Modeling the transverse tensile and compressive failure behavior of triaxially braided composites

Composites Science and Technology 172 (2019) 96–107

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Modeling the transverse tensile and compressive failure behavior of triaxially braided composites

T

Zhenqiang Zhaoa,b,c, Peng Liua,b,c, Chunyang Chena,b,c, Chao Zhanga,b,c,∗, Yulong Lia,b,c,∗∗ a

Department of Aeronautical Structure Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China Shaanxi Key Laboratory of Impact Dynamics and Its Engineering Applications, Xi'an, Shaanxi 710072, China c Joint International Research Laboratory of Impact Dynamics and Its Engineering Applications, Xi'an, Shaanxi 710072, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: A. Textile composites B. Mechanical properties C. Damage mechanics C. Finite element analysis (FEA) Meso-scale model

The complex failure behavior of triaxially braided composites under in-plane transverse load conditions is investigated through quasi-static experiments and meso-scale finite element (FE) simulations. A three-dimensional (3D) progressive damage model for the fiber tows is integrated with a cohesive model for the interfaces to simulate the initiation, accumulation and propagation behavior of damage in braided composites. The mesoscale FE model predicts well the global stress–strain responses, and the predicted strain distribution contours compare well with the experimental results captured by digital image correlation. The fully validated FE model is subsequently adopted to investigate the failure mechanism of a triaxially braided composite under transverse tensile and compressive loads. Numerical parametric studies are implemented to evaluate the effect of interface strength on the effective properties of the material and to identify the appropriate definition of through-thickness boundary conditions in the meso-FE simulation. The model presented in this study shows fairly good accuracy in predicting the failure behavior of a triaxially braided composite under different loadings, and it can be further employed to study the mechanical performance of similar materials.

1. Introduction Fiber-reinforced textile/braided composites are increasingly used as structural materials in aerospace, automotive and marine industrial applications, due to their efficiency in providing reinforcement along multiple directions using a single layer of fabrics, their ability to conform to surfaces with complex curvatures [1], and their superior damage tolerance as compared to traditional laminated composites. Another advantage of textile composites is their suitability for integrated molding of large-scale structures, which reduces the manufacturing cost and fabrication time [2]. For example, a two-dimensional triaxially braided composite (2DTBC) was successfully employed in the design of an engine fan case for the GEnx aero-engine to prevent impact-induced damage from debris or broken turbine blades. As the foundation of a composite structural design, the mechanical properties and failure mechanisms for the studied composite materials under different loading conditions must be established in advance. However, the complex architecture of textile composites in the micro/meso scale is a combination of the mechanical response and failure process, which further increases the complexity in testing and analysis. Lomov et al. and Ivanov

∗

et al. [3,4] conducted uniaxial tensile testing of triaxially braided composites along different characteristic directions of textile reinforcement, accompanied with acoustic emission detection and fullfield strain measurement on the surface to investigate damage initiation and propagation during the loading process. Wehrkamp-Richter et al. [5] performed uniaxial tensile tests to characterize the tensile properties along the axial fiber tow and the bias tow as well as in the orientation perpendicular to axial tows. It is reported that the triaxial braided architecture contributes greatly to the high damage tolerance of the composites, due to the relief of crack extension by the intertwined fiber bundles. Relatively few studies in the literature focus on the compression behavior of 2DTBCs. Quek et al. [6] studied the compressive failure mechanism of a 2DTBC and reported that tow buckling and kinking along with matrix inelasticity are the dominant failure modes. Comprehensive tests of triaxially braided composite were later conducted by Littell [7]. The in-plane mechanical properties (including tension, compression and shear) were obtained under quasi-static loading rates, and the results indicated that the toughness of the matrix has a significant influence on the effective properties of the 2DTBC. In addition,

Corresponding author. Department of Aeronautical Structure Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China Corresponding author. Department of Aeronautical Structure Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China E-mail addresses: [email protected] (C. Zhang), [email protected] (Y. Li).

∗∗

https://doi.org/10.1016/j.compscitech.2019.01.008 Received 17 October 2018; Received in revised form 23 December 2018; Accepted 11 January 2019 Available online 14 January 2019 0266-3538/ © 2019 Elsevier Ltd. All rights reserved.

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Littell also found that free-edge damage, which is apparent in transverse tension tests, leads to premature failure and underestimation of the transverse strength. To overcome the influence of free-edge effect on the characterization of tensile strength, Kohlman [8] designed notched end tube specimens for the purpose of obtaining more representative results; however, unsatisfactory results were obtained due to the presence of undesirable failure modes for these non-standard specimens. The experimental studies indicated that the mechanical response of the 2DTBC along the transverse direction is very complicated and needs further investigation. Experimentally, it is difficult to monitor the internal failure behavior and specify the failure mechanism of braided composites due to the limitation of available techniques for damage monitoring. However, meso-scale finite element (FE) modeling has proven to be a useful approach for predicting the local response and damage progression behavior of composites having a complex textile architecture [9–11]. Waas and coworkers [12,13] developed a meso-scale FE model to simulate the compressive response of a 2DTBC that considers fiber tow undulation; they concluded that geometrical imperfections in axial fiber tows and the nonlinear behavior of the matrix material are the most significant factors affecting the compressive strength. In a more recent study, Wehrkamp-Richter et al. [14] generated a meso-FE model with an automated simulation workflow, which does an excellent job of capturing the internal geometry of triaxially braided unit cells and is able to provide good predictions of the elastic properties under tensile loading. Ren and co-workers [15,16] also studied the failure behavior of 2DTBCs under quasi-static tensile loads using meso-scale FE models. In most of the previous studies [13–17], the researchers intended to use either a single unit cell or a small number of unit cells to model the mechanical response of coupon-type specimens, incorporating symmetrical or periodic boundary conditions. However, none of these models is able to capture the free-edge effect and free-edge damage–induced premature failure for triaxially braided composites under transverse loading. In light of this, Zhang et al. [18,19] established a three-dimensional (3D) meso-scale FE framework with translational symmetry boundary conditions that explicitly models the width of straight-sided coupon specimens; this model was found to provide an excellent simulation of the free-edge effect and capture its influence on transverse tensile failure behavior. Most prior studies of 2DTBCs focus mainly on the tensile failure behavior. Few experimental or numerical studies are reported that systematically explore the progressive failure behavior of a 2DTBC under compressive loads, especially for transverse compression. On the other hand, for a multi-layer specimen, the definition of the throughthickness boundary condition is an issue considering the computational cost and model accuracy. Considering this, the study presented in this paper is a numerical and experimental investigation of the progressive damage behavior of a 2DTBC under transverse tension and compression. The effective stress–strain responses, full-field surface strain distribution, and internal damage progression are studied in order to identify the main damage features of this material. Numerical parametric studies are conducted to investigate the effect of interface strength on the effective responses of a 2DTBC and identify the appropriate boundary conditions across the thickness direction when modeling a multi-layer specimen.

temperature of 130 °C. The composite panel is eight-layer with a thickness of 4.5 mm and a fiber volume ratio of 56%. The composite materials used in this study were provided by Sinoma Science & Technology Co. (Nanjing, China). Table 1 lists the mechanical properties of the components; the properties for the matrix were obtained from Cheng et al. [20], while the properties for the fiber are described in Li et al. [21]. Straight-sided coupon specimens were cut from the 2DTBC panel using water jets. Fig. 1 shows the specimen dimensions and a diagram of the experimental setup. The length for both tensile and compressive specimens is defined as being perpendicular to the axial fiber tows. Two kinds of specimens are represented: transverse tension (TT) specimens and transverse compression (TC) specimens. Details regarding the design of the test specimens can be found in the work of Littell [6], and in ASTM standards D3039 [22] and D3410 [23]. For the tensile specimens, the grip region is designed to be long enough to prevent sliding between the specimen and the fixture. The gauge region of the compressive specimens is designed to be relatively shorter so that buckling failure can be avoided. The width for the tensile and compressive specimens is 38.5 mm, and each specimen contains seven unit cells through its width direction [19]. As shown in Fig. 1(b), the tests were conducted on an Instron 8803 hydraulic testing machine with displacement-controlled loading rates of 8.4 mm/min and 1.8 mm/min for tension and compression, respectively, ensuring a strain rate of approximately 1e-3/s. 3D digital image correlation (DIC) technology was adopted to measure the full-field displacement and strain distributions. The commercially GOM optical measurement system consists of two stereo digital cameras connected to a computer for simultaneous image capturing as well as analysis using ARAMIS DIC software. A speckle pattern was painted on the surface of each specimen, and calibration of the DIC measurement process was performed prior to each test. During the calibration procedure, the cameras were calibrated using the software ARAMIS by taking pictures of a specialized calibration block in size of 55 × 44 mm2 in various orientations. The bottom picture of Fig. 1(b) shows a schematic of the calibration volume of approximately 96 × 70 × 40 mm3. The resolution of captured image is 2351 × 1727 pixels and test precision of this GOM system is about 1 μm, which is precise enough to obtain the displacement field and strain distribution for the test samples. In order to generate the stress–strain data, an optical strain extensometer introduced in the study by Littell [6] was positioned at the center of the optical strain image to measure the average strain, and the stress was calculated by dividing the resistance force by the cross-sectional area of the specimen.

2. Experimental testing program

3. Meso-scale finite element model

The two-dimensional triaxially braided fabrics used in this study is manufactured with three intertwined fiber tows that are aligned along the axial (0°) and bias ( ± 60°) directions. The axial tow is composed of 24,000 fiber filaments, and the bias tow contains 12,000 fiber filaments. A toughened epoxy resin, the 3266 epoxy resin system, was injected to a T700 carbon fiber braided architecture using closed-mold resin transfer molding (RTM) technology, under a pressure of 1 MPa. The curing temperature profile for T700/3266, was 3 h at a maximum

The meso-scale FE model is suitable for analyzing the failure mechanisms of braided or textile composite, as it has the ability to capture the complex braided architecture of the material. In a meso-FE model, the fiber bundles and the surrounding pure matrix region are distinguished and can be generated separately. The impregnated fiber tows are considered as unidirectional composite lamina and are modeled as a transverse-isotropic material. The Hashin criterion [24] and the Hou criterion [25] are combined to examine the damage initiation

Table 1 Mechanical properties of composite components.

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Property

Fiber

Matrix

Material type Axial modulus (GPa) Transverse modulus (GPa) Shear modulus (GPa) Tensile strength (MPa) Density (g/cm3)

T700 230 15 24 4900 1.8

3266 epoxy 2.5 2.5 0.96 89 1.2

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Fig. 1. (a) Fabrics architecture of 2DTBC and diagrams showing the dimensions of tensile and compressive specimens; (b) Test apparatus.

In the finite element mesh, due to the usage of hexahedral elements, the boundaries of fiber bundles are not smooth. However, the numerical results of the present model were found to be more reasonable compared with models using tetrahedrons elements, because of the uniform mesh size and the absence of singularity. Fig. 2(b) shows the meso-scale FE models for the 8-layer coupon specimen of the 2DTBC considered in this study. In Fig. 2(b), seven unit cells are aligned between the two free edges, and displacements are applied on the opposite surfaces (“LEFT” and “RIGHT”) to simulate the tension or compression loads. To simulate the long coupon tensile specimens, translational-symmetrical boundary conditions are applied along the loading direction (transverse direction, x axis). The control equations for the translational-symmetrical boundary condition can be written as follows:

Table 2 Strength and fracture toughness of the interface. tn0 (MPa)

ts0 (MPa)

tt0 (MPa)

Gnc (mJ/mm2)

Gsc (mJ/mm2)

Gtc (mJ/mm2)

122

136

136

0.268

1.45

1.45

in the fiber tows, and Murakami–Ohno damage theory [26] is employed to model the damage evolution process after the failure criteria are satisfied. A user-defined subroutine VUMAT is implemented in ABAQUS/EXPLICIT to simulate the failure behavior of fiber tows in this 2DTBC. A more detailed introduction of the progressive damage model can be found in Appendix A or previous publication by Zhao et al. [27]. The pure matrix part of the composite, on the other hand, is modeled as an elastic–perfectly-plastic material for the purpose of simplification [6]. In addition, a cohesive element model [28] implemented in ABAQUS is introduced to simulate the tow-to-tow and tow-to-matrix delamination. Since no data is available for the interface properties of the studied T700/3266 system, the calibrated parameters of a similar material system (T700/E862) reported in other studies [21,29] are adopted in this work; the parameters are listed in Table 2. Here, t0 represents the interface strength and G0 is the fracture toughness of interface, the subscripts n, s, and t denote the normal, first and second shear directions, respectively. Systematic parametric studies were conducted to examine the effect of interface strength on the mechanical response of the 2DTBC, as will be discussed in Section 4.3.1. Fig. 2(a) presents the microscopic cross-section images of coupon specimens, and geometry and finite element representation of 2DTBC unit cell. The representative geometry model of a unit cell was established using an open-source software TexGen, based on the design parameters of the triaxially braided architecture. As shown in Fig. 2(a), WA denotes the width of the axial fiber tow, WB indicates the distance between every two neighbor axial fiber tows, and Wb indicates the width of bias fiber tows. W and L are the width and length of a unit cell, respectively. Table 3 summarizes the geometric parameters of the unit cell model. The mesoscale FE mesh shown in the bottom of Fig. 2(a) can be generated automatically through TexGen. Each unit cell contains a total of 13,520 eight-node hexahedral elements. More specifically, there are 52 elements along the width direction, 26 elements along the axial direction and 10 elements through the thickness of the unit cell. Fig. 2(a) also shows the cross-section images of the geometry model and FE mesh. The cross-section shapes of fiber bundles are close to ellipse.

LEFT − UxRIGHT = δx ⎧Ux ⎪ LEFT Uy − UyRIGHT = 0 ⎨ LEFT ⎪Uz − UzRIGHT = 0 ⎩

(1)

where U denotes node displacement, and the superscripts LEFT and RIGHT stand for the node set of the two opposite loading surfaces shown in Fig. 2(b). The subscripts x, y and z represent the coordinate directions, while δx indicates the applied transverse tensile displacements. For the compressive test, the gauge region is 30 mm long containing less than two complete unit cells along the loading direction. To enhance the computational efficiency and be consistent with the tensile model, the compressive FE model uses one unit cell (19 mm) along the loading direction. This simplification was reported to have little influence on the numerical simulation results [13], and is further confirmed by the simulation results of this work as will be shown in a later section. Displacement-controlled compression loads are directly applied in a symmetrical manner on the two opposite loading surfaces. In Section 4.4 of this article, periodic boundary condition is applied on a singlelayer FE model along the through-thickness direction to predict the mechanical response of an infinite-thick panel. The control equations can be written as Eq. (2), where TOP and BOTTOM represent the node set of the top and bottom surfaces, respectively.

UiTOP − UiBOTTOM = 0, i = x , y, z

(2)

The determination of mechanical parameters for axial and bias fiber tows is an important step for meso-FE simulation. Due to the difficulty in testing an impregnated fiber bundle, micromechanical theories are 98

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Fig. 2. Meso-scale FE model for (a) a unit cell and (b) an 8-layer coupon specimen of 2DTBC.

microbuckling, the Chamis model [31] is employed to update the longitudinal compression strength, as suggested by Zhao et al. [27]. Table 4 summarizes the mechanical properties of both axial and bias fiber bundles for the studied T700/3266 2DTBC. The fiber volume fraction of axial tow and the bias tow are determined through correlating the initial modulus against experimental measurement for both axial and transverse loading conditions, while constraining the global Vf to be the same as the realistic specimen (Vf = 56%). It worth noting that the calibrated fiber volume fraction of axial tow (86%) and bias tow (80%) are relatively higher than that of the realistic specimens. This is due to the larger volume of pure matrix region in the idealized meso-scale FE model while the fabrics in the realistic specimen is significantly extruded (see the microscopic image in Fig. 2(a)).

Table 3 Architecture parameters of unit cell and fiber bundle. W (mm)

L (mm)

WA (mm)

WB (mm)

Wb (mm)

19

5.5

5.5

4

3.5

Table 4 Mechanical properties of axial and bias fiber tows for a T700/3266 2DTBC.

Fiber volume fraction Vf E11 (GPa) E22, E33 (GPa) G12, G13 (GPa) G23 (GPa) v12, v13 v23 S1t (MPa) S1c (MPa) S2t, S3t (MPa) S2c, S3c (MPa) S12, S13, S23 (MPa)

Axial fiber tows

Bias fiber tows

86% 198.15 10.81 8.37 3.68 0.28 0.47 4221 2288 63 119 98

80% 184.50 9.56 6.39 3.27 0.28 0.46 3931 2148 63 121 94

4. Results and discussion 4.1. Experimental results and FE model validation The presented meso-scale FE model is first validated by comparing the experimentally measured and numerically predicted global stress–strain curves for a 2DTBC under both transverse tension and transverse compression loading conditions. As shown in Fig. 3, the FE model does well in predicting the TT and TC stress–strain responses, capturing both the initial stiffness and damage-induced nonlinearity behavior. The predicted strength in the TC condition is slightly higher than the experimental value, due to the possible overestimation of compression strength as discussed by Li et al. [21]. In a realistic specimen, the presence of fiber bundle undulation is likely to result in a reduction of compressive strength when compared to that of the

generally applied to predict the mechanical properties of the fiber tows. In this study, Huang's bridging model [30] is adopted to calculate the stiffness and strength of the fiber tows based on the properties of the components listed in Table 1 as well as the fiber volume ratio of each fiber tow. Considering the possible failure modes of fiber tows under an axial compressive load such as fiber rupture, delamination/shear, or

Fig. 3. Comparison of numerical predicted and experimental measured stress–strain curves for the eight-layer coupon specimen under (a) transverse tension and (b) transverse compression. 99

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Fig. 4. Comparison of numerical prediction and experimental results for strain contours for TT condition.

higher (resin-rich regions between the bias tows) and accumulates locally at the corresponding areas, resulting in architecture-dependent axial and shear strain patterns. Compared with the axial strain contours, the shear strain contours show more significant strain concentration distributing periodically along the free edges, which is also captured by the meso-scale FE model. For transverse compression, the numerically predicted strain contours compare well with experimental results at a global strain level of 0.9%, as shown in Fig. 5. The surface strain distributions of transverse compression also show obvious architecture dependency. More importantly, the free-edge effect–induced shear strain concentration has been also characterized using DIC during the compression test, which has not been reported elsewhere. The free-edge damage induced shearstrain concentration is also been captured by the meso-scale FE model, as well as the strain concentrate among the resin-rich regions where bias fiber tows intersect with each other. The free-edge effect causes tensile-torsion coupled deformation, which can generate shear strain/stress concentration, leading to premature interface failure of the bias fiber tow and finally resulting in a shear failure of the bias fiber tows. Fig. 6(a) shows the zig-zag path of crack propagation in the TT specimen. The axial fiber bundles play a role in resisting the crack propagation, where the crack propagates along the bias direction, is stopped by the axial fiber tow and further propagates along the reverse bias direction, forming a serrated zig-zag fracture surface. This phenomenon is similar to the transverse tension failure behavior of a single-layer 2DTBC reported by Zhang [19]. The complex path of the crack propagation in the 2DTBC under such a high stress status absorbs a large amount of external energy, thus demonstrating the excellent damage tolerance of this material. As discussed by Zhang et al. [19], the presence of free-edge effect will result in size-dependent properties for the transverse tension coupon specimens. The observation of free-edge effect during the transverse compression test further confirms that free-edge effect is an inherent behavior related to the braided architecture and will be present wherever fiber tows terminate at free edges, regardless of the type

theoretical prediction [32]. In addition, the interface properties may also affect the effective response, as will be discussed in a later section. The nonlinear behavior of the stress–strain curves indicate that the 2DTBC specimens suffer damage continuously during the tensile and compressive tests, resulting from the inelastic behavior of pure matrix material, transverse cracking of the fiber bundles, and shear failure of the interfaces. For both TT and TC simulations, the unloading of predicted curves begins after a peak value is reached and is followed by fiber breakage damage in the bias tows. A primary advantage in using a meso-scale FE model is its capability for predicting the local deformation and damage behavior for materials with a complex architecture. Thus, to demonstrate the accuracy and predictability of the model, it is imperative to show the capability of the model in predicting local responses (for example, surface strain distributions at different loading stages). Fig. 4 compares the numerical prediction and the experimental measured results for normal strain (εy) and in-plane shear strain (εxy) distributions at two different global strain levels (0.7% and 1.4%). The simulated results are duplicated and translationally shifted along the loading direction to match the size of the DIC results, where the strain at the boundaries of the FE model are continuous due to the application of a translational symmetrical boundary condition in the presented meso-FE model. As can be noticed from Fig. 4, a significant concentration of strain is present at the free edges in locations where bias fiber bundles are terminated, and it propagates along the direction of the bias fiber bundles. The overall strain distributions show obvious dependency on the braided architecture, and the axial and shear strain in the center area concentrates at locations where bias tows intersect with each other. The numerical simulations capture not only the characteristics of local deformation but also the progressive failure process of the coupon specimens. For the TT condition, the external applied load is perpendicular to the axial fiber tows, and load is mainly transferred by the matrix and the tow–matrix interface, due to the lack of continuous fiber tows crossing the two ends of the coupon specimen. Thus, strain concentration presents at locations where the matrix volume fraction is 100

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Fig. 5. Comparison of numerical prediction and experimental strain contours for TC condition.

4.2. Progressive failure behavior of a 2DTBC Once confidence in the accuracy of the meso-FE model has been established, the model can then be used to study the damage mechanism of this braided composite, such as damage evolution in fiber tows and interface failure between fiber tows or fabrics layers. For the damage contour plots of the fiber tows and the interface layer (shown in Figs. 7 and 8, respectively), the different colors correspond to the value of the specific damage variable, which range from 0 to 1. A value of 0 (areas shown in blue) indicates that damage has not yet occurred, while a value of 1 (areas shown in red) indicates that the element is totally damaged. The primary damage modes for a 2DTBC under transverse tension are fiber breakage, fiber bundle splitting and subsurface delamination. Fig. 7 shows the damage contours from the numerical simulation at a global strain level of 0.7% and at the instant after fracture for a 2DTBC under transverse tension, as well as images showing the fracture morphologies for the tested specimen. From the simulation results at a strain level of 0.7%, it can be noticed that matrix damage occurs mainly in the axial fiber tows, while interface damage occurs at the intersection of bias tows. This damage then leads to a reduction in global stiffness and to nonlinearity in the tensile stress–strain curves. In addition, fiber damage was not observed, and the interfaces between layers are nearly intact, except for some minor damage at the free edges. At the instant after specimen failure, fiber damage is observed in the bias fiber tows, indicating that the breakage of bias fiber tows is the ultimate failure mode, which can also be seen in the fracture morphologies of the tested specimen. In addition, matrix cracking in both axial and bias tows, as well as edge-initiated delamination damage, are also well simulated using the meso-FE model. Meanwhile, interface failure between the bias tows extends throughout the gage region along the axial direction, and the delamination between layers spreads from the free edges to the central region. Compared with the previous work on single-layer coupon specimens [19], the free-edge effect still plays a critical role on the failure of multi-layer coupon specimens, where the premature interface damage produces significant concentration of shear stress in the bias fiber tows, resulting in the shear failure of the fiber bundles. Fig. 8 shows the damage contours of the fiber tows from the simulation at a global strain level of 0.7% and the instant after material failure under transverse compression, as well as images showing the fracture morphologies of the test specimens. For the predicted results at a strain level of 0.7%, the damage status of fiber tows and interface layers are similar to those in the transverse tensile condition: matrix damage accumulates mainly in the axial fiber tows, interface damage

Fig. 6. Failure process of 2DTBC under (a) transverse tension; and (b) transverse compression.

of external loading condition. The results also suggest that the measured compressive properties may not reflect the realistic properties of this material, which is consistent with the experimental observations of Kohlman [7], where the transverse compressive modulus and strength for a tube specimen (without a free edge) is higher than that of a coupon specimen having a similar shape. Similar to the TT condition, the shear strain concentration along the free edges causes premature delamination failure of the bias fiber tows and forms a similar zigzag pattern of crack propagation across the entire gauge section of the compression specimen, as shown in Fig. 6(b). Layer-to-layer delamination can also be found, as shown in Fig. 6(b).

101

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Fig. 7. Simulated damage contours and images showing failure of 2DTBC test specimens under transverse tension.

4.3. Effect of interface strength

presents slightly at the free edges and the intersection of bias tows, and fiber damage is not observed. The initial tensile and compressive stiffness are similar, and the compressive curve exhibits more nonlinearity due to the microbuckling of the fibers and the more dominant role of the matrix material during compression. At the instant after failure, matrix damage spreads to the bias fiber tows, which corresponds to the matrix cracking found in bias tows of the tested specimen. In addition, fiber damage propagates across the bias fiber tows, indicating fiber breakage failure of the bias tows, which can also be found in the fracture morphologies of the tested specimen. Similar to that of TT condition, the free-edge effect causes edge-initiated delamination and separation of the bias fiber tows in the surface layer of the specimens. The subsequent propagation of interface failure leads to the severe bending of constrained bias tows and eventually causes breakage of the fiber tows.

The interface properties used in the current meso-scale FE model (listed in Table 2) are adopted from previous work on a similar material [21,29], due to the lack of experimental data for the 3266 resin system. It is known that interface properties can greatly affect the performance of a composite material [33], especially for transverse loading conditions, where the applied load does not follow the direction of the axial fiber bundles [29]. Thus, parametric studies are necessary to investigate the sensitivity of the effective transverse properties for 2DTBC on the interface strength and the associated mechanisms. In this section, the strength properties of the interface listed in Table 2 are introduced as base values, and two additional cases are considered: one using half (0.5 times) the base value for all interfaces and another using twice (2 times) the base value. In view of the traction–separation based

Fig. 8. Simulated damage contours and images showing failure of 2DTBC test specimens under transverse compression. 102

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Fig. 9. Effect of interface properties on global stress–strain response and damage behavior of a 2DTBC under transverse tension.

Fig. 10. Effect of interface properties on global stress–strain response and damage behavior of a 2DTBC under transverse compression.

displacement contours in Fig. 9 also confirm that the free-edge effect–induced deformation can be reduced by an increase in interface strength. In a previous work by Littell [6], the mechanical properties of triaxially braided composites with the same T700S fabrics and four different epoxy resins (PR520, E862, 3502 and 5208) were characterized and analyzed, where the 2DTBC made from the resin with better toughness shows higher tensile strength properties. The results of the numerical parametric study in this work are consistent with Littell's observations. In addition, it is interesting to note that the predicted strength of the case in this study that uses twice the base interface properties exhibits a similar strength (600 MPa) to that of Littell's results for a 2DTBC made with PR520, which is known to have the best toughness among the four different resins considered. These results further prove the accuracy and the predictability of the meso-FE model presented in this study. Moreover, a stronger interface can partly overcome the nonlinearity of the tensile stress–strain curve and enhance the damage tolerance of the braided composite structure. Fig. 10 shows the global stress–strain curves and damage contours

modeling approach for a cohesive element, the fracture toughness listed in Table 2 is changed by the same magnitude as the strength property in order to stabilize the damage evolution process of the interface layers. Figs. 9 and 10 compare the stress–strain curves and the damage contours from the numerical predictions for the three numerical cases. The stress–strain curves shown in Fig. 9 indicate that the interface strength has an obvious influence on the ultimate strength under transverse tension, where a weaker interface strength corresponds to a relatively low ultimate strength of the material. This may be explained in two ways. First, under a transverse tensile load condition, the interface and matrix materials contribute greatly to the transmission of load, and a stronger interface will enable the composites to sustain more load. In addition, the enhanced interface property can decrease the occurrence of interface failure as confirmed by the damage contours in Fig. 9, where the area and extent of interface damage are obviously minor for models having a higher interface strength. For the case that uses two times the base interface properties, the layer-to-layer delamination is completely eliminated and tow-to-tow debonding is negligible at a global strain level of 1.28%. Moreover, the out-of-plane 103

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infinitely thick specimen presents the highest strength value, which can be treated as the theoretical strength of the material considering that there is no free edge for a large structural component. The single-layer specimen exhibits the most significant free-edge warping, which results in more obvious nonlinearity in the stress–strain curve and a much lower strength as compared to other specimens. The global stress–strain curves for specimens with 2, 4, 6 and 8 layers nearly coincide with each other, and the areas for the free-edge warping zone are quite close in size. From a modeling perspective, based on the stress–strain response and the out-of-plane deformation contour, we propose that using a model with four or more layers is capable of simulating the transverse tension response for an eight-layer coupon specimen. For transverse compression, the change in specimen thicknesses has a more obvious influence on the failure behavior, as it affects not only the extent of free-edge warping but also the deformation mode of the coupon specimens. From the displacement contours shown in Fig. 11(b), it can be observed that extensive out-of-plane deformation is distributed on specimens with 1, 2 and 4 layers, which suggests that instability governs the compressive deformation and the specimens are not exposed to a realistic compressive load. Thus, the models with an insufficient number of layers cannot be correlated with the results from a compressive experiment. For the specimens having 6 and 8 layers, the two predicted stress–strain curves overlap, and the predicted deformation contour also shows similar local warping along the free edges. For the infinite thickness condition, similar to that of transverse tension condition, the predicted compressive strength is relatively higher than that of specimens with a finite thickness, due to the elimination of out-of-plane displacement. As discussed in a previous work [18], the proper definition of boundary conditions is a critical step for developing a meso-FE model. Combining the parametric studies in this section, we come to the conclusion that six is the minimum number of layers needed for a meso-FE model to simulate the transverse tensile and compressive response of a 2DTBC coupon specimen, and it is also the recommended specimen thickness for fundamental experimental characterization of this type of material.

of transverse compression specimens with different interface strengths. In contrast to the TT condition, the increase in interface strength exhibits little influence on the ultimate compression strength; however, decreasing the interface strength leads to an obvious reduction in the ultimate compression strength. However, the mechanism that controls the sensitivity of the TC response on the interface properties is similar to that of the TT response, which is attributed to the critical role of interface damage on transverse tension and compression failure behavior. Increasing the interface strength can help reduce the occurrence of interface damage inside the braided composite and can increase the strength accordingly. As for the damage contours at a global strain level of 0.95%, obvious interface damage between the fiber tows and freeedge delamination between different layers can be seen for the model with the lowest interface properties (the case using half the base values). However, there is no apparent interface failure for the other two cases. This suggests that the reference interface properties are good enough for resisting interface failure for a 2DTBC under transverse compression. Thus, any further increase in the interface properties may not increase the transverse compression strength by a significant amount. The out-of-plane displacement contours of the three cases are compared in Fig. 10. As can be noticed from this figure, the model with the lowest interface strength exhibits more serious free-edge warping, while the warping in the other two models is more or less the same (both in terms of amplitude and area of the out-of-plane deformation). It should be noted that free-edge warping is a kind of elastic deformation, which is related to the braided architecture but is independent of the interface properties [19]. The relatively larger deformation for the model having the lowest interface strength is due to the presence of free-edge delamination (see the damage contour plots in Fig. 10), which promotes out-of-plane deformation. Overall, improving the interface properties will not eliminate the free-edge effect (out-of-plane warping along the free edges), but it can help to mitigate the free-edge effect–induced damage for coupon specimens. Furthermore, varying the interface strength has a negligible influence on the damage of fiber tows, as can be noticed from the contour plots for fiber damage and matrix damage shown in Fig. 10. This also explains the high coincidence of the stress–strain curves prior to unloading.

5. Conclusion Quasi-static tensile and compressive failure behavior of a triaxially braided composite under in-plane transverse loading conditions were investigated by experimental and numerical studies. The developed meso-scale FE method shows fairly good correlation with the experimental results in terms of global stress–strain curves and full-field strain distributions. The meso-FE model shows the capability to predict the internal damage evolution process of fiber tows and interfaces. The main failure modes of a braided composite subjected to transverse loads are fiber breakage in the bias tows and transverse matrix cracking in axial tows, as well as tow-to-tow and layer-to-layer delamination. Freeedge effect is identified to occur under both transverse tensile and transverse compressive loading conditions in the form of periodic outof-plane warping along the free edges, which can lead to premature edge-initiated interface failure. Numerical parametric studies were conducted to examine the influence of interface strength and specimen thickness on the mechanical response of this braided composite. Both the transverse tension and transverse compression strength increase with an increase in interface strength. However, the reference interface properties of the current material system (i.e., the base values) are sufficiently high to resist interface failure under transverse compression. Thus, any further increase in the interface strength does not increase the ultimate compressive strength to a significant extent. The specimen thickness has an obvious influence on the global response of this braided composite, mainly due to the free-edge effect–induced warping and interface delamination. The increase of thickness can partially restrict the out-ofplane warping but cannot eliminate it. A minimum of six layers is recommended to properly study the transverse tensile and compressive

4.4. Comparison between models having a different number of layers The computational cost of the meso-FE model is of critical concern for its application in the engineering design of braided composite structures. The development of proper boundary conditions to reduce the number of unit cells in a meso-FE simulation has attracted a great deal of attention [19,34,35]. However, previous studies focus mainly on in-plane dimensions, and there is little understanding on the behavior in the through-thickness direction, which could be very important for the compression simulations. In this section, specimens having a different number of layers (1, 2, 4, 6, 8 and infinite) are subjected to transverse tension, compression loads are analyzed using the presented meso-scale FE model, and the numerical results are compared to aid in identifying the number of layers needed to achieve an acceptable prediction. The infinite number of layers is satisfied by applying a periodic boundary condition along the thickness direction on a single-layer model. Fig. 11 shows the comparison of mechanical response for meso-FE models with different numbers of layers under transverse tension and compression. The out-of-plane displacement contours at the moment of specimen fracture are plotted to understand the effect of thickness on free-edge warping. From Fig. 11(a), it can be seen that the magnitude and area of free-edge warping decrease with an increase in the number of layers. For the infinitely thick specimen (labeled as “1 layer + PBC” in the figure), the free-edge warping is negligible due to the continuity between the top and bottom surfaces, which can suppress the out-ofplane deformation. Because of the elimination of free-edge effect, the 104

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Fig. 11. Comparison of out-of-plane deformation and global stress–strain response for meso-FE model using a different number of layers: (a) transverse tension; (b) transverse compression.

failure behavior of a multi-layer coupon specimen of 2DTBC. Foundation of China (Grant No. NSFC 11772267) and Fundamental Research Funds for the Central Universities.

Acknowledgement This work was supported by the National Natural Science Appendix A

The progressive damage model is adopted to describe the damage initiation and damage evolution of the fiber tows. Four distinct failure modes are considered in this model which combines Hashin's [24] and Hou's [25] criteria and integrates with continuum damage laws. The damage initiation criteria are formulated as follows. For fiber tension failure (σ11 > 0 ), 2

2

σ +σ31 ⎞ σ eft = ⎛ 11 ⎞ + α ⎛ 12 ≥1 ⎝ F1t ⎠ ⎝ Fls ⎠ ⎜

⎟

⎜

⎟

(A-1)

For fiber compression failure (σ11 < 0 ), 105

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σ efc = ⎛ 11 ⎞ ≥ 1 ⎝ F1c ⎠ ⎜

⎟

(A-2)

For matrix tension failure (σ22 > 0 ), 2

2

2

σ σ σ emt = ⎛ 22 ⎞ + ⎛ 12 ⎞ + ⎛ 23 ⎞ ≥ 1 ⎝ F2t ⎠ ⎝ Fls ⎠ ⎝ Fts ⎠ ⎜

⎟

⎜

⎟

⎜

⎟

(A-3)

For matrix compression failure (σ22 < 0 ), 2

emc =

2

F 2 σ22 1 ⎛ −σ22 ⎞ σ σ + 2c2 − 22 + ⎛ 12 ⎞ ≥ 1 4 ⎝ Fls ⎠ F2c 4Fls F2c ⎝ Fts ⎠ ⎜

⎟

⎜

⎟

(A-4)

In the above equations, eft, efc, emt and emc are damage variables corresponding to the four kinds of failure mode. F1t, F1c, F2t, F2c, Fls, and Fts are axial tensile strength, axial compressive strength, transverse tensile strength, transverse compressive strength, longitudinal shear strength and transverse shear strength, respectively. A coefficient ɑ is introduced to consider the influence of shear stress on the fiber tension failure behavior, which has been calibrated to be 0.06 for a similar material system (T700/E862) reported in Literatures [21,29] will be followed in this study. Murakami–Ohno [26] damage theory is used to predict the evolution process after satisfaction of failure criteria. The evolution of each damage variable is governed by an equivalent displacement expressed by the following equation:

dI =

δIf, eq (δI , eq − δI0, eq) δI , eq (δIf, eq − δI0, eq)

, I= ft , fc, mt and mc (A-5)

where δI , eq represents the equivalent displacement of the current incremental step,

δIf, eq

is the final equivalent displacement of corresponding failure

mode, and δI0, eq is the initial equivalent displacement when the failure criterion just satisfies. δIf, eq and δI0, eq can be computed by the following equations:

δIf, eq =

δI0, eq =

2GI σI0, eq

(A-6)

δI , eq eI

(A-7)

δIf, eq

where, σI0, eq denotes the initial equivalence stress. The detailed algorithm equations used to compute the equivalent displacement and stress in different damage modes can be found in our previous work [29]. eI is the value of each damage criteria. The fracture energy GI which is a material property that must be specified, the values for the fracture energies of fiber bundles used in this study were obtained from the literature [21]. The effective stiffness matrix of the damaged element will then degrade accordingly based on the current damage variables. A second-order symmetric tensor is used to describe the damage state for the fiber bundles. The corresponding damaged stiffness matrix C(d) is obtained as follows:

⎡ df E11 (1 − dm ν23 ν32) df dm E11 (ν21 + ν23 ν31) df E11 (ν31 + ν21 ν32) ⎤ ⎢ ⎥ dm E22 (1 − df ν13 ν31) dm E22 (ν32 + df ν12 ν31) ⎢ ⎥ E33 (1 − df dm ν12 ν21) 1⎢ ⎥ C (d ) = ⎢ ⎥ Δ d d G f m 12 ⎢ ⎥ ⎢ ⎥ dm G23 ⎢ df G13 ⎥ ⎣ ⎦

(A-8)

df = (1 − dft )(1 − dfc )

(A-9)

dm = (1 − dmt )(1 − dmc ) Δ=

(A-10)

1 1 − df dm ν12 ν21 − dm ν23 ν32 − df ν13 ν31 − 2df dm ν12 ν13 ν23

(A-11)

df and dm are global damage variables associated with fiber failure and matrix failure, respectively. While dft, dfc, dmt and dmc are the damage evolution variables associated with four different damage modes. The constitutive and damage model as described above is implemented using a user subroutine “VUMAT” compatible with ABAQUS/Explicit solver. Apart from a 3D progressive damage model for composite fiber tows, the tow-to-tow and tow-to-matrix interfaces are simulated by using the cohesive-zone modeling approach, which has been embedded into ABAQUS as an optional element type. The responses of cohesive elements are governed by a typical bilinear traction-separation law, and a quadratic nominal stress criterion is used to describe interfacial damage initiation. Besides, a power-law criterion is adopted, which claims that failure under mixed-mode conditions is governed by a second-order power law interacting of the energies required to cause failure in the individual (normal and two shear) modes. The quadratic nominal stress criterion for damage initiation and a power-law criterion for failure are represented in Eq. (A-12) and (A-13). 2

2

2

⎛⎜ 〈tn 〉 ⎞⎟ + ⎜⎛ ts ⎟⎞ + ⎜⎛ tt ⎟⎞ = 1 0 0 0 ⎝ ts ⎠ ⎝ tt ⎠ ⎝ tn ⎠ 2

2

(A-12)

2

⎛⎜ Gn ⎞⎟ + ⎜⎛ Gs ⎟⎞ + ⎜⎛ Gt ⎟⎞ = 1 c c c ⎝ Gn ⎠ ⎝ Gs ⎠ ⎝ Gt ⎠

(A-13)

Where tn denotes the traction normal stress, and tt and ts denote shear stresses. The Macaulay brackets are used to signify that a pure compressive deformation or stress state does not initiate damage. Similarly, Gn, Gs, and Gt refer to the work done by the traction and its conjugate relative 106

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displacement in the normal, first, and second shear directions, respectively; and Gnc , Gsc , and Gtc are critical fracture energies required to cause failure in each of the three directions. Detailed formulations of the mixed-mode cohesive zone model can be found in the ABAQUS user's manual.

[17] Tsinuel N. Geleta, Kyeongsik Woo, Multi-scale progressive failure analysis of triaxially braided textile composites, International Journal of Aeronautical and Space Sciences 18 (3) (2017) 436–449. [18] C. Zhang, W.K. Binienda, R.K. Goldberg, et al., Meso-scale failure modeling of single layer triaxial braided composite using finite element method, Compos. Appl. Sci. Manuf. 58 (58) (2014) 36–46. [19] C. Zhang, W.K. Binienda, R.K. Goldberg, Free-edge effect on the effective stiffness of single-layer triaxially braided composite, Compos. Sci. Technol. 107 (2015) 145–153. [20] Q. Cheng, J. Wang, K. Jiang, et al., Fabrication and properties of aligned multiwalled carbon nanotube-reinforced epoxy composites, J. Mater. Res. 23 (11) (2008) 2975–2983. [21] X. Li, W.K. Binienda, R.K. Goldberg, Finite element model for failure study of two dimensional triaxially braided composite, J. Aero. Eng. 24 (2) (2010) 170–180. [22] D.3039 ASTM, Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials, ASTM International, West Conshohocken, Pa., 2008. [23] D.3410 ASTM, Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials with Unsupported Gage Section by Shear Loading, ASTM International, West Conshohocken, Pa., 2003. [24] Z. Hashin, Failure criteria for unidirectional fiber composites, J. Appl. Mech. 47 (2) (1980) 329–334. [25] J.P. Hou, N. Petrinic, C. Ruiz, et al., Prediction of impact damage in composite plates, Compos. Sci. Technol. 60 (2) (2000) 273–281. [26] Zdeněk P. Bažant, B.H. Oh, Crack band theory for fracture of concrete, Matériaux et Construction 16 (3) (1983) 155–177. [27] Z. Zhao, H. Dang, C. Zhang, et al., A multi-scale modeling framework for impact damage simulation of triaxially braided composites, Compos. Appl. Sci. Manuf. 110 (2018) 113–125. [28] D.C.S. Simulia, ABAQUS 6.14 Analysis User's Manual, (2014) Abaqus 6.14 Documentation. [29] C. Zhang, N. Li, W. Wang, et al., Progressive damage simulation of triaxially braided composite using a 3D meso-scale finite element model, Compos. Struct. 125 (2015) 104–116. [30] Z.M. Huang, The mechanical properties of composites reinforced with woven and braided fabrics, Compos. Sci. Technol. 60 (4) (2000) 479–498. [31] C.C. Chamis, Simplified composite micromechanics equations for strength, fracture toughness and environmental effects, SAMPE Q. 15 (4) (1984) 41–55. [32] H.J. Chun, J.Y. Shin, I.M. Daniel, Effects of material and geometric nonlinearities on the tensile and compressive behavior of composite materials with fiber waviness, Compos. Sci. Technol. 61 (1) (2001) 125–134. [33] C. Zhang, J.L. Curiel-Sosa, T.Q. Bui, A novel interface constitutive model for prediction of stiffness and strength in 3D braided composites, Compos. Struct. 163 (2017) 32–43. [34] Z. Xia, C. Zhou, Q. Yong, et al., On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites, Int. J. Solid Struct. 43 (2) (2006) 266–278. [35] S. Jacques, I.D. Baere, W.V. Paepegem, Application of periodic boundary conditions on multiple part finite element meshes for the meso-scale homogenization of textile fabric composites, Compos. Sci. Technol. 92 (3) (2014) 41–54.

References [1] O. Bacarreza, P. Wen, M.H. Aliabadi, Micromechanical modelling of textile composites, in: M.H. Aliabaki (Ed.), Computational and Experimental Methods in Structures, Volume 6: Woven Composites, World Scientific Publishing Co., Hackensack, N.J., 2015, pp. 1–74. [2] T.W. Chou, F. Ko, Textile Structural Composites, Elsevier Science Pub. Co., Amsterdam, 1988. [3] S.V. Lomov, D.S. Ivanov, T.C. Truong, et al., Experimental methodology of study of damage initiation and development in textile composites in uniaxial tensile test, Compos. Sci. Technol. 68 (12) (2008) 2340–2349. [4] D.S. Ivanov, F. Baudry, B.V.D. Broucke, et al., Failure analysis of triaxial braided composite, Compos. Sci. Technol. 69 (9) (2009) 1372–1380. [5] T. Wehrkamp-Richter, R. Hinterhölzl, S.T. Pinho, Damage and failure of triaxial braided composites under multi-axial stress states, Compos. Sci. Technol. 150 (2017) 32–44. [6] S.C. Quek, A.M. Waas, K.W. Shahwan, et al., Compressive response and failure of braided textile composites: Part 1—experiments, Int. J. Non Lin. Mech. 39 (4) (2004) 635–648. [7] J.D. Littell, The Experimental and Analytical Characterization of the Macromechanical Response for Triaxial Braided Composite, Ph.D. Dissertation, The University of Akron, Akron, Ohio, 2008. [8] L.W. Kohlman, Evaluation of Test Methods for Triaxial Braid Composites and the Development of a Large Multiaxial Test Frame for Validation Using Braided Tube Specimens, Ph.D. Dissertation, The University of Akron, Akron, Ohio, 2012. [9] S.V. Lomov, D.S. Ivanov, I. Verpoest, et al., Meso-FE modelling of textile composites: road map, data flow and algorithms, Compos. Sci. Technol. 67 (9) (2007) 1870–1891. [10] S. Daggumati, W.V. Paepegem, J. Degrieck, et al., Local damage in a 5-harness satin weave composite under static tension: Part II – meso-FE modelling, Compos. Sci. Technol. 70 (13) (2010) 1934–1941. [11] S.A. Tabatabaei, S.V. Lomov, I. Verpoest, Assessment of embedded element technique in meso-FE modelling of fibre reinforced composites, Compos. Struct. 107 (1) (2014) 436–446. [12] C.Q. Shu, A.M. Waas, K.W. Shahwan, et al., Compressive response and failure of braided textile composites: Part 2—computations, Int. J. Non Lin. Mech. 39 (4) (2004) 649–663. [13] S. Song, A.M. Waas, K.W. Shahwan, et al., Braided textile composites under compressive loads: modeling the response, strength and degradation, Compos. Sci. Technol. 67 (15) (2007) 3059–3070. [14] T. Wehrkamp-Richter, N.V.D. Carvalho, S.T. Pinho, A meso-scale simulation framework for predicting the mechanical response of triaxial braided composites, Compos. Appl. Sci. Manuf. 107 (2018) 489–506. [15] Y. Ren, S. Zhang, H. Jiang, et al., Meso-scale progressive damage behavior characterization of triaxial braided composites under quasi-static tensile load, Appl. Compos. Mater. (1) (2017) 1–18. [16] H. Jiang, Y. Ren, S. Zhang, et al., Multi-scale finite element analysis for tension and ballistic penetration damage characterizations of 2D triaxially braided composite, J. Mater. Sci. 53 (5) (2018) 1–24.

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