Tensile impact damage behaviors of co-woven-knitted composite materials with a simplified microstructure model

Abstract

The tensile damage behaviors of co-woven-knitted (CWK) composite materials under high strain rates were investigated with a simplified microstructure model. In the microstructure model, the knitted structure is homogenized with the resin in the CWK composite. Then the CWK composite was simplified as a woven structure reinforced homogenized knitted composite. In the woven composite, the mechanical properties of the resin were calculated from a homogenized, knitted-structure, reinforced composite. With this homogenization of the knitted component, the CWK composite is transformed as a woven composite. The tensile behaviors of the CWK composite under high strain rates were predicted with the finite element method (FEM) at the woven composite level. The stress–time curves and the damage fractures of the CWK composite along the warp direction (0°), the diagonal direction (45°) and the weft direction (90°) under various strain rates were obtained and compared with those from the experiments. The influences of the strain rate on the tensile behaviors of the CWK composites were analyzed both from the experiments and the FEM results. It was shown that FEM results and experimental results were in good agreement in terms of strength and failure morphology prediction. It was concluded that the simplified model and the methodology developed are suitable for the design of other kinds of hybrid-structure composite under both quasi-static and impact loading by homogenizing some parts of the reinforcing materials.

Keywords

co-woven-knitted (CWK) composite tensile impact high strain rates simplified microstructure model finite element method (FEM)

Three-dimensional (3-D) textile structural composites have been applied increasingly in engineering because of their high fracture toughness and impact damage tolerance. There are various types of 3-D textile structures, such as 3-D woven, 3-D knitted and 3-D braided structures which can be combined to form a hybrid structure for optimizing the mechanical behaviors.

Co-woven-knitted (CWK) fabric is a novel textile structure in which the woven and knitted structures are manufactured simultaneously on a modified knitting machine.¹ This structure combines the woven and knitted structure together and has the advantages of both stability of the woven fabric and formability of the knitted fabric. Since the invention of this hybrid structure by Hu et al.,¹ the CWK composite has become an interesting topic and has attracted many investigators. Hu et al.² investigated the influence of microstructure geometry parameters on the tensile, bending and impact properties of the CWK composite. Xu et al.³ analyzed the geometrical model of the CWK composites. Ma et al.⁴ investigated strain-rate sensitivity of the tensile behaviors of the CWK composites. Ma et al.^5–7 analyzed the damage mechanisms of CWK composites under tensile impact with fast Fourier transform, Hilbert-Huang transform and system function methods. Chen et al.^8,9 have studied the electromagnetic and electrostatic shielding properties of CWK composites. However, the tensile behaviors of CWK composites under high strain rates have not been studied at the microstructure level so far.

The present research continues our previous work^4–7 to numerically investigate the tensile behaviors of the CWK composite under various strain rates with a simplified geometrical model based on the CWK composite microstructure. The CWK composite geometrical model is regarded as a parallel structure composed of a woven fabric and a knitted fabric. The CWK composite is then simplified as a woven composite. With the simplified model, the CWK composite is transformed to a woven composite. In the woven composite, the mechanical properties of the resin were calculated from homogenized, knitted-structure, reinforced composite. By combining the knitted structure and the resin, the stiffness matrix of the equivalent resin was derived from the knitted-structure, reinforced composite. Based on the mechanical behaviors of the woven composite, the tensile properties of the CWK composite under various strain rates was calculated with the finite element method (FEM) and compared with those from the experimental results. Compared with our previous work^4–7 on characterizing the tensile behaviors of CWK composites in the frequency domain, here we will focus on the tensile behaviors and stress–strain curves prediction from the simplified CWK composite microstructure.

Experimental preparation and testing

Sample preparation

Figure 1 shows the CWK fabric, where 0° is the warp direction and 90° is the weft direction. The woven and knitted structures were interwoven simultaneously during fabric manufacture. The warp and weft yarns were the E-glass filament tows, and the knitted yarns were the polyester fiber tows. Table 1 lists the specification of the fabric.

Figure 1.

Structure of the CWK fabric.

Table 1.

Specification of the CWK fabric

Yarns	Material	Fineness	Yarn density	Fabric density
Warp yarn	E-glass filament	2400 tex	26 end/10 cm	71.65 g/(10 cm)²
Weft yarn	E-glass filament	900 tex	94 end/10 cm
Knitted yarn	High tenacity polyester filament	83.3 tex × 2	7 end/25.4 mm

The CWK composite was consolidated with an epoxy resin through a vacuum-aided, resin transfer molding (VARTM) process. A mixture of Bisphenol A epoxy (Type 618) and agent Tri-methyl-hexamethylene-diamine (Type 593) (both supplied by Shanghai Resin Factory of China) in a volume proportion of 3:1 was first injected into the CWK fabric by a VARTM system, and then cured at 110℃ for 60 minutes.

The fiber volume fraction of the CWK composite was approximately 40%. Figure 2 shows the size and the shape of the composite coupon for the tensile tests under quasi-static conditions and the high strain-rate tensile tests.

Figure 2.

Size and shape of the CWK composite: (a) size of the composite coupon and (b) the composite coupon along the warp direction.

Tensile impact tests

The composite specimens along three directions (0°, 90° and 45°) were prepared for testing according to the size and shape shown in Figure 2(a). Figure 2(b) illustrates the composite specimen along the warp direction (0°).

The tensile tests along three directions (0°, 90° and 45°) were conducted under high strain rates, from 1589/s to 2586/s on the split Hokinson tension bar (SHTB) apparatus shown in Figure 3. For each strain rate, at least three tests were conducted.

Figure 3.

Experimental setup of the SHTB apparatus.

Simplified microstructure model of the CWK composite

Simplified principle

As shown in Figure 1, the CWK fabric contains the paralleled woven structure and knitted structure. The stable and rigid woven structure leads to the higher stiffness and strength of the CWK fabric. The knitted structure is flexible and deformable under low loading. Compared with the woven fabric, the path of the knitted yarn is more complex. It is more difficult to establish a microstructure model of the knitted structure than the woven structure at the yarn level. Furthermore, owing to the microstructure irregularity of the knitted composite, the number of elements will be numerous during finite element analysis (FEA) meshing. In order to establish a geometrical model of the CWK composite, the CWK fabric will be simplified into two parallel structures. Figure 4 shows the sketch of a simplified model. First the knitted structure was combined with the resin to be homogenized as a knitted composite. The stiffness matrix of the knitted composite was derived and equated to the resin of a woven composite. Because the woven geometrical structure is relatively simple to established, the tensile–impact behaviors of the CWK composite could be easily investigated at the woven structure level. Therefore, the mechanical behaviors of the CWK composite model were derived from the woven structure and the behaviors of the equivalent resin, i.e. homogenized knitted composite.

Figure 4.

Simplified method.

Stiffness matrix of the equivalent resin

The equivalent resin is the resin reinforced by the knitted structure as shown in Figure 5. The representative volume element (RVE) of the knitted composite contains two knitted yarns. The RVE was partitioned into a series of sub-cells along the wale direction which is supposed to have two parts. Each sub-cell was considered to be a unidirectional, fiber-reinforced composite and had the same fiber volume fraction as the RVE. Upon determining the constitutive relationship of each unidirectional composite and according to Huang et al.’s¹⁰ theory, the overall compliance matrix of the RVE was obtained by assembling the contributions of all the unidirectional composites in the global coordinate system. The axes of the global coordinate for the RVE were the wale, course and thickness direction of the knitted yarns, respectively.

Figure 5.

Schematic diagram of the rib stitch and the RVE of the equivalent resin.

Volume fraction of the knitted yarn and resin

The volume fraction of the knitted yarn (V_f) and resin (V_m) in the equivalent resin was calculated from the volume fraction of the knitted yarn (V_k), the warp yarn ( $V warp$ ), the weft yarn ( $V weft$ ) and the resin (V_r) in the CWK composite. The calculated results showed that the knitted yarn (V_f) and the resin (V_m) in the equivalent resin were 20% and 80%, respectively.

Stiffness matrix of each sub-cell

The knitted yarn and the resin were assumed as transverse-isotropic and isotropic materials, respectively. As mentioned above, the sub-cell was regarded as a unidirectional composite. The local Cartesian coordinates ( $1 - 2 - 3$ ) of each sub-cell were the longitudinal, transverse and thickness direction of the yarn. The local coordinate sub-cell compliance matrix $[S ij]$ and the stiffness matrix $[C ij]$ were calculated from the following formula

[S ij] = [\frac{1}{E 11} - \frac{ν 12}{E 11} - \frac{ν 13}{E 11} 000 - \frac{ν 12}{E 11} \frac{1}{E 22} - \frac{ν 23}{E 22} 000 - \frac{ν 13}{E 11} - \frac{ν 23}{E 22} \frac{1}{E 33} 000000 \frac{1}{G 23} 000000 \frac{1}{G 13} 000000 \frac{1}{G 12}]

(1)

[C ij] = [S ij] - 1

(2)

E 11 = V f E f + V m E m

(3)

ν 12 = ν 13 = V f ν f + V m ν m

(4)

E 22 = E 33 = \frac{E m}{1 - V f (1 - E m / E f)}

(5)

G 12 = G 13 = \frac{G m}{1 - V f (1 - G m / G f)}

(6)

ν 23 = \frac{ν 12 (1 - ν 12 E 22 / E 11)}{1 - ν 12}

(7)

G 23 = \frac{E 22}{2 (1 + ν 23)}

(8)

where, E is the Young’s modulus, V is the volume fraction, ν is the Poisson’s ratio and G is the shear modulus; subscript f means knitted yarn, m means the resin and 1, 2, 3 mean the three directions of the local coordinate system.

The compliance and the stiffness matrix of each sub-cell was transformed into the global coordinate system through the following formula

[\bar{S ij}] n K = [T ij] S [S ij] [T ij] S T

(9)

[\bar{C ij}] n K = [T ij] C [C ij] [T ij] C T

(10)

where the superscript K refers to the yarn number (the first or the second yarn) in the RVE, the subscript n denotes the yarn segment (shown in Figure 6), and

[T ij] S

and

[T ij] C

are the coordinate transformation matrices, given by

pr [T ij] s = [l 12 l 22 l 32 l 2 l 3 l 3 l 1 l 1 l 2 m 12 m 22 m 32 m 2 m 3 m 3 m 1 m 1 m 2 n 12 n 22 n 32 n 2 n 3 n 3 n 1 n 1 n 2 2 m 1 n 1 2 m 2 n 2 2 m 3 n 3 m 2 n 3 + m 3 n 2 n 3 m 1 + n 1 m 3 m 1 n 2 + m 2 n 1 2 n 1 l 1 2 n 2 l 2 2 n 3 l 3 l 2 n 3 + l 3 n 2 n 3 l 1 + n 1 l 3 l 1 n 2 + l 2 n 1 2 l 1 m 1 2 l 2 m 2 2 l 3 m 3 l 2 m 3 + l 3 m 2 l 1 m 3 + l 3 m 1 l 1 m 2 + l 2 m 1]

(11)

[T ij] C = [l 12 l 22 l 32 2 l 2 l 3 2 l 3 l 1 2 l 1 l 2 m 12 m 22 m 32 2 m 2 m 3 2 m 3 m 1 2 m 1 m 2 n 12 n 22 n 32 2 n 2 n 3 2 n 3 n 1 2 n 1 n 2 m 1 n 1 m 2 n 2 m 3 n 3 m 2 n 3 + m 3 n 2 n 3 m 1 + n 1 m 3 m 1 n 2 + m 2 n 1 n 1 l 1 n 2 l 2 n 3 l 3 l 2 n 3 + l 3 n 2 n 3 l 1 + n 1 l 3 l 1 n 2 + l 2 n 1 l 1 m 1 l 2 m 2 l 3 m 3 l 2 m 3 + l 3 m 2 l 1 m 3 + l 3 m 1 l 1 m 2 + l 2 m 1]

(12)

where the direction cosines are defined as

l 1 = \cos (α) \sin (β), l 2 = - \sin (α), l 3 = - \cos (α) \cos (β)

m 1 = \sin (α) \sin (β), m 2 = \cos (α), m 3 = - \sin (α) \cos (β)

n 1 = \cos (β), n 2 = 0, n 3 = \sin (β)

Figure 6.

Yarn segment orientation in the global coordinate system.

Assembly of the stiffness matrix for the equivalent resin

The contribution of each sub-cell was combined with the volume averaging method to obtain the overall response characteristics of the RVE. The stiffness matrix $[\bar{C ij}]$ and compliance matrix $[\bar{S ij}]$ of the RVE were given by

[\bar{C ij}] = \sum V n K [\bar{C ij}] n K

(13)

[\bar{S ij}] = \sum V n K [\bar{S ij}] n K

(14)

where

V n K

is the volume fraction of each sub-cell in the RVE.

The stiffness¹¹ and compliance¹² methods are two conventional methods used to integrate the stiffness and compliance matrices together. In this study, the two conventional methods were combined in the Huang method et al.^10,13 to calculate the stiffness and the compliance matrices. The overall compliance matrix $[S ij] G$ and stiffness matrix $[C ij] G$ were determined as

[S ij] G = α c ([\bar{C ij}] - 1) + α s ([\bar{S ij}])

(15)

[C ij] G = [S ij] - 1

(16)

where

α c

and

α s

are the contribution ratio of the stiffness approach and the compliance approach, respectively.

As discussed by Huang et al.,¹⁰ the contribution ratio depends on the type of composite being investigated, i.e. types of resin, fibers and fabric structures. Huang et al.¹⁰ listed a series of the contribution ratio to find the influence of $α c$ on the engineering moduli of the knitted composite. The value of $α c$ should be further studied. Here, we use $α c = 0.9$ for the calculation. Then $α s = 1 - α c$ .

The detailed calculation processes have been shown in the Appendix 1.

Simplified model

Figure 7 shows the simplified model of the CWK composite. The simplified model contains the plain woven structure and the equivalent resin. The mechanical properties of the equivalent resin are the same as the knitted composite mentioned above. The woven composite would be regarded as the simplified CWK composite.

Figure 7.

Simplified model of the CWK composite: (a) the woven structure, (b) the equivalent resin and (c) the simplified model.

FEA

All the FEA were conducted in the following environments:

OS platform: Windows XP

FEA software: ABAQUS 6.10

Finite element models

The finite element models included the incident bar and transmission bar of the split SHTB apparatus and the composite specimen as shown in Figure 8. As shown in Figure 7 and Figure 8, all the mentioned parts have the same size as those in the experiment. The incident bar and transmission bar were assumed to be an isotropic elastic body which has the mechanical parameters of steel. The interaction between the yarns and the equivalent resin was defined as “Tie”; the surface of the equivalent resin was the master surface while the surface of the yarns was the slave surface. Meanwhile, the composite sample was tied to the incident bar and the transmission bar. The original impact stress wave in the incident bar was inputted as the initial loading condition. Figure 9 shows the initial stress wave of 2407/s along the 0° direction. On the assumption of the perfect bonding between yarns and the equivalent resin, the elements of the yarns and the resin shared the same nodes and surface in the mesh. The bars and the yarns were meshed with a C3D8R hexahedron solid element, while the equivalent resin was meshed with a C3D4 tetrahedron solid element. The mesh schemes of the CWK composite along the three directions (0°, 90° and 45°) can be seen in Figure 10. The total element numbers of the three directions (0°, 90° and 45°) are 47399, 121650 and 59071, respectively. For the 45° direction, because of the special angle of the warp yarn and the weft yarn and the difficulty to mesh the yarns, a C3D4 solid element was adopted. For the 0° and 90° direction, the element type and size were the same. The global coordinate ( $x - y - z$ ) of the homogenized, knitted component (the so called ‘equivalent resin’) was in the wale, course and thickness direction of the knitted structures, respectively. The local coordinates ( $1 - 2 - 3$ ) of the equivalent resin was defined as always along the warp, the weft and the thickness direction of the CWK composite.

Figure 8.

Sketch diagrams of the SHTB apparatus with the composite sample: (a) sketch diagram of the SHTB apparatus; (b) connection among the composite coupon, the incident bar and the transmission bar; (c) principle of the SHTB apparatus testing, positions of strain gages and stress wave propagation; and (d) photograph of the connections of the sample with the bars, left: front view, right: top view.

Figure 9.

Initial stress wave of 2407/s along the 0° direction.

Figure 10.

Mesh scheme of each component in three directions: (a) warp direction (0°), diagonal direction (45°) and (c) weft direction (90°).

In addition, the mesh sensitivity was not investigated in the current study. For the calculations along the 0° and 90° directions, the mesh schemes were the same. The C3D8R element was used for meshing. For the calculations along 45°, owing to the shape irregularity of the resins and fibers, the C3D4 element was used. The current mesh scheme satisfied the required precision.

Failure criteria

Resin

The epoxy matrix was modeled by an elasto-plastic, behavior material with strain-rate sensitivity. In addition, a failure criterion was implemented based on a maximum equivalent strain defined by a failure strain $ɛ f$ . This strain-rate dependent behavior of the epoxy resin has been studied by Gilat et al.¹⁴ and Tay et al.¹⁵ Gilat et al.¹⁴ tested the tensile behaviors of epoxy resin up to the strain rate of 460/s. Tay et al.¹⁵ obtained the compressive behaviors of epoxy resin under high strain rates. Figure 11 shows the tensile behaviors of the epoxy resin under the different strain rates (up to 700/s). From Figure 11, the stresses fluctuate significantly and the curves are concentrated in the narrow region. The stress–strain curves of the epoxy resin are different from those in the quasi-static state. In the FEM calculations, the data of the failure strain $ɛ f$ from the strain rate at 700/s were referred to instead of the strain rates from 1589/s to 2586/s used in the current study. This introduced some errors in the calculations. However, as far as the authors’ knowledge, the tensile behaviors of the pure epoxy resin under a strain rate of more than 1500/s are not available. The data from the strain rate of 700/s is an approximation. The epoxy resin analyzed in the aforementioned work was not exactly the same as the one used by Gilat et al.,¹⁴ but it was very similar in the properties and composition.

Figure 11.

Tensile stress–strain curves for epoxy resin at different strain rates.¹⁴

The constitutive model presented is based on Bodner’s¹⁶ internal state variable constitutive equations, which were originally developed to analyze the deformation of metals above one half of the melting temperature. The model was modified to analyze the strain rate and hydrostatic stress-dependent, nonlinear deformation of polymeric materials. Inelastic strains were assumed to be present at all values of stress (there was no yield condition), and state variables, which evolve with stress and inelastic strain, were defined to represent the average effects of the deformation mechanisms. All of the nonlinearity and strain-rate dependence was assumed to be due to inelastic deformation, where in reality the nonlinearity could be due to a mixture of deformation and damage.

For the epoxy resin, the small strain theory was assumed to apply where the total strain rate, $\overset{\cdot}{ɛ} ij$ , is composed from elastic, $\overset{\cdot}{ɛ} ij e$ , and inelastic, $\overset{\cdot}{ɛ} ij I$ , parts

\overset{\cdot}{ɛ} ij = \overset{\cdot}{ɛ} ij e + \overset{\cdot}{ɛ} ij I

(17)

The elastic strain rate is given by Hooke’s law, and the inelastic strain rate is written in the form

\overset{\cdot}{ɛ} ij I = λ \frac{\partial f}{\partial σ ij}

(18)

where λ is a scalar rate function of the internal state variables, and f is a flow potential. In the Von Mises yield criterion of the isotropic resin, for the three principal stresses

σ 1, σ 2 and σ 3

, taking

σ 1 > σ 2 > σ 3

, then the yield criterion is

f = J 2 - k 2

, where k is the yield stress of the material in pure shear. At the onset of yielding, the magnitude of the shear yield stress in pure shear is

\sqrt{3}

times lower than the tensile yield stress in the case of simple tension.

And

\frac{\partial J 2}{\partial σ 1} = \frac{\partial}{\partial σ 1} \frac{1}{6} [(σ 1 - σ 2) 2 + (σ 2 - σ 3) 2 + (σ 3 - σ 1) 2] = \frac{2}{3} [σ 1 - \frac{1}{2} (σ 2 + σ 3)]

(19)

The flow potential f is a scalar function which, when differentiated with respect to the stresses, gives the plastic strains. It is also called the plastic potential. The flow potential is assumed to be independent of the third stress invariant. It consists of the Mises cylinder in compression with an ellipsoidal cap in tension. The value of the potential depends on the current stress point.

The finite element code used, ABAQUS/Explicit, incorporates this type of material behavior by defining a number of $σ - ɛ$ curves over a range of strain rates. The linear elastic segment of the stress–strain behavior was assumed to be independent of strain rate. Beyond the elastic limit, the strain rate will have an effect on stress–strain curves. The failure was also included in the material behavior, which exhibited a softening post-failure behavior based on a linear energy dissipation rule. In the rule, a parameter for the energy dissipation per unit area G_I is needed. This value and the rest of the material parameters needed for the simulation have been presented in Gilat et al.¹⁴

Equivalent resin

Appendix 1 describes the stiffness of the equivalent resin, i.e. the knitted-fiber, tow-reinforced, epoxy composite. The maximum strain failure criterion was used in the calculation. Considering that the tensile behaviors of the equivalent resin cannot be tested, the failure strain of the epoxy under high strain rates was used instead of the failure strain of the equivalent resin. This is not precise since the equivalent resin (knitted-fiber, tow-reinforced resin) should be stiffer than the pure resin, which should have lower strain to failure. This approximation will lead to the lower failure strain for the equivalent resin. In the FEM calculation based on the meso-scale model, the strain of the equivalent resin could be obtained and compared with the maximum failure strain. If the strain is greater than the maximum failure strain, the elements with the maximum strain are deleted.

Fiber tows

The E-glass warp and weft fiber tows were treated as transversely-isotropic materials. Table 2 and Table 3 list the mechanical parameters. The tensile modulus and the strength were obtained from that in high strain-rate tension as provided by Wang et al.¹⁷ The strain rates in the current study vary from 1589/s to 2586/s and the stress–strain curve of 1700/s was adopted to predict the failure stress of the different strain rate in the current study. The maximum stress failure criterion was used to determine the failure in the FEM calculation. When the stress in the microstructure model of the woven composite was greater than the maximum tensile strength of the E-glass fiber tow, the elements with the maximum stress were deleted.

Table 2.

Stiffness of glass fiber tows

Yarn	$E 11$ (GPa)	$E 22$ (GPa)	$E 33$ (GPa)	$G 12$ (GPa)	$G 13$ (GPa)	$G 23$ (GPa)	$ν 12$	$ν 13$	$ν 23$
warp	85	80	80	28	28	27	0.25	0.25	0.2
weft	88	80	80	28	28	27	0.25	0.25	0.23

Table 3.

Strength of glass fiber tows

Yarn	XT (MPa)	XC (MPa)	YT (MPa)	YC (MPa)	SS12 (MPa)	SS23 (MPa)
warp	4650	−4650	3550	−3550	1400	1200
weft	5156	−5156	4150	−4150	1700	1400

XT: tensile strength along the x-direction; XC: compressive strength along the x-direction; YT: tensile strength along the y- direction; YC: compressive strength along y-direction; SS: shear strength.

Results and discussion

Results of comparisons along the warp direction (0°)

Figure 12 shows the stress–time curves obtained from the experiment and the FEA calculations. The stress–time curves are highly sensitive to the strain rate, and the failure stress, time and the initial modules are increased with the strain rate. The FEA results and the experimental results have the same tendency in Figure 12. Herein, the tensile curve under the quasi-static condition is also presented. The FEA calculation of the quasi-static tension is different from the high strain rate in the defining load, i.e. applying the constant displacement. As shown in Figure 12, the experimental and FEA results of the quasi-static tension are in good agreement in strength.

Figure 12.

Stress–time curves of the simplified model along the 0° direction.

In the analyses and comparisons of tensile damage both in the FEA and the experiment, the geometrical models of the woven fabric and the entire CWK composite were presented for comparisons. Figure 13 shows the fabric structure model in the CWK composite and Figure 14 shows the geometrical model of the entire CWK composite. Both the geometrical models in Figure 13 and Figure 14 are from a same FEM model. The tensile damage evolution of yarns under a 2586/s strain rate is shown in Figure 13. It clearly shows the damage process of the yarns. Figure 14 is the fractograph of the CWK composite under a 1840/s strain rate obtained from both the experiment and the FEA calculation. As shown in Figure 14, the FEA result is in good agreement with the experimental result.

Figure 13.

Tensile damage evolution of yarns under a strain rate of 2586/s along the 0° direction: (a) 8 µs, (b) 45 µs, (c) 57 µs and (d) 74 µs.

Figure 14.

Damage morphology of the CWK composite under a strain rate of 1840/s along the 0° direction: (a) experiment and (b) FEA.

Results of comparisons along the diagonal direction (45°)

Figure 15 illustrates the stress–time curves obtained from both the experiment and the finite simulation along the diagonal direction (45°). The stress–time curves are different from the curves along the 0° direction. The shear failure of the resin was the main failure of the composite along the 45° direction. As shown in Figure 15, the stresses of the FEA calculation are a little higher than those in the experiment. The main reason is that the knitted yarns and the resin were regard as the equivalent resin in the FEA calculation. The degradation of the stiffness and strength of the knitted yarns were neglected in the equivalent resin. The yarns and the resins in the CWK composites along the 45° direction only were meshed with a C3D4 solid element because of the irregular shape of the yarns and resins. However, the yarns were meshed with a C3D8R solid element along the 0° and 90° directions.

Figure 15.

Stress–time curves of the simplified model along the 45° direction.

The tensile damage evolution of yarns under a 2253/s strain rate is illustrated in Figure 16. The warp yarns are damaged first, and then followed by the weft yarns being damage because the cross-section of the weft yarn was larger than the warp yarn. Figure 17 shows the fractograph of the CWK composite under a 2336/s strain rate obtained from both the experiment and the FEA calculations. As shown in Figure 17, the damage of the warp and weft yarns as well the resin of the FEA result agrees well with the experimental result.

Figure 16.

Tensile damage evolution of yarns under a strain rate of 2253/s along the 45° direction: (a) 8 µs, (b) 36 µs, (c) 70 µs and (d) 78 µs.

Figure 17.

Damage morphology of the CWK composite under a strain rate of 2336/s along the 45° direction: (a) experiment and (b) FEA.

Results of comparisons along the weft direction (90°)

Figure 18 compares the stress–time curves between the FEA and the experiment along weft direction (90°). As shown in Figure 18, the stresses along the 90° direction are higher than those along the 0° and 45° directions.

Figure 18.

Stress–time curves of the simplified model along the 90° direction.

The tensile damage evolution of the yarns under a 1932/s strain rate is illustrated in Figure 19. The fractograph of the weft yarns along the 90° direction is different from the warp yarns along the 0° direction. Figure 20 is the fractograph of the CWK composite under a 2159/s strain rate obtained both from the experiment and the FEA calculation.

Figure 19.

Tensile damage evolution of the yarns under a strain rate of 1932/s along the 90° direction: (a) 8 µs, (b) 36 µs, (c) 70 µs and (d) 78 µs.

Figure 20.

Damage morphology of the CWK composite under a strain rate of 2159/s along the 90° direction: (a) experimental and (b) FEA.

From the comparison of the stress–time curves between the FEA and the experiment along the 0°, 45° and 90° directions, it was found that the simplified model was appropriate to calculate the tensile behaviors of the CWK composite. However, the fibers were assumed as a transversely-isotropic material and the slippage between the yarns and the resin was ignored. In fact, some fibers had probably been broken during the fabric weaving process and the distance of the closed weft yarns also existed. Additionally, after homogenization of the knitted structure reinforcing the epoxy resin, the homogenized knitted component becomes an anisotropic material which led to uneven transmission of the stress wave. The reflection of the stress waves from the interface also led to the further interface damage. These factors led to the difference between the results obtained from the experiment and the FEA calculations.

The reasons for the drastic response differences in Figures 12, 15 and 18, between the strain rates in the range of 2200–2400/s and the range of 1550–1850/s are from the strain-rate effect of glass fiber tows, the resins and the CWK structures. The rate-dependent behaviors of the fiber tows and resins will contribute to the rate sensitivity of the CWK composite. More importantly, the CWK fabric structure will influence the tensile behaviors of the CWK composite significantly, just as the tensile behaviors of 3-D woven fabrics are influenced under various strain rates.^18,19 The tensile behaviors of the CWK fabric under various strain rates will be tested further to verify the influence of the fabric structure on the tensile properties.

Fracture of the equivalent resin

The stiffness matrix for the homogenized, knitted component is presented in Appendix 1. The fractographs of the equivalent resin along three directions under different strain rates are shown in Figure 21. The damage locations and morphologies of the three directions are very different. Damage locations along the 0° direction and the 90° direction are center-left and center-right while along the 45° direction it is in the middle; fracture morphology along the 0° direction is orderly, but is irregular along the other two directions especially along the 45° direction. All of the above reflects the anisotropic properties of the equivalent resin.

Figure 21.

Damage morphologies of the equivalent resin: (a) along the 0° direction under a strain rate of 2220/s, (b) along the 45° direction under a strain rate of 1722/s and (c) along the 90° direction under a strain rate of 2390/s.

Conclusions

The tensile impact properties of a CWK composite at various strain rates along the 0°, 45° and 90° directions were studied with FEA calculations and experimental investigation. In the experiments, the stress–strain curves of the CWK composite are sensitive to the strain rate. The failure stress and strain increase with the strain rate. The rate-dependent behaviors of the CWK composite attribute to the rate sensitivities of both the fiber tows, resins and fabric structures. In the FEA calculation, a simplified microstructure model which regards the CWK composite as a woven composite was established. The mechanical properties of the knitted structures were combined with that of the resin and converted to the equivalent resin. With this homogenization of the knitted component, the CWK composite is transformed as a woven composite. Based on the simplified microstructure model and the FEA calculation, the tensile stress–strain curves of the CWK composite under various strain rates have been obtained. From the comparisons of the stress–time curves at the different strain rates and directions between the FEM and the experiment, it was demonstrated that such a simplified method could be employed to calculate the tensile impact behavior of the CWK composite under various strain rates effectively.

This is one of the efforts to simplify complex 3-D textile composites into a continuum or into simple textile structural composites. The simplified model and the methodology can be extended to the design of other kinds of hybrid-structure composites by converting the reinforced phase into resin.

Footnotes

Funding

The authors acknowledge the financial support from the National Science Foundation of China (Grant No. 11272087). The financial support from the Foundation for the Author of the National Excellent Doctoral Dissertation (FANEDD) of the People’s Republic of China (FANEDD, No. 201056), the Shanghai Rising-Star Program (11QH1400100) and the Fundamental Research Funds for the Central Universities of China are also gratefully acknowledged.

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