A new analytical method for progressive failure analysis of two-dimensional triaxially braided composites

Composites Science and Technology 186 (2020) 107936

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A new analytical method for progressive failure analysis of two-dimensional triaxially braided composites Haoyuan Dang a, b, c, Zhenqiang Zhao a, b, c, Peng Liu a, b, c, Chao Zhang a, b, c, *, Liyong Tong d, Yulong Li a, 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 Joint International Research Laboratory of Impact Dynamics and Its Engineering Applications, Xi’an, Shaanxi, 710072, China d School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW, 2006, Australia b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Strength Mechanical properties Analytical modeling Damage mechanics Textile composites

A new analytical model is developed for predicting progressive damage, equivalent stiffness and strength of a two-dimensional triaxially braided composite (2DTBC) under tensile and compressive loading. In this model, an alternative concept of equivalent lamina elements (ELEs) is first established for the four subcells parallel to the axial direction in a representative unit cell of the 2DTBC; Next, the ELEs are laminated in the thickness direction and integrated via a series-parallel model to establish a two-way stress–strain response within the unit cell, that allows failure initiation and propagation in the ELEs to be examined and ultimate failure strength to be predicted. The analytical results are validated against a finite element simulation of an infinite plate as well as experimental results. Numerical case studies are conducted to assess the sensitivity of analytical results against compressive strength of ELEs. The results demonstrate the capability and efficiency of the present model for predicting the mechanical responses of single-layer and multi-layer 2DTBC specimens under different loading conditions.

1. Introduction Fiber-reinforced composites, including two-dimensional (2D) and three-dimensional (3D) braided composites, are widely used in aero­ space, transportation and civil engineering applications because of their high specific stiffness and strength and good resistance to impact and corrosion. As an example, a 2D triaxially braided composite (2DTBC), which is manufactured using three intertwined fiber tows that are aligned along the axial (0� ) and bias (�θ� ) directions, is used in fan containment systems for aero-engine. To facilitate the application of braided composites in engineering structures, it is necessary to develop reliable computational models and efficient predictive tools that can effectively predict the mechanical properties and failure responses of the materials. This work intends to develop a novel analytical model with the capability of predicting the progressive failure, effective strength and stiffness of a 2DTBC. Previous studies have focused on testing and modeling of the me­ chanical properties and failure behavior for 2DTBCs. Littell [1] measured the in-plane effective mechanical properties (axial tension and

compression, transverse tension and compression, and in-plane shear) of four different material systems for a 2DTBC and concluded that weak interfaces cause premature failures and interlayer delamination and contribute to the non-linearity of the resulting stress–strain curves. Kohlman [2] determined that notched coupon specimens exhibit a higher transverse tensile strength than straight-sided coupon specimens due to the elimination of free-edge induced premature failure. Wehrkamp-Richter et al. [3] found that 2DTBC specimens exhibit severe nonlinearities before final failure due to progressive matrix cracking and delamination accompanied by fiber bundle pull-out. The complicated failure behavior of 2DTBCs present a significant challenge for the development of modeling methods in terms of exploring the internal progressive failure mechanisms and predicting the effective response and ultimate failure under various loads and complex loading conditions. Numerical modeling methods, in particular the finite element anal­ ysis (FEA) method, have been widely used to study the mechanical behavior of 2DTBCs. Quek et al. [4,5] developed a finite element model that captures the braid architecture and predicts the compressive

* Corresponding author. Department of Aeronautical Structure Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, China. E-mail address: [email protected] (C. Zhang). https://doi.org/10.1016/j.compscitech.2019.107936 Received 15 June 2019; Received in revised form 25 October 2019; Accepted 21 November 2019 Available online 22 November 2019 0266-3538/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. (a) Two-dimensional triaxially braided fabric, (b) top view and (c) side view of the representative unit cell, along with (d) the through-thickness scheme of the subcell components, (e) geometric model of a triaxially braided fabric, with (f) the scheme for the coordinate systems and bias fiber bundle undulation and (g) a cross-sectional view of the bias fiber bundle undulation in the YZ plane.

strength of a 2DTBC under a range of conditions. Cheng [6] proposed a through-the-thickness method for considering a braided textile (also known as a subcell model), where the representative unit cell is dis­ cretized into laminated plates having different angles and fiber volume fractions in the finite element. The subcell model was continuously enhanced by Littell [1], Blinzler [7], Cater et al. [8] and Sorini et al. [9] to enhance its efficiency for simulating impacts. Zhang et al. [10,11] and Zhao et al. [12] established meso-scale and multi-scale FE frameworks to study the mechanical failure behavior of 2DTBC coupon specimens under quasi-static and impact loading conditions, and the resulting models demonstrate good accuracy in simulating the global stress–strain response of the specimens as well as their damage progression behavior under different loading conditions. Although FE analysis has shown great success in simulating the progressive failure of 2DTBCs, it is computationally intensive due to geometric modeling and meshing, and it may not be able to provide a quick and effective solution. To address this, it is important to develop a simple and efficient analytical model that is capable of predicting the mechanical properties for 2DTBC and that not only predicts the effective stiffness but also captures the progressive failure and ultimate strength. Previous studies that have investigated the prediction of effective stiff­ ness properties [13–16] have mainly used the volume-averaging method. Byun [13] presented explicit mathematical equations for the geometry of a 2D braided textile composite and developed an analytical model based on the principle of volume-averaging. Quek et al. [14] and Zhang et al. [15] studied the effect of braided angle and fiber tow un­ dulation on the effective stiffness. Shokrieh and Mazloomi [16] simpli­ fied a triaxially braided composite as a series of laminated cross-ply composites and calculated the effective stiffness of the braided com­ posite using the volume-averaging method. Despite these recent advances, developing an analytical model that provides efficient prediction of the progressive failure and strength of a 2DTBC remains a challenge, and only a few extremely limited studies demonstrate models with simple, effective and comprehensive predic­ tive capabilities. Deng et al. [17] established a two-way scale coupling method for predicting the mechanical properties of a 2DTBC and

predicted the progression of axial tensile failure of the 2DTBC using a volume-averaging method. Liu et al. [18] utilized the Multi Scale Generalized Method of Cells (MSGMC) method to analyze the stress–­ strain response of 2DTBC where the representative volume element at different scales are discretized into subcell elements. The MSGMC method provides good accuracy in predicting stress-strain relationship of unidirectional lamina, but has limitations in predicting the progres­ sive failure of textile composites due to their more complicated failure behavior [18]. On the other hand, the involvement of Euler integration formulation leads to high computational cost and very complicated iterative process [18]. The modeling frameworks presented in these two studies [17,18] are limited, as they can only partially predict progressive failure and strength for the 2DTBC under tensile loading. To the best of our knowledge, no studies report the prediction of the tensile and compressive progressive failure and strength of textile composites in both the axial and transverse directions. The main chal­ lenge in developing an analytical model for failure analysis of a 2DTBC lies in building a two-way reversible relationship between a global (unitcell level) stress–strain response and a local (fiber-tow level) stress–­ strain response, due to the inherent complexity of the meso-scale fabric structure. To overcome this difficulty, this work adopts the subcell concept [6] to consider 2DTBC consisting of multiple representative lamina elements aligned in a parallel and sequential manner, which allows the establishment of explicit equations to connect the global and local stress–strain responses. The structure of this paper is organized as follows: Section 2 in­ troduces the analytical model for a 2DTBC, including a geometrical representation of the fabric architecture, the modeling strategy and a flowchart of the geometrical discretization and reassembly for the 2DTBC. Section 3 briefly introduces the finite element model for an infinite plate. In Section 4, model validation and numerical case studies using the proposed analytical model are presented. The conclusions are presented in Section 5.

2

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0� fiber bundle and θ� fiber bundles, the gap between two adjacent θ� fiber bundles can be calculated as g ¼ W cosðθÞ Wb . An additional parameter δ as shown in Fig. 1(b) is introduced to define the relative location of þθ� fiber bundles, which are assumed to have a gap g that is the same as that between the θ� fiber bundles. We note that the value of δ only affects the bias fiber arrangement but has no influence on the mechanical performance due to the periodic nature of the fabric struc­

Table 1 Architectural properties of 2DTBC specimens. Parameter

Six-layer

Single-layer

Parameter

Six-layer

Single-layer

θ( )

60

60

(mm)

0.1575

0.201

W (mm)

9.05

9.05

h60 A

0.2665

0.31

WA (mm)

5.50

5.50

h60 B (mm)

62.89

48.81

Wb (mm)

3.50

3.50

h (mm)

0.533

0.62

L (mm)

5.22

5.22

h0A (mm)

0.218

0.218

�

Vþ60 A (%) V0A (%) 60

VA (%) Vþ60 B (%)

VB 60 (%)

77.03

77.03

62.89

48.81

37.18

31.81

37.18

31.81

ture. The parameters h0A , hθA and hθB denote the thickness of the 0� ELE in Subcell A, �θ� ELE in Subcell A, and �θ� ELE in Subcell B, respectively, as shown in Fig. 1(d). A global XYZ coordinate system is used for the entire unit cell as shown in Fig. 1. In addition, two local coordinate systems, xyz and 123, are used for an undulated θ� fiber bundle and its projection on the XY plane, as depicted in Fig. 1(f and g). Based on geometric analysis, the equivalent fiber volume fractions of different ELEs in the subcell can be determined by � � θ 2 V þθ 4LhθA sinðθÞ (1) A ¼ V A ¼ π d Nfb

2. Analytical model Consider a typical 0� /�θ� triaxially braided composite fabric without resin as depicted in Fig. 1(a) and (e), in which the þθ� bias fiber bundles cross over and under θ� bias fiber bundles and vice versa, whereas the 0� straight fiber bundles are inserted and intertwined with the bias fiber bundles. For a perfectly fabricated composite, one can choose the representative repetitive volume or unit cell as depicted in Fig. 1(b) (which is the top view of one of the rectangles enclosed by red dash lines in Fig. 1(a); a side view of this unit cell is shown in Fig. 1(c)). As shown in Fig. 1(d), the unit cell can be divided into four subcells, labeled as “A”, “B”, “C” and “D”, where each subcell can be further modeled using an equivalent lamina element (ELE). Subcells A and C consist of three ELEs ( θ� /0� /þθ� ) while Subcells B and D have two ELEs (þθ� / θ� ). In previous studies by Cater et al. [8], Subcells B and D were divided into three ELEs to capture the tension-twist deformation of finite-size specimen under transverse-tensile load. However, Cater et al. [8] also found that the division method has negligible effect on the prediction of effective properties. Thus, to facilitate the computational efficiency, this work still considers Subcells B and D as two ELEs. A new analytical model is developed for the unit cell (as shown in Fig. 1(b)–1 (g)) for predicting its equivalent stiffness and failure strength based on the series-parallel model in the in-plane directions and the classical laminate plate theory (CLT) in the through thickness direction.

V 0A ¼ πd2 Nfa

V þθ B

θ

�

¼ VB ¼

4WA h0A

�

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � �2 �2 � W WA � θ þ h0A þ hθA sinðθÞ WA hB

πd2 Nfb 4L W

(2) (3)

where Nfa and Nfb are the number of filaments in each axial fiber bundle and bias fiber bundle, respectively, and d denotes the diameter of a single fiber. More detailed equations for computing the volume fractions are presented in Appendix A. In this work, two types of specimens are studied to demonstrate the efficiency of model: one type of specimen is made from six-layers of fabric with a fiber volume ratio of 56%, and the other is made from a single-layer of fabric with a fiber volume ratio of 48%. The measured dimensions of the unit cell architecture and the calculated geometry parameters are listed in Table 1. Quasi-static inplane tests of the six-layer and single-layer specimens using an MTS universal testing machine under displacement control were performed by Littell [1] and Kohlman [2], respectively. 2.2. Micromechanics model of ELEs Fig. 2 shows a flowchart of the geometric discretization and reas­ sembly process for a 2DTBC unit cell model. Based on the antisymmetrical feature, the mechanical properties of 2DTBC can be rep­ resented by a half unit cell to simplify the calculation, which is composed of Subcells A and B. Each subcell is simplified as a stack of ELEs with different angles, and the mechanical properties of each ELE in

2.1. Unit cell geometry and fiber volume fractions When considering the top view of the unit cell as shown in Fig. 1(b), let W denote the braiding space or distance between two adjacent 0� axial fiber bundles. The unit cell has a width of twice the braiding space (2W) and a length L ¼ W=tanðθÞ. If WA and Wb are the width of

Fig. 2. Scheme of geometric discretization and reassembly for a 2DTBC unit cell. 3

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local coordinate system (xyz) can be directly determined using micromechanical models. Following which, the effective stiffness of each ELE in global coordinate system (XYZ) can be calculated after two rounds of coordinate transformation, which are then further assembled using the CLT and series-parallel model to determine the effective stiffness matrix of the unit cell. For each ELE, four elastic parameters (Exx, Eyy, Gxy, and ѵxy) and five strength parameters (Xt, Xc, Yt, Yc, and S), can be determined by adopting micromechanical models, such as the Chamis model [19], which provides simple and explicit equations to calculate the mechan­ ical properties of composites, and reasonable prediction of strength properties [19]. However, the proposed method is not limited to Chamis model and is compatible with other micromechanical models. The calculated mechanical properties of ELEs are listed in Table C-1 and Table C-2 of Appendix C, where the axial compressive strength is cali­ brated against the FE results (a more detailed discussion is included in Section 4). By using the four elastic constants, one can determine the stiffness matrix of each ELE in the local coordinate system (xyz), as shown in Eq. (4). � 3 1 2 � v� 0 1=Exx xy Eyy � j� � j� 1 5 ​ i ¼ A ​ ​ ​ j ¼ θ;0;þθ 4 C i ¼ Si ¼ 1 Eyy �0 i ¼ B ​ ​ ​ j ¼ θ;þθ symmetric 1 Gxy

2

m2 ½R1 � ¼ 4 0 0

0 1 0

3 0 05 m

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and m ¼ cos βðYÞ ¼ 1= 1 þ tan2 βðYÞ, in which tan βðYÞ, is given by Eq. (7). Following the procedure shown in Fig. 2, the global stiffness matrices of þθ� and θ� ELE in Subcells A and B in the global coordinate system (XYZ), can be obtained by � � � � � j i ¼ A ​ ​ ​ ​ j ¼ θ; þθ Qi ¼ ½R2 � Qji ½R2 �T ​ ​ ​ (10) i ¼ B ​ ​ ​ ​ j ¼ θ; þθ where 2

p2 ½R2 � ¼ 4 q2 pq

3 q2 2pq 2 p 2pq 5 pq p2 q2

2.3. Effective stiffness model Subcell A and Subcell B can be modeled as three-ply or two-ply laminates based on CLT [20]. Since no out-of-plane bending was observed during the tests, Littell [1] and Goldberg et al. [21] suggested that tension-bending can be ignored in order to simplify the analysis of the 2DTBC. Therefore, the stress–strain relationship of Subcell A (or Subcell B) in the global coordinate system can be written as 2 3 3 32 3 3 2 2 2 σ iX εiX εiX N iX Ai11 Ai12 0 6 i 7 16 i 7 16 i 76 i 7 6 i 7 i 6 σ 7¼ 6 N 7¼ 6A 76 7 7 6 4 Y 5 h 4 Y 5 h 4 21 A22 0 54 εY 5 ¼ ½Ci �4 εY 5 ​ ​ i ¼ A; B i i i i 0 0 A66 σXY N XY γXY γ iXY

The stiffness matrix of each ELE in the local coordinate system (123) has the following structural form: 3 2 � 0 Q11 Q12 � j� i ¼ A ​ ​ ​ j ¼ θ; 0; þθ 4 Qi ¼ Q21 Q22 (5) 0 5​ i ¼ B ​ ​ ​ j ¼ θ; þθ 0 0 Q66 A coordinate transformation is then conducted to determine the global stiffness matrices of each ELE. For the 0� ELE, all the three co­ ordinate systems are the same (Fig. 2) and coordinate transformation is not necessary. Thus, the stiffness matrix of the 0� ELE in the global

(12)

0

system is given by ½QA � ¼ ½Q0A � ¼ ½C0A �. To determine the global stiffness matrices of the θ� ELE, two rounds of coordinate transformations are needed considering the undulation and braided angle of the bias fiber bundles. For the first round of co­ ordinate transformation, we can determine the undulation angle through projecting the bias fiber bundle onto YZ plane, as shown in Fig. 1(g)), where the elliptical-shaped cross sections of the axial fiber bundles are in blue and the undulated θ� bias fiber bundles are in pink. The black dashed line in Fig. 1(g) is used to describe the midline of the projected undulation of the bias fiber bundle, which can be expressed as � � h0 þ hθA Y Z¼ A π (6) cos W 2

where ( �� � þθ � � 0� � θ� ½CA � ¼ QA � hθA þ QA � h0A þ QA � hθA h � �� � þθ � � θ ½CB � ¼ QB � hθB þ QB � hθB h

2.4. Strength prediction The constitutive relationships derived for the ELE, the subcells and the unit cell can be further used to predict the strength of the 2DTBC by considering stiffness degradation. In this study, Hoffman failure crite­ rion is employed for damage initiation together with a stiffness evolu­ tion rule that is implemented to model the process of progressive failure. 2.4.1. Determination of local stress In this work, local stress components in the ELEs are explicitly determined and are used to study the failure progression of the 2DTBC. Considering an external load [NX, NY, NXY] applied to the unit cell, the strain can be given by 2 3 2 3 � NX εX 4 εY 5 ¼ 1½Cmn � 1 4 NY 5 ​ ​ m ¼ 1; 2; 6 (14) n ​ ¼ 1; 2; 6 h γXY NXY

The first round of coordinate transformation calculates the stiffness matrices of þθ� and θ� ELE in Subcells A and B in the local coordinate system (123), with equations expressed as

> > > � j� > > > > QB ¼ :

� � ½R1 � CjA ½R1 �T dY

, WA

WA =2 W ZWA =2

​ ​ ​ j¼

, � � ½R1 � CjB ½R1 �T dY

ðW

θ; þθ

(13)

Based on the series-parallel model [22] (see Appendix B for more details), the effective stiffness matrix (Cmn, where m,n ¼ 1,2,6) of the unit cell can be obtained.

As shown in Fig. 1(f) and (g), the projected and actual undulation angles of the bias fiber bundle denoted as αðYÞ and βðYÞ can be calcu­ lated by � � � π h0A þ hθA dZ Y tan βðYÞ ¼ tan αðYÞsinðθÞ ¼ sinðθÞ ¼ sin π sinðθÞ (7) 2W dY W

W ZA =2

(11)

and where p ¼ cosðθÞ, and q ¼ sinðθÞ.

(4)

8 > > > �Qj � ¼ > > A > > <

(9)

(8)

Based on the series-parallel model presented in Appendix B, the strain components of Subcells A and B can be obtained (using Equation (B5) in Appendix B). Furthermore, the local stresses in the ELEs in Subcells A and B based on CLT are given by

WA Þ

WA =2

where 4

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2

3

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3

2

σi

εi

6 1i 7 � �6 X 7 6 σ 7 ¼ ½R3 � Qj 6 εi 7 ​ ​ i 4 Y 5 4 25 τi12 γ iXY where 2

p2 ½R3 � ¼ 4 q2 pq

q2 p2 pq

�

i¼A​ ​ ​j¼ i¼B​ ​ ​j¼

θ; 0; þθ θ; þθ

3 2pq 2pq 5 p2 q2

Once the failure criterion is satisfied (see Eq. (17)), the calculation process will continue to Step 4; 4) Failure mode classification and stiffness degradation: The failure mode is determined by identifying the maximum value of stress to strength ratios and the stress status. Calculation of the stiffness degradation is then conducted, based on the observed mode of fail­ ure [25]. 5) Examination of the critical failure: The iteration stops with the occurrence of critical failure; for example, fiber failure of the 0� ELE in Subcell A under axial loading conditions and matrix-type failure for all �θ� ELEs under transverse loading conditions. The critical failure will cause a significant reduction in resistance for the corre­ sponding loading conditions and will result in complete failure of the material at the current loading level, which will be recorded as the corresponding strength. If no critical failure occurs, the current iteration continues to Step 6; 6) Examination of new damage: Following the calculation of the stiff­ ness degradation, the local stress tensor of the ELEs will be updated, and a new round of failure examination is needed. If there are new failure events happened immediately after the previous damage, the calculation will return to Step 4 for subsequent calculation of the stiffness degradation and stress analysis. Otherwise, a new iteration will commence.

(15)

(16)

2.4.2. Failure examination and stiffness degradation After calculating the local stresses in each ELE, a failure criterion can be introduced to determine the initiation and progression of damage to the composite material during the loading process. As an example, the following Hoffman failure criterion [23] is used: � σ2 f σij ¼ 1 Xt Xc

σ1 σ2 Xt Xc

þ

σ22 Yt Yc

þ

Xc Xt Y Yt τ2 σ1 þ c σ 2 þ 122 Xt Xc Yt Yc S

(17)

An iterative computational program was developed to study the failure initiation and to update the stiffness during the loading process. If fðσ ij Þ < 1, no failure occurs and there is no degradation in stiffness; in this case, the load is increased in a subsequent iteration. If the failure criterion is satisfied, the damage mode is determined by finding the maximum value among the stress to strength ratios ðσ 21 =Xt Xc ; ​ σ22 =Yt Yc ; ​ ​ τ212 =S212 Þ as suggested by Zako et al. [24], and the stiffness of the corresponding ELE will be degraded according to the failure mode. For example, if σ21 =Xt Xc is the maximum, then fiber tensile failure (σ1 > 0) or compressive failure (σ1 < 0) occurs. The elastic constants E11, E22 and G12 are set to be zero and the damaged ELE can carry no further in-plane loads [25]. If σ 22 =Yt Yc or τ212 =S212 is the maximum, then matrix tensile/compressive failure (tensile: σ2 > 0, compressive: σ2 < 0) or matrix shear failure occurs, which means the matrix broken and the fiber can still bear loads. Then stiffness E22 and G12 are set to be zero while the E11 remains unchanged [25]. The stiff­ ness degradation will change the global stiffness matrix of the subcell elements and the distribution of stress in the unit cell. Once damage occurs, the local stress tensor of the ELEs will be updated, and the updated stresses will be used to conduct another examination of the failure status of the degraded material under current external loading conditions. New failure events may occur immediately after previous damage has been initiated, which would result in further stiffness degradation and stress redistribution. If no subsequent failure occurs, then the external load will be increased, and another iteration of calculation will be performed.

3. Finite element model of the 2DTBC A finite element simulation is introduced to validate the proposed analytical model. As a demonstration, a 0� /�60� triaxially braided composite is selected. This composite is made from T700s fiber and E862 epoxy matrix with braided angles of 0� /�60� and is processed through resin transfer molding. The fabrics are fabricated with 24 K (0� ) fiber tows in the axial direction and 12 K (�60� ) fiber tows in the bias di­ rection. The T700s carbon fiber has an axial modulus of 230 GPa [15] and a transverse modulus of 15 GPa [15], while the tensile and compressive strength of the fiber filament are 4900 MPa [15] and 2400 MPa [27], respectively. The E862 epoxy resin is isotropic with Young’s modulus of 2.7 GPa and shear modulus of 1 GPa, while the tensile, compressive and shear strength are 61 MPa, 92 MPa and 45 MPa [1], respectively. The periodicity of the 2DTBC allows selection of different unit cells. Therefore, in order to facilitate the application of loading and boundary conditions, the unit cell model for finite element modeling is translated horizontally by a half width of subcell A, as shown in Appendix C. The corresponding meso-scale FE model constructed by Zhang et al. [10,11] and Zhao et al. [12] has been well validated against experimental results in previous studies, and displays excellent capacity in predicting the stress–strain response and the progressive failure behavior of a 2DTBC. Thus, in this work, the previously well-correlated meso-scale finite element model is employed to validate the analytical model by imposing periodical boundary conditions for both in-plane directions. The infinite plate FE model is established using Texgen software (as shown in Fig. 1(e)) and discretized by solid elements (C3D8R). Both sixlayer and single-layer specimens are studied to validate the analytical model. Each model contains a single unit cell in-plane with periodic boundary conditions along the axial and transverse directions, as shown in Fig. C-1 of Appendix C. The translational symmetrical boundary conditions of axial tension and transverse tension are introduced in a previous work [11], while the periodic boundary conditions are applied perpendicular to the loading directions for axial and transverse compression.

2.4.3. Iteration procedure The proposed analytical model is programmed in MATLAB R2014a [26]. To facilitate a better understanding of the calculation process, the detailed iteration procedure for strength prediction is summarized below: 1) Calculation of the effective stiffness matrices: The effective stiffness matrices of subcell elements and the unit cell are determined based on CLT and the series-parallel model (see Eqs. (12) and (13) as well as Eqs. (B1) – (B4) in Appendix B); 2) Determination of the local stress–strain relationship: The stress–­ strain tensors of each ELE, the subcell elements and the unit cell can be individually determined through a bottom-up procedure for axial/transverse loading conditions (see Eqs. (14) – (16) and Eq. (B5) in Appendix B). 3) Failure examination: For each time step, the stress in each ELE will be substituted into the failure criterion for failure examination. If no failure occurs, the program returns to Step 2 and another iteration begins with an incremental load of σ X ¼ σX � ΔσX or σY ¼ σ Y � Δσ .

4. Results and discussion In this section, the proposed model is validated by comparing the predictions with the FEA and the experimental results. In the compari­ son, particular attention is paid to the equivalent elastic stiffness, the 5

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stress–strain curves, and the progressive damage behavior under different loading conditions.

Table 2 Comparison of analytical, numerical and experimental results on the effective stiffness of six-layer and single -layer 2DTBCs. Methods

FE simulation of coupon specimen Experiment (coupon specimen) FE prediction of infinite plate Analytical prediction (infinite plate)

Six-layer

4.1. Prediction of elastic stiffness

Single-layer

EXX (GPa)

EYY (GPa)

EXX (GPa)

EYY (GPa)

49.6

41.8

42.8

38.9

46.9 50.5 49.9

41.6 49.5 48.6

40.6 43.8 42.7

38.5 43.1 41.6

As mentioned earlier, for the sake of consistency, the analytical model needs to be validated against the FE predictions for an infinite plate. Table 2 presents the experimentally measured as well as the numerically and analytically predicted axial and transverse moduli for the six-layer and single-layer specimens. As can be noticed from Table 2, the FE model results for the coupon specimen correlate well with the experimental results, showing the accuracy of the FE model. The FE prediction for the axial modulus is slightly higher than the experimental results [1,2], which is attributed to the presence of axial fiber bundle undulation in the real specimen, while the analytical results for the single-layer and six-layer specimens compare well with the numerical results for the infinite plate. It is also found that the transverse modulus of the infinite plate is

Fig. 3. Comparison of experimental, numerical and analytical results for a six-layer T700/E862 triaxially braided composite under different loading conditions: (a) axial tension, (b) transverse tension, (c) axial compression, and (d) transverse compression.

Fig. 4. Comparison of experimental, numerical and analytical results for a single-layer T700/E862 triaxially braided composite under different loading conditions: (a) axial tension, and (b) transverse tension. 6

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H. Dang et al.

and numerical results, as shown in Figs. 3 and 4. The strength and failure strain properties were extracted from the experimental and computa­ tional curves and compared in Table 3. For the case of axial tension (AT), as shown in Fig. 3(a) and Fig. 4(a), the predicted and experimental stress–strain curves all correlate well with each other in terms of initial stiffness and linearity of the curves. It is found that the predicted strength of the six-layer coupon specimen is higher than that for the single-layer specimen, which is reasonable considering the higher fiber volume fraction of the six-layer specimen. For the case of transverse tension (TT), the stress–strain curves of the analytical model and the FE simulation of the infinite plate model are in close agreement but predict a higher stiffness than the experimental and numerical results obtained for the coupon specimen, as shown in Fig. 3 (b) and Fig 4(b). This observation is attributed to the presence of sig­ nificant free-edge effects during transverse loading of the coupon spec­ imen, which induces out-of-plane warping and results in size-dependent effective properties. As discussed in previous studies [10,28], the elim­ ination of free edges will generate obviously larger stiffness and strength in the specimen, corresponding to the analytical prediction and the simulation of infinite plate FE model. From Figs. 3 and 4, it can be noticed that the numerical predicted stress–strain curves compare well with the experimental results of the coupon specimens for both tension and compression, suggesting the accuracy of the numerical model. For the presented analytical model, the predicted stress-strain curves match well with the numerical simu­ lation results of an infinite plate for different loading cases. The errors � � � � ​ of ​ infinite ​ plale Analytical ​ predicion� (��FE ​ prediction �) for all the predictions are less FE ​ prediction ​ of ​ infinite ​ plale

Table 3 Comparison of experimental and computational strength and failure strain properties for 2DTBCs under various loading conditions. Methods Strength (MPa)

Failure Strain (%)

Six-layer FE simulation of coupon specimen Experiment (coupon specimen) FE prediction of infinite plate Analytical prediction (infinite plate) FE simulation of coupon specimen Experiment (coupon specimen) FE prediction of infinite plate Analytical prediction (infinite plate)

Single-layer

AT

TT

AC

TC

AT

TT

909

487

327

318

797

437

800

462

327

304

800

452

906

536

418

424

789

492

933

584

430

410

798

536

2.05

1.37

1.22

1.04

1.95

1.19

1.78

1.44

1.01

0.87

1.98

1.20

1.96

1.21

0.93

1.01

1.94

1.29

2.10

1.31

0.92

0.88

2.10

1.39

noticeably higher than that of the 2DTBC coupon specimens. This is due to the presence of free-edge effect in the coupon specimens, which re­ sults in a reduction in the measured effective stiffness [11]. Overall, the proposed analytical model is able to accurately predict the elastic properties of the 2DTBC.

than 9%, where the maximum error is 8.94% for the transverse-tensile strength of the six-layer specimen. For axial compression (AC) and transverse compression (TC) (see Figs. 3(c) and (d), respectively), the stress–strain curves of coupon specimens show significant nonlinearity, due to the presence of pro­ gressive interface failure during the loading process as reported in other studies [1,12]. However, because of the plane-stress assumption (σ3 ¼

4.2. Prediction of strength Following the iteration procedures, the mechanical response of sixlayer and single-layer 2DTBC specimens under tensile and compres­ sive loading conditions were predicted and compared with experimental

Fig. 5. Comparison of progressive failure prediction using analytical model and FE model for six-layer infinitely large 2DTBC panel under different loading con­ ditions: (a) axial tension, (b) transverse tension, (c) axial compression, and (d) transverse compression. The shaded rectangles correspond to failed ELEs. 7

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Composites Science and Technology 186 (2020) 107936

Fig. 6. Comparison of progressive failure prediction using the analytical model and an FE model for a single-layer infinitely large 2DTBC panel under different loading conditions: (a) axial tension, (b) transverse tension. The shaded rectangles correspond to failed ELEs.

τ13 ¼ τ23 ¼ 0) for the CLT theory, the proposed analytical model has

Compared with the FE results, the matrix tensile damage starts at the intersection of the fiber bundles and spreads into the �60� fiber bundles, eventually causing the 0� fiber tensile fracture failure. For the transverse tension loadings, (as shown in Figs. 5(b) and 6(b)), unlike the case of axial tension, the matrix tensile failure first occurs in the 0� ELE of Subcell A, after which matrix tensile failure takes place in the �60� ELEs of Subcell A, propagates into the �60� ELEs of Subcell B and results in the failure of the unit cell. As from the FE prediction, the matrix damage initiates at locations where bias fiber bundles intersect, due to the local shear strain concentration. The composite eventually breaks due to fiber tensile failure caused by local shear stress concen­ tration at the intersection of the bias fiber bundles. For axial compression loading (see Fig. 5(c)), the analytical results show that the matrix compressive failure initiates only in the �60� ELEs of Subcell A. However, the FE simulation results suggest that matrix compression failure first occurs in the intersection zone of Subcell B and propagates to the �60� ELEs of Subcell A. Ultimately, for both the analytical and numerical prediction results, the composite specimen fails due to fiber compressive failure in the 0� ELE of Subcell A. For transverse compression (see Fig. 5(d)), the analytical results show that the matrix compressive failure starts in the �60� ELEs of Subcell A and propagates to the �60� ELEs of Subcell B. As for the FE prediction, the matrix failure continues to propagate and accumulates in the region where bias fiber bundles cross, and the composite fails due to fiber compressive failure at the intersection zone. Compared to the damage initiated under axial compression, the damage under transverse compression starts earlier, which results in more significant nonlinearity of the stress/strain response. Overall, the proposed analytical model provides a good prediction for the failure strength, strain and progression of the 2DTBC, especially for predictions of the tensile strength. The predictions for compressive

limitations in predicting delamination failure, which results in the over-prediction of analytical results. In addition, the plastic deformation of the matrix is not considered in the analytical calculation for purposes of simplification due to its incompatibility with the CLT and series-parallel method, which could be another contributing factor to the discrepancy between the analytical results and the FEA predictions. 4.3. Prediction of the progressive failure process To further demonstrate the capability of the proposed analytical model, the predicted progressive failure processes of 2DTBC under various loading conditions are summarized and compared with those of the numerical predictions for an infinite plate, as shown in Figs. 5 and 6. Both the failure mode (MT represents matrix tensile failure, MC repre­ sents matrix compressive failure, FT represents fiber tensile failure, and FC represents fiber compressive failure) and the corresponding global strain are recorded to facilitate a comparison. The shaded rectangles correspond to failed ELEs. For both axial and transverse tension loading conditions, the pro­ posed analytical model provides a good prediction of the failure modes, failure progression and corresponding critical strain levels for six-layer and single-layer specimens, in comparison with the numerical predic­ tion of an infinite plate. At the same time, it is found that the main failure modes are not affected by the difference in volume fraction, as shown in Figs. 5 and 6. For the axial tension loadings (as shown in Figs. 5(a) and 6(a)), with the increase of external load, the analytical model predicts that the matrix tensile failure first occurs in the �60� ELEs of Subcell A, followed by matrix tensile failure of the �60� ELEs in Subcell B; finally, the 0� ELE fails by fiber tension corresponding to the fracture of the unit cell.

Fig. 7. Effect of normalized axial compressive strength of ELEs on (a) axial tension and transverse tension strength, and (b) axial compression and transverse compression strength. 8

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Composites Science and Technology 186 (2020) 107936

response are not as good as those for tensile response, due to the inability of the current model in predicting delamination failure.

ultimate failure mode in the transverse compression test. It is noticed that the axial compressive strength also affects the matrix compressive failure, due to the quadratic and interactive nature of the failure criteria. Through the parametric study results shown in Fig. 7, the normalized axial compressive strength of the presented analytical model is deter­ mined to be 0.8 for the studied material; the exact values are listed in Appendix C.

4.4. Parametric study of the compressive strength of ELEs Undulation in the axial fiber bundle is an inevitable defect that originates from the manufacturing process. As discussed earlier, this phenomenon may result in lower longitudinal modulus and strength than the theoretical expectations for textile composites. Kohlman [2] proposed that fiber tow undulation creates a pre-buckled tow geometry, which will redistribute the axial compressive loads into through-thickness loads and will result in delamination rather than classic micro-buckling. Moreover, the direct characterization of compressive properties of fiber tows remains a challenge because of their irregular shape and small size. In a previous work, Quek et al. [4] concluded that matrix cracking and buckling are the main failure modes in compression based on experimental results. Li et al. [29] determined the axial compressive strength of fiber tows as the averaged value of three possible failure modes (fiber rupture, buckling or kinking, and delamination) considered in the Chamis model. In the proposed analytical model in this work, the compressive strength is obtained by correlating the analytical results against FE re­ sults through a parametric study, as presented in Fig. 7. This figure shows the evolution of predicted tensile and compressive strength values against the normalized axial compressive strength of ELEs for a six-layer 2DTBC specimen. The averaged value obtained from the Chamis model (same as Li et al. [29]) is taken as the reference value for the normali­ zation and the compressive strength of all ELEs are reduced in the same ratio. As can be noticed from Fig. 7(a), the compressive strength has a negligible influence on the prediction of the tensile strengths. The transverse tensile strength reduces slightly with an increase in the compressive strength, due to the mitigation of compressive damage to shear damage. As can be determined from Eq. (17), shear damage will occur earlier with the increase of compressive strength. The axial compressive and transverse compressive strengths increase noticeably with the increase in compressive strength, due to the decaying of both matrix and fiber compressive failure. This is consistent with the failure mechanism illustrated in Fig. 5(c) and (d), where fiber compression failure in the 0� ELE is the ultimate failure mode in the axial compression test and matrix-compression failure in the bias ELEs is the

5. Conclusions In this paper, a novel analytical model was developed to predict inplane progressive failure behavior and strength of a 2DTBC through a discretization and reassembly process based on an integrated lamination and series-parallel model. A meso-scale finite element model for an infinitely large plate was introduced to validate the accuracy of the proposed analytical model. The proposed model shows good accuracy in predicting the effective stiffness, damage progression, and failure strength of the 2D triaxially braided composite under both tensile and compressive loading conditions. The simulation results indicate that shear stress plays an important role in the damage development, a finding consistent with the results of previous FE modeling [10–12]. It is also found that the determination of compressive strength for the ELEs has an obvious effect on the strength prediction; thus, further study is needed to characterize the in-situ compressive strength of the fiber tows in textile composites. The pro­ posed analytical model provides useful insights on the stiffness and strength as well as failure progression in a 2DTBC, which will be useful in the design and optimization of large-scale structures such as aero-engine fan cases and automobile front rail assemblies. The pro­ posed model can also be applied to study the progressive failure and strength of other textile composites, based on the concept of series-parallel integration of laminates. Acknowledgment This work is supported by National Natural Science Foundation of China (NSFC) under grant number 11772267, Fundamental Research Funds for the Central Universities under grant number 3102018jcc005, and the Shaanxi Key Research and Development Program for Interna­ tional Cooperation and Exchanges under grant number 2019KW-020.

Appendix A The analytical model for determining the fiber volume fractions of different ELEs is presented here. The overall fiber volume fraction of the unit cell can be measured and can also be calculated as Vf ¼ VF =ðWLhÞ, in which the total volume of fiber filaments in half unit cell can be calculated as VF ¼ V 0FA þ V θFA þ V θFB ¼

πd2 4

Nfa L þ 2Nfb lbA þ 2Nfb lbB

�

(A-1)

where V 0FA , V θFA and V θFB are the volumes of fiber filaments in the 0� fiber bundles, the �θ� fiber bundles in Subcell A, and the �θ� fiber bundles in Subcell B, respectively; lbA and lbB denote the length of bias fiber bundles in Subcell A and Subcell B, respectively, which are estimated as lbA ¼ WA = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sinðθÞand lbB ¼ ½ðW WA Þ=sinðθÞ�2 þ ðh0A þ hθA Þ , respectively, using the bias fiber bundle shown in Fig. 1(f). Let V 0 and V θ denote the fiber volume fraction of the 0� and the θ� fiber bundle composites, respectively. Assuming that the axial fiber bundle has a

rectangular cross section, the height of the 0� ELE and the �θ� ELEs in Subcell A can be determined by h0A ¼ V 0FA =ðV0 WA LÞ and hθA ¼ ðh respectively. Furthermore, the height of the �θ ELEs in Subcell B can also be determined as different ELEs in the subcell can be determined using the following equations: �

VAþθ ¼ VA θ ¼ V 0A ¼

θ VFA π d2 Nfb ¼ 2LWA hθA 4LhθA sinðθÞ

hθB

h0A Þ= 2,

¼ h=2. Hence, the equivalent fiber volume fractions of

(A-2)

V 0FA πd2 Nfa ¼ 0 LWA hA 4WA h0A

(A-3)

9

H. Dang et al.

θ V þθ B ¼ VB ¼

Composites Science and Technology 186 (2020) 107936

V θFB πd2 Nfb � θ¼ � 2L W WA hB 4L W WA hθB

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � �2 �2 � W WA þ h0A þ hθA sinðθÞ

(A-4)

It is worth noting that the volume of each homogenized ELE must be not less than the volume of the corresponding fiber bundle composites. For the 0� fiber bundle composites in Subcell A, this condition can be readily satisfied by the dimensions and fiber volume fractions. For the bias fiber bundle composites in Subcell A, this condition can be expressed as 2WA LhθA � V θFA =V θ ; for the bias fiber bundle composites in Subcell B, this condition can be expressed as 2ðW

WA ÞLhθB � V θFB =V θ .

Appendix B The series-parallel model [21] for calculating the homogenized properties is presented here. According to the series-parallel assumption, the in-plane stress and strain response of the subcells and the unit cell are assumed to satisfy the following relationship: (B-1)

εAX ¼ εBX ¼ εX ; σAY ¼ σBY ¼ σY ; σ AXY ¼ σ BXY ¼ σ XY which leads to the following three equilibrium and compatibility conditions:

(B-2)

kA σ AX þ kB σ BX ¼ σX ; kA εAY þ kB εBY ¼ εY ; kA γAXY þ kB γ BXY ¼ γXY

where kA and kB denote the effective width ratios of Subcell A and Subcell B, respectively. Solving these equations yields the following formuli for the effective stiffness matrix (Cmn, where m,n ¼ 1,2,6) of the unit cell: 8 � � > > > C11 ¼ kA CA11 þ kB CB11 þ kA kB p1 CA12 CB12 CB21 CA21 > > > � > A B B A > C12 ¼ p1 kB C C þ kA C C > 22 12 22 12 < � (B-3) C21 ¼ p1 kB CA22 CB21 þ kA CB22 CA21 > > A B > > C ¼ p C C 22 1 22 22 > > > A B > > : C66 ¼ p2 C66 C66

where p1 ¼

1 1 ; p2 ¼ kA CB22 þ kB CA22 kA CB66 þ kB CA66

(B-4)

The strain components in Subcells A and B can be expressed as 8 > > > > εAX ¼ εBX ¼ εX > > � > > > εAY ¼ p1 CB22 εY þ kB p1 CB21 CA21 εX < � εBY ¼ p1 CA22 εY þ kA p1 CA21 CB21 εX > > > > γAXY ¼ p2 CB66 γXY > > > A B > > : γXY ¼ p2 C66 γXY

(B-5)

Appendix C Fig. C-1 presents a schematic diagram of an infinite plate model under different loading conditions for a six-layer and a single-layer 2DTBC. Tables C-1 and C-2 represent the elastic moduli and strength properties, respectively, of different ELEs of the six-layer and single-layer 2DTBC specimens.

10

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Composites Science and Technology 186 (2020) 107936

Fig. C-1. (a) Top view and side views of the meso-scale model for six-layer and single-layer 2DTBCs. Meso-scale model for an infinite plate under different loading conditions: (b) axial and transverse tension, (c) axial compression and (d) transverse compression. Table C-1 Elastic moduli of ELEs of six-layer and single-layer 2DTBC specimens Specimen

ELE

Exx (GPa)

Eyy (GPa)

Gxy (GPa)

ѵxy

Six-layer

A60� A0� B60� A60� A0� B60�

145.6 177.8 87.6 113.6 177.8 75.5

7.7 9.6 5.4 6.3 9.6 5.0

4.2 6.3 2.4 3.0 6.3 2.2

0.31 0.30 0.33 0.32 0.30 0.34

Single-layer

Table C-2 Strength of ELEs of six-layer and single-layer 2DTBC specimens Specimen

ELE

Xt (MPa)

Xc (MPa)

Yt (MPa)

Yc (MPa)

S (MPa)

Six-layer

A60� A0� B60� A60� A0� B60�

3082 3774 1829 2392 3774 1568

1215 1660 787 950 1660 722

53 56 49 50 56 49

80 84 74 76 84 73

38 40 35 36 40 34

Single-layer

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