Transverse tensile damage behaviors of three-dimensional five-directional braided composites by meso-scale finite element approach

Abstract

In this paper, the meso-scale finite element method is used to evaluate the transverse tensile damage behaviors of three-dimensional five-directional braided composites based on the representative volume cell. Finite element models with two kinds of interior braiding angles, 30° and 45°, are established on the basis of the realistic geometry shape of each yarn. A continuum damage mechanics model, which is implemented as a user-defined material subroutines in the ABAQUS commercial finite element code, is used to capture the complete damage initiation/evolution in yarns and matrix. The local responses and the failure mechanisms are analyzed. Numerical results show that the transverse tensile strength and the failure mechanisms are significantly influenced by the braiding angle. The predicted stress–strain responses and final damage morphology agree well with the experimental data, verifying the applicability of the meso-scale finite element method.

Keywords

Three-dimensional preform textile composites transverse tensile mechanical properties finite element analysis

Introduction

Via new materials and novel weaves recently developed, three-dimensional (3D) braided preforms, which are formed through multidirectional yarns interlaced with each other, have been increasingly incorporated into aerospace, military, and civil engineering industry because of their excellent transverse strength, structural integrity, and relatively low production cost for complex-shaped components.^1–4 However, compared with other 3D textile composites, such as 3D angle-interlock woven, 3D orthogonal woven, 3D triaxial braided, a major limitation in 3D braiding is that most braiding machines are difficult to produce the braided preforms of large dimensions along the transverse direction.⁵ As the braiding technology and numerical techniques are improved, therefore, the mechanical responses of 3D braided composites under transverse loads need to be systematically studied for a better understanding of the structure–property relationships.

A number of experimental studies have been conducted to understand the mechanical behaviors and damage mechanism of 3D braided composites. For example, Kalidindi and Abusafieh⁶ studied the longitudinal and transverse compression moduli and strength of 3D braided graphite/epoxy composites with a range of braiding angles from 0° to 30°. Sun et al.⁷ tested the uniaxial tensile properties of 3D braided E-Glass/epoxy composites at quasi-static and high strain rates (up to 2800 s^–1) by using Mechanical test and simulation (MTS) and split hopkinson tension bar (SHTB), respectively. The result showed that 3D braided composites are rate sensitive and exhibit a brittle mode at high strain rates. Li et al.⁸ focused on the effect of cut edge on the longitudinal tensile and compressive properties of 3D braided composites. Li et al.⁹ investigated experimentally the uniaxial compressive stress–strain responses of 3D braided carbon/phenolic composites at high rates from 350 to 1600 s^–1. It was concluded that the damage modes include matrix cracking yielding and falling off, interface debonding, migration, local buckling, and shear fracture of the fibers. Song and Li¹⁰ studied the effect of heat accelerated aging on the tensile properties of 3D four-directional braided carbon/epoxy composites. Carvelli et al.¹¹ examined the quasi-static and fatigue tensile responses of 3D braided carbon/epoxy composites. Li et al.¹² studied the nonlinear buckling and postbuckling behaviors of 3D braided composites under axial compression in thermal environments, and the results revealed that the temperature and the fiber volume fraction have a significant effect on the nonlinear buckling and postbuckling properties.

In the past decade, the finite element method based on the meso-scale representative volume cells (RVCs) has been gaining popularity in simulating the elastic deformation and failure response.^13–17 Li et al.¹⁸ described a micromechnical prediction procedure to evaluate the axial stiffness and the strength properties of 3D five-directional braided composites based on the three kinds of microstructural unit cell models. Fang et al.^19,20 developed a damage theory to predict the progressive damage behaviors of 3D four-directional braided composites subjected to longitudinal tensile and compression loading. Numerical results showed that the strength of 3D braided composites with different braiding angles is determined by the different microscopic failure modes, including transverse tension, shear breakage of yarns, and matrix cracking. Lu et al.²¹ evaluated the effect of the interfacial properties on the longitudinal tensile behaviors of 3D braided composites using a new nonlinear model. They confirmed that the interface damage is one of the critical factors resulting in the nonlinearity of the stress–strain responses. Jiang et al.²² proposed a theoretical model based on the helix geometry unit cell to simulate the effective elastic constants and the failure strength of 3D braided composites under longitudinal loading. Li et al.²³ proposed a new approach for testing and predicting the longitudinal tensile modulus of 3D braided composites based on the structure characteristics and the stiffness average volume formula. Wu et al.²⁴ reported the three-point bending fatigue behavior of four-step three-dimensional braided composites with three types of representative unit cell model. Xu et al.^25,26 developed a multiscale approach and micromechanics of failure (MMF) based progressive damage model to predict the elastic constants and the ultimate strength of biaxial and trixial braided textile composites in which the braiding angle ranges from 15° to 75°. They concluded that since MMF-based model begins with the fundamental constituents of materials, once the properties of constituents are obtained, type of textile structures can be analyzed under different conditions.

From the literature study, it is evident that information about the transverse properties of 3D braided composites is very limited.^27–30 Here, a RVC is chosen to evaluate the damage behaviors of 3D five-directional braided composites subjected to transverse tensile loading. First, based on the realistic geometry cross sections, the RVCs of 3D five-directional braided composites with two kinds of internal braiding angles, 30° and 45°, are established in the next section 2. Next, a damage model with respect to the damage initiation criteria, the damage evolvement model, and the damage stiffness matrix is introduced in “Progressive damage models” section. Subsequently, the periodic boundary conditions of RVCs are defined in “Periodic boundary condition” section. The experimental details including the materials properties, composite specimen preparation, and the transverse tensile test are described in “Experimental details” section. Furthermore, a commercial finite element software ABAQUS/Standard is employed to capture the damage initiation/evolution and the stress distribution in yarns and matrix. The predicted results are described and compared with the experimental data in “Results and discussion” section. Finally, some valuable conclusions are summarized in the final section.

RVC of 3D five-directional braided composites

Meso-scale finite element model instead of the whole 3D five-directional braided composites is usually developed to analyze the effective mechanical properties. At this scale, the topological structure and the yarn cross sections determined by the braiding technology are considered. Figure 1(a) presents 3D five-directional braiding process using four-step 1 × 1 braiding pattern on a machine bed (x–y plane). After a machine cycle, all the yarns are interwined to form a pitch length of preform along the z-direction under the guidance of yarn carrier movements. According to the above braiding process, the orientation and the position of each yarn in the space can be defined, then the topological structure of interior RVC can be established (see Figure 1(b)). In addition, the yarn cross sections of braided yarns and axial yarns become complicated due to their mutual squeezing. On the basis of microscopic image analysis in Li et al.³¹ (see Figure 1(c)), the following assumptions are proposed: (1) the cross section of braiding yarn is uniform and can be idealized as hexagon, as shown in Figure 1(d); (2) the cross section dimension of axial yarn is a variable and the initial shape can be considered as quadrangle (see Figure 1(d)); (3) the braided structure is uniform; and (4) the interior RVC is chosen to analyze the whole structure. The related geometric relationships in the interior RVC are given as follows³¹

W = T = 2 \sqrt{2} (e + \frac{k}{2})

(1)

S a = \frac{λ a}{ρ a \times ɛ a}

(2)

S b = \frac{λ b}{ρ b \times ɛ b}

(3)

S a = e 2

(4)

S b = 2 (\frac{k}{2} + e) \times k \times \cos λ - \frac{k 2}{2} \cot \frac{θ}{2}

(5)

h = \frac{W}{\tan γ}

(6)

θ = 2 arc \sin [\frac{1}{\sqrt{1 + \cos 2} γ}]

(7)

\tan γ = \sqrt{2} \tan ϕ

(8)

where

W, T, and h

are the width, thickness, and height of RVC, respectively.

θ, k, and l

are the section parameters of braided yarns, and e is the side length of axial yarns, as shown in Figure1(d).

γ and ϕ

are the interior braiding angle and the ridge braiding angle, respectively.

S_a,

λ a

ρ a

, and

ɛ a

are the sectional areas, linear density, fiber density, and yarn packing factor of axial yarns, respectively.

S_b,

λ b

ρ b

, and

ɛ b

are the sectional areas, linear density, fiber density, and yarn packing factor of braided yarns, respectively.

The detailed geometry parameters of RVCs are summarized in Table 1. Note that V_a and V_b in Table 1 indicate the fiber volume fraction of axial yarns and braided yarns, respectively. Figure 2 illustrates the meso-scale geometric models of 30° RVC and 45° RVC, respectively.

Table 1.

Detailed geometry parameters of RVCs.

γ (°)	ϕ (°)	l (mm)	k (mm)	θ (°)	e (mm)	W (mm)	T (mm)	H (mm)	V_a (%)	V_b (%)
30	22.20	1.60	0.38	98.20	0.56	2.70	2.70	6.50	24.96	34.58
45	35.26	1.38	0.46	109.20	0.52	2.80	2.80	4.00	21.91	38.18

Progressive damage models

Damage initiation criterion

Usually, failure mechanisms of 3D braided composites can be divided into three different types: yarn breaking, matrix cracking, and interface debonding.³² In this paper, taking account of the different damage modes including yarns’ longitudinal failure (L direction as shown in Figure 3), yarns’ transverse failure (T, Z direction as shown in Figure 3), yarns’ shear failure (LT, LZ, and TZ direction), and matrix cracking. Three-dimensional Hashin failure criterion³³ and maximum stress criterion are employed to define the damage initiation of yarns and matrix, respectively. The interface between yarns and matrix is assumed to be excellent, and the interface debonding is ignored. The corresponding initiation criterion under different damage modes can be expressed as follows:

Yarn tensile failure in L direction ( $σ L \geq 0$ ):³³

δ_{fL}^{t} = (\frac{σ L}{F_{L}^{t}}) 2 + β (\frac{σ LT}{S LT}) 2 + β (\frac{σ ZL}{S ZL}) \geq 1

(9)

Yarn compression failure in L direction ( $σ L < 0$ )³³

δ_{fL}^{c} = (\frac{σ L}{F_{L}^{c}}) 2 \geq 1

(10)

Yarn tensile and shear failure in T and Z direction ( $σ T + σ Z \geq 0$ )³³

δ_{fT (Z)}^{t} = (\frac{σ L + σ Z}{F_{T}^{t}}) 2 + \frac{(σ_{TZ}^{2} - σ T σ Z)}{(S TZ) 2} + (\frac{σ LT}{S LT}) 2 + (\frac{σ ZL}{S LZ}) 2 \geq 1

(11)

Yarn compression and shear failure in T and Z direction ( $σ T + σ Z < 0$ )³³

δ_{fT (Z)}^{c} = \frac{1}{F_{T}^{c}} ((\frac{F_{T}^{c}}{2 S TZ}) 2 - 1) (σ T + σ Z) + \frac{(σ T + σ Z) 2}{(2 S TZ) 2} + (\frac{σ_{TZ}^{2} - σ T σ Z}{S_{TZ}^{2}}) + (\frac{σ LT}{S LT}) 2 + (\frac{σ ZL}{S LZ}) 2 \geq 1

(12)

In the above equations, $F_{L}^{t}$ and $F_{T}^{t}$

are L and T directional tensile strengths of yarns, respectively.

F_{L}^{c}

and

F_{T}^{c}

are L and T directional compression strengths of yarns, respectively.

S LT

S LZ

, and

$σ T \geq 0$ are LT, LZ, and TZ shear strengths of yarns, respectively.

is the contribution factor in the each failure mode.

Matrix tensile failure criterion ( $σ_{m}^{t} \geq 0$ )

δ_{m}^{t} = (\frac{σ_{m}^{t}}{F_{m}^{t}}) 2 \geq 1

(13)

Matrix compression failure criterion ( $σ_{m}^{c} < 0$ )

δ_{m}^{c} = (\frac{σ_{m}^{c}}{F_{m}^{c}}) 2 \geq 1

(14)

where

F_{m}^{t}

and

F_{m}^{c}

are the tensile and compression strength of matrix, respectively.

σ_{m}^{t}

and

σ_{m}^{c}

are the max tensile and compression stress of matrix, respectively.

Damage evolution model

When the damage initiation of yarns or matrix is reached, the damage development will follow the damage evolution law, which is associated with the element strain, element characteristic length, and material fracture density. Note that the damage variables

d I (L, T, Z)

are introduced to define the damage modes of L, T, and Z directions for yarns and matrix. Moreover, according to Fang et al.,^19,20 each damage mode can be described by its corresponding equivalent displacement and stress, which are listed in Table 2. The damage evolution law can be expressed as^34–36

d I = 1 - \frac{X_{eq}^{Ii} (X_{eq}^{If} - X_{eq}^{I})}{X_{eq}^{I} (X_{eq}^{If} - X_{eq}^{Ii})}

(15)

where

X_{eq}^{li}

and

X_{eq}^{lf}

are the initiation and full damage equivalent displacement of failure mode I, respectively.

Table 2.

The equivalence displacements and stresses corresponding to different failure modes.

Type	Direction	Failure modes	Equivalence displacement	Equivalence stresses
MIFB	L	$σ L \geq 0$ (tensile)	$X_{eq}^{Lt} = l \sqrt{á ɛ L ñ 2 + β ɛ_{LT}^{2} + β ɛ_{ZL}^{2}}$	$l (〈 σ L 〉〈 ɛ L 〉 + β σ LT ɛ LT + β σ LZ ɛ LZ) / l (〈 σ L 〉〈 ɛ L 〉 + β σ LT ɛ LT + β σ LZ ɛ LZ) X_{eq}^{Lt} X_{eq}^{Lt}$
	L	$σ L \geq 0$ (compression)	$X_{eq}^{Lc} = l á - ɛ L ñ$	$l 〈 - σ L 〉〈 - ɛ L 〉 / l 〈 - σ L 〉〈 - ɛ L 〉 X_{eq}^{Lc} X_{eq}^{Lc}$
	T	$σ T \geq 0$ (tensile)	$X_{eq}^{Tt} = l \sqrt{á ɛ T ñ 2 + β ɛ_{LT}^{2} + β ɛ_{ZL}^{2}}$	$l (〈 σ T 〉〈 ɛ T 〉 + β σ LT ɛ LT + β σ ZT ɛ ZT) / l (〈 σ T 〉〈 ɛ T 〉 + β σ LT ɛ LT + β σ ZT ɛ ZT) X_{eq}^{Tt} X_{eq}^{Tt}$
	T	$σ T \geq 0$ (compression)	$X_{eq}^{Tc} = l á - ɛ T ñ$	$l 〈 - σ T 〉〈 - ɛ T 〉 / l 〈 - σ T 〉〈 - ɛ T 〉 X_{eq}^{Tc} X_{eq}^{Tc}$
	Z	$σ Z \geq 0$ (tensile)	$X_{eq}^{Zt} = l \sqrt{á ɛ Z ñ 2 + β ɛ_{TZ}^{2} + β ɛ_{LZ}^{2}}$	$l (〈 σ Z 〉〈 ɛ Z 〉 + β σ TZ ɛ TZ + β σ ZL ɛ ZL) / l (〈 σ Z 〉〈 ɛ Z 〉 + β σ TZ ɛ TZ + β σ ZL ɛ ZL) X_{eq}^{Zt} X_{eq}^{Zt}$
	Z	$σ Z \geq 0$ (compression)	$X_{eq}^{Zc} = l á - ɛ Z ñ$	$l 〈 - σ Z 〉〈 - ɛ Z 〉 / l 〈 - σ Z 〉〈 - ɛ Z 〉 X_{eq}^{Zc} X_{eq}^{Zc}$
Matrix	–	–	$X_{eq}^{mt} = l \| ɛ L \|$	$l \| ɛ L \| \| σ L \| / l \| ɛ L \| \| σ L \| X_{eq}^{mt} X_{eq}^{mt}$
Matrix	–	–	$X_{eq}^{mc} = l \| ɛ Z \|$	$l \| ɛ Z \| \| σ Z \| / l \| ɛ Z \| \| σ Z \| X_{eq}^{mc} X_{eq}^{mc}$

$á x ñ = (x + | x |) / 2$ .

Damaged stiffness matrix

In the present study, Murakami–Ohno damage model³⁷ is used to characterize the damage process of yarns and matrix. Moreover, three principal damage variables are used to express the damage states. It can be defined as

D = \sum D i n i \otimes n i i = (L, T, Z)

(16)

where D_i and

n i

are the principal damage value and the principal unit vector of damage tensor, respectively. Note that D_i combining with

d I (L, T, Z)

is described as:

For yarns at L, T, and Z direction

D L = \max (d Lt, d Lc) D T = \max (d Tt, d Tc) D Z = \max (d Zt, d Zc)

(17)

For matrix

D L = D T = D Z = \max (d Rt, d Rc)

(18)

By definition, the effective stress $σ *$ can be expressed as

σ * = \frac{1}{2} [(I - D) - 1 • σ + σ • (I - D) - 1] = M (D) : σ

(19)

where σ is the undamaged stress.

C (D) = [(1 - D L) 2 C 11 (1 - D L) (1 - D T) C 12 (1 - D L) (1 - D Z) C 13 000 (1 - D T) 2 C 22 (1 - D T) (1 - D Z) C 23 000 (1 - D Z) 2 C 33 000 (\frac{2 (1 - D L) (1 - D T)}{2 - D L - D T}) 2 00 sym (\frac{2 (1 - D L) (1 - D Z)}{2 - D L - D Z}) 2 0 (\frac{2 (1 - D T) (1 - D Z)}{2 - D T - D Z}) 2]

(20)

In order to introduce the damage variable into the undamaged stiffness matrix, Cordebois–Sidoroff energy assumptions³⁸ are used, and the detailed damage stiffness matrix $C (D)$ can be expressed as follows³⁹ where $C ij, (i, j = L, T, Z)$ is the undamage stiffness matrix items.

Periodic boundary condition

In order to save the time and reduce the computational cost, meso-level finite element simulation usually involves single RVC instead of the whole composite specimen. However, it should be noted that it is not isolated from its adjacent RVCs in the composites, and the boundary effects from the adjacent RVCs should be taken into account. Thus, the applied periodic boundary conditions in the present work are to keep the continuity of displacements on the opposite faces of RVC.

The macroscopic displacement field on a pair of parallel opposite boundary surfaces denoted by “ $σ Z \geq 0$ ” and “0” is written as^40–43

u i (l) = \bar{ɛ} ik x k (l) + u_{i}^{*}

(21)

u i (0) = \bar{ɛ} ik x k (0) + u_{i}^{*}

(22)

where

\bar{ɛ} ik

is the average strains of RVC, x_k is the coordinates of arbitrary point in RVC,

u_{i}^{*}

is the periodic part of the displacement components on the boundary surfaces.

The total displacement at the opposite boundary surface is expressed by the above two equations

u i (l) - u i (0) = \bar{ɛ} ik (x k (l) - x k (0)) = \bar{ɛ} ik Δ x k

(23)

Since $Δ x k$ is a constant for each pair of the boundary surface, the right side of the equation (23) becomes constant once $\bar{ɛ} ik$ is provided. Then the above equations can be carried out easily in the finite element analysis by setting the multipoint constraint conditions.

Experimental details

Material properties

Three-dimensional braided composites studied in the present paper are composed of Carbon T700-12 K braided and axial yarns and epoxy matrix. Braided and axial yarns are thought to be matrix-impregnated fiber bundles, which are assumed to be transversely isotropic. Epoxy matrix is assumed to be isotropic. The mechanical parameters of the constituents are summarized in Table 3, which is used to simulate the damage initiation and propagation of 3D five-directional braided composites in the subsequent sections.

Table 3.

Materials properties of yarns and epoxy matrix.

	Parameters	Constant	Value
T700-12 K yarns	Longitudinal modulus (GPa)	E_L	175.17
	Transverse modulus (GPa)	$E T = E Z$	9.92
	Longitudinal Poisson’s ratio	$ν LT = ν LZ$	0.29
	Longitudinal shear modulus (GPa)	$G LT = G LZ$	6.89
	Transverse shear modulus (GPa)	$G TZ$	3.34
	Longitudinal tensile strength (MPa)	$F_{L}^{t}$	2880
	Longitudinal compression strength (MPa)	$F_{L}^{c}$	1470
	Transverse tensile strength (MPa)	$F_{T}^{t} = F_{Z}^{t}$	66
	Transverse compression strength (MPa)	$F_{T}^{c} = F_{Z}^{c}$	210
	Shear strength (MPa)	$S LT = S LZ$	104
	Shear strength (MPa)	$S ZT$	98
Epoxy matrix	Elastic modulus (GPa)	E_m	3.5
	Poisson’s ratio	ν	0.35
	Tensile strength (MPa)	$F_{m}^{t}$	80
	Compression strength (MPa)	$F_{m}^{c}$	241
	Shear strength (MPa)	S_m	60

Composite specimen preparation

Three-dimensional five-directional braided preforms are fabricated using the 3D braided machine at the Institute of Composite Materials of Tianjin Polytechnic University (China). As described earlier, the braiding technology and machine is currently the major factor impeding wider investigations of the broad width preform. As shown in Figure 4, the braiding machine width

L'

can be expressed as

L' = \frac{(2 W f \times L)}{w}

(24)

where

W f, L, w

are the fabric width, yarn carrier’s spacing, and braided pitch width, respectively. Here, W_f is 360 mm, while

L, w

are 20 and 2.5 mm, respectively. By computing, the braiding machine width (

L'

) is equal to 5760 mm, which obviously exceeds the fabric width (360 mm). This result means that when a braided yarn runs from the point A to the point B (see Figure 4), internal tension in braided yarn changes distinctly, resulting in the inhomogeneity of the braiding parameter in braiding preform. In order to improve the phenomenon, the elastic yarn is used, and its two ends are fixed in the braided yarn and the yarn carrier (see Figure 4), respectively. In the braiding process, the internal tension in all the braided yarns should be kept uniform by changing the length of the elastic yarns. However, the preform with large braiding angle is still difficult to fabricate due to the increasing braided pitch width. Afterward, the 3D five-directional braided carbon/epoxy composites of size 360 mm × 160 mm are cured by the process of the resin transfer molding. Here, an important issue is that the experimental samples of 3D braided composites are generally composed of three types of regions, interior region, surface region, and corner region, while meso-scale finite element model contains only interior RVCs under periodic boundary conditions. Therefore, the significant differences between the meso-scale finite element model and the experiment arise from the fact that whether the free edges exist or not. Therefore, for comparison, the cutting samples of 150 mm × 15 mm, in which the yarns at the edges are discontinuous, are designed, as shown in Figure 5. In addition, it should also be noted that the defects at the edge of specimens can be formed because of cutting.⁸ This phenomenon results in the decreasing of the mechanical properties to a certain extent. Specifications of test samples are listed in Table 4.

Table 4.

Specifications of carbon/epoxy composite samples.

Specimen	γ (°)	ϕ (°)	Thickness (mm)	Fiber volume fraction (%)
TC1	32.14	23.96	3.14	57.82
TC2	31.56	23.48	4.10	58.10

Transverse tensile tests

Since there are no definite standards of transverse tensile tests for 3D braided composites, the test procedures are conducted according to the standard of ASTM D3039. The transverse tensile tests are conducted by SHIMADZU AG-250KNE universal material machine setup in the room temperature, as shown in Figure 6. In order to reduce the error of the cut-edge effect, the transverse tensile loadings are applied on five samples for each type of thickness. Further, the stress–strain curves and the failure modes of the specimens are recorded.

Results and discussion

After generating the meso-scale finite element models with two kinds of braiding angles, a commercial finite element software ABAQUS/Standard is used to study the transverse tensile responses of 3D braided composites. In order to implement the damage evolution model described earlier, a user-defined material subroutine (UMAT) is compiled and incorporated in ABAQUS. The predicted macroscopic stress–strain curves are compared with the experimental results to validate the accuracy of the models. Furthermore, the damage cloud pictures and the stress distribution are captured to investigate the damage developments, the local response, and the failure mechanism.

The macroscopic stress–strain curves

Figure 7 illustrates the predicted and tested stress–strain responses of 3D five-directional braided composites. Clearly, the predicted curve tendency is essentially in accordance with the experimental results. Moreover, it is found that the max stress of 30° RVC (44.55 MPa) exceeds slightly that of TC1 (43.09 MPa) by 3.39% and TC2 (42.85 MPa) by 3.97%. Two factors should be considered to explain the phenomena. A possible explanation for this is due to the cut-edge effect as mentioned earlier. On the other hand, for meso-scale finite element model, an interior RVC is usually picked out and developed instead of the whole 3D five-directional braided composites to analyze the effective mechanical properties when column number and row number of yarn carriers are large enough. However, in this paper, tested sample is narrow and only contains a few of RVCs, leading to a lower strength. This result is confirmed by ASTM standard transverse tension results.⁴⁴ Zhang et al.⁴⁵ suggest that the number of RVC affects the ultimate strength to a certain extent.

Additionally, Figure 7 illustrates the damage initiation points of yarns and matrix for 30° RVC. Here, it can be seen that the predicted stress–strain curves increase linearly until the initiation damage in matrix. Similarly, Fang et al.¹⁹ and Lu et al.²¹ also described such phenomena by examining the longitudinal tensile properties of 3D four-directional braided composites and 3D full five-directional braided composites, respectively. It is noticed that T and Z directional damages occur first in axial yarns, followed by braided yarns. However, this result fails to lead to the nonlinear response of the stress–strain curve for 30° RVC. Fang²⁰ suggested that the behaviors are related to the stiffness reduction theory. Moreover, no L directional damage points are observed for both axial yarns and braided yarns, suggesting that L directional properties in all yarns have a little effect on the transverse tensile failure of 30° RVC.

Figure 8 shows the predicted stress–strain curve and the damage initiation points of 45° RVC. It can be observed form the graphs that as compared to 30° RVC, there are three obvious differences. Firstly, the curve increases linearly until the initiation damage in matrix, and later exhibits remarkable nonlinear responses before reaching the max stress, which can be attributed to the transition of the failure modes, as well as the changes of the specimen stiffness. Secondly, the max stress (82.47 MPa) of 45° RVC is 85.12% higher than that of 30° RVC (44.55 MPa). Finally, there exists L directional damage in 45° RVC during the transverse tensile process, while this phenomenon is not found in 30° RVC. For aforementioned results, it may be reasonably concluded that the transverse tensile behaviors of 3D five-directional composites are very sensitive to the braiding angles.

Damage development

Three state damage variables (SDVs), including SDV1 (L directional damage), SDV2 (T directional damage), and SDV3 (Z directional damage), are set in UMAT. Note that the SDVs are governed by the damage equivalent strains and evolved from 0 to 1 irreversibly after the failure initiation. The detailed analysis is as follows.

In order to make better interpretation for the damage development, the damage cloud pictures at three key strain points, A, B, and C (see Figures 7 and 8), are chosen for further study. Figures 9 to 11 present the detailed damage developments in 30° RVC. Here, we only focus on the T and Z directional damage evolution in all yarns according to the above results. In Figure 9, at ɛ = 0.50% (point A in Figure 7), it can be seen clearly that little visible damage is found in braided yarns. With the external load reaching point B (ɛ = 0.68%), some shear damages are observed in braided yarns and propagate along the T and Z direction until point C (ɛ = 1.01%). It should be noted that the max values of SDV2 and SDV3 in braided yarns are close to 0.65, representing that there are no catastrophic damages. As to the graphs in Figure 10, discrete damage points occur initially at the edge of axial yarns perpendicular to the loading direction at ɛ = 0.50% with the increasing of the tensile strain (ɛ = 1.01%), T directional damages in axial yarns have little change, while Z directional damages propagate along the joint areas neighboring braided yarns and matrix, and subsequently become larger. In Figure 11, the graphs show that the matrix damages first appear near the point of the max stress (ɛ = 0.68%) and develop in the stress concentration regions where yarns contact each other.

Figures 12 to 14 show the detailed damage developments in 45° RVC. The damage evolutions in axial yarns and matrix of 45° RVC are similar with that of 30° RVC, as shown in Figures 13 and 14. However, unlike 30° RVC condition, L, T, and Z directional braided yarns of 45° RVC present significant damage before reaching the max stress, as shown in Figure 12. The difference proves that with internal braiding angle varying from 30° to 45°, the stress distribution and the failure mechanism change correspondingly.

Local response and failure mechanism

To further characterize the local response and the failure mechanism, the typical damage elements, whose final SDV is maximum in the corresponding components, are selected. For 30° RVC, the stress versus strain responses of the typical damage elements E288765, E198378, and E67188, which are located, respectively, in braided yarns, axial yarns, and matrix, are shown in Figure 15. Herein, we only consider the key stress components related to the transverse tensile damages according to the above analysis of damage developments. For E288765 in braided yarns (Figure 15(a)), the max values of the stress components, S11, S22, S33, S12, S13, and S23, are respectively far less than the limit values of $F_{L}^{t}, F_{T}^{t}, F_{Z}^{t}, S LZ, and S TZ$ . The results once again imply that braided yarns have no obvious influence on the transverse tensile strength of 30° RVC. As displayed in Figure 15(b), for E198378 in axial yarns, it is found that the max value of S33 (Z direction) exceeds 66 MPa ( $F_{L}^{c}$ ) near ɛ = 0.7%, while that of shear stress approaches zero. Thus, it can be concluded that for axial yarns of 30° RVC, the effect of the axial tensile stress is much more significant than the shear stress, confirming that the transverse tensile damages of axial yarns are the dominant failure mode. As shown in Figure 15(c), for E67188 in matrix, S11 (L direction) is greater than 80 MPa ( $F_{m}^{t}$ ), which is also responsible for the ultimate failure of 30° RVC.

Figures 16 and 17 show the final damage morphologies of 30° RVC and TC2 specimen, respectively. It can be noticed that the damage modes from the numerical results (Figure 16) are totally dominated by axial yarn fracture, matrix failure, crack between axial yarn and matrix, and braided yarn pull-out, which are in good agreement with the experimental fracture modes as shown in Figure 17. Therefore, it considerably demonstrates the validity of the meso-scale finite element model. Moreover, these observations together with the stress-stain curves as noted earlier indicate that for 30° braided composites, the cracks initiate in axial yarn oriented perpendicularly to the loading direction. Subsequently, it expands and grows into the matrix-rich areas. Furthermore, a reasonable inference can be refined that the final strength of 30° braided composites is largely governed by the matrix yielding and the transverse strength of axial yarn.

Also, for 45° RVC, the typical damage elements E53471, E131327, and E162473 are selected, respectively, and the corresponding stress versus strain curves are described in Figure 18. In comparison with 30° RVC, the local stress distributions of 45° RVC have an obvious difference, namely, S11 and S13 in braided yarns significantly increases, especially for S13 which is near the shear strength of yarns ( $S LZ$ ). Besides, the stress distributions of axial yarn and matrix in 45° RVC are similar with that in 30° RVC, as shown in Figure 18(b) and (c).

Figure 19 shows the finial damage morphologies of 45° RVC. It can be seen clearly from the graph that the dominant failure modes are most consistent with the 30° RVC one, except the longitudinal tensile damage (Figure 19(d)). The main reason is that the embedding length of braided yarn increases with the increasing of the braiding angle when the sample width is given. Thus, as compared to 30° RVC, 45° RVC is able to withstand a longer pull-out process after the initiation damage in matrix, and share more axial and shear loads, leading to more serious damages in braided yarns. This also explains why the transverse tensile strength of 45° RVC is higher than that of 30° RVC as described earlier. Hence, for 45° RVC, the final strength is mainly decided by the transverse shear strength of braided yarns.

Conclusions

In the current study, the experimental and numerical investigations are carried out to evaluate the progressive damage behaviors of 3D five-directional braided composites subjected to the transverse tensile loading. Following conclusions are drawn from the study:

From the comparisons of the stress–strain curves and the damage modes, the predicted results are most consistent with the experimental data. The failure of 30° RVC and TC2 specimen is mainly dominated by axial yarn fracture, matrix failure, crack between axial yarn and matrix, and braided yarn pull-out.

Under the transverse tensile loading, there is considerable increase in the max stress for 45° RVC as compared to 30° RVC. This is because that braided yarns in 45° RVC bear a longer pull-out process after the initiation damage in matrix, thereby leading to higher stress levels.

The stress distributions and the failure mechanisms are different for 30° RVC and 45° RVC. The ultimate strength of 30° RVC is mainly influenced by the matrix yielding and the transverse tensile strength of axial yarns, while that of 45° RVC is largely governed by the transverse shear strength of braided yarns.

Further experimental studies are planned to examine the mechanical responses of 3D braided composites with large braiding angles. In addition, optimizing the interfacial effect and developing a new damage/constitutive are promising to be exploited for the reasonable mechanics system for 3D braided composites.

Footnotes

Funding

This work is supported by the National Natural Science Foundation of China (No. 11102133 and 11072175) and High Technology Research and Development Program of China (No. 2012AA03A201).

Conflict of interest

None declared.

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