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Equivalent initial flaw size model for fracture strength prediction of optical fibers with indentation flaws
Engineering Fracture Mechanics 215 (2019) 36–48
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Equivalent initial flaw size model for fracture strength prediction of optical fibers with indentation flaws
T
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Yuxuan Cui, Yunxia Chen, Jingjing He
Beihang University, School of Reliability and Systems Engineering, No. 37, Xueyuan Road, Haidian District, Beijing 100191, China Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Surface flaw Crack growth Residual strength
This work presents an equivalent initial flaw size (EIFS) model for indentation flaws to quantify the fracture strength of optical fibers. Three geometry parameters of the indentation flaws, namely, the total projection length, the average crack length and the total flaw depth are used to establish the EIFS model. Indentation flaws of different dimensions are made using a Berkovich indenter by adjusting the load force. A stereo microscopic imaging system is employed to acquire the geometry parameters of the indentation flaws. Tensile testing is then performed using 16 commercial optical fiber specimens. A groove fixture is employed to ensure the reliability of data acquisition. The experimental data are used for model development and performance validation. Furthermore, the proposed model is compared with existing models to demonstrate the effectiveness of the method. The proposed EIFS model gives more accurate prediction of fracture strength in current investigation.
1. Introduction Silica optical fibers are widely applied in the fields of telecommunication and structure health monitoring due to the outstanding performance of sensitivity, speed, and flexibility [1]. The mechanical integrity and the strength of optical fibers are critical to the reliability of the optical-fiber-based sensing systems [2,3]. Fracture is the most common failure mode of optical fibers under the combination of mechanical loads and harsh environments [4]. The mechanical strength of the optical fibers can reach up to 10GPa when they are in pristine state; however, surface flaws and impurities can reduce the strength to roughtly one order of magnitude [5,6]. Therefore, it is of great importance to quantify the influence of the surface flaws on the mechanical properties of optical fibers. Two types of flaws commonly exist in optical fibers. The first one is the contamination in glass interior and surface [7]. For example, the gritty particles originate from the impurities of ingredient or from the drawing environment conditions are typical contaminations [8]. The other type of flaws are introduced before the fiber is coated during the drawing process [9,10]. Although severe flaws could be eliminated by the screening procedure, those survived flaws can still pose a great risk for critical applications [6]. In this work the surface flaws are mainly studied to investigate their effect on the fracture strength of optical fibers. Artifical surface flaws have been reported to investigate the mechanical properties of optical fibers [11]. Mechanical abrasion with particles and indentation are two commonly used methods to create artificial flaws. The former one can best represent a natural flaw as the process is very similar to the naturally developed flaws [12]. However, it is difficult to control the flaw size and the residual strength of an optical fiber [13]. Indentation is an alternative method to create surrogate flaws and has been used for the analysis of fatigue ⁎ Corresponding author at: Beihang University, School of Reliability and Systems Engineering, No. 37, Xueyuan Road, Haidian District, Beijing 100191, China. E-mail address: [email protected] (J. He).
https://doi.org/10.1016/j.engfracmech.2019.04.021 Received 5 December 2018; Received in revised form 15 April 2019; Accepted 16 April 2019 Available online 07 May 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
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behavior in glass and optical fiber materials [14–16]. It is convenient to control the residual strength over a wide range by adjusting the indentation load. Sharp pyramid indenters including Vickers, Berkovich and cube-corner are widely applied for inducing indentation, for their advantages on geometrical similarly and residual strength consistency. Vickers indenters induce rectangular pyramid shape indentation flaws while Berkovich and cube-corner indenters result in triangular pyramid shape indentation flaws with different centerline-to-face angles. Despite of indentation induced by sharp indenters, cracks may nucleate due to the remnant plastic deformation and propagate along the radial direction under higher indentation loads [13]. Existing literatures have shown that flaws with radial cracks would cause the fracture strength of optical fibers more than quintuple smaller than that of flaws without radial cracks [10,13]. The influence of indentation flaws and resulting radial cracks on glass and optical fibers have been extensively studied. Nascimento et al. [17] carried out a statistical study on fracture of optical fibers with the indentation flaws, and observed that both Vickers and Berkovich indentation flaws would reduce the strength of optical fiber significantly. Marshall and Lawn reported that the residual contact stress had a detrimental effect on the fracture behavior of Vickers indentation flaws with radial cracks. The authors proposed an indentation load dependent model for the prediction of residual-stress-sensitive strength [18]. The work of Dabbs et al. showed that the indentation flaws without well-developed radial cracks lead to a corner-to-corner failure. In addition, the radial cracks can significantly increase the crack velocity exponent in the crack propagation process [19,20]. Experimental results reported by Jakus et al. [21] indicate that the Weibull median fracture strength decreases significantly as the length of the indentation flaw length increases from 1 μm to 10 μm. It can be seen that both the radial cracks and the size of flaws play important roles in the strength prediction. The mechanical properties of glass and optical fiber materials subject to different shapes of indentation flaws have been studied. Lathabai developed a shear-fault-microcrack model to correlate the stress intensity factor and the geometry parameters of indentation flaws on glass plate [22], which contributes to the strength prediction in inert environment. Jung et al. [23] carried out experiments to measure the strength of brittle material plate containing indentation flaws. A model was developed to describe the correlation between the strength and the geometry parameters such as the characteristic contact radius and the radial crack length. In their study, the critical shear fault was simplified as a mode I crack with a length of the radial crack in the fracture mechanics analysis. Lin and Matthewson [10] compared the inert fracture strength of optical fibers containing Vickers indentation flaws using the fracture mechanics models. Moreover, the inert fracture strength of optical fibers with indentation flaws could be modelled by a crack of size equal to the flaw size. The similar idea can be found in Refs. [21] and [24]. In Ref. [19], the influence of radial crack on the fracture strength of optical fiber is included by increasing the crack velocity exponent. In their study, the crack velocity exponent is 20 for indentation flaws without radial cracks and 30 for indentation flaws with radial cracks. But the correlation between the accurate length of radial crack and the crack velocity exponent is not included in Ref. [19]. In real engineering scenarios, the crack formation during the indentation process has inherent uncertainties due to environment, instrumentation, and operator etc. It is known that the same load can result in indentation flaws with different geometry features, which can obviously induce different crack propagation procedure and further impact the fracture strength of optical fibers [25,26]. Therefore, the morphology of the radial crack is another factor that should be studied for more precise fracture strength evaluation. In the present work, an equivalent initial flaw size (EIFS) model is proposed to predict the fracture strength of optical fiber with indentation flaws. The proposed model can deal with indentation flaws with and without radial cracks. Three geometry quantities of the indentation flaws, namely, the total projection length, the average crack length and the depth of indentation flaw are used to formulate an equivalent initial flaw size (EIFS) model. The EIFS model is used to correlate the fracture strength and the morphology of indentation flaws. A total number of 16 indented optical fiber specimens are used in the tensile test for model development and validation. The remainder of the study is organized as follows: Section 2 describes the development of the equivalent initial flaw size (EIFS) model. The geometry quantities used to build the model are described in details, and a linear model is proposed based on the geometry quantities. The indentation procedure and tensile testing are introduced in Section 3. The indentation flaws induced by the Berkovich indenter are measured using a microscopy system. Tensile testing is carried out for the optical fiber specimens with a constant strain rate of 5%/min. Following that, validations are made by comparing the model prediction of fracture strength and experimental data in Section 4. 2. Equivalent initial flaw size (EIFS) modeling 2.1. Fracture strength evaluation The failure of optical fibers due to strength reduction caused by flaws is dominated by the crack initiation and propagation process. For optical fibers under the tensile testing, the linear elastic fracture mechanics is used to describe the crack growth. A power law form relation is normally assumed between the time-dependent crack growth rate c ̇ (t ) and the stress intensity factor at the tip of the crack [27]:
c ̇ (t ) = A·[KI (t ) ]n ,
(1)
where A is an environmental dependent parameter and n is the crack velocity exponent. A linear relationship between logA and n is determined from the experimental results [28]. The applied stress intensity factor KI (t ) for a crack of depth c(t) subjected to the applied stress σa (t ) at time t in tensile test is given by [13]
KI (t ) = Y[c (t )]·[σa (t )]·[c (t ) ]1/2 ,
(2)
where Y[c(t)] is the geometric correction factor and is also a time-dependent parameter related to the crack depth c(t). The 37
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Fig. 1. Diagram of the cross section of the glass core with an initial crack.
relationship between the geometric correction factor and the crack depth can be found in Ref. [29]. It is known that the surface cracks in optical fibers propagate in a semi-elliptical pattern [27,30]. A semi-elliptical crack is assumed to represent the initial flaw as shown in Fig. 1. The symbols in Fig. 1 denote the following meanings. r is the radius of the glass core, c(t0 = 0) is the depth of the initial crack and w is half of the initial crack width on the outer surface. The condition t0 = 0 represents the initial state of the crack. Failure is considered when the current stress intensity factor (SIF) exceeds the fracture toughness KIC given in [31]
KIC = Y [c (t f )]·σf ·[c (t f )]1/2 ,
(3) 1/2
for KIC is used in this where σf is the fractured stress and c(tf) is the crack depth at the fracture time tf. The value of 0.58 MPa·m study. Because the optical fiber used in the study is mainly composed of silica glass, which is the same material in Ref. [32]. The same value of KIC has also been adopted in strength evaluation of optical fiber in Ref. [33]. During the tensile test, the total differential of the applied stress to the crack depth can be obtained from Eqs. (1) and (2),
d [σa (t )] σȧ σȧ = = , d [c (t )] A·[KI (t )]n A·Y [c (t )]n ·[σa (t )]n ·[c (t )]n/2
(4)
where σ̇a is the stress rate. By separating the variables in Eq. (4) and integrating versus time from the initial moment t0 = 0 to the fracture time tf, the following equation can be obtained.
1 σ̇ ·[σ fn + 1 − (σa (0))n + 1] = a · n+1 A
c (t f )
∫c (t =0) 0
d [c (t )] , Y [c (t )]n ·[c (t )]n /2
(5)
where σa(0) is the initial applied stress which equals 0, and the depth of the initial crack c(t0 = 0) is used to represent the equivalent initial flaw size. In this model, σf and σ̇a can be obtained from the tensile testing. Next the geometry representation of the indentation flaw and the process of equivalent initial flaw size modeling is described. 2.2. Equivalent initial flaw size (EIFS) modeling In this study, a Berkovich indenter with 65.3° centerline-to-face angle is adopted in the experimental procedure to create indentation flaws. The cracking threshold for the Berkovich indenter is similar to that of Vickers indenter, around 1000 mN to 1500 mN for fused silica [32,34]. Other experimental conditions including surface condition, relative humidity, and indentation duration also have an effect on the formation of radial cracks. The radial cracks can increase the stress concentration at the tip and drastically reduce the fracture strength of optical fibers [13]. Fig. 2 shows both diagrams of the triangular pyramid shape indentation flaws with and without radial cracks. It has been reported for the cube-corner indenter, which also induces triangular pyramid shape indentation similar with the Berkovich indenter, the localized tangential stress during loading encourages the formation of radial cracks at the corners (from B to C in Fig. 2(a)). More importantly, these radial cracks intersect with each other beneath the plastic deformation zone (intersecting at E in Fig. 2(a)), making the total flaw depth (h in Fig. 2(a)) larger than the depth of the indent (h′ in Fig. 2(a)). According to the focused ion beam tomography reported in Refs. [35,36], the cracks on glass induced by cube-corner indenters can be depicted as quarter-ellipse. On the other hand, the finite element analysis results in Ref. [37] have also shown that the radial crack created by Berkovich indentation can be approximately depicted as quarter-ellipse cracks. Therefore, the same assumption of quarterellipse crack has been adopted in current study. The symbols in Fig. 2 denote the following meanings. x is the axial direction of the optical fiber, y is the diametrical direction, and d is the diameter of the optical fiber glass core. The term e is the total projection length including crack and a′ is the projection lengths of the triangular pyramid indent along the x-axis. For flaws with radial cracks, the values of e equal to the projection length including both the triangular pyramid indent (a′) and radial cracks. For flaws without radial cracks, the values of e equal to those of a′. l is the average crack length from the indent corner to the crack tip. h′ and h are the depth of indent and the total flaw depth including cracks. For flaws without radial cracks, the values of h equal to those of h′, which are obtained from the indentation load-depth curve. Both the failure of indentation flaws with and without radial cracks are controlled by the combination of applied stress and 38
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(a) Dimension diagrams of an indentation flaw with radial cracks
(b) Dimension diagrams of an indentation flaw without radial cracks Fig. 2. Dimension diagrams of the indentation flaws. (a) dimension diagrams of an indentation flaw with radial cracks, (b) dimension diagrams of an indentation flaw without radial cracks.
residual stress induced during the indentation process. For the case of flaws with radial cracks, the crack propagation process will contribute to the most part of the fracture process of optical fibers with the absence of crack initiation. As the crack gradually propagates, longer crack length and deeper crack depth would increase the stress intensity factor and further accelerate the fracture process until the critical threshold [38]. On the other hand, for the case of flaws without radial cracks, the fracture process includes the crack initiation and the following crack propagation. Under the influence of the near-field residual stress, cracks initiate at the corners of the flaw. In contrast, the pressure exerted by the plastic zone on the surrounding matrix results in the far-field residual stress and further controls the crack arrest [21]. The strength of flaws without cracks would be much higher than that of flaws with radial cracks, which is confirmed by both the experiment in this study and Ref. [10]. The flaws without radial cracks are also characterized by the flaw length on the surface according to Ref. [39] and the flaw depth based on experiments in section 3. Therefore, both the total length e and the flaw depth h are selected to quantify the equivalent initial flaw size for flaws. The model form in Ref. [40] for charactering the flaw geometry is adopted in this work. The proposed equivalent initial flaw size model is expressed as following.
A1·(e·h)1/2 + B1, with radial cracks c (t 0 = 0) = ⎧ 1/2 ⎨ A ⎩ 2·(e·h) + B2, without radial cracks
(6)
The depth of the initial crack c(t0 = 0) is used to represent the equivalent initial flaw size at time t0 = 0, and A1, A2, B1 and B2 are model parameters obtained through experimental data. Based on the assumption of crack geometry, the flaw depth h can be expressed with the average crack length l multiplying a coefficient k. The parameter k as well as the model parameters A1, A2, B1 and B2 can be obtained by the least square method. And the flaw depth h equals to the indent depth h′ for flaws without cracks. As shown in Eq. (6), the initial flaw size c(t0 = 0) in Fig. 1 can be expressed as a function of flaw geometry parameters. By this means, the influences of the flaw dimension and radial cracks are incorporated for more accurate fracture strength evaluation. It is worth mentioning that both the material property and the experimental condition may affect the values of these parameters. The proposed method provides a general framework of fracture strength prediction including the initial flaw influence. Therefore, the value of the model parameters used in this paper may need to be adjusted for different scenarios. 39
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Fig. 3. The overall framework of the modeling and experimental process.
3. Experiments for model development Section 2 proposes an equivalent initial flaw size model to establish a quantitative relation between the flaw geometries and fracture strength. The overall framework of the modeling and experimental work is shown in Fig. 3. First, indentation flaws are induced using a Berkovich indenter. Next, the measurements of the geometry parameters are acquired through an optical microscopic imaging system, and the geometry parameters are used to obtain the EIFS model parameters with the tensile testing data. The experimental procedures are introduced in detail as following.
3.1. Material and specimen preparation The specimens used in the experiments (shown in Fig. 4) are single-mode optical fibers with a 125 μm-diameter glass core and a 245 μm-diameter polymer coating layer. Both ends of the specimens are glued to carbon fiber sheets with epoxy resin as shown in Fig. 4a. Plastic tapes are adhesived parallel to the optical fiber to provide extra support. Next, the polymer coatings are removed for the indentation procedure. To avoid causing mechanical damage to the glass core, about 1 cm of the central section of the optical fiber is immersed in tetrahydrofuran for a duration of 180 seconds. After that the specimens are rinsed in acetone to remove the residual tetrahydrofuran. The partial enlarged detail of the processed optical fiber specimen is shown in Fig. 4b. Total 16 specimens are used in the pre-indentation procedure and the subsequent tensile testing. The surface flaws are introduced by the nanoindentation method. One of the most important issues is to keep the circular optical fiber stable during the indentation process. The indenter would slip to one side of the fiber once the optical fiber moves. Before the indentation process, the uncoated parts of optical fiber samples are put on the glass substrate with V-shape grooves and then fixed. The schematic of a substrate used for 125 μm optical fiber cores is shown in Fig. 5a. It is also very important to ensure that the glass core is not damaged during the experiments. Therefore, the ethyl acrylate with low viscosity is smeared on the grooves of the substrate to provide extra protection. While the ethyl acrylate is half cured, the uncoated glass sections of the fibers are put in the grooves with an optical microscope. After the ethyl acrylate is fully solidified, it fixes the bare glass core and provided an elastic buffer to protect it from the mechanical damage (Fig. 5b).
Fig. 4. Specimen for indentation: (a) optical fiber specimen fixed to carbon fiber sheet; (b) partial enlarged detail of an uncoated optical fiber specimen. 40
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Fig. 5. Grooved substrates used to stabilize optical fibers. (a) the side view diagram of the glass substrate and the V-shape grooves with an optical fiber, and (b) the top view microscopic image of the grooves and an optical fiber.
3.2. Indentation procedure The indentation process is carried out by a TI950-triboindenter indentation instrument as shown in Fig. 6. After the indentation location is chosen through an in-situ optical microscopy, a Berkovich indenter is used to create surface flaws [33,41]. The indenter can create tetrahedral shape flaws with extended radial cracks originating from the three corners. The angle between the faces of the indenter and the axis of the symmetry is 65.3°. A constant loading rate of 2.5% per second (P′/P = 0.025 s−1) is applied during the loading procedure. The indentation loads are set between 500mN and 2200mN to induce both flaws with and without radial cracks. The indenter is unloaded at a half speed of the loading after reaching the preset load. During the whole indentation process, the load and displacement are updated and recorded in real time. Fig. 7 presents a typical load-depth response curve of the indentation procedure. The depth of the indent h′ can be obtained from the load-depth curve. 3.3. Indentation flaw measurement The geometry parameters of the indentation flaws are acquired with a calibrated stereo microscope. Typical flaw morphologies are shown in Fig. 8. Parameters characterizing the flaws include the total length e, the average crack length l and the total depth h for flaws with cracks; the total length a′ and the indent depth h for flaws without cracks. The detailed geometry parameters of the flaws on each specimens are presented in Table 1. Radial cracks can originate from the corners of the indentations in a random pattern, as reported in Refs. [13,42]. 3.4. Tensile testing The tensile testing is performed using the IBTC-5000 testing machine in this work. Specimens are preconditioned in the testing environment and the testing is conducted in air environment (25 °C and 40% relative humidity) as shown in Fig. 9. Tensile loads with a constant strain rate of 5%/min are applied on the optical fibers until completely fractured. The loading information and the testing results are listed in Table 2. The geometry parameters of indentation flaws and tensile
Fig. 6. Indentation procedure. 41
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Fig. 7. Typical indentation load-depth curve.
Fig. 8. Morphologies of the indentation flaws. (a) an indentation flaw with radial cracks, and (b) an indentation flaw without radial cracks. Table 1 Geometry parameters of the flaws. Specimen No.
Indentation load P/mN
Flaw length e/μm
Indent depth h'/μm
Average crack length l/μm
With radial cracks
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
500 600 700 800 900 1000 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200
18.618 14.580 13.342 14.099 15.968 13.348 33.050 32.971 42.169 33.025 17.030 41.856 43.676 49.632 27.175 29.214
– – – 2.314 1.325 2.658 – – – – 3.783 – – – 6.597 4.063
4.51 3.67 2.73 – – – 11.74 20.98 22.19 29.39 – 25.65 22.98 29.98 – –
Y Y Y N N N Y Y Y Y N Y Y Y N N
42
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Fig. 9. Tensile testing setup. (a) tensile testing machine and computer, and (b) sample and fixtures. Table 2 Experimental results of tensile testing. Specimen No.
σf /MPa
t f /s
σ̇a /MPa/s
Usage
With radial cracks
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
111.72 137.47 146.03 675.20 513.70 427.24 107.81 76.03 94.36 78.80 366.69 83.12 78.15 79.37 291.40 263.77
9.9 10.6 10.6 49.9 38.2 34.5 7.4 12.9 7.3 9.1 27.7 6.6 6.9 7.5 22.3 20.4
11.285 12.969 13.776 13.531 13.464 12.384 14.569 5.894 12.926 8.659 13.237 12.593 11.326 10.582 13.038 12.939
Regression Regression Validation Regression Regression Validation Validation Regression Regression Regression Regression Validation Regression Regression Validation Regression
Y Y Y N N N Y Y Y Y N Y Y Y N N
testing results are used for EIFS modeling. The experimental data are divided into two sets, one for EIFS model parameter estimation and the other for model validation, as identified in Table 2. The parameter estimation procedure of the EIFS model is described in the following steps: 1) Calculate the critical crack depth cf using Eq. (3); 2) Calculate the EIFS c(t0 = 0) using Eq. (5). The integral formula can be numerically solved by substituting the fracture stress σf and stress rate σ̇a from the experiments and critical crack depth cf in step 1). The values of the crack velocity exponent n are 20 for flaws without cracks and 30 for flaws with cracks; and 3) Obtain model parameters using the least square method. Using the above described procedure and data in Table 1, the model parameters of the proposed EIFS model can be obtained: A1 = 7.743 × 10−3, B1 = −1.447 × 10−2, A2 = 6.287 × 10−4, B2 = −2.645 × 10−3. The calculated values and regression values of the equivalent initial flaw sizes are plotted in Fig. 10 and it can be seen that regression values are in good agreement with the calculated values. Besides, the results indicate that the radial cracks have a significant impact on the equivalent initial flaw size. According to the tensile test results, the sizes of the equivalent initial flaw c(t0 = 0) with radial cracks are in the range from 1 × 10−2μm to 8 × 10−2μm, while the sizes of the equivalent initial flaw c(t0 = 0) of flaws without radial cracks are in the range from 1 × 10−3μm to 4 × 10−3μm. In this study, experimental results show that the difference in EIFS results in great disparity of fracture strengths. Based on the proposed EIFS model, larger values of the geometry parameters would lead to great equivalent intial flaw size, and the data in Tables 1 and 2 also indicate that optical fibers with larger flaws have lower fracture strength. This is physically reasonable since the greater radial crack length and depth would cause the greater stress intensity and give rise to stress concentration at the crack tip for flaws with radial cracks. And for flaws without cracks, larger indent size would increase the nearfield residual stress promoting the crack initiation and on the other hand decrease the sectional area under tensile stress which further increase the stress intensity. As a contrast, the flaw type has much greater effect on the fracture strength. The EFISs of most flaws with 43
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Fig. 10. The relationship between regression values and calculated values of c(t0 = 0).
radial cracks are in the range from 1 × 10−2 μm to 9 × 10−2 μm, while the values of flaws without cracks are between 9 × 10−4 μm and 5 × 10−3 μm according to the proposed model. It is also shown in Table 2 that most of the fracture strengths of flaws without radial cracks are over 300 MPa, whereas a majority of the fracture strengths of flaws with radial cracks are around 100 MPa. Similarly, experimental results in Refs. [10,13] indicated strengths of flaws without radial cracks are five times higher than those of flaws with radial cracks using the Vickers indentation method. Therefore, indentation flaws with or without radial cracks are considered as two different physical models in this study to better describe their influences on the fracture strength. The proposed method provides a framework to calculate the EIFS of flaws and to evaluate the corresponding tensile strength under the given test condition. It provides a general tool to evaluate the fracture strength of optical fibers with initial flaw for quality control purpose. For example, the relationship between the EIFS and the tensile strength can be obtained as Fig. 11 given the stress rate equals to 13 MPa/s and the crack velocity exponent equals 20 It is convenient to predict the strength using the relationship described in the Fig. 11 once the flaw geometry parameters are measured and the EIFS is calculated.
4. Methodology validation Five specimens are randomly chosen to validate the effectiveness of the proposed EIFS model, as shown in Table 2. Firstly, the EIFS of the validation specimens are calculated using the proposed EIFS model. Next, the fracture strengths are predicted using the resulting EIFS model. The calculated EIFS, the predicted fracture strength, and the actual fracture strength are listed in Table 3. The model prediction and actual values of the fracture strength are compared and shown in Fig. 12. A scatter factor of 1.2 is used to quantify the performance of the model without loss of generality [43]. It can be seen from Fig. 12
Fig. 11. The relationship between the predicted fracture strength and EIFS c(t0 = 0). 44
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Table 3 Results comparison between experimental and predicted tensile strength. Specimen No.
c(t0 = 0)/μm
Predicted fracture strength/MPa
Actual fracture strength/MPa
With radial cracks
3 6 7 12 15
0.010418 0.001100 0.044293 0.076668 0.003840
164.29 482.92 106.85 86.39 276.25
146.03 427.24 107.81 83.12 291.40
Y N Y Y N
Fig. 12. The comparison between the predicted and actual values of the fracture strength.
that all the data points are within the region of the scatter limit (shown with dashed lines). Fig. 12 also demonstrates that the radial crack has a significant effect on the strength of optical fibers. It can increase the stress concentration around the crack tip, making the optical fiber fracture more easily. According to the established EIFS model, the predicted fracture strengths of indentation flaws without radial cracks are 3 to 7 times larger than that of flaws with radial cracks. The result is consistent with Lin and Matthewson’s observation [10]. In addition, the proposed model is compared with existing models for the prediction of fracture strengths. Jung et al. proposed indentation load dependent fracture strength prediction models for triangular indentation flaws (Berkovich indenter) with and without radial cracks [23]. For indentation flaws without radial cracks, the fracture strength is
1 α·H 1/4 ⎞ ·KIC − κ·H⎤, σf = ⎛ ⎞·⎡ ⎛ ⎢ ⎥ ⎝ λ ⎠ ⎣⎝ P ⎠ ⎦
(7)
where λ and κ are crack-geometry coefficients, α the indent center-to-corner dimension, H the material hardness and P is the indentation load. For a Berkovich indenter, the values of α is 33/2/4 [23]. The values of λ and κ for soda-lime glass are adopted here, which are 0.27, 0.015 respectively [44], because the soda-lime glass has the same shear fault configuration as fused silica glass used in this work. The hardness of fused silica is 7GPa [10]. For indentation flaws with radial cracks, the fracture strength is 1/3
K4 3 1 σf = ⎜⎛ ⎟⎞·⎛⎜ ⎞⎟·⎛⎜ IC ⎞⎟ 4 ψ 4 χ ⎝ ⎠⎝ ⎠⎝ P ⎠
,
(8)
where χ is a dimensionless indenter/specimen constant, ψ is the dimensionless crack shape parameter and P is the indentation load. For a Berkovich indenter, the value of χ is 0.1450 [18,44], and the value of ψ is 0.4497 [10,45]. In Jung’s model, the parameters λ , κ and ψ are crack-geometry parameters, α and χ the indenter dependent parameters and H the material parameter. One of the advantage of Jung’s model is that it does not need to fit the model parameters for different experimental conditions or materials. It takes information of material, indenter, flaw shape and load to predict the strength with Jung’s model. Although the proposed EIFS model needs data of flaw geometry to fit the parameters, the advantage is that the uncertainty of environment and instrumentation is contained during the parameter fitting process. Therefore the proposed model could provide more accurate prediction. The predicted values of fracture strengths of optical fibers calculated from the proposed EIFS model and the 45
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Fig. 13. Comparisons of the model prediction between the proposed model and existing models.
two mentioned models are plotted in Fig. 13 together with the validation experimental data. Fig. 13 signifies that the proposed EIFS model provides more accurate predictions of the fracture strength than Jung’s model. In particular, Jung’s model tends to produce conservative results compared with experimental data. It may cause by the direct simplification of this critical shear faults as mode I cracks which ignores the crack propagation process under different angles. Besides, Jung’s model fails to predict the strength of glass with flaws around the cracking threshold load as shown in Ref. [23]. Fig. 14 shows the relative prediction errors between predicted and actual value of the fracture strength for Jung’s model and the proposed EIFS model, where an overall smaller relative error can be achieved using the proposed model. The maximum relative errors are less than 13% for both flaws with and without radial cracks. 5. Conclusion This paper presents an equivalent initial flaw size model to investigate the influence of indentation flaws on the fracture behavior of optical fiber. In particular, three geometry parameters including the total projection length, average crack length and total flaw depth are used to establish a linear model to correlate with the EIFS. Artificial indentations are induced in 16 optical fiber specimens using the Berkovich indenter. The geometry parameters are subsequently acquired using a stereo microscopy imaging system. Tensile testing are conduced using commercial single mode optical fibers, and experimental data are used for model parameter estimation and model validation. Following conclusions are drawn based on the current investigation. (1) The proposed EIFS model can incorporate the radial cracks in a universal model format. The effectiveness and accuracy of the proposed model are validated using tensile testing data. Results indicate that the proposed method can predict the fracture
Fig. 14. Comparisons of relative error between the proposed model and Jung’s model. 46
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strength reliably, and the independent validation points are within a scatter band of a factor of 1.2. (2) The performance of the proposed EIFS model is compared with existing models. The results show that the proposed model is able to predict the fracture strength more accurately in terms of the relative error. In this work several geometry parameters including the total length, the average crack length, the crack depth and the indent depth are adopted to characterize the flaws and evaluate the strength. In order to quantitatively analyse the effect of these parameters on the fracture strength, experiments with different values of the above geometry parameters are needed to provide a sensitive model. Besides, currently the typical Berkovich indenter is used in the study. Ideally, the EIFS model could be applied on different kinds of indenters including the pyramid indenter and the cube-corner indenter, since the proposed method provides a general framework to quantitatively characterize the indentation flaw geometry and predict the strength with EIFS. However, comprehensive research including both theoretical and experimental work are also required to confirm the feasibility and robustness of the proposed method under different applications. Acknowledgments This work is supported by the National Key R&D Program of China (Grant No. 2016YFB0100400) and the National Natural Science Foundation of China (Grant No. 51675025). The supports are greatly acknowledged. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.04.021. References [1] Zhang Z, Ansari F. Crack tip opening displacement in micro-cracked concrete by an embedded optical fiber sensor. Engng Fract Mech 2005;72:2505–18. [2] Colaizzi J, Matthewson MJ. Mechanical durability of ZBLAN and aluminum fluoride-based optical fiber. J Lightwave Technol 2002;12:1317–24. [3] Zinck P, Gérard JF, Wagner HD. On the significance and description of the size effect in multimodal fracture behavior. Experimental assessment on E-glass fibers. Engng Fract Mech 2002;69:1049–55. [4] Evano N, Abdi RE, Poulain M. Lifetime modeling of silica optical fiber in static fatigue test. J Appl Res Technol 2016;14:278–85. [5] Kurkjian CR. Strength of inorganic glass. New York: Plenum Press; 1985. [6] Foray G, Descamps-Mandine A, R’Mili M, Lamon J. Statistical flaw strength distributions for glass fibres: Correlation between bundle test and AFM-derived flaw size density functions. Acta Mater 2012;60:3711–8. [7] Sonnenfeld C, Berghmans F, Thienpont H, Skorupski K, Gomina M, Makara M, et al. Mechanical strength of microstructured optical fibers. J Lightwave Technol 2014;32:2193–201. [8] Yoshida K, Satoh T, Enomoto N, Yagi T, Hihara H, Oku M. Fracture origins of optical fibers fabricated by hybridized process. J Lightwave Technol 2002;14:2506–12. [9] Glaesemann GS. Advancements in mechanical strength and reliability of optical fibers. Proc SPIE: Int Soc Opt Eng 1999:1029502–35. [10] Lin B, Matthewson MJ. Inert strength of subthreshold and post-threshold Vickers indentations on fused silica optical fibres. Philos Mag A 1996;74:1235–44. [11] Semjonov SL, Glaesemann GS. Fatigue behavior of silica fibers with different defects. Proc SPIE - Int Soc Opt Eng 2001;4215:28–35. [12] Sakaguchi S, Hibino Y. Fatigue in low-strength silica optical fibres. J Mater Sci 1984;19:3416–20. [13] Matthewson MJ. Optical fiber reliability models. Critical Review Collection: International Society for Optics and Photonics 1993:3–31. [14] Glaesemann G. The mechanical behavior of large flaws in optical fiber and their role in reliability predictions. Proc Inter Wire Cable Symp 1992:689–704. [15] Zhu X, Ying Y, Li L, Du Y, Feng L. Identification of interfacial transition zone in asphalt concrete based on nano-scale metrology techniques. Mater Des 2017;129:91–102. [16] Jiang WG, Su JJ, Feng XQ. Effect of surface roughness on nanoindentation test of thin films. Engng Fract Mech 2008;75:4965–72. [17] Nascimento EMD, Lepienski CM. Mechanical properties of optical glass fibers damaged by nanoindentation and water ageing. J Non-Cryst Solids 2006;352:3556–60. [18] Marshall DB, Lawn BR. Flaw characteristics in dynamic fatigue: the influence of residual contact stresses. J Am Ceram Soc 1980;63:532–6. [19] Dabbs TP, Lawn BR. Strength and fatigue properties of optical glass fibers containing microindentation flaws. J Am Ceram Soc 1985;68:563–9. [20] Donaghy FA, Dabbs TP. Subthreshold flaws and their failure prediction in long-distance optical fiber cables. J Lightwave Technol 1988;6:226–32. [21] Jakus K, Jr JER, Choi SR, Lardner T, Lawn BR. Failure of fused silica fibers with subthreshold flaws. J Non-Cryst Solids 1988;102:82–7. [22] Lathabai S, Rödel J, Dabbs T, Lawn BR. Fracture mechanics model for subthreshold indentation flaws. J Mater Sci 1991;26:2157–68. [23] Jung YG, Pajares A, Banerjee R, Lawn BR. Strength of silicon, sapphire and glass in the subthreshold flaw region. Acta Mater 2004;52:3459–66. [24] Chen J, Diao B, He J, Pang S, Guan X. Equivalent surface defect model for fatigue life prediction of steel reinforcing bars with pitting corrosion. Int J Fatigue 2018;110:153–61. [25] Mikowski A, Serbena FC, Foerster CE, Lepienski CM. Statistical analysis of threshold load for radial crack nucleation by Vickers indentation in commercial sodalime silica glass. J Non-Cryst Solids 2006;352:3544–9. [26] Sellappan P, Rouxel T, Celarie F, Becker E, Houizot P, Conradt R. Composition dependence of indentation deformation and indentation cracking in glass. Acta Mater 2013;61:5949–65. [27] Wiederhorn SM. Subcritical crack growth in ceramics. Boston: Springer; 1974. [28] Kalish D, Tariyal BK. Static and dynamic fatigue of a polymer-coated fused silica optical fiber. J Am Ceram Soc 1978;61:518–23. [29] Muraoka M, Abe H. Subcritical crack growth in silica optical fibers in wide range of crack velocities. J Am Ceram Soc 1996;79:51–7. [30] Muraoka M, Abé H, Aizawa N. The distribution of the stress intensity factor along the front of the growing crack in an optical glass fiber. J Electron Packag 1992;114:403–6. [31] Matthewson MJ. Optical fiber reliability models. Proc Soc Photo-Opt Instrum Eng Crit Rev 1993:3–31. [32] Pharr GM, Harding DS, Oliver WC. Mechanical properties and deformation behavior of materials having ultra-fine microstructures. The Netherland: Kluwer; 1993. [33] Semjonov SL, Kurkjian CR. Strength of silica optical fibers with micron size flaws. J Non-Cryst Solids 2001;283:220–4. [34] Pharr GM. Measurement of mechanical properties by ultra-low load indentation. Mater Sci Engng, A 1998;253:151–9. [35] Schiffmann KI. Determination of fracture toughness of bulk materials and thin films by nanoindentation: comparison of different models. Phil Mag 2011;91:1163–78. [36] Cuadrado N, Seuba J, Casellas D, Anglada M, Jiménez-Piqué E. Geometry of nanoindentation cube-corner cracks observed by FIB tomography: Implication for
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Equivalent initial flaw size model for fracture strength prediction of optical fibers with indentation flaws
T
⁎
Yuxuan Cui, Yunxia Chen, Jingjing He
Beihang University, School of Reliability and Systems Engineering, No. 37, Xueyuan Road, Haidian District, Beijing 100191, China Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Surface flaw Crack growth Residual strength
This work presents an equivalent initial flaw size (EIFS) model for indentation flaws to quantify the fracture strength of optical fibers. Three geometry parameters of the indentation flaws, namely, the total projection length, the average crack length and the total flaw depth are used to establish the EIFS model. Indentation flaws of different dimensions are made using a Berkovich indenter by adjusting the load force. A stereo microscopic imaging system is employed to acquire the geometry parameters of the indentation flaws. Tensile testing is then performed using 16 commercial optical fiber specimens. A groove fixture is employed to ensure the reliability of data acquisition. The experimental data are used for model development and performance validation. Furthermore, the proposed model is compared with existing models to demonstrate the effectiveness of the method. The proposed EIFS model gives more accurate prediction of fracture strength in current investigation.
1. Introduction Silica optical fibers are widely applied in the fields of telecommunication and structure health monitoring due to the outstanding performance of sensitivity, speed, and flexibility [1]. The mechanical integrity and the strength of optical fibers are critical to the reliability of the optical-fiber-based sensing systems [2,3]. Fracture is the most common failure mode of optical fibers under the combination of mechanical loads and harsh environments [4]. The mechanical strength of the optical fibers can reach up to 10GPa when they are in pristine state; however, surface flaws and impurities can reduce the strength to roughtly one order of magnitude [5,6]. Therefore, it is of great importance to quantify the influence of the surface flaws on the mechanical properties of optical fibers. Two types of flaws commonly exist in optical fibers. The first one is the contamination in glass interior and surface [7]. For example, the gritty particles originate from the impurities of ingredient or from the drawing environment conditions are typical contaminations [8]. The other type of flaws are introduced before the fiber is coated during the drawing process [9,10]. Although severe flaws could be eliminated by the screening procedure, those survived flaws can still pose a great risk for critical applications [6]. In this work the surface flaws are mainly studied to investigate their effect on the fracture strength of optical fibers. Artifical surface flaws have been reported to investigate the mechanical properties of optical fibers [11]. Mechanical abrasion with particles and indentation are two commonly used methods to create artificial flaws. The former one can best represent a natural flaw as the process is very similar to the naturally developed flaws [12]. However, it is difficult to control the flaw size and the residual strength of an optical fiber [13]. Indentation is an alternative method to create surrogate flaws and has been used for the analysis of fatigue ⁎ Corresponding author at: Beihang University, School of Reliability and Systems Engineering, No. 37, Xueyuan Road, Haidian District, Beijing 100191, China. E-mail address: [email protected] (J. He).
https://doi.org/10.1016/j.engfracmech.2019.04.021 Received 5 December 2018; Received in revised form 15 April 2019; Accepted 16 April 2019 Available online 07 May 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
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behavior in glass and optical fiber materials [14–16]. It is convenient to control the residual strength over a wide range by adjusting the indentation load. Sharp pyramid indenters including Vickers, Berkovich and cube-corner are widely applied for inducing indentation, for their advantages on geometrical similarly and residual strength consistency. Vickers indenters induce rectangular pyramid shape indentation flaws while Berkovich and cube-corner indenters result in triangular pyramid shape indentation flaws with different centerline-to-face angles. Despite of indentation induced by sharp indenters, cracks may nucleate due to the remnant plastic deformation and propagate along the radial direction under higher indentation loads [13]. Existing literatures have shown that flaws with radial cracks would cause the fracture strength of optical fibers more than quintuple smaller than that of flaws without radial cracks [10,13]. The influence of indentation flaws and resulting radial cracks on glass and optical fibers have been extensively studied. Nascimento et al. [17] carried out a statistical study on fracture of optical fibers with the indentation flaws, and observed that both Vickers and Berkovich indentation flaws would reduce the strength of optical fiber significantly. Marshall and Lawn reported that the residual contact stress had a detrimental effect on the fracture behavior of Vickers indentation flaws with radial cracks. The authors proposed an indentation load dependent model for the prediction of residual-stress-sensitive strength [18]. The work of Dabbs et al. showed that the indentation flaws without well-developed radial cracks lead to a corner-to-corner failure. In addition, the radial cracks can significantly increase the crack velocity exponent in the crack propagation process [19,20]. Experimental results reported by Jakus et al. [21] indicate that the Weibull median fracture strength decreases significantly as the length of the indentation flaw length increases from 1 μm to 10 μm. It can be seen that both the radial cracks and the size of flaws play important roles in the strength prediction. The mechanical properties of glass and optical fiber materials subject to different shapes of indentation flaws have been studied. Lathabai developed a shear-fault-microcrack model to correlate the stress intensity factor and the geometry parameters of indentation flaws on glass plate [22], which contributes to the strength prediction in inert environment. Jung et al. [23] carried out experiments to measure the strength of brittle material plate containing indentation flaws. A model was developed to describe the correlation between the strength and the geometry parameters such as the characteristic contact radius and the radial crack length. In their study, the critical shear fault was simplified as a mode I crack with a length of the radial crack in the fracture mechanics analysis. Lin and Matthewson [10] compared the inert fracture strength of optical fibers containing Vickers indentation flaws using the fracture mechanics models. Moreover, the inert fracture strength of optical fibers with indentation flaws could be modelled by a crack of size equal to the flaw size. The similar idea can be found in Refs. [21] and [24]. In Ref. [19], the influence of radial crack on the fracture strength of optical fiber is included by increasing the crack velocity exponent. In their study, the crack velocity exponent is 20 for indentation flaws without radial cracks and 30 for indentation flaws with radial cracks. But the correlation between the accurate length of radial crack and the crack velocity exponent is not included in Ref. [19]. In real engineering scenarios, the crack formation during the indentation process has inherent uncertainties due to environment, instrumentation, and operator etc. It is known that the same load can result in indentation flaws with different geometry features, which can obviously induce different crack propagation procedure and further impact the fracture strength of optical fibers [25,26]. Therefore, the morphology of the radial crack is another factor that should be studied for more precise fracture strength evaluation. In the present work, an equivalent initial flaw size (EIFS) model is proposed to predict the fracture strength of optical fiber with indentation flaws. The proposed model can deal with indentation flaws with and without radial cracks. Three geometry quantities of the indentation flaws, namely, the total projection length, the average crack length and the depth of indentation flaw are used to formulate an equivalent initial flaw size (EIFS) model. The EIFS model is used to correlate the fracture strength and the morphology of indentation flaws. A total number of 16 indented optical fiber specimens are used in the tensile test for model development and validation. The remainder of the study is organized as follows: Section 2 describes the development of the equivalent initial flaw size (EIFS) model. The geometry quantities used to build the model are described in details, and a linear model is proposed based on the geometry quantities. The indentation procedure and tensile testing are introduced in Section 3. The indentation flaws induced by the Berkovich indenter are measured using a microscopy system. Tensile testing is carried out for the optical fiber specimens with a constant strain rate of 5%/min. Following that, validations are made by comparing the model prediction of fracture strength and experimental data in Section 4. 2. Equivalent initial flaw size (EIFS) modeling 2.1. Fracture strength evaluation The failure of optical fibers due to strength reduction caused by flaws is dominated by the crack initiation and propagation process. For optical fibers under the tensile testing, the linear elastic fracture mechanics is used to describe the crack growth. A power law form relation is normally assumed between the time-dependent crack growth rate c ̇ (t ) and the stress intensity factor at the tip of the crack [27]:
c ̇ (t ) = A·[KI (t ) ]n ,
(1)
where A is an environmental dependent parameter and n is the crack velocity exponent. A linear relationship between logA and n is determined from the experimental results [28]. The applied stress intensity factor KI (t ) for a crack of depth c(t) subjected to the applied stress σa (t ) at time t in tensile test is given by [13]
KI (t ) = Y[c (t )]·[σa (t )]·[c (t ) ]1/2 ,
(2)
where Y[c(t)] is the geometric correction factor and is also a time-dependent parameter related to the crack depth c(t). The 37
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Fig. 1. Diagram of the cross section of the glass core with an initial crack.
relationship between the geometric correction factor and the crack depth can be found in Ref. [29]. It is known that the surface cracks in optical fibers propagate in a semi-elliptical pattern [27,30]. A semi-elliptical crack is assumed to represent the initial flaw as shown in Fig. 1. The symbols in Fig. 1 denote the following meanings. r is the radius of the glass core, c(t0 = 0) is the depth of the initial crack and w is half of the initial crack width on the outer surface. The condition t0 = 0 represents the initial state of the crack. Failure is considered when the current stress intensity factor (SIF) exceeds the fracture toughness KIC given in [31]
KIC = Y [c (t f )]·σf ·[c (t f )]1/2 ,
(3) 1/2
for KIC is used in this where σf is the fractured stress and c(tf) is the crack depth at the fracture time tf. The value of 0.58 MPa·m study. Because the optical fiber used in the study is mainly composed of silica glass, which is the same material in Ref. [32]. The same value of KIC has also been adopted in strength evaluation of optical fiber in Ref. [33]. During the tensile test, the total differential of the applied stress to the crack depth can be obtained from Eqs. (1) and (2),
d [σa (t )] σȧ σȧ = = , d [c (t )] A·[KI (t )]n A·Y [c (t )]n ·[σa (t )]n ·[c (t )]n/2
(4)
where σ̇a is the stress rate. By separating the variables in Eq. (4) and integrating versus time from the initial moment t0 = 0 to the fracture time tf, the following equation can be obtained.
1 σ̇ ·[σ fn + 1 − (σa (0))n + 1] = a · n+1 A
c (t f )
∫c (t =0) 0
d [c (t )] , Y [c (t )]n ·[c (t )]n /2
(5)
where σa(0) is the initial applied stress which equals 0, and the depth of the initial crack c(t0 = 0) is used to represent the equivalent initial flaw size. In this model, σf and σ̇a can be obtained from the tensile testing. Next the geometry representation of the indentation flaw and the process of equivalent initial flaw size modeling is described. 2.2. Equivalent initial flaw size (EIFS) modeling In this study, a Berkovich indenter with 65.3° centerline-to-face angle is adopted in the experimental procedure to create indentation flaws. The cracking threshold for the Berkovich indenter is similar to that of Vickers indenter, around 1000 mN to 1500 mN for fused silica [32,34]. Other experimental conditions including surface condition, relative humidity, and indentation duration also have an effect on the formation of radial cracks. The radial cracks can increase the stress concentration at the tip and drastically reduce the fracture strength of optical fibers [13]. Fig. 2 shows both diagrams of the triangular pyramid shape indentation flaws with and without radial cracks. It has been reported for the cube-corner indenter, which also induces triangular pyramid shape indentation similar with the Berkovich indenter, the localized tangential stress during loading encourages the formation of radial cracks at the corners (from B to C in Fig. 2(a)). More importantly, these radial cracks intersect with each other beneath the plastic deformation zone (intersecting at E in Fig. 2(a)), making the total flaw depth (h in Fig. 2(a)) larger than the depth of the indent (h′ in Fig. 2(a)). According to the focused ion beam tomography reported in Refs. [35,36], the cracks on glass induced by cube-corner indenters can be depicted as quarter-ellipse. On the other hand, the finite element analysis results in Ref. [37] have also shown that the radial crack created by Berkovich indentation can be approximately depicted as quarter-ellipse cracks. Therefore, the same assumption of quarterellipse crack has been adopted in current study. The symbols in Fig. 2 denote the following meanings. x is the axial direction of the optical fiber, y is the diametrical direction, and d is the diameter of the optical fiber glass core. The term e is the total projection length including crack and a′ is the projection lengths of the triangular pyramid indent along the x-axis. For flaws with radial cracks, the values of e equal to the projection length including both the triangular pyramid indent (a′) and radial cracks. For flaws without radial cracks, the values of e equal to those of a′. l is the average crack length from the indent corner to the crack tip. h′ and h are the depth of indent and the total flaw depth including cracks. For flaws without radial cracks, the values of h equal to those of h′, which are obtained from the indentation load-depth curve. Both the failure of indentation flaws with and without radial cracks are controlled by the combination of applied stress and 38
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(a) Dimension diagrams of an indentation flaw with radial cracks
(b) Dimension diagrams of an indentation flaw without radial cracks Fig. 2. Dimension diagrams of the indentation flaws. (a) dimension diagrams of an indentation flaw with radial cracks, (b) dimension diagrams of an indentation flaw without radial cracks.
residual stress induced during the indentation process. For the case of flaws with radial cracks, the crack propagation process will contribute to the most part of the fracture process of optical fibers with the absence of crack initiation. As the crack gradually propagates, longer crack length and deeper crack depth would increase the stress intensity factor and further accelerate the fracture process until the critical threshold [38]. On the other hand, for the case of flaws without radial cracks, the fracture process includes the crack initiation and the following crack propagation. Under the influence of the near-field residual stress, cracks initiate at the corners of the flaw. In contrast, the pressure exerted by the plastic zone on the surrounding matrix results in the far-field residual stress and further controls the crack arrest [21]. The strength of flaws without cracks would be much higher than that of flaws with radial cracks, which is confirmed by both the experiment in this study and Ref. [10]. The flaws without radial cracks are also characterized by the flaw length on the surface according to Ref. [39] and the flaw depth based on experiments in section 3. Therefore, both the total length e and the flaw depth h are selected to quantify the equivalent initial flaw size for flaws. The model form in Ref. [40] for charactering the flaw geometry is adopted in this work. The proposed equivalent initial flaw size model is expressed as following.
A1·(e·h)1/2 + B1, with radial cracks c (t 0 = 0) = ⎧ 1/2 ⎨ A ⎩ 2·(e·h) + B2, without radial cracks
(6)
The depth of the initial crack c(t0 = 0) is used to represent the equivalent initial flaw size at time t0 = 0, and A1, A2, B1 and B2 are model parameters obtained through experimental data. Based on the assumption of crack geometry, the flaw depth h can be expressed with the average crack length l multiplying a coefficient k. The parameter k as well as the model parameters A1, A2, B1 and B2 can be obtained by the least square method. And the flaw depth h equals to the indent depth h′ for flaws without cracks. As shown in Eq. (6), the initial flaw size c(t0 = 0) in Fig. 1 can be expressed as a function of flaw geometry parameters. By this means, the influences of the flaw dimension and radial cracks are incorporated for more accurate fracture strength evaluation. It is worth mentioning that both the material property and the experimental condition may affect the values of these parameters. The proposed method provides a general framework of fracture strength prediction including the initial flaw influence. Therefore, the value of the model parameters used in this paper may need to be adjusted for different scenarios. 39
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Fig. 3. The overall framework of the modeling and experimental process.
3. Experiments for model development Section 2 proposes an equivalent initial flaw size model to establish a quantitative relation between the flaw geometries and fracture strength. The overall framework of the modeling and experimental work is shown in Fig. 3. First, indentation flaws are induced using a Berkovich indenter. Next, the measurements of the geometry parameters are acquired through an optical microscopic imaging system, and the geometry parameters are used to obtain the EIFS model parameters with the tensile testing data. The experimental procedures are introduced in detail as following.
3.1. Material and specimen preparation The specimens used in the experiments (shown in Fig. 4) are single-mode optical fibers with a 125 μm-diameter glass core and a 245 μm-diameter polymer coating layer. Both ends of the specimens are glued to carbon fiber sheets with epoxy resin as shown in Fig. 4a. Plastic tapes are adhesived parallel to the optical fiber to provide extra support. Next, the polymer coatings are removed for the indentation procedure. To avoid causing mechanical damage to the glass core, about 1 cm of the central section of the optical fiber is immersed in tetrahydrofuran for a duration of 180 seconds. After that the specimens are rinsed in acetone to remove the residual tetrahydrofuran. The partial enlarged detail of the processed optical fiber specimen is shown in Fig. 4b. Total 16 specimens are used in the pre-indentation procedure and the subsequent tensile testing. The surface flaws are introduced by the nanoindentation method. One of the most important issues is to keep the circular optical fiber stable during the indentation process. The indenter would slip to one side of the fiber once the optical fiber moves. Before the indentation process, the uncoated parts of optical fiber samples are put on the glass substrate with V-shape grooves and then fixed. The schematic of a substrate used for 125 μm optical fiber cores is shown in Fig. 5a. It is also very important to ensure that the glass core is not damaged during the experiments. Therefore, the ethyl acrylate with low viscosity is smeared on the grooves of the substrate to provide extra protection. While the ethyl acrylate is half cured, the uncoated glass sections of the fibers are put in the grooves with an optical microscope. After the ethyl acrylate is fully solidified, it fixes the bare glass core and provided an elastic buffer to protect it from the mechanical damage (Fig. 5b).
Fig. 4. Specimen for indentation: (a) optical fiber specimen fixed to carbon fiber sheet; (b) partial enlarged detail of an uncoated optical fiber specimen. 40
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Fig. 5. Grooved substrates used to stabilize optical fibers. (a) the side view diagram of the glass substrate and the V-shape grooves with an optical fiber, and (b) the top view microscopic image of the grooves and an optical fiber.
3.2. Indentation procedure The indentation process is carried out by a TI950-triboindenter indentation instrument as shown in Fig. 6. After the indentation location is chosen through an in-situ optical microscopy, a Berkovich indenter is used to create surface flaws [33,41]. The indenter can create tetrahedral shape flaws with extended radial cracks originating from the three corners. The angle between the faces of the indenter and the axis of the symmetry is 65.3°. A constant loading rate of 2.5% per second (P′/P = 0.025 s−1) is applied during the loading procedure. The indentation loads are set between 500mN and 2200mN to induce both flaws with and without radial cracks. The indenter is unloaded at a half speed of the loading after reaching the preset load. During the whole indentation process, the load and displacement are updated and recorded in real time. Fig. 7 presents a typical load-depth response curve of the indentation procedure. The depth of the indent h′ can be obtained from the load-depth curve. 3.3. Indentation flaw measurement The geometry parameters of the indentation flaws are acquired with a calibrated stereo microscope. Typical flaw morphologies are shown in Fig. 8. Parameters characterizing the flaws include the total length e, the average crack length l and the total depth h for flaws with cracks; the total length a′ and the indent depth h for flaws without cracks. The detailed geometry parameters of the flaws on each specimens are presented in Table 1. Radial cracks can originate from the corners of the indentations in a random pattern, as reported in Refs. [13,42]. 3.4. Tensile testing The tensile testing is performed using the IBTC-5000 testing machine in this work. Specimens are preconditioned in the testing environment and the testing is conducted in air environment (25 °C and 40% relative humidity) as shown in Fig. 9. Tensile loads with a constant strain rate of 5%/min are applied on the optical fibers until completely fractured. The loading information and the testing results are listed in Table 2. The geometry parameters of indentation flaws and tensile
Fig. 6. Indentation procedure. 41
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Fig. 7. Typical indentation load-depth curve.
Fig. 8. Morphologies of the indentation flaws. (a) an indentation flaw with radial cracks, and (b) an indentation flaw without radial cracks. Table 1 Geometry parameters of the flaws. Specimen No.
Indentation load P/mN
Flaw length e/μm
Indent depth h'/μm
Average crack length l/μm
With radial cracks
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
500 600 700 800 900 1000 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200
18.618 14.580 13.342 14.099 15.968 13.348 33.050 32.971 42.169 33.025 17.030 41.856 43.676 49.632 27.175 29.214
– – – 2.314 1.325 2.658 – – – – 3.783 – – – 6.597 4.063
4.51 3.67 2.73 – – – 11.74 20.98 22.19 29.39 – 25.65 22.98 29.98 – –
Y Y Y N N N Y Y Y Y N Y Y Y N N
42
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Fig. 9. Tensile testing setup. (a) tensile testing machine and computer, and (b) sample and fixtures. Table 2 Experimental results of tensile testing. Specimen No.
σf /MPa
t f /s
σ̇a /MPa/s
Usage
With radial cracks
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
111.72 137.47 146.03 675.20 513.70 427.24 107.81 76.03 94.36 78.80 366.69 83.12 78.15 79.37 291.40 263.77
9.9 10.6 10.6 49.9 38.2 34.5 7.4 12.9 7.3 9.1 27.7 6.6 6.9 7.5 22.3 20.4
11.285 12.969 13.776 13.531 13.464 12.384 14.569 5.894 12.926 8.659 13.237 12.593 11.326 10.582 13.038 12.939
Regression Regression Validation Regression Regression Validation Validation Regression Regression Regression Regression Validation Regression Regression Validation Regression
Y Y Y N N N Y Y Y Y N Y Y Y N N
testing results are used for EIFS modeling. The experimental data are divided into two sets, one for EIFS model parameter estimation and the other for model validation, as identified in Table 2. The parameter estimation procedure of the EIFS model is described in the following steps: 1) Calculate the critical crack depth cf using Eq. (3); 2) Calculate the EIFS c(t0 = 0) using Eq. (5). The integral formula can be numerically solved by substituting the fracture stress σf and stress rate σ̇a from the experiments and critical crack depth cf in step 1). The values of the crack velocity exponent n are 20 for flaws without cracks and 30 for flaws with cracks; and 3) Obtain model parameters using the least square method. Using the above described procedure and data in Table 1, the model parameters of the proposed EIFS model can be obtained: A1 = 7.743 × 10−3, B1 = −1.447 × 10−2, A2 = 6.287 × 10−4, B2 = −2.645 × 10−3. The calculated values and regression values of the equivalent initial flaw sizes are plotted in Fig. 10 and it can be seen that regression values are in good agreement with the calculated values. Besides, the results indicate that the radial cracks have a significant impact on the equivalent initial flaw size. According to the tensile test results, the sizes of the equivalent initial flaw c(t0 = 0) with radial cracks are in the range from 1 × 10−2μm to 8 × 10−2μm, while the sizes of the equivalent initial flaw c(t0 = 0) of flaws without radial cracks are in the range from 1 × 10−3μm to 4 × 10−3μm. In this study, experimental results show that the difference in EIFS results in great disparity of fracture strengths. Based on the proposed EIFS model, larger values of the geometry parameters would lead to great equivalent intial flaw size, and the data in Tables 1 and 2 also indicate that optical fibers with larger flaws have lower fracture strength. This is physically reasonable since the greater radial crack length and depth would cause the greater stress intensity and give rise to stress concentration at the crack tip for flaws with radial cracks. And for flaws without cracks, larger indent size would increase the nearfield residual stress promoting the crack initiation and on the other hand decrease the sectional area under tensile stress which further increase the stress intensity. As a contrast, the flaw type has much greater effect on the fracture strength. The EFISs of most flaws with 43
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Fig. 10. The relationship between regression values and calculated values of c(t0 = 0).
radial cracks are in the range from 1 × 10−2 μm to 9 × 10−2 μm, while the values of flaws without cracks are between 9 × 10−4 μm and 5 × 10−3 μm according to the proposed model. It is also shown in Table 2 that most of the fracture strengths of flaws without radial cracks are over 300 MPa, whereas a majority of the fracture strengths of flaws with radial cracks are around 100 MPa. Similarly, experimental results in Refs. [10,13] indicated strengths of flaws without radial cracks are five times higher than those of flaws with radial cracks using the Vickers indentation method. Therefore, indentation flaws with or without radial cracks are considered as two different physical models in this study to better describe their influences on the fracture strength. The proposed method provides a framework to calculate the EIFS of flaws and to evaluate the corresponding tensile strength under the given test condition. It provides a general tool to evaluate the fracture strength of optical fibers with initial flaw for quality control purpose. For example, the relationship between the EIFS and the tensile strength can be obtained as Fig. 11 given the stress rate equals to 13 MPa/s and the crack velocity exponent equals 20 It is convenient to predict the strength using the relationship described in the Fig. 11 once the flaw geometry parameters are measured and the EIFS is calculated.
4. Methodology validation Five specimens are randomly chosen to validate the effectiveness of the proposed EIFS model, as shown in Table 2. Firstly, the EIFS of the validation specimens are calculated using the proposed EIFS model. Next, the fracture strengths are predicted using the resulting EIFS model. The calculated EIFS, the predicted fracture strength, and the actual fracture strength are listed in Table 3. The model prediction and actual values of the fracture strength are compared and shown in Fig. 12. A scatter factor of 1.2 is used to quantify the performance of the model without loss of generality [43]. It can be seen from Fig. 12
Fig. 11. The relationship between the predicted fracture strength and EIFS c(t0 = 0). 44
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Table 3 Results comparison between experimental and predicted tensile strength. Specimen No.
c(t0 = 0)/μm
Predicted fracture strength/MPa
Actual fracture strength/MPa
With radial cracks
3 6 7 12 15
0.010418 0.001100 0.044293 0.076668 0.003840
164.29 482.92 106.85 86.39 276.25
146.03 427.24 107.81 83.12 291.40
Y N Y Y N
Fig. 12. The comparison between the predicted and actual values of the fracture strength.
that all the data points are within the region of the scatter limit (shown with dashed lines). Fig. 12 also demonstrates that the radial crack has a significant effect on the strength of optical fibers. It can increase the stress concentration around the crack tip, making the optical fiber fracture more easily. According to the established EIFS model, the predicted fracture strengths of indentation flaws without radial cracks are 3 to 7 times larger than that of flaws with radial cracks. The result is consistent with Lin and Matthewson’s observation [10]. In addition, the proposed model is compared with existing models for the prediction of fracture strengths. Jung et al. proposed indentation load dependent fracture strength prediction models for triangular indentation flaws (Berkovich indenter) with and without radial cracks [23]. For indentation flaws without radial cracks, the fracture strength is
1 α·H 1/4 ⎞ ·KIC − κ·H⎤, σf = ⎛ ⎞·⎡ ⎛ ⎢ ⎥ ⎝ λ ⎠ ⎣⎝ P ⎠ ⎦
(7)
where λ and κ are crack-geometry coefficients, α the indent center-to-corner dimension, H the material hardness and P is the indentation load. For a Berkovich indenter, the values of α is 33/2/4 [23]. The values of λ and κ for soda-lime glass are adopted here, which are 0.27, 0.015 respectively [44], because the soda-lime glass has the same shear fault configuration as fused silica glass used in this work. The hardness of fused silica is 7GPa [10]. For indentation flaws with radial cracks, the fracture strength is 1/3
K4 3 1 σf = ⎜⎛ ⎟⎞·⎛⎜ ⎞⎟·⎛⎜ IC ⎞⎟ 4 ψ 4 χ ⎝ ⎠⎝ ⎠⎝ P ⎠
,
(8)
where χ is a dimensionless indenter/specimen constant, ψ is the dimensionless crack shape parameter and P is the indentation load. For a Berkovich indenter, the value of χ is 0.1450 [18,44], and the value of ψ is 0.4497 [10,45]. In Jung’s model, the parameters λ , κ and ψ are crack-geometry parameters, α and χ the indenter dependent parameters and H the material parameter. One of the advantage of Jung’s model is that it does not need to fit the model parameters for different experimental conditions or materials. It takes information of material, indenter, flaw shape and load to predict the strength with Jung’s model. Although the proposed EIFS model needs data of flaw geometry to fit the parameters, the advantage is that the uncertainty of environment and instrumentation is contained during the parameter fitting process. Therefore the proposed model could provide more accurate prediction. The predicted values of fracture strengths of optical fibers calculated from the proposed EIFS model and the 45
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Fig. 13. Comparisons of the model prediction between the proposed model and existing models.
two mentioned models are plotted in Fig. 13 together with the validation experimental data. Fig. 13 signifies that the proposed EIFS model provides more accurate predictions of the fracture strength than Jung’s model. In particular, Jung’s model tends to produce conservative results compared with experimental data. It may cause by the direct simplification of this critical shear faults as mode I cracks which ignores the crack propagation process under different angles. Besides, Jung’s model fails to predict the strength of glass with flaws around the cracking threshold load as shown in Ref. [23]. Fig. 14 shows the relative prediction errors between predicted and actual value of the fracture strength for Jung’s model and the proposed EIFS model, where an overall smaller relative error can be achieved using the proposed model. The maximum relative errors are less than 13% for both flaws with and without radial cracks. 5. Conclusion This paper presents an equivalent initial flaw size model to investigate the influence of indentation flaws on the fracture behavior of optical fiber. In particular, three geometry parameters including the total projection length, average crack length and total flaw depth are used to establish a linear model to correlate with the EIFS. Artificial indentations are induced in 16 optical fiber specimens using the Berkovich indenter. The geometry parameters are subsequently acquired using a stereo microscopy imaging system. Tensile testing are conduced using commercial single mode optical fibers, and experimental data are used for model parameter estimation and model validation. Following conclusions are drawn based on the current investigation. (1) The proposed EIFS model can incorporate the radial cracks in a universal model format. The effectiveness and accuracy of the proposed model are validated using tensile testing data. Results indicate that the proposed method can predict the fracture
Fig. 14. Comparisons of relative error between the proposed model and Jung’s model. 46
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strength reliably, and the independent validation points are within a scatter band of a factor of 1.2. (2) The performance of the proposed EIFS model is compared with existing models. The results show that the proposed model is able to predict the fracture strength more accurately in terms of the relative error. In this work several geometry parameters including the total length, the average crack length, the crack depth and the indent depth are adopted to characterize the flaws and evaluate the strength. In order to quantitatively analyse the effect of these parameters on the fracture strength, experiments with different values of the above geometry parameters are needed to provide a sensitive model. Besides, currently the typical Berkovich indenter is used in the study. Ideally, the EIFS model could be applied on different kinds of indenters including the pyramid indenter and the cube-corner indenter, since the proposed method provides a general framework to quantitatively characterize the indentation flaw geometry and predict the strength with EIFS. However, comprehensive research including both theoretical and experimental work are also required to confirm the feasibility and robustness of the proposed method under different applications. Acknowledgments This work is supported by the National Key R&D Program of China (Grant No. 2016YFB0100400) and the National Natural Science Foundation of China (Grant No. 51675025). The supports are greatly acknowledged. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.04.021. References [1] Zhang Z, Ansari F. Crack tip opening displacement in micro-cracked concrete by an embedded optical fiber sensor. 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