0.000 & 0.000 \\ 0.000 & 1.672 & 0.000 \\ 0.000 & 0.000 & 1.577 \end{matrix} \end{matrix}]$ GPa

Composite sample

Stiffness matrix

Plain weft-knitted

C_{i j} = [\begin{matrix} \begin{matrix} 9.589 & 3.193 & 1.822 \\ 3.123 & 10.117 & 3.282 \\ 2.710 & 2.906 & 8.501 \end{matrix} \begin{matrix} 3.008 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \\ \begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \begin{matrix} 1.135 & 0.000 & 0.000 \\ 0.000 & 1.582 & 0.000 \\ 0.000 & 0.000 & 1.482 \end{matrix} \end{matrix}]

GPa

Biaxial weft-knitted

C_{i j} = [\begin{matrix} \begin{matrix} 10.646 & 3.361 & 2.363 \\ 3.401 & 11.370 & 3.124 \\ 3.511 & 3.372 & 8.653 \end{matrix} \begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \\ \begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix} \begin{matrix} 1.245 & 0.000 & 0.000 \\ 0.000 & 1.672 & 0.000 \\ 0.000 & 0.000 & 1.577 \end{matrix} \end{matrix}]

GPa

The values of Table 6 could be used for the linear behavior of each composite structure. For modeling the nonlinear and failure behavior of the composites, failure parameters are required. These parameters could be obtained from the meso model as mentioned before. Figure 5 shows the failure behavior of the plain and biaxial weft-knitted composites under tensile load through the course direction.

Figure 5.

Failure behavior of the biaxial and plain weft-knitted composite under tensile load through the course direction in different strain values.

For the plain weft-knitted composite, the loop head and feet show more stress concentration than other parts on the threshold of the failure (Figure 5(a)). The curvature of the loop geometry causes the reinforcement part to not reinforce the matrix as well as straight yarns in the biaxial weft-knitted composite. So, an extreme failure occurs after the failure threshold (Figure 5(b) and (c)). On the other hand, the presence of straight yarns (warp and weft yarns) within the composite improves the mechanical properties of the composite under tensile load. In Figure 5(d), the weft yarn plays an important role in the tensile strength of the composite through the course direction and resists more stress in comparison with other parts. After the failure threshold, some cracks could be observed on the matrix and the weft yarn starts to fail, but a complete fracture could not be seen at this moment (Figure 5(e)). By increasing the strain value, the cracks propagate to other spots of the matrix and failure occurs on loop yarns (Figure 5(f)). The failure mechanism of the plain and biaxial weft-knitted composites under tensile loads through the wale direction is shown in Figure 6.

Figure 6.

Failure behavior of the biaxial and plain weft-knitted composite under tensile load through the wale direction in different strain values.

According to Figure 6(a), loop legs are under more stress than other loop parts at the failure threshold. But, as mentioned before, because of the curvature of the loop geometry, they cannot show a significant effect on the failure mechanism of the composite. So, by increasing the strain value in the wale direction, the loop legs and matrix start to fail (Figure 6(b)). In Figure 6(c), the complete fracture could be observed in matrix and yarns although the failure of loop head and feet is not completed because they are not aligned through the load direction. For the biaxial weft-knitted composite, the warp yarn that aligned through the load is under more stress at the failure threshold (Figure 6(d)). As the strain value is increased, the warp yarn starts to fail and a crack propagates in the matrix at this location (Figure 6(e)). Moreover, the loop legs are under maximum stress after the failure of the warp yarn. In Figure 6(f), the crack grows inside the matrix and divides the matrix into two parts. At this strain, the loop legs start to fail and fracture will be completed. Table 8 shows the number of failed elements in strain 0.028.

Table 8.

The number of failed elements in strain 0.028 for biaxial and plain weft-knitted composites.

Model	Failed elements for tensile through the course direction	Failed elements for tensile through the wale direction
Biaxial composite	1170	1175
Plain composite	2900	4131

According to the failure behavior of the composites in the meso model, the failure parameters for Hashin’s criterion were obtained and are given in Table 9.

Table 9.

Failure parameters for each composite structure.

Model	$X_{T}$ (MPa)	$X_{C}$ (MPa)	$Y_{T}$ (MPa)	$Y_{C}$ (MPa)	$S_{L}$ (MPa)	$S_{T}$ (MPa)
Plain weft-knitted	94.32	81.73	105.12	90.71	23.06	24.19
Biaxial weft-knitted	111.36	86.14	128.73	92.55	31.57	33.22

Parameters of Table 9 were used for the macro model and investigation of bending behavior of the composite samples. Figure 7 shows bending deformation, and fiber tensile and compressive damage based on the Hashin failure criterion for SU and RSU samples under bending load at the macro scale.

Figure 7.

Deformation and fiber tensile and compressive damage contours of SU and RSU composites under bending load, (a) fiber compressive damage of SU, (b) fiber tensile damage of SU, (c) fiber compressive damage of RSU, and (d) fiber tensile damage of RSU.

Once the bending load is applied to the composite, the connection layers start to buckle under the bending load. This action increases the energy absorption of the 3D composite. For both SU and RSU samples, fiber compressive damage can be observed on the top face and the connected layer. However, the fiber tensile damage due to the biaxial structure for RSU is lower than SU (Figure 7(b) and (d)). The deformation, and fiber tensile and compressive damage contours based on the Hashin failure criterion for DU and RDU composites are shown in Figure 8.

Figure 8.

Deformation and fiber tensile and compressive damage contours of DU and RDU composites under bending load, (a) fiber compressive damage of DU, (b) fiber tensile damage of DU, (c) fiber compressive damage of RDU, and (d) fiber tensile damage of RDU.

According to Figure 8, once a bending load is applied to the 3D composites, the areas near the loading noise are under compressive stress. Therefore, fiber compressive damage contours are more than other areas. Moreover, the connection points between horizontal layers and connection layers are the critical points for the fiber compressive damage. However, for the fiber tensile damage mode, the connection layers and especially the connection layer near the loading bar have great potential to damage, and failure occurs at this point. In order to compare numerical results with experimental, the force–deflection curves for experimental and numerical results are compared and shown in Figure 9.

Figure 9.

Force-deflection curves for SU, RSU, DU, and RDU samples.

According to Figure 9, the first regime of the curve is linear and it shows the top layer is under applied load. As the bending load is increased the trend of the curve change to nonlinear. The reason for this phenomenon is the buckling of the connection layers and stiffness reduction of the top layer. Moreover, a sudden drop in the load value is not seen after this point because the connection and bottom layers resist the external load. This behavior of the composite increases the energy absorption during the bending process. Considering the FE curves in Figure 9, it could be found that the numerical model has a good agreement for predicting elastic behavior. Some deviation from the experimental results can be observed for the nonlinear regime; however, the trend of the curves is generally acceptable. The numerical and experimental failures are compared in Table 10.

Table 10.

Comparison between numerical and experimental results at the failure point.

Sample	Experimental (N)	FE (N)	Error (%)
SU	369	406	10
RSU	906	970	7
DU	605	670	9.7
RDU	1441	1517	5.2

As stated in Table 10, the maximum error is 10% which is shown by the proposed model’s ability to predict the failure point and failure behavior of the 3D composites with acceptable accuracy. This point shows that the model could be used for optimizing the composite structure by changing the structural parameters such as the yarn diameter or yarn type to improve the mechanical properties of the composite based on the final function. Moreover, utilizing the model leads to reduction of the production costs because the mechanical behavior of the composite can be predicted before production and the weak points of the composite found and eliminated.

Some reasons can be brought for the error between numerical and experimental results. During the fabrication process of the knitted fabric, some glass fibers are damaged. This phenomenon reduces the bending strength of the composite. But, in the finite element model, this issue was not considered, and it could be observed that the failure load for the numerical model is more than that of the experimental model. Another reason is that the loop density near the connection points between layers is changed during the manufacturing process, and this case affects mechanical properties of the composite. However, in this study, the loop density through each layer was assumed to be the same.

Conclusion

In this study, a multi-scale finite element model was proposed to predict the bending behavior of the 3D multi-cell weft-knitted composites under bending load. The numerical model was utilized for the linear and nonlinear behavior of the composite. According to the results at the meso scale, using straight yarns such as weft and warp yarns within the composite structure improves the tensile strength of the composite under tensile load through the wale and course directions. Moreover, the stiffness of the composite is also increased by the presence of the warp and weft yarns. For weft-knitted composites, the loop yarn, because of its curvature, cannot show maximum ability under applied load, and the tensile strength of this structure is lower than that of the biaxial weft-knitted composite. The outputs from the meso models were used for the macro model. The results of the bending test at the macro model were compared with the experimental, and a good agreement between numerical and experimental results was found. The proposed model could predict the linear and nonlinear behavior of the composites under bending load. Moreover, utilizing a biaxial structure for fabricating a 3D composite improves the mechanical properties of the composite and increases the bending strength and energy absorption of the composite. According to numerical results, it could be concluded that the proposed model can not only be used in the 3D weft-knitted composite to predict bending behavior of this structure, but also could be applied for other structures to achieve the optimum structure based on the final function.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by The Science and Technology Project of Jiangxi Provincial Department of Education “Research on Clothing Design Based on Outdoor Sports Monitoring Function” (No: GJJ181045).

ORCID iD

Amir Reza Eskenati

References

Bilisik

. Multiaxis 3D weaving: comparison of developed two weaving methods (tube-rapier weaving versus tube-carrier weaving) and effects of bias yarn path to the preform properties. Fibers Polym 2010; 11: 104–114.

Mouritz

Bannister

Falzon

, et al. Review of applications for advanced three-dimensional fibre textile composites. Compos A: Appl Sci Manuf 1999; 30: 1445–1461.

Abounaim

. Process development for the manufacturing of flat knitted innovative 3D spacer fabrics for high performance composite applications, Bangladesh, 2010.

Abounaim

Hoffmann

Diestel

, et al. Development of flat knitted spacer fabrics for composites using hybrid yarns and investigation of two-dimensional mechanical properties. Textile Res J 2009; 79: 596–610.

Abounaim

Hoffmann

Diestel

, et al. Thermoplastic composite from innovative flat knitted 3D multi-layer spacer fabric using hybrid yarn and the study of 2D mechanical properties. Compos Sci Technol 2010; 70: 363–370.

Liu

. Stiffness prediction of multilayer-connected biaxial weft-knitted fabric-reinforced composites. J Reinf Plast Compos 2015; 34: 1113–1125.

Hessami

Alamdar Yazdi

Mazidi

. The effect of loop density on the tensile behavior of biaxial weft knitted composites using both experimental tests and numerical method. J Ind Text 2021; 51(1): 48–67.

Fan

White

Zhu

, et al. Estimation of the molding thickness of knitted fabric reinforced thermoplastic composites produced by hot-pressing. J Thermoplast Compos Mater 2007; 20: 487–497.

Hessami

Yazdi

Mazidi

. Investigation of tensile and flexural behavior of biaxial and rib 1×1 weft-knitted composite using experimental tests and multi-scale finite element modeling. J Compos Mater 2019; 53: 3201–3215.

10.

Khondker

Herszberg

Leong

. An investigation of the structure-property relationship of knitted composites. J Compos Mater 2001; 35: 489–508.

11.

Ramakrishna

Hull

. Tensile behaviour of knitted carbon-fibre-fabric/epoxy laminates-Part I: experimental. Compos Sci Technol 1994; 50: 237–247.

12.

Rios

Ogin

Lekakou

, et al. A study of damage development in a weft knitted fabric reinforced composite. Part 1: experiments using model sandwich laminates. Compos A: Appl Sci Manuf 2007; 38: 1773–1793.

13.

Shekarchizadeh

Abedi

Jafari Nedoushan

. Prediction of elastic behavior of plain weft-knitted composites. J Reinforced Plast Compos 2016; 35: 1613–1622.

14.

Dinh

Weeger

Kaijima

, et al. Prediction of mechanical properties of knitted fabrics under tensile and shear loading: mesoscale analysis using representative unit cells and its validation. Compos B: Eng 2018; 148: 81–92.

15.

Vassiliadis

Provatidis

Kallivretaki

. Geometrical modeling of plain weft knitted fabrics. Indian J Fibre Text Res 2007; 32: 62–71.

16.

Hamedi

Hasani

Dibajian

. Numerical simulating the flexural properties of 3D weft-knitted spacer fabric reinforced composites. J Compos Mater 2017; 51: 1887–1899.

17.

Omrani

Hasani

Dibajian

. Multi-scaled modeling the mechanical properties of tubular composites reinforced with innovated 3D weft knitted spacer fabrics. Appl Compos Mater 2018; 25: 145–161.

18.

Hosseini

S-S

Hasani

Ajeli

. Effect of profile and reinforcing the surface layers on bending behavior of 3D multi-cell spacer weft-knitted composites. J Ind Text 2019; 1528083719892920.

19.

Masoumi

Abdellahi

Hejazi

. Investigation flexural behavior of hybrid-reinforced layered filament wound pipes using experimental tests and numerical model. J Ind Text 2021: 15280837211034244.

20.

Rostami

Zarrebini

Abdellahi

, et al. Investigation of flexural performance of concrete reinforced with indented and fibrillated macro polypropylene fibers based on numerical and experimental comparison. Struct Concr 2021; 22: 250–263.

21.

Dehkordi

SSH

Ghane

Abdellahi

, et al. Numerical modeling of the air permeability of knitted fabric using computational fluid dynamics (CFD) method. Fibers Polym 2017; 18: 1804–1809.

22.

Shekarchizadeh

Jafari Nedoushan

Dastan

, et al. Experimental and numerical study on stiffness and damage of glass/epoxy biaxial weft-knitted reinforced composites. J Reinf Plast Compos 2021; 40: 70–83.

23.

Weeger

Sakhaei

Tan

, et al. Nonlinear multi-scale modelling, simulation and validation of 3D knitted textiles. Appl Compos Mater 2018; 25: 797–810.

24.

Lomov

Ivanov

Verpoest

, et al. Meso-FE modelling of textile composites: road map, data flow and algorithms. Compos Sci Technol 2007; 67: 1870–1891.

25.

Chamis

. Simplified composite micromechanics equations for strength, fracture toughness and environmental effects. Cleveland: NASA Lewis Research Center 1984; 15: 41–55.

26.

Puck

Schürmann

. Failure analysis of FRP laminates by means of physically based phenomenological models. Compos Sci Technol 2002; 62: 1633–62.

27.

Tabatabaei

Lomov

. Eliminating the volume redundancy of embedded elements and yarn interpenetrations in meso-finite element modelling of textile composites. Comput Struct 2015; 152: 142–154.

28.

Barbero

. Finite element analysis of composite materials using AbaqusTM. Boca Raton: CRC Press, 2013.

29.

Kodagali

Tessema

Kidane

. Progressive failure analysis of a composite lamina using puck failure criteria. In: Proceedings of the American society for composites—Thirty-second technical conference 2017, Purdue University, West Lafayette, Indiana, October 23–25.

30.

Rong

Shujian

Pengnan

, et al. Effect of fiber orientation angles on the material removal behavior of CFRP during cutting process by multi-scale characterization. Int J Adv Manufact Technol 2020; 106: 5017–5031.

31.

Lee

C-S

Lee

J-M

. Failure analysis of reinforced polyurethane foam-based LNG insulation structure using damage-coupled finite element analysis. Compos Struct 2014; 107: 231–245.

32.

Lee

C-S

Yoo

B-M

Kim

M-H

, et al. Viscoplastic damage model for austenitic stainless steel and its application to the crack propagation problem at cryogenic temperatures. Int J Damage Mech 2013; 22: 95–115.

33.

Hashin

. Failure Criteria for Unidirectional Fiber Composites. ASME. J. Appl. Mech. June 1980; 47(2): 329–334.