In situ measurement of strains at different locations in 3-D braided composites with FBG sensors

Composite Structures 230 (2019) 111527

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In situ measurement of strains at different locations in 3-D braided composites with FBG sensors

T

Shuo Yan, Junjie Zhang, Baozhong Sun, Bohong Gu

⁎

College of Textiles, Donghua University, Shanghai 201620, China

ARTICLE INFO

ABSTRACT

Keywords: 3-D braided composite In situ strain Fiber Bragg Grating (FBG) sensor Microstructure Finite element analysis (FEA)

Due to the significance of the structural health monitoring in terms of the application 3-D braided composite materials, we utilized Fiber Bragg Grating (FBG) sensors embedded in various regions in 3-D braided composite to measure the in-situ strains under the uniaxial tensile test. We also developed a finite element analysis (FEA) model to calculate the strains and compared with those measured by the FBG sensors. We found that the internal strain varies in the regions of different distances to the sample end and of various unit-cells. In the region with a distance of 60 mm to the end (R-60), the strain of the interior unit-cell is greater than that both of the surface and corner unit-cell. While in the region with 100 mm to the end (R-100), the strains of all these three unit-cells are larger than those of the R-60, though the strains of these three unit-cells are even in this region. The different insitu strains within a 3-D braided composite sample under uniaxial tension infer to that an appropriate failure criterion should be used for strength design.

1. Introduction On account of the high specific stiffness, good structural integrity, and high energy absorption performance [1], three-dimensional (3-D) braided composites are widely used as engineering members. With a near-net-shaping feature [2,3], 3-D braided preform is capable of forming some irregular shapes of the profile for its applications in structural parts, such as tail shafts, I-shape beams and booms [3,4]. Hence, 3-D braided composites show a wide application in aviation, transport and construction [5–7]. During its working time, it is pivotal to monitor its response to the exterior field in real-time and in situ, e.g., load, temperature, magnetic, etc. Many researchers focus on the in-situ measurement of its behaviors in order to realize the structural health monitoring (SHM) [7,8]. Optical fiber sensors, especially the embedded Fiber Bragg Grating (FBG) sensors, have been attracted widely attention in SHM for advanced composites [8–11]. FBG sensors can accurately measure the in-situ strain and temperature of 3-D braided composites, and can realize online monitoring [8,12,13]. They provide basic data for the early warning system. Once the in-situ strain or temperature within the structure reaches the critical value, the warning system is issued. What’ s more, they can be used for structural strength design. The 3-D braided composites differ from the laminates due to its braided structures of preform. As a result of this structure, the strain responses vary in different regions, viz. interior unit-cell, surface unit-

⁎

cell, and corner unit-cell. It is important to study this structural effect on its strain response and reveal the mechanism between the structure and the deformation response. There have been many reports on the macroscopic tensile behaviors of the 3-D braided composites. The tensile failure of the 3-D braided composite is a progressive process [14], including matrix cracking, interface debonding, fiber drawing out and fracturing [15]. Bogdanovich et al. [16,17] investigated the tensile properties of the 3-D braided composites, and they discovered that tensile strain-to-failure was insensitive to the epoxy resin. Zheng et al. [18] investigated the effects of the 3-D braided composites on the tensile response, and the results illustrated that the tensile properties of the composites with carbon fibers as axial yarns were better than with aramid fibers. At present, the macroscopic tensile performances of the 3-D braided composites have been researched widely. The macroscopic tensile behaviors and failure mechanism have been revealed. However, the local and internal properties of the 3-D braided composites are rarely investigated. The embedded FBG sensors can be applied for this field. Ramakrishman et al. [7,19] embedded the FBG sensors in laminates by both the hand-up and the prepreg layup method, which means to place the FBG sensors directly in the desired positions when laying the layers. However, this method is just suitable for laminate composites and not for 3-D braided composites. Haentzsche et al. [20] placed the FBG sensor along the weft direction in the weaving process, but this

Corresponding author. E-mail address: [email protected] (B. Gu).

https://doi.org/10.1016/j.compstruct.2019.111527 Received 30 April 2019; Received in revised form 26 September 2019; Accepted 4 October 2019 Available online 05 October 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

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method is suitable for 3-D angle-interlocked woven composite, not for 3-D braided composite. As for 3-D braided composites, Li et al. [21] and Yuan et al. [22] proposed the co-braid method for embedding the FBG sensor, i.e., one is that the optical fiber is braided along the axial direction, and another is that the optical fiber is braided along the braiding yarn. In the actual braiding process, the brittle optical fiber (FBG) is easy to break if the optical fiber (FBG) is directly braided with yarns. In this paper, we propose a hollow capillary tube assist method to protect the optical fiber. Currently, many scholars utilized FBG sensors to investigated the process of composite materials’ curing [7,8,23,24]. Zhang et al. [25] investigated the strain distribution of the carbon fiber composites during the curing process with the FBG sensors, and discovered that there was uneven residual stress in the cured composites. Kang et al. [26], Ding et al. [27] and Rao et al. [28] utilized FBG sensors to measure the strains and temperature simultaneously of the composites, and obtained the history of the thermal strains as well as the corresponding temperatures. Jung et al. [29] investigated the expansion and contraction behaviors during the curing process, and measured the internal strains under compressive and bending loads using the embedded FBG sensors. Kim et al. [30] utilized FBG sensors to investigated the strain evolution and curing reaction, which demonstrated that FBG sensors can meet the requirements for composites. There are many applications of the FBG sensors for monitoring the curing process and measuring the thermal strain of the 3-D braided composites. Although these studies investigated the thermal strain and temperature of 3-D braided composites, they rarely involved the mechanical strains. Meng et al. [31] investigated the effect of the embedded FBG sensors on the structural strength, and the results illustrated that the strength dropped little. Yuan et al. [22] and Li et al. [21] investigated the internal strains under tensile load with the co-braid FBG sensors experimentally and numerically, and the results illustrated that the strain value increased linearly with increasing tensile loading. Matveenko et al. [32] investigated the strains measured with the FBG sensors under tension, and proposed a interrelation model between Bragg wavelength peak shift and the strain of the optical fiber. Song et al. [33] placed the FBG sensors between the first layer and second layer of the laminates, and obtained the strain distribution under various loads. Kocaman et al. [34] used a type of embedded FBG sensor system for SHM within the sandwich composites, and investigated its failure mode. Wang et al. [35] stuck the FBG sensors on the surface of the composites to measure the strains, and proposed a strain transfer model. Hu et al. [36] and Jang et al. [37] utilized the FBG sensors to

monitor the low-speed impact response, and derived the relationship between impact energy and wavelength shift. These researches are concentrated on the strain measurement of laminates and the surface of 3-D braided composites. Although the theoretic methods of measuring the tensile internal strain of the 3-D braided composites have been proposed [7,21,22], there are not many reports on the tensile internal strains actually. From studies listed above, most researchers focus mainly on surface and overall strain responses while they rarely concentrate on the internal local strain under uniaxial tension. None of these focuses on the structural effect on the local strain responses of 3-D braided composites. Fig. 1 is the sketch of the current investigation. In this investigation, we embedded the FBG sensors inside the 3-D braided specimens axially and transversely to measure its internal local strain responses in-situ in the linear elastic region under the uniaxial tensile loading. Besides that, two kinds of finite element analysis (FEA) models have been established to investigate the internal local stress-strain response. The results reveal the structural effect on the internal local strains. 2. The fundamentals of FBG sensor 2.1. The fundamentals The FBG sensor with a small size and high precision is considered as the ideal sensing element. The Bragg grating region of the FBG sensor is normally written by UV beam. Only the light of the particular wavelength is reflected, and others transmit when it travels inside the FBG sensor. The period and refractive index of the FBG sensor change with the strain, temperature and pressure, thereby, the reflectance and transmission spectra will change. Fig. 2 shows the principle of the FBG sensor [38,39]. It can measure static and dynamic physical quantities at a special position, such as strain, temperature and pressure. The Bragg condition is given by [29], B

= 2neff

(1)

where, B is the Bragg wavelength of the FBG sensor. neff is the effective refractive index of the fiber core, and is the Bragg period of the FBG sensor. Once the effective refractive index neff or the Bragg period changes under external loading, the Bragg wavelength B will drift. The wavelength drift caused by the changes of mechanical strain and temperature can be expressed as

Fig. 1. Investigation sketch.

2

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Fig. 2. Principle of FBG sensor.

B

= 2neff

2 neff

1

2

( )

ε is the strain along the FBG sensor. 1, 2 and 3 are the principle strains of the structure. 1, 2 and 3 are the angles between the FBG sensor and the structure. Suppose that the deviation angles are 1, 2 and 3 . The strain in the FBG sensor can be expressed as

dn eff

·[p12

(p11 + p12 )] · +

+

dT

neff

· T

(2) where, B is the drift variation in Bragg wavelength. p11, p12 are the components of the photo-elastic coefficient of the optical fiber. ν is the Poisson’ s ratio of the optical fiber. ε is the axial strain of the optical fiber. α is the thermal expansion coefficients of the optical fiber. T is the temperature and ΔT is the change of the temperature. Eq. (2) can be expressed into the same form: B

= (1

pe )· + ( + )· T = K e· + ( + )· T

B

= 1·cos2 (

n2 ·[p12 2

(3)

·(p11 + p12 )]

=

3 ]·

cos 1 0 0 0 2 0 · cos cos 0 0 3

2

+

cos2

+ cos2

3

cos

2 1cos 1

+

2

2

cos

cos2

3

3

1

+ cos2

2

2

+

2)

+ 3·cos2 (

3

+

3)

(5)

3. Materials and tests 3.1. Material and samples Table 1 lists the properties of the carbon fiber tows (T700-12K), epoxy resin (JC-02A), hardener (JC-02B) and FBG sensors [21,40]. The 3-D carbon fiber/epoxy braided rectangular preforms were manufactured by 1 × 1 four-step method with yarn array of 13 × 5. Fig. 4 shows the scheme of the initial state of the yarns. The braided yarns in the preform can be divided into the corner, surface and interior yarns according to the yarn paths. The yarns of the different paths have different spatial structures. Therefore, the FBG sensor was programmed along or vertical to the axial direction of the preform at a predetermined position in the braiding process.

1 2 3

(4)

where

cos2

+ 2·cos2 (

To verify the strain sensitivity of the FBG sensors, the FBG sensors used were calibrated prior to the test. The tensile calibration tests [42] were conducted by the universal testing machine, as shown in Fig. 3(a). (= i The calibrated results are showed in Fig. 3(b), where 0 ) is the Bragg wavelength difference and µ (=(µ ) i (µ )0) is the strain difference. As shown in Fig. 3(b), the strain sensitivity coefficient of the FBG sensors is about 1.20 pm/με, and it approximates to the theoretical value. In this study, the silica FBG sensors with center wavelength 1550 nm and Bragg grating region length l = 10 mm were applied. B The temperature component was ignored as the tests were carried out at the Standard Reference Atmospheric Condition (T = 20 °C, P = 1 atm) [43]. The direction of the FBG sensor was along the direction desired.

pe is the photo-elastic coefficient of the optical fiber. ξ is the thermooptic coefficient of the optical fiber and K e is the strain sensitivity factor of the FBG sensor. According to Eq. (3), the two parts are independent of each other. When the temperature change is small enough or negligible, the wavelength shifts with the axial strain. The FBG sensor can be considered as a strain sensor. If the wavelength drift is only caused by the temperature change, the FBG sensor can be considered as a temperature sensor [40]. If the strain and temperature loading cannot be ignored, the coupling of the two must be considered. For a silica FBG sensor with a center wavelength B 1550 nm , the strain and temperature sensitivity are approximately 1.2 pm/με and 13.6 pm/°C [41], respectively. According to [40], 1

1)

2.2. Sensor calibration

= (dneff / dT )/neff

= [ cos

+

According to Eq. (5), in order to measure the principle strain in a certain direction of the composite, the simplest method is to place the FBG sensor along the direction to be measured.

where

pe =

1

=1

3

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Fig. 3. Sensor calibration (a) the Tester, (b) Calibration curves. Table 1 Mechanical parameters of fiber tows and resin. Materials

E11 (GPa)

E22 = E33 (GPa)

G12 = G13 (GPa)

G23 (GPa)

ν12 = ν13

ν23

Density (g/cm3)

Carbon fiber tows (T700-12 K) Resin (JC-02A/B) Optical fiber

230 2.4 70

15 2.4 /

9 0.89 29.9

5 0.89 /

0.25 0.33 0.17

0.3 0.33 /

1.79 1.13 2.20

Fig. 4. Initial state of the braiding yarns.

The biggest problem is that the optical fiber (FBG sensor) is easy to break. In this study, the optical fiber (FBG sensor) was embedded by a hollow capillary tube assisted method. Fig. 5 shows the schemes of the embedded method of the axial FBG sensors. Fig. 6 shows the scheme of the relative positions between the FBG sensors and the different type yarns. Fig. 7 shows the scheme of the relative positions of the axial FBG sensors on the cross section of the preform. Except for the placed directions of the FBG sensors, the specific locations of the Bragg grating region should be considered. The size of the tensile specimens is 200 mm (length) × 15 mm (width) × 5 mm (thickness), referring to the ASTM D3039/D3039M-17. The gage length of the specimen is 80 mm, and the length positions of the Bragg grating region are at 60 mm (R-60 mm) and 100 mm (R-100 mm) as shown in Fig. 8. The resin was impregnated into the preform by vacuum assisted resin transfer molding (VARTM) method. The weight ratio of the epoxy and hardener is 10:8. Fig. 9 shows the VARTM method. Table 2 lists the basic parameters of the 3-D braided composites. In this study, there are 6 types of the specimens, FBG_C_R-60, FBG_C_R-100, FBG_S_R-60, FBG_S_R-100, FBG_I_R-60 and FBG_I_R-100 (where C, S and I mean corner, surface and interior unit-cell, respectively). R-60 means the region with a distance of 60 mm to the end and R-100 means the region with a distance of 100 mm to the end. Fig. 10 shows the scheme of the FBG_C_R-100 and Fig. 11 shows the photograph of the FBG_C_R-100.

Fig. 5. Scheme of the hollow capillary tube assisted method (a) 4-step braiding (b) samples with FBG sensors.

The axial and transverse internal local strains were measured in the interior unit-cell simultaneously. As for the corner and surface unit-cell, only axial internal local strain was measured.

4

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Fig. 8. Scheme of the length positions the Bragg grating sensor embedded.

Fig. 9. The impregnation process by means of VARTM. (a) Molding (b) samples with FBG sensors.

Fig.6. Relative positions between the FBG sensors and yarns. (a) The axial and transverse FBG sensors in the interior unit-cell, (b) The axial FBG sensors in the surface unit-cell, and (c) The axial FBG sensor in the corner unit-cell.

sm125 unit (as shown in Fig. 12(b)) was used to demodulate optical signals and modulate the grating wavelength of the FBG sensors. Fig. 13 shows the schematic diagram of the test process. 4. Numerical analysis 4.1. Fully-homogenized model The internal local strains measured with the embedded FBG sensors in the tests are for the 3-D braided composites, neither for the yarn nor for the resin. To verify the results of the FBG sensors measured and reveal the structural effect on its internal strain distribution, a fullyhomogenized model is presented, as shown in Fig. 14. The 3-D braided composite is considered as a transversely isotropic material in this model.

Fig. 7. Scheme of the relative positions of the axial FBG sensors on the cross section of the preform.

4.1.1. The properties of the fully-homogenized model In order to obtain the properties of the homogeneous model, the properties of the yarns and unit-cells must be calculated firstly.

3.2. Tests Quasi-static uniaxial tensile tests were conducted in constant temperature conditions with the 810 Material Test System (MTS) (as shown in Fig. 12(a)) at a speed of 2 mm/min. An optical sensing interrogator

4.1.1.1. The mechanical performance of the yarn. The cross section of the yarn is considered as a regular hexagon, because the carbon fiber tows 5

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Table 2 Micro-scale parameters of the 3-D braided composites. Fiber volume fraction Vf

Braiding angle, α

Corner braiding angle, β

Surface braiding angle, θ

Interior braiding angle, γ

Braiding patch length, h

Width of unit-cell, Wx = Wy

40%

40.8°

17.7°

17.7°

50.6°

7.5 mm

2.3 mm

Fig. 10. Scheme of the embedded FBG sensor under corner yarns.

Fig. 13. Schematic diagram of the test process. Fig. 11. Photograph of the sample.

Fig. 14. Boundary conditions of the fully-homogenized model.

Fig. 15. Cross section of the yarn model.

[S f ] =

Fig. 12. Photographs of the equipment. (a) MTS material tester (b) The sm125 unit.

f S11 S12f S12f

0

0

0

S12f S22f S23f

0

0

0

S12f S23f S22f

0

0

0

0

0

0

0

0

S44f

0

0

0

0

S55f

0

0

0

0

0

0

S55f

f 12 f , E11

S22f =

where,

squeeze each other in the braiding process. The yarn consists of carbon fiber tows and resin as shown in Fig. 15. The fiber tows are considered transversely isotropic and the resin is isotropic. The compliance matrix of the carbon fiber tows and resin can be obtained according to Eqs. (6) and (7), respectively.

S11f =

1

f , E11

S12f =

f , E22

1

(6)

S23f =

f 23 f , E22

S44f =

1 f G 23

, S55f =

1 f G12

[S f ] is the compliance matrix of the fiber tows. E11f , E22f are the elastic modulus of the longitudinal and transverse direction, respectively. 12f , f 23 are the Poisson’ s ratios of the longitudinal and transverse direction, respectively. G12f , G23f are the shear modulus of the fiber tows. 6

Composite Structures 230 (2019) 111527

S. Yan, et al. m m m S11 S12 S12 0 0 0 m m m S12 S11 S12 0 0 0 m m m S12 S12 S11 0 0 0 m [S ] = m 0 0 0 S44 0 0 m 0 0 0 0 S44 0 m 0 0 0 0 0 S44

(7)

where, m 1 1 m m m S11 = Em , S12 = Gm = Em , S44 [S m] is the compliance matrix of the resin. E m and Gm are the elastic and shear modulus of the resin, respectively. m is the Poisson’ s ratio of the resin. The yarn is considered as a unidirectional composite. In this study, a bridging model [44] is employed. In this model, the constitutive relationships of the fiber tows and resin in the unidirectional composites are as shown in Appendix, in an incremental form Eqs. (A.1)–(A.4) [44]). The overall compliance matrix of the yarn can be calculated via Eq. (8) [44].

[S y ] = (Vf [S f ] + Vm [S m][Aij ])(Vf [I ] + Vm [Aij ])

1

(8)

Fig. 17. Relationship between the local and global coordinate system.

where is the compliance matrix of yarn. [I ] is a unit matrix. In the bridging model, [Aij ] is the bridging matrix as shown in Eq. (A.5) in Appendix. According to Eq. (9), the stiffness matrix of the yarn can be obtained.

[S y ]

[C y ] = [S y]

1

two [48] is showed in Fig. 17. According to Eq. (10) [44], in the global coordinate system, the compliance matrix of the yarn in the interior unit cell can be obtained.

[SUI ]g = [Tij ]s [SUI ]I [Tij ]Ts

(9)

where [SUI ]g

(10)

are the compliance matrix in the global and local coordinate systems, respectively. [Tij ]s [48] is the transformation matrix, which is shown in Appendix. The angles of the transformation for the three type unit cells are listed in Table 3. The compliance and stiffness matrix of the interior unit cell can be obtained according to Eqs. (11) and (12) [48].

4.1.1.2. Mechanical behaviors of unit-cells. According to Fig. 6, there are three type unit-cells, which include interior, surface and corner unitcell. According to Eqs. (A.6)–(A.8) [45,46] in Appendix, the volume fractions of the three-type unit-cells can be calculated. In this study, [m × n] is [13 × 5], so PI, PS and PC are 64.86%, 32.43% and 2.70%, respectively. Hence the surface and corner unit-cell cannot be neglected. Fig. 16 shows the three unit-cell model. The different orientation yarns are considered as different unidirectional composites in the unit-cells. The yarn volume fractions in each unit-cell can be calculated by Eqs. (A.10)–(A.12) [47] in Appendix. The following only illustrates the mechanical properties of the interior unit-cell. As shown in Fig. 16(b), there are four orientations of the yarns. They have the same volume and yarn volume fraction (VI). According to the bridging model above, in the local coordinate system, the compliance matrix [SUI ] of the unidirectional composite in the interior unit-cell can be calculated. However, the mechanical properties of the 3-D braided composites are based on the global coordinate system. Therefore, the transformation of the two coordinate systems is required. The relationship of the

and [SUI ]I

[S ]cell =

1 V

[C ]cell =

[S ]cell1

Vi

[S i] dV +

Vi + 1

[S i + 1] dV +…+

Vk

[Sk ] dV +

Vm

[S m] dV (11) (12)

where [S ]cell and [C ]cell are the compliance and stiffness matrix of the unit-cell, respectively. V is the volume of the unit-cell. [Si] is the compliance matrix of the unidirectional composite in the unit-cell. Vi is the volume of the unidirectional composite. k is the number of the unidirectional composites. [Sm] is the compliance matrix of the resin. Vm is the volume of the resin in the unit-cell and the suffix m indicates resin. Similarly, the compliance matrix of the other two type unit-cells can be calculated.

Fig. 16. Unit-cell model. (a) The model of the preform, (b) Interior unit-cell, (c) Surface unit-cell and (d) Corner unit-cell. 7

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homogeneous structure are the same with the fully-homogenized model. Therefore, the properties of the full-size semi-micro model are easily obtained.

Table 3 Transformation angles. Type

x

Interior unit-cell Surface unit-cell Corner unit-cell

± /4 0, /2 0, /2

z

4.2.2. Mesh generation and boundary conditions There are three components of the model: yarns, resin and homogeneous body, respectively. The mesh generation of the yarns and homogeneous bodies is with mesh software of ‘HyperMesh’ and the resin is with ABAQUS, in order to make the nodes equivalent among the structures, as shown in Fig. 19. Table 4 lists the mesh properties. All contact conditions are with TIE and the boundary conditions are the same as the fully-homogenized model. Micro model-1 and Micro model2 are analyzed with ABAQUS/Explicit Model; meanwhile, a VUMAT file is used. Hashin’s failure criterion is for the yarn and homogeneous body. Ductile damage and shear damage failure criteria are for the resin.

±γ ±θ ±β

4.1.1.3. Mechanical behaviors of homogeneous part in 3-D braided composite. According to Eq. (13) [49], the stiffness matrix of the homogeneous structure can be calculated.

[C ]total =

pi [Ci ]cell i

(13)

where [C ]total is the stiffness matrix of the homogeneous body. pi is the volume fraction of the unit-cell. [Ci ]cell are the stiffness of the unit-cell and the subscript i indicates interior, surface and corner unit-cell, respectively.

5. Results and discussion 5.1. Results measured with material tester MTS

4.1.2. Mesh and boundary conditions The quasi-static simulation of the fully-homogenized model is with ABAQUS 6.12. The mesh type, size and number are C3D8R, 0.5 mm and 120000, respectively. As shown in Fig. 14, one end of the model is fixed, and the other is loaded.

The engineering load-displacement and stress-strain curves of the 3D braided composites measured with materials tester MTS are shown in Fig. 20(a)–(b). Table 5 shows the peak load, stress and the absolute value of errors. Fig. 20 and Table 5 illustrate that the embedded FBG sensors have little effect on the 3-D braided composites in the tensile linear elastic region. The peak load and stress are about 75 kN and 850 MPa. The axial elastic modulus of the 3-D braided composite is about 30 GPa. The curves show linearity clearly, and exhibit plasticity as the engineering strain reaches around 3% from Fig. 20(b).

4.2. FEA model To analyze and predict the mechanical properties of the yarns and resin with the same strain as the FBG sensor, a full-size micro-scale model is presented. According to Fig. 8, the size of the tensile specimen is large, but the Bragg grating region is just 10 mm long. To simplify the model, a full-size semi-micro model is established on the basis of the full-size micro-scale model. This model consists of the micro and homogeneous structures. The micro-scale structural component (20 mm long) consists of the yarns and resin, and the homogeneous structure is transversely isotropic. Due of the small diameter (250 μm) of the optical fiber, it was ignored. The location of the micro structure corresponds to the Bragg grating region position of the placed FBG sensor. Fig. 18 shows the full-size semi-micro model.

5.2. Local behaviors under axial tensile tests FBG sensor can accurately measure the in-situ strain of 3-D braided composite under small deformation (about 1%), and the excessive deformation can cause data distortion and FBG sensor failure. Therefore, we only investigated the in-situ strains at the linear-elastic region in this paper. 5.2.1. Internal local strains of different samples In order to reduce the errors caused by the dispersion of the samples and reduce the accidental error, the repetitive experiments were conducted. The results are showed in Fig. 21(a)–(f), where A means axial direction and T means transverse direction.

4.2.1. Full-size semi-micro model The micro structure includes the yarns and resin. The properties of the yarn can be obtained according to Section 4.1.1.1, and the properties of the resin are listed in Table 1. The properties of the

Fig. 18. Full-size semi-micro model. (a) The location of the micro structure lies in the middle (named as Micro model-1), (b) The components, (c) The location of the micro structure lies off the middle (names as Micro model-2). 8

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Fig. 19. The mesh generation of the full-size semi-micro model. (a) The local mesh, (b) The meshes of the homogeneous body, yarns and resin.

micro model, respectively. From Fig. 22(a), the internal local strain of the interior unit-cell is largest, and the strains of the surface and corner unit-cells are the same at R-60. From Fig. 23, the elements extracted from the fully-homogenized model show the same trends. As shown in Fig. 24(a) and (b), there are regular patterns of the stress distributions, which the internal stress of the inner yarns and resin is larger than the surface and corner. Obviously, the internal local strain and stress fields of the 3-D braided composite are affected by the structures. The structures of the threetype unit-cells are different apparently as shown in Figs. 6 and 16. What’s more, the internal yarns intertwine with each other and the volume fraction (PI) is large (64.86%). On the contrary, the volume fractions of the surface and corner yarns are small and their constraints are fewer. The interior unit cell is subjected to a greater strain and stress than the surface and corner unit-cell. From Fig. 22(b), the internal strains of the three-type unit-cells are the same almost in the linear-elastic region at R-100. The internal stress field distributions are even as shown in Figs. 23 and 24(c), (d). R-100 is the main bearing region of 3-D braided composite under uniaxial tension through the experiments and numerical calculation. The effect of the main bearing exceeds the structural effect of the unit-cell at R-100 in the linear-elastic region. Therefore, the stress-strain responses of three-type unit cells are consistent and R-100 shows even deformation as shown in Figs. 23 and 24. To illustrate the even distribution of stress further, the nodal stresses of the whole cross section R-60 and R-100 were extracted as shown in Fig. 25. The average nodal stresses and standard deviations at three moments (t1, t2, t3) are showed in Fig. 26. Fig. 26 (a) shows the

Table 4 Mesh schemes of the full-size semi-microstructure model. Model

Structure

Type

Size (mm)

Number

Micro model-1

Homogeneous body Yarn Resin Homogeneous body Yarn Resin

C3D6 C3D10M C3D4 C3D6 C3D10M C3D4

0.6 × 3 0.6 0.6 0.6 × 2 0.6 0.5

45720 + 45000 47,782 94,216 30480 + 114800 47,176 94,822

Micro model-2

From Fig. 21, the three curves of the same type of the unit-cells are almost coincident except the ones in Fig. 21(a). As shown in Fig. 21(a), curve 2 and curve 3 are nearly identical, and the difference between curve 1 and curve 2, 3 may be caused by accidental error in the experiment. There are two reasons why these curves are consistent. One is that all 3-D braided composite samples are processed in the same batch, so the difference is small among the samples. And the other is that only the strains in linear-elastic region were measured, so the fluctuation of the curves is small. Only one of each type of samples is selected for analysis below. 5.2.2. Internal tensile properties of different unit-cells It is significant to investigate the internal local tensile strain responses of the different unit-cells. Fig. 22(a) and (b) are the strain-time curves at R-60 and R-100, respectively. Figs. 23 and 24 are the numerical results of the fully-homogenized model and the full-size semi-

Fig. 20. The tensile curves. (a) Engineering load-displacement curve (b) Engineering stress-strain curve. 9

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Table 5 Peak load, stress and among the different type specimens. No FBG

Peak Error/%

Load/kN Stress/MPa Load Stress

71.78 874.54 / /

FBG C_R-60

C_R-100

S_R-60

S_R-100

I_R-60

I_R-100

75.87 846.37 5.69 3.22

75.15 886.45 4.69 1.36

76.50 869.16 6.57 0.61

68.92 829.81 3.98 5.11

76.04 863.89 5.93 1.21

75.83 830.78 5.64 5.00

Fig. 21. Internal local strain-time curves among the different samples. (a) Corner unit-cell at R-60, (b) Corner unit-cell at R-100, (c) Surface unit-cell at R-60, (d) Surface unit-cell at R-100, (e) Interior unit-cell at R-60, (f) Interior unit-cell at R-100. 10

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Fig. 22. Internal local strain-time curves of different unit cells measured with FBG sensors (a) At R-60 and (b) At R-100.

Fig. 23. The internal stress distributions of the different unit-cells.

average stresses of yarns in different regions at R-100 and R-60. Fig. 26(b), (c) are show the average stress of resin at R-100 and R-60, respectively. From Fig. 26(a), the average stress of region 1 (interior yarns) is larger than the one of region 2 (surface and corner yarns) at R-60, and the fluctuation ranges are small. This is consistent with the result of Fig. 24(a). For the R-100, the average stresses of region 1 and region 2

are almost equal, and their fluctuation ranges are small as well. This means that the stress distribution of yarns for the whole section is even. This is consistent with the result as shown in Fig. 24(c). From Fig. 26(b) and (c), the average stress of resin is much smaller than the one of yarns. And the stress fluctuation ranges of resin are small at R-100 and R-60. Therefore, the stress distribution for the whole R-100 is even.

Fig. 24. Internal stress distributions of the different cross sections from the full-size semi-micro models. (a) The stress distribution of the yarn structure at R-60, (b) The stress distribution of the resin at R-60, (c) The stress distribution of the yarn structure at R-100, (d) The stress distribution of the resin at R-100.

11

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Fig. 25. Scheme of the nodes at R-60/R-100 (a) Yarn, (b) Resin.

Although the stress of region 1 is larger than the one of region 2 at R-60, the stress distributions of region 1 and region 2 are even, respectively.

5.3. Internal tensile responses of the yarns and resin In order to investigate the internal tensile responses of the yarns and resin, assuming that the 3-D braided composite is an approximate spring model in the tensile linear region, and the axial tensile modulus at each moment (Ei) is considered to be the same in the global and local region. The axial tensile modulus at each moment can be calculated according to Fig. 20(b). The local stress can be obtained via Eq. (14).

5.2.3. Internal tensile properties of different sections To verify that the middle section is the main bearing region further, the internal local strains of the same type unit-cell at R-60 and R-100 were measured and calculated as shown in Fig. 27. The differences in the corner and surface unit-cells are apparent, and the strain at R-100 is larger than at R-60. The strain of the interior unit-cell at R-100 is slightly larger than at R-60, since the interior unit-cell is the main bearing unit-cell. The colors of the numerical simulations (as shown in Figs. 23 and 24) and Fig. 26(a) also verified these conclusions.

local

= Ei ·

local

(14)

where local is the internal local stress, Ei is the axial tensile modulus and local is the internal local strain. As shown in Fig. 28, Micro means full-size semi-micro model and

Fig. 26. The nodal average stress of cross section. (a) the nodal stresses of yarns at R-100 and R-60, (b) the nodal stress of resin at R-100, (c) the nodal stress of resin at R-60. 12

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Fig. 27. Internal local strain-time curves at R-60 and R-100. (a) Corner unit-cell, (b) Surface unit-cell, (c) Interior unit-cell and (d) The numerical results in the linear region of the fully-homogenized model.

Homo means fully-homogenized model. The numerical results of the fully-homogenized model are good agreements with the experimental results as shown in Fig. 28. Actually, the in-situ strain measured with FBG sensor is an average strain of a certain small region, because the length of Bragg grating region is 10 mm. Further, the fully-homogenized model is built by the homogenization method [48], which reduces the strain deviation arisen from the heterogeneous constituents and braided structure. So, the numerical results relatively agree well with the experimental results as shown in Fig. 28. From the macroscopic view of point, the average in-situ strains measured represent the overall mechanical strain responses of the 3-D braided composite, not for the yarn or the resin individually instead. This means that the strain measured by FBG sensor is different from any separate strain of the yarn or resin in the full-size semi-micro model. So, Fig. 28 shows the different response patterns between the yarn and resin. The yarn bears all tensile loading almost and the resin bears little loading. From the data of Fig. 28(a) and (b), the internal local stress of the surface yarn is smaller than the one of the interior unit cell at R-60, and the internal local stress of the surface yarn reaches the yield point firstly. This also illustrates that the interior unit-cell (internal yarn) is the main bearing structure at R-60. From the data of Fig. 28(c) and (d), the internal local stress-strain curves of the surface and interior yarn are similar with each other. This also illustrates that the middle section is the main bearing region.

5.4. Internal Poisson’ s ratio The Poisson’s ratio of 3-D braided composite is close to 0.3 according to [50–52]. The conventional method measured Poisson’s ratio is sticking two strain gauges to the surface. In this study, two FBG sensors were embedded into the interior of the 3-D braided composite as shown in Fig. 29. The internal Poisson’s ratio reflects the properties more accurately than surface. According Fig. 21(e), (f) and Eq. (15), the curves of internal Poisson’s ratios can be obtained, as shown in the lower area of Fig. 30. The internal Poisson’s ratios are respectively about 0.2 and 0.3 at R-60 and at R-100. The difference in Poisson’s ratio between R-60 and R-100 may be due to the difference in locations in the tensile gage section (see Fig. 8). The average internal Poisson’s ratio is about 0.3, which is consistent with the one in [50–52]. xy

=

yy xx

(15)

where, xy is Poisson’s ratio. xx , yy are the axial and transverse strain, respectively. In order to confirm the consequence measured by the FBG sensors further, we stuck two strain gauges which were perpendicular to each other on the outer surface of the specimens. The outer surface Poisson’s ratio is about 0.48 as shown in Fig. 30. The outer surface Poisson’s ratio is slightly larger than the internal one from Fig. 30. The strain gauges were stuck to the outer surface of the specimen, so the axial strain measured by the axial strain gauge was 13

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Fig. 28. Internal local stress-time curves between the experiments and numerical methods (a) The specimen named FBG_S_R-60, (b) The specimen named FBG_I_R-60, (c) The specimen named FBG_S_R-100 and (d) The specimen named FBG_I_R-100.

Fig. 29. Method of measuring Poisson’s ratio.

close to the one measured by the axial FBG sensor under uniaxial tension. However, the transverse strain measured by the transverse strain gauge was slightly larger than the one measured by the transverse FBG sensor. Thence, the Poisson’s ratio measured by the strain gauges is a little larger. Internal Poisson’s ratio measured by FBG sensors is reliable according to the above tests. And there is a slight difference in Poisson’s ratio between interior and outer surface.

Fig. 30. Poisson’ s ratio.

6. Conclusion

region of interior unit-cell is the principle deformed area within the 3-D braided composites since the strain of it is much larger than that both of surface and corner unit-cell at R-60. But at R-100, the strain values of these three unit-cells are almost the same. In addition, the deformation within the R-100 distributes more evenly than that in the R-60. Based both on the simulation results of the two FEA models, we found that the stress distribution at R-100 is more even than that in R60, which verifies that of experiments. According both to the experimental and numerical results, the strain responses agree well with each other. Hence, these two models are able to reveal the mechanism between the structure and deformation response of 3-D braided composites in macro-scale and micro-scale separately.

To realize the in-situ measurement of internal local strain responses under uniaxial tensile loading, the FBG sensors were embedded into the 3-D carbon fiber/epoxy braided composite. Also, we have developed two different FE models, i.e. a full-size semi-micro model and a fullyhomogenized model, to reveal its mechanism between the structure and the deformation response. From the experimental results, it is found that the introducing of the embedded FBG sensors has a bare effect on the tensile behaviors of the specimens. The variance in terms of its modulus and strength between the specimens with FBG and without FBG sensor is approximately 5%, which can be neglected during its loading process. The Poisson’s ratio of 3-D braided composite is about 0.3. Apart from that, we found that the 14

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Declaration of Competing Interest

Acknowledgements

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors acknowledge the financial supports from the Chang Jiang Scholars Program and National Science Foundation of China (Grant Number 51875099 and 51675095).

Appendix

{d

m i } f

{d i } =

[Sijf

m i }

[Sijf

{d

f j}

= [Aij ]{d

=

]{d

f j}

]{d

m j }

{d i} = Vf {d

i

f

(A.1) (A.2) (A.3)

+ Vm }{d

m i }

(A.4)

where,

{d i} = {d

11

{d i } = {d 11 Vm = 1

d d

d

22 22

d

33

d 33

2d

23 23

d

13

2d 13

d

T 12 }

2d 12 }T

Vf

The suffixes f and m indicate fiber and matrix (resin), respectively. Vf and Vm are the fiber and resin volume fractions of the unidirectional composite.

a11 a12 a13 0 0 0 0 a22 0 0 0 0 0 0 a33 0 0 0 [Aij ] = 0 0 0 a44 0 0 0 0 0 0 a55 0 0 0 0 0 0 a66 where,

a11 =

Em f E11

a55 = a66 =

, a12 = a13 =

f (S12

(A.5)

m)·(a S12 11

f (S11

m) S11

a22 )

1

, a22 = a33 = a44 = 2 · 1 +

Em f E22

and

1 Gm · 1+ f 2 G12

PI =

2·(m 1)·(n 1) 2·m · n + m + n

(A.6)

Ps =

3·(m + n + 2) 2· m ·n + m + n

(A.7)

PC =

4 2· m ·n + m + n

(A.8)

where PI, PS and PC are the volume fractions of the interior, surface and corner unit-cell, respectively. One of m and n is odd at least. m and n are the numbers of yarn arrangement.

[Tij ]s

l12

l22

l32

l2 l3

l3 l1

l1 l2

m12 n12

m 22 n22

m32

m2 m3

m3 m1

m1 m2

n32 n2 n3 n3 n1 n1 n2 2m1 n1 2m2 n2 2m3 n3 m2 n3 + m3 n2 n3 m1 + n1 m3 m1 n2 + m2 n1 2n1 l1 2n2 l2 2n3 l3 l2 n3 + l3 n2 n3 l1 + n1 l3 l1 n2 + l2 n1 2l1 m1 2l2 m2 2l3 m3 l2 m3 + l3 m2 l1 m3 + l3 m1 l1 m1 + l2 m1

(A.9)

where, l1 = cos( x )·sin( z ) , l2 = sin( x ), l3 = cos( x )·cos( z ) , m1 = sin( x )·sin( z ) , m2 = cos( x ) , m3 = sin( x )·cos( z ) , n1 = cos( z ) , n2 = 0 , n3 = sin( z )

VI =

/ 3 · 8

Vs =

cos · 6 1 + 3· 32 cos

(A.10)

·

(A.11)

15

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VC =

cos 3 3· · 4(1 + 2 2 ) cos

(A.12)

where VI , Vs and VC are the yarn volume fractions of the interior, surface and corner unit-cell, respectively.γ, θ and β are the braided angles of the interior, surface and corner unit-cell, respectively. ε is the fiber volume fraction of the yarn (ε = 0.785). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.compstruct.2019.111527.

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