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Critical bending load of CFRP panel with shallow surface scratch determined by a tensile strength model
Composites Science and Technology 191 (2020) 108072
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Critical bending load of CFRP panel with shallow surface scratch determined by a tensile strength model Bingyan Yuan , Yunsen Hu , Xiaozhi Hu * Department of Mechanical Engineering, The University of Western Australia, Perth, WA, 6009, Australia
A R T I C L E I N F O
A B S T R A C T
Keywords: Composite fracture Shallow surface scratch “Composite” tensile strength Non-LEFM model Normal distribution
Carbon fibre reinforced polymer (CFRP) composite structures may experience light surface damage such as shallow surface scratch around 50 μm or less in applications. In this study, critical bending loads of 2D woven CFRP panels with and without shallow surface scratches were assessed under three-point-bending (3-p-b) con ditions and then modelled by a simple strength theory for composite fracture. This closed-form tensile strength solution is a non- Linear Elastic Fracture Mechanics (LEFM) model derived specifically for heterogeneous composites. Direct tensile tests of the CFRP panels (without shallow surface scratch) were also performed, and measured tensile strength was compared with that from 3-p-b tests. The relative error between the two different tests was only around 6.7%. The ply thickness around 140 μm was selected as the controlling composite microstructure parameter or characteristic composite unit Cch in this non-LEFM model. This simple model shows that a suitable composite characteristic microstructure measurement such as Cch ¼ ply thickness can significantly simplify composite fracture analysis. A simple statistical analysis using a 2D normal distribution methodology for strength measurements was also introduced so that the experimental scatters can be analysed. The 95% reli ability band can be conveniently used in safe design of composite structures.
1. Introduction Carbon fibre reinforced polymer (CFRP) composites enable signifi cant weight saving and complex shape design in aircraft manufacturing through their usage not only in secondary structures including stabil isers and trailing edge panels but also in primary structures such as fu selages and wings [1,2]. Nowadays, CFRP has increasingly been used for the front air blades in aircraft gas turbine engines, as shown in Fig. 1(a). GE has used the 4th-generation carbon fibre composite fan blades in GE9X engine for lighter and more durable structures instead of metal blades, and has thus reduced the fuel consumption [3]. Protective metal alloy thin plates are used at the leading edges of CFRP blades in order to prevent blade edge erosion due to potential high-speed impacts of micro-debris. However, shallow surface scratches, around 50 μm or less in depth, may still be induced along the blade surface with prolonged service time. This kind of surface damage can also occur in other large thin plate-like CFRP structures through surface contacts. Here lies a practical and potentially important issue of engineering challenge, i.e., to predict bulk mechanical properties of CFRP panel structures with shallow surface crack/damage as shown in Fig. 1(b),
where both the depth and width of a shallow scratch on a CFRP panel surface are around 50 μm or less. The classic Linear Elastic Fracture Mechanics (LEFM) and traditional Strength of Materials (SoM) limited to homogeneous and isotropic materials cannot be used for fracture anal ysis of a shallow crack in CFRP panel with complicated hierarchical microstructures. For instance, a shallow surface scratch less than 50 μm in depth is well within one carbon fibre ply around 120 μm in thickness, containing woven carbon fibres around 5 μm in diameter. Yet, it seems there is no simple closed-form solution available in literature for failure analysis of CFRP panels or carbon fibre composites in general [4–12]. Various micro-mechanics models have been proposed for fracture analysis of laminar composites, considering micro-damage such as delamination and crack bridging [13–15]. While micro-mechanics modelling is specifically useful for a good understanding of detailed fracture mechanisms, a few assumptions made for material properties relevant to micro-cracking and damage are common, and a closed-form solution normally cannot be derived for macro or bulk composite properties. In this study, we adopt a macro-mechanics approach so that a simple closed-form model recently developed for quasi-brittle fracture of het erogeneous solids can be tested for the first time to determine the
* Corresponding author. E-mail address: [email protected] (X. Hu). https://doi.org/10.1016/j.compscitech.2020.108072 Received 24 July 2019; Received in revised form 7 February 2020; Accepted 13 February 2020 Available online 15 February 2020 0266-3538/© 2020 Elsevier Ltd. All rights reserved.
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Composites Science and Technology 191 (2020) 108072
in Fig. 2(a) is chosen in this study to illustrate how Pmax is linked to the “composite” tensile strength ft. It should be emphasized that ft of CFRP defined in this study is a material constant. Common strength estima tions from fracture loads of CFRP samples using classic SoM (limited only to homogeneous materials) are not constant, but sample thickness and size dependent, which is particularly true for 3-p-b conditions. The new non-LEFM model adopted in this study uses the same “composite” tensile strength ft (a material constant) to model both 3-p-b fracture and direct tensile failure of CFRP. Furthermore, it is based on one of the most fundamental relations in solid mechanics for material failure, i.e., Fracture Load ¼ Tensile Strength x Area (cross-section area for uniform tensile fracture). The model has defined an equivalent area Ae for 3-p-b conditions so that the relation still holds, i.e., Fracture Load ¼ Tensile Strength x Ae [21].
Nomenclature a0 ae Y(α) a*ch Δafic W B S
α
Cch Ae Pmax ft KIC
σn βav
μ σ σf
initial crack/scratch depth equivalent crack depth geometry factor characteristic crack fictitious crack sample thickness sample width span length ratio of a0 and W characteristic composite unit equivalent area fracture load “composite” tensile strength “composite” fracture toughness nominal strength average discrete number mean value standard deviation flexural strength
2. Fracture modelling of CFRP panel with shallow surface scratch Fracture of composites of any kind including CFRP panels cannot be modelled by LEFM due to the heterogeneous composite structures. Extensive non-LEFM modelling has been done for brittle heterogeneous solids such as rock and concrete [22,23]. In those cases, the average grain size G or aggregate size dav is the characteristic composite unit Cch, while ply thickness is Cch for laminated CFRP panel in spite of its com plex structural composition. The key equations and the logic in model ling are summarized below. Unlike homogeneous material whose brittle fracture is defined by fracture toughness of LEFM, heterogeneous composites possessing hi erarchical microstructures typically displace quasi-brittle fracture behaviour determined by both “composite” tensile strength ft and “composite” fracture toughness KIC, as shown in Fig. 3. The comparable ratio of crack-size/composite-microstructure means that LEFM is hardly relevant to composite failure, i.e., the strength region and quasi-brittle fracture region (both are strongly influenced by composite structures) should be considered for CFRP. For simplicity and due to the limited size of Δafic at Pmax, a constant cohesive stress distribution is assumed and marked by the nominal strength σn as in Fig. 2(a). The relationship between nominal strength σ n at the micro-cracking zone Δafic in Fig. 2(a) and “composite” tensile
fracture loads of CFRP panels with shallow surface scratches under three-point-bending (3-p-b) conditions. This non-LEFM model specially derived for heterogeneous composites has been successfully used to analyse fracture of bone-like bamboo composites [16], concrete and granite [17–19], and polysilicon micro-components with nano-defects [20]. Despite completely different micro-mechanisms of cracking and fracture existing in laminar CFRP and granular/particle composites, this macro-mechanics approach considers a combined non-LEFM region through a combined fictitious crack length Δafic in front of the initial crack tip a0 as shown in Fig. 2(a), so that the overall effects of different micro-damage and cracking activities in various composites can be modelled through specific properties over the non-linear zone Δafic. The maximum fracture load Pmax of a CFRP panel with a shallow surface scratch is the primary concern. The common 3-p-b test geometry
Fig. 1. (a) GE9X engine, with fan case and 16 fan blades made from CFRP [3]. While the CFRP blades are protected at the leading edge, shallow scatch on the blade surface is still possible. (b) Shallow surface scratch in CFRP panel, dark blue indicating the depth of scratch and yellow areas showing surface lift due to edge delamination, as measured by Altisurf 520 (ALTIMET, France). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 2
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Composites Science and Technology 191 (2020) 108072
Fig. 2. (a) Stress distribution of CFRP panel under 3-p-b condition with a shallow surface scratch or initial crack a0. The standing-out ply thickness is selected as the characteristic com posite unit Cch. (b) Through-thickness micro-cracking in 2D woven fibre structures in front of the initial crack a0, modelled by the fictitious crack Δafic, covering the crack-tip damage zone. (c) X-ray μCT images of longitudinal section of a g-2.09-notched specimen after 3-p-b test.
� �2 KIC � 3Cch a*ch ¼ 0:25 ft
(3)
By combining Eqs. (1) and (3), σn is calculated as Eq. (4). f
σ n ¼ qffiffiffiffiffitffiffiffiffiffiffiffiffiffi 1 þ 3Caech
The fictitious crack Δafic in front of the initial crack tip a0 is linked to Cch through a discrete number βav as Eq. (5). The discrete number βav (e. g. 0, 0.5, 1.0, 1.5, 2.0) is used to model the discontinuous crack growth in composites. For a heterogeneous polycrystalline solid with an average grain size G, variation in the discrete number βav is due to randomly distributed coarse aggregate structures varying from specimen to spec imen [22]. For 2D twill-weaved CFRP panel, the uncertainty of βav is attributed to the fact that under an trans-laminar Mode-I fracture load, crack from a0 propagates much more effortlessly through transverse fi bres than through longitudinal fibres, as demonstrated in Fig. 2(b). In this study, we test the βav solution for polycrystalline solids, i.e., βav ¼ 1.5 [16,19], and check the applicability of the following equation to CFRP using experimental results from both 3-p-b and direct tensile tests.
Fig. 3. Non-LEFM fracture of CFRP panels with shallow surface scratch, which is strongly influenced by heterogeneous composite microstructures. LEFM fracture of homogeneous material defined by KIC is not applicable [4,16].
Δafic ¼ βav ⋅Cch ¼ 1:5Cch
strength ft (a material constant) is calculated by Eq. (1), where ae is the equivalent crack size and a*ch is the characteristic crack. f
1 þ a*e
ch
The equivalent crack size ae is defined by Eq. (2), where a0 is initial crack depth, α is ratio of a0 and W (sample thickness) and Y(α) is geo metric function. Since we concentrate on shallow surface scratch in this study, whose α is far smaller than 1, Y(α) is fairly close to 1.12. ae ¼
� ð1
�2
αÞ2 ⋅YðαÞ 1:12
⋅a0 ¼ ð1
αÞ4 ⋅a0
(5)
Following Fig. 2(a), σ n can be calculated by the equilibrium condi tions and the result is given by Eq. (6), where Pmax is fracture load, S is span length and B is sample width, considering the strain condition along the crack plane and two equilibrium conditions for bending stress and bending moment after Δafic is determined [23]. �� 1:5 BS ⋅Pmax 3Pmax ⋅S �¼ σn ¼ (6) ðW a0 Þ⋅ðW a0 þ 3Cch Þ 2BðW a0 Þ⋅ W a0 þ 2Δafic
(1)
σ n ¼ qffiffiffiffitffiffiffiffiffiaffiffiffi
(4)
By combining Eqs. (4) and (6), a linear relationship between Pmax and ft can be obtained as Eq. (7), where the equivalent area Ae is fully determined by W, α and Cch, i.e., Fracture load ¼ Tensile Strength x Ae. The closed-form solution of Eq. (7) on the linear relation between Pmax and Ae is easy to use as all parameters are well defined. The “composite” tensile strength ft determines the slope of the straight line.
(2)
The characteristic crack a*ch as given in Eq. (3) is a material constant related to the bulk composite material properties ft and KIC, which is the intersection point of strength criterion and toughness criterion in Fig. 3. And it is explicitly linked to characteristic composite unit Cch, according to previous study [16]. 3
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Composites Science and Technology 191 (2020) 108072
� � W 2 ⋅ð1 αÞ⋅ 1 α þ 3CWch � � qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pmax ¼ ft ⋅ Ae ðW; α; Cch Þ ¼ ft ⋅ 1:5 BS ⋅ 1 þ 3Caech
(7)
After ft is determined, the “composite” fracture toughness KIC (also a material constant) can be determined using ft and Cch as Eq. (8) ac cording to Eq. (3), so that the failure of CFRP with or without (α ¼ 0) shallow surface scratch can be predicted by the same strength criterion. (8)
KIC ¼ 2ft √ð3Cch Þ
It should be emphasized that a thin plate such as the 3-p-b sample illustrated in Fig. 2(a) can never be used to directly measure KIC following LEFM, which requires a sufficiently long crack and a suffi ciently thick CFRP panel to ensure the homogeneous material assump tion. The non-LEFM model adopted in this study has effectively removed the stringent requirements of LEFM, making theoretical prediction of the fracture load Pmax of a CFRP panel with a shallow surface scratch possible.
Fig. 4. (a) Dimensions (in mm) of dumbbell-shaped specimen for direct tensile tests (without shallow surface scratch). (b) Typical fracture location, indicating stress concentration from machining (only slight edge grinding using 120 Grit Sandpaper).
3. Sample preparation and experimental methods
specimen was easily determined by using Eq. (7), as listed in Table 2. Experimental scatters were noticed among specimens of different thickness and notch condition, so that a simple statistical analysis methodology (2D normal distribution) as in refs. [16,19] was adopted for ft measurements. Eq. (9) gives the basic formula, where μ is the mean value and σ is the standard deviation of the normal distribution.
2D woven CFRP panels (CarbonWiz Technology Limited, China) in 2.09 mm and 1.44 mm thick were used, whose ply thickness was around 140 μm as provided by the manufacturer and was adopted as the char acteristic composite unit Cch in the non-LEFM model. Panels were cut into 15 mm wide beams, whose length was 90 mm and 62 mm respec tively. The initial notch or surface scratch a0 was manually introduced along the centre line in the bottom surface (as illustrated in Fig. 2(a)) using a cutter. The average notch depth was 0.03 mm although actual depth varied, measured by Altisurf 520 (ALTIMET, France) as shown in Fig. 1(b). Corresponding notch-thickness ratios were listed in Table 1. 3-p-b tests were conducted using an Instron 5982 universal me chanical testing machine with a 100 kN load cell and 2.5 mm/min loading speed according to ASTM D 7264. Span-to-thickness ratio was 32. 10 samples were tested for each group, i.e., g-2.09, g-1.44, g-2.09notched and g-1.44-notched. Panels were also cut into dumbbell-shaped samples (dimensions given in Fig. 4(a)) for direct tensile tests (without shallow surface scratch), conducted using the same machine with 1 mm/ min loading speed. 10 specimens were tested for 2.09 mm panel and 5 specimens for 1.44 mm panel. A g-2.09-notched specimen with a0 ¼ 22 μm after 3-p-b test was scanned at 40 kV and 74 μA using an X-ray microcomputed tomography (X-ray μCT) system (Versa 520, Zeiss, Pleasanton, CA, USA) and images of longitudinal section were given in Fig. 2(c). Trans-laminar Mode-I crack growth from a0 was seen, around 2Cch long, so that the assumption of Δafic ¼ 1.5Cch before Pmax is reasonable.
1 f ðxÞ ¼ pffiffiffiffiffi e 2πσ
ft values from all 40 Pmax measurements and normal distribution analysis of them were plotted in Fig. 5(a), where μ ¼ 841.33 MPa and σ ¼ 106.77 MPa. Slope of the mean Pmax - Ae curve is ft ¼ μ ¼ 841.33 MPa, and that of the upper and lower Pmax - Ae curves are ft ¼ μ � 2σ ¼ 841.33 � 2*106.77 MPa. Commonly used linear curve fitting was also conducted, whose ft ¼ 857.03 MPa was close to that from normal distribution analysis. However, confidence bands were not predictable by curve fitting. According to the normal distribution analysis of ft, the non-linear σn – ae relation in Eq. (1) was illustrated in Fig. 5(b) as a predictive tool for safe design of CFRP fracture. An initial notch of 4 μm was assumed for g2.09 and g-1.44 to characterize machining defect. Trans-laminar "Composite" fracture toughness KIC ¼ 34.48 MPa√m was obtained based on Eq. (8). This estimation is very close to the values obtained by compact tension tests, KIC ¼ 38.9 MPa√m calculated from the “propa rffiq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 Ex Ex gation value” GIC ¼ 133 kJ/m2 using GIC ¼ pffiffiffiICffiffiffiffiffiffi vxy and Ey þ 2Gxy 2Ex Ey
KIC ¼ 32.3 MPa√m calculated from the “initiation value” GIC ¼ 91.6 kJ/ m2 [24,25]. Non-LEFM fracture applies to CFRP with shallow surface scratch, and LEFM fracture applies to very thick CFRP panel with deep notch. Although Pmax measurements of g-1.44 is smaller than that of g-1.44notched, which might be attributed to the poor sample preparation process, the average ft from 30 Pmax measurements (without g-1.44) is only 4.7% deviated from that of all 40 Pmax measurements. Furthermore, the lower limit of ft of g-1.44 and g-1.44-notched is consistent, consid ering the 95% reliability band predicted by the normal distribution.
With the analytically established Ae and experimentally measured Pmax from 3-p-b tests, the “composite” tensile strength ft value from each
Table 1 Specimen detailsa of 3-p-b tests with and without shallow surface scratch. g-1.44
g-2.09-notched
g-1.44-notched
Thickness W (mm) Notch Depth a0 (mm) Notch-Thickness Ratio α Span S (mm) Number of sample
2.09 0 0 66 10
1.44 0 0 46 10
2.09 0.03 0.014 66 10
1.44 0.03 0.021 46 10
a
(10)
Pmax ¼ ðμ � 2σ Þ⋅Ae ðW; α; Cch Þ
4.1. Determination of ft from 3-p-b fracture loads and normal distribution analysis
g-2.09
(9)
Upper and lower bounds with 95% reliability could be obtained by Eq. (10). In this way, experimental scatters were considered in this simple non-LEFM tensile strength model.
4. Results and discussion
Group
ðx μÞ2 2σ 2
4.2. Comparison between ft from 3-p-b tests and ft from tensile tests ft predicted from 3-p-b tests (40 specimens) using non-LEFM tensile
Four different types of 3-p-b specimens. 4
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Composites Science and Technology 191 (2020) 108072
Table 2 α, Pmax and ft from 3-p-b tests using non-LEFM tensile strength model. g-2.09
1 2 3 4 5 6 7 8 9 10
g-1.44
g-2.09-notched
g-1.44-notched
α
Pmax (N)
ft (MPa)
α
Pmax (N)
ft (MPa)
α
Pmax (N)
ft (MPa)
α
Pmax (N)
ft (MPa)
0 0 0 0 0 0 0 0 0 0
695 762 738 683 700 698 799 780 739 718
815.65 900.65 882.53 805.63 828.46 852.95 943.03 892.49 909.31 880.90
0 0 0 0 0 0 0 0 0 0
368 424 443 449 411 347 392 434 381 389
681.67 773.43 769.88 744.29 744.61 642.48 700.11 733.99 675.26 704.64
0.011 0.011 0.009 0.009 0.013 0.067 0.023 0.026 0.018 0.029
590 680 657 702 649 583 710 637 651 730
900.76 896.35 927.64 874.32 909.26 975.76 936.09 915.85 902.64 1012.29
0.014 0.018 0.017 0.029 0.029 0.010 0.017 0.028 0.016 0.029
475 457 462 434 344 581 544 598 385 350
881.19 884.94 845.24 863.93 709.01 987.58 961.42 1048.99 697.16 640.78
Fig. 6. Comparison between ft from 3-p-b tests (40 specimens), and curve fitted ft from 15 tensile tests. Ae ¼ A (cross-section area) for tensile tests.
tests with cross-section area ranging from 13 to 32 mm2. Average ft obtained from tensile tests was 784.82 MPa and the slope of linear curve fitting with fixed intercept 0 turned out to be 790 MPa. It is evident that although most of the Pmax measurements from 3-p-b tests gathered slightly above the linear fitted curve of tensile tests, indicating a small deviation in ft values, a marvellous overall consistency was achieved for tensile strength from two different testing and calculation approaches. Therefore, through the introduction of the composite microstructure measurement Cch in the closed-form model, the “composite” tensile strength ft from tensile tests can be used to describe 3-p-b failure of CFRP panel, and vice versa. Normal distribution of “composite” tensile strength ft from 3-p-b tests and tensile tests (total 55 measurements) was shown in Fig. 7(a), and mean curve, upper and lower bounds with 95% reliability were given in Fig. 7(b), where μ ¼ 825.92 MPa and σ ¼ 98.42 MPa. Magnified plotting of 40 Pmax measurements from 3-p-b tests was given in Fig. 7(c), which were well described by ft predicted from 55 measurements with 95% confidence band. Table 3 gave relative error between average ft estimated from 3-p-b tests using non-LEFM model and that from tensile tests, which was merely 6.7%. The flexural strength σf from 3-p-b tests were also evaluated by Eq. (11), where a0 ¼ 0 for g-2.09 and g-1.44, based on the classic SoM for homogeneous materials.
Fig. 5. (a) Predictions of ft from the non-LEFM model with normal distribution and its comparison with curve fitting (bold dotted line) based on all Pmax - Ae data from 3-p-b tests. (b) Non-LEFM prediction of CFRP fracture for safe design based on Eq. (1). All data are in strength and quasi-brittle fracture regions, specified by Fig. 3.
strength model was compared to ft obtained directly from tensile tests (15 specimens), as shown in Fig. 6, where Ae represented cross-section area for tensile tests. Coordinate axises were broken down so that left bottom corner showed Pmax measurements from 3-p-b tests with Ae ranging from 0 to 1 mm2 and right top corner presented that from tensile
σf ¼
3Pmax ⋅S 2BðW
a0 Þ2
(11)
Clearly, Cch is not involved in Eq. (11) as it is not for composites. According to the fundamental principles of SoM for homogeneous 5
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Table 3 Comparison of ft from 3-p-b tests using non-LEFM model, ft obtained directly from tensile tests, and flexural strength from 3-p-b tests using classic SoM. Fracture strength (MPa)
Minimum
Average
Maximum
Relative error (%)
non-LEFM model (3-pb tests)
All
640.79
841.33
1048.99
0
g-2.09 g-1.44 g-2.09notched g-1.44notched
805.63 642.48 874.32
871.16 717.04 925.09
943.03 773.43 1012.29
3.55 - 14.77 9.96
640.79
852.03
1048.99
1.27
tensile tests Strength of Materials ASTM D 7264 (3-p-b tests)
681.88
784.82
845.12
- 6.7
All
784.58
1026.46
1286.02
22
g-2.09 g-1.44 g-2.09notched g-1.44notched
978.26 817.71 1041.78
1057.84 912.59 1081.32
1145.11 984.36 1167.96
25.7 8.5 28.5
784.58
1054.09
1286.02
25.3
materials, both ft from direct tensile tests and σ f from bending tests should be the same. However, in most cases, experimental results indi cate σf > ft, which shows the inadequacy of SoM for composite fracture analysis. The flexural strength σf was 22% higher than estimated “composite” tensile strength ft, accordant to the flexural-tensile strength ratio for fibre composites, which is from 1.04 to 1.49 [26–28]. According to Fig. 2 (a), flexural strength or tensile strength in outer layer was derived as per linear stress distribution in tensile region under 3-p-b condition. Thus, it is higher than the estimated ft without consideration of the fictitious crack Δafic or crack-tip damage/micro-cracking. Eq. (7) with the consideration of micro-cracking shows both σf and ft are equal. 4.3. Failure mechanism analysis Specimens for 3-p-b tests and tensile tests were cut manually in lab with only slight edge grinding using 120 Grit Sandpaper. It was possible that some minor machining defects still existed, and cutting of curved sections might not be perfect. For example, tensile test specimens broke mainly in the junction area where stress concentration might exist due to variations in cross-section or polishing condition, as shown in Fig. 4(b). Therefore, tensile strength measurement from both test methods were underestimated while still comparable. It is noteworthy from Table 3 that ft estimation from g-1.44 was the smallest and that from g-1.44-notched saw largest scatter among all four groups, which might due to different failure modes as shown in Fig. 8, besides the aforementioned machining defects. Failure Mode 1 was defined with obvious fibre kinking on top surface, fibre tensile breakage on bottom surface, and trans-laminar crack growth in thickness direc tion, applied to specimens in large ellipse in Fig. 8. Failure Mode 2 saw less evident fibre fracture on both top and bottom surfaces, but distinct delamination on side face, for specimens in small ellipse. Therefore, relatively lower Pmax due to interfacial debonding was obtained for specimens failed by Mode 2 and thus tensile strength was slightly underestimated. 4.4. Extended application of non-LEFM model Fig. 7. (a) Normal distribution of “composite” tensile strength ft based on all Pmax - Ae data from all different 3-p-b tests and tensile tests, different to a normal distribution of scatters from identical specimens. (b) Normal distribution analysis of “composite” tensile strength ft based on all Pmax - Ae data from 3-p-b tests and tensile tests. (c) Comparison between “composite” tensile strength ft measurements and predictions with 95% reliability based on all tensile and bending tests.
The simple closed-form non-LEFM solution presented in this study was derived in recent years for quasi-brittle fracture of heterogeneous solids such as coarse-grained ceramics, rocks and concrete with the average grain size G as Cch [18,21], based on the previous boundary effect model [29–31]. This non-LEFM model has been well tested by a numerous test results from different sources for polycrystalline solids and concrete-like particle composites. 6
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composite unit Cch and “composite” tensile strength ft, i.e., KIC ¼ 2ft√(3Cch). (5) It is worthwhile emphasized again that the linear relation Pmax ¼ ft x Ae applies to non-LEFM fracture of CFRP with shallow surface scratch and ideal LEFM fracture of very thick CFRP structures with deep cracks/notches (which may never happen in reality). Since CFRP panels made from carbon fibre pre-pregs have compli cated hierarchical microstructures, further studies with systematic tests of CFRP with various woven structures are warranted. Declaration of competing interest None. CRediT authorship contribution statement Bingyan Yuan: Investigation, Formal analysis, Writing - original draft, Writing - review & editing. Yunsen Hu: Investigation. Xiaozhi Hu: Conceptualization, Methodology, Writing - review & editing, Supervision.
Fig. 8. “Composite” tensile strength ft estimated from 3-p-b tests as affected by failure modes.
The recent study on bone-like bamboo composites [16] has extended the non-LEFM solution to composites with fibre bundle microstructures. The current study on the carbon fibre composites made from 2D woven fabrics has further extended the model into fracture analysis of laminar CFRP structures made from carbon fibre pre-pregs, which is commonly used in aeronautical and aerospace industries. Probably, the extension of this closed-form solution to fracture of CFRP is the most important application of this non-LEFM model. It is encouraging to see that the simple closed-form model has worked well for all those different com posites with distinctly different microstructures. The characteristic composite unit Cch plays a significant role in modelling of composite fracture as shown in Eq. (7). Neither LEFM nor SoM contains Cch as they are strictly limited to homogeneous materials. Potentially, Eq. (7) with the characteristic composite unit Cch can open the door for future frac ture research of other composite systems as long as Cch responsible for bulk fracture properties such as “composite” tensile strength ft and “composite” fracture toughness KIC can be clearly identified.
Acknowledgement B. Yuan and Y. Hu thank the financial supports from Australian Government through “Australian Government Research Training Pro gram Scholarship”. References [1] A.R. Ravindran, R.B. Ladani, S. Wu, A.J. Kinloch, C.H. Wang, A.P. Mouritz, Multiscale toughening of epoxy composites via electric field alignment of carbon nanofibres and short carbon fibres, Compos. Sci. Technol. (2018), https://doi.org/ 10.1016/j.compscitech.2018.07.034. [2] C. Soutis, Fibre reinforced composites in aircraft construction, Prog. Aero. Sci. 41 (2) (2005) 143–151, https://doi.org/10.1016/j.paerosci.2005.02.004. [3] https://www.geaviation.com/commercial/engines/ge9x-commercial-aircraft-e ngine accessed May 2019. [4] A.G. Atkins, Y.-W. Mai, Elastic and Plastic Fracture: Metals, Polymers, Ceramics, Composites, Biological Materials, 1985. [5] P.D. Soden, M.J. Hinton, A.S. Kaddour, A comparison of the predictive capabilities of current failure theories for composite laminates, Compos. Sci. Technol. 58 (7) (1998) 1225–1254, https://doi.org/10.1016/S0266-3538(98)00077-3. [6] M.J. Hinton, A.S. Kaddour, P.D. Soden, A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence, Compos. Sci. Technol. 62 (12–13) (2002) 1725–1797, https://doi.org/ 10.1016/S0266-3538(02)00125-2. [7] R.M. Christensen, Failure criteria for fiber composite materials, the astonishing sixty year search, definitive useable results, Compos. Sci. Technol. (2019) 107718, https://doi.org/10.1016/j.compscitech.2019.107718. [8] R.M. Christensen, Timoshenko medal award paper—completion and closure on failure criteria for unidirectional fiber composite materials, J. Appl. Mech. 81 (1) (2013), 011011, https://doi.org/10.1115/1.4025177, 2014. [9] R.M. Christensen, K. Lonkar, Failure theory/failure criteria for fiber composite laminates, J. Appl. Mech. 84 (2) (2017), 021009, https://doi.org/10.1115/ 1.4035119. [10] R.M. Christensen, Lamination theory for the strength of fiber composite materials, J. Appl. Mech. 84 (7) (2017), 071007, https://doi.org/10.1115/1.4036825. [11] S.S.R. Koloor, M.R. Khosravani, R.I.R. Hamzah, M.N. Tamin, FE model-based construction and progressive damage processes of FRP composite laminates with different manufacturing processes, Int. J. Mech. Sci. 141 (2018) 223–235, https:// doi.org/10.1016/j.ijmecsci.2018.03.028. [12] P.F. Liu, J.Y. Zheng, Recent developments on damage modeling and finite element analysis for composite laminates: a review, Mater. Des. 31 (8) (2010) 3825–3834, https://doi.org/10.1016/j.matdes.2010.03.031. [13] X. Hu, Y. Mai, Mode I delamination and fibre bridging in carbon-fibre/epoxy composites with and without PVAL coating, Compos. Sci. Technol. 46 (2) (1993) 147–156, https://doi.org/10.1016/0266-3538(93)90170-L. [14] D. Shu, Y. Mai, Delamination buckling with bridging, Compos. Sci. Technol. 47 (1) (1993) 25–33, https://doi.org/10.1016/0266-3538(93)90092-U. [15] B.F. Sørensen, E.K. Gamstedt, R.C. Østergaard, S. Goutianos, Micromechanical model of cross-over fibre bridging–Prediction of mixed mode bridging laws, Mech. Mater. 40 (4–5) (2008) 220–234, https://doi.org/10.1016/j. mechmat.2007.07.007. [16] W. Liu, Y. Yu, X. Hu, X. Han, P. Xie, Quasi-brittle fracture criterion of bamboobased fiber composites in transverse direction based on boundary effect model,
5. Conclusion A simple closed-form solution for bending failure of CFRP panels with or without shallow surface scratch has been presented, and confirmed by preliminary tensile and bending test results presented in this study. A characteristic composite unit Cch has been identified as the key microstructure measurement for composite fracture, which is the ply thickness for CFRP made from pre-pregs. The following conclusions can be drawn from the current study: (1) The “composite” tensile strength ft of CFRP can be determined from both direct tensile tests and 3-p-b tests, but notched bending tests are much easier to perform. (2) Similar to the common relation for direct tensile tests, i.e., Fracture Load ¼ Strength x Area, an equivalent area Ae has been defined for 3-p-b conditions so that the fundamental relation that Fracture Load ¼ Strength x Ae still holds. (3) The closed-form non-LEFM model has previously been used suc cessfully for “particle composites” or heterogeneous poly crystalline solids such as coarse-grained ceramics, rocks and concrete. Without any modification, its direct application to CFRP panels appears to be successful based on the results ana lysed in this study. (4) The “composite” fracture toughness KIC of CFRP through the thickness direction is linked to the ply thickness or characteristic
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[17] [18] [19] [20] [21] [22] [23]
Composites Science and Technology 191 (2020) 108072 [24] S.T. Pinho, P. Robinson, L. Iannucci, Fracture toughness of the tensile and compressive fibre failure modes in laminated composites, Compos. Sci. Technol. 66 (13) (2006) 2069–2079, https://doi.org/10.1016/j.compscitech.2005.12.023. [25] M.J. Laffan, S.T. Pinho, P. Robinson, A.J. McMillan, Translaminar fracture toughness testing of composites: a review, Polym. Test. 31 (3) (2012) 481–489, https://doi.org/10.1016/j.polymertesting.2012.01.002. [26] R.E. Bullock, Strength ratios of composite materials in flexure and in tension, J. Compos. Mater. 8 (2) (1974) 200–206, https://doi.org/10.1177/ 002199837400800209. [27] J.M. Whitney, M. Knight, The relationship between tensile strength and flexure strength in fiber-reinforced composites, Exp. Mech. 20 (6) (1980) 211–216, https://doi.org/10.1007/BF02327601. [28] M.R. Wisnom, The relationship between tensile and flexural strength of unidirectional composites, J. Compos. Mater. 26 (8) (1992) 1173–1180, https:// doi.org/10.1177/002199839202600805. [29] X. Hu, F. Wittmann, Size effect on toughness induced by crack close to free surface, Eng. Fract. Mech. 65 (2–3) (2000) 209–221, https://doi.org/10.1016/S0013-7944 (99)00123-X. [30] K. Duan, X. Hu, Specimen boundary induced size effect on quasi-brittle fracture, Strength, Fract. Complex. 2 (2) (2004) 47–68. [31] X. Hu, K. Duan, Size effect and quasi-brittle fracture: the role of FPZ, Int. J. Fract. 154 (1–2) (2008) 3–14, https://doi.org/10.1007/s10704-008-9290-7.
Compos. Struct. 220 (2019) 347–354, https://doi.org/10.1016/j. compstruct.2019.04.008. J. Guan, X. Hu, Q. Li, In-depth analysis of notched 3-pb concrete fracture, Eng. Fract. Mech. 165 (2016) 57–71, https://doi.org/10.1016/j. engfracmech.2016.08.020. Y. Wang, X. Hu, L. Liang, W. Zhu, Determination of tensile strength and fracture toughness of concrete using notched 3-pb specimens, Eng. Fract. Mech. 160 (2016) 67–77, https://doi.org/10.1016/j.engfracmech.2016.03.036. X. Han, Y. Chen, X. Hu, W. Liu, Q. Li, S. Chen, Granite strength and toughness from small notched three-point-bend specimens of geometry dissimilarity, Eng. Fract. Mech. 216 (2019) 106482, https://doi.org/10.1016/j.engfracmech.2019.05.014. R. Xu, X. Hu, Effects of nano-grain structures and surface defects on fracture of micro-scaled polysilicon components, J. Am. Ceram. Soc. (2020) 1–6, https://doi. org/10.1111/jace.17032, 00. X. Hu, J. Guan, Y. Wang, A. Keating, S. Yang, Comparison of boundary and size effect models based on new developments, Eng. Fract. Mech. 175 (2017) 146–167, https://doi.org/10.1016/j.engfracmech.2017.02.005. J. Guan, P. Yuan, X. Hu, L. Qing, X. Yao, Statistical analysis of concrete fracture using normal distribution pertinent to maximum aggregate size, Theor. Appl. Fract. Mech. (2019), https://doi.org/10.1016/j.tafmec.2019.03.004. Y. Wang, X. Hu, Determination of tensile strength and fracture toughness of granite using notched three-point-bend samples, Rock Mech. Rock Eng. 50 (1) (2017) 17–28, https://doi.org/10.1007/s00603-016-1098-6.
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Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
Critical bending load of CFRP panel with shallow surface scratch determined by a tensile strength model Bingyan Yuan , Yunsen Hu , Xiaozhi Hu * Department of Mechanical Engineering, The University of Western Australia, Perth, WA, 6009, Australia
A R T I C L E I N F O
A B S T R A C T
Keywords: Composite fracture Shallow surface scratch “Composite” tensile strength Non-LEFM model Normal distribution
Carbon fibre reinforced polymer (CFRP) composite structures may experience light surface damage such as shallow surface scratch around 50 μm or less in applications. In this study, critical bending loads of 2D woven CFRP panels with and without shallow surface scratches were assessed under three-point-bending (3-p-b) con ditions and then modelled by a simple strength theory for composite fracture. This closed-form tensile strength solution is a non- Linear Elastic Fracture Mechanics (LEFM) model derived specifically for heterogeneous composites. Direct tensile tests of the CFRP panels (without shallow surface scratch) were also performed, and measured tensile strength was compared with that from 3-p-b tests. The relative error between the two different tests was only around 6.7%. The ply thickness around 140 μm was selected as the controlling composite microstructure parameter or characteristic composite unit Cch in this non-LEFM model. This simple model shows that a suitable composite characteristic microstructure measurement such as Cch ¼ ply thickness can significantly simplify composite fracture analysis. A simple statistical analysis using a 2D normal distribution methodology for strength measurements was also introduced so that the experimental scatters can be analysed. The 95% reli ability band can be conveniently used in safe design of composite structures.
1. Introduction Carbon fibre reinforced polymer (CFRP) composites enable signifi cant weight saving and complex shape design in aircraft manufacturing through their usage not only in secondary structures including stabil isers and trailing edge panels but also in primary structures such as fu selages and wings [1,2]. Nowadays, CFRP has increasingly been used for the front air blades in aircraft gas turbine engines, as shown in Fig. 1(a). GE has used the 4th-generation carbon fibre composite fan blades in GE9X engine for lighter and more durable structures instead of metal blades, and has thus reduced the fuel consumption [3]. Protective metal alloy thin plates are used at the leading edges of CFRP blades in order to prevent blade edge erosion due to potential high-speed impacts of micro-debris. However, shallow surface scratches, around 50 μm or less in depth, may still be induced along the blade surface with prolonged service time. This kind of surface damage can also occur in other large thin plate-like CFRP structures through surface contacts. Here lies a practical and potentially important issue of engineering challenge, i.e., to predict bulk mechanical properties of CFRP panel structures with shallow surface crack/damage as shown in Fig. 1(b),
where both the depth and width of a shallow scratch on a CFRP panel surface are around 50 μm or less. The classic Linear Elastic Fracture Mechanics (LEFM) and traditional Strength of Materials (SoM) limited to homogeneous and isotropic materials cannot be used for fracture anal ysis of a shallow crack in CFRP panel with complicated hierarchical microstructures. For instance, a shallow surface scratch less than 50 μm in depth is well within one carbon fibre ply around 120 μm in thickness, containing woven carbon fibres around 5 μm in diameter. Yet, it seems there is no simple closed-form solution available in literature for failure analysis of CFRP panels or carbon fibre composites in general [4–12]. Various micro-mechanics models have been proposed for fracture analysis of laminar composites, considering micro-damage such as delamination and crack bridging [13–15]. While micro-mechanics modelling is specifically useful for a good understanding of detailed fracture mechanisms, a few assumptions made for material properties relevant to micro-cracking and damage are common, and a closed-form solution normally cannot be derived for macro or bulk composite properties. In this study, we adopt a macro-mechanics approach so that a simple closed-form model recently developed for quasi-brittle fracture of het erogeneous solids can be tested for the first time to determine the
* Corresponding author. E-mail address: [email protected] (X. Hu). https://doi.org/10.1016/j.compscitech.2020.108072 Received 24 July 2019; Received in revised form 7 February 2020; Accepted 13 February 2020 Available online 15 February 2020 0266-3538/© 2020 Elsevier Ltd. All rights reserved.
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in Fig. 2(a) is chosen in this study to illustrate how Pmax is linked to the “composite” tensile strength ft. It should be emphasized that ft of CFRP defined in this study is a material constant. Common strength estima tions from fracture loads of CFRP samples using classic SoM (limited only to homogeneous materials) are not constant, but sample thickness and size dependent, which is particularly true for 3-p-b conditions. The new non-LEFM model adopted in this study uses the same “composite” tensile strength ft (a material constant) to model both 3-p-b fracture and direct tensile failure of CFRP. Furthermore, it is based on one of the most fundamental relations in solid mechanics for material failure, i.e., Fracture Load ¼ Tensile Strength x Area (cross-section area for uniform tensile fracture). The model has defined an equivalent area Ae for 3-p-b conditions so that the relation still holds, i.e., Fracture Load ¼ Tensile Strength x Ae [21].
Nomenclature a0 ae Y(α) a*ch Δafic W B S
α
Cch Ae Pmax ft KIC
σn βav
μ σ σf
initial crack/scratch depth equivalent crack depth geometry factor characteristic crack fictitious crack sample thickness sample width span length ratio of a0 and W characteristic composite unit equivalent area fracture load “composite” tensile strength “composite” fracture toughness nominal strength average discrete number mean value standard deviation flexural strength
2. Fracture modelling of CFRP panel with shallow surface scratch Fracture of composites of any kind including CFRP panels cannot be modelled by LEFM due to the heterogeneous composite structures. Extensive non-LEFM modelling has been done for brittle heterogeneous solids such as rock and concrete [22,23]. In those cases, the average grain size G or aggregate size dav is the characteristic composite unit Cch, while ply thickness is Cch for laminated CFRP panel in spite of its com plex structural composition. The key equations and the logic in model ling are summarized below. Unlike homogeneous material whose brittle fracture is defined by fracture toughness of LEFM, heterogeneous composites possessing hi erarchical microstructures typically displace quasi-brittle fracture behaviour determined by both “composite” tensile strength ft and “composite” fracture toughness KIC, as shown in Fig. 3. The comparable ratio of crack-size/composite-microstructure means that LEFM is hardly relevant to composite failure, i.e., the strength region and quasi-brittle fracture region (both are strongly influenced by composite structures) should be considered for CFRP. For simplicity and due to the limited size of Δafic at Pmax, a constant cohesive stress distribution is assumed and marked by the nominal strength σn as in Fig. 2(a). The relationship between nominal strength σ n at the micro-cracking zone Δafic in Fig. 2(a) and “composite” tensile
fracture loads of CFRP panels with shallow surface scratches under three-point-bending (3-p-b) conditions. This non-LEFM model specially derived for heterogeneous composites has been successfully used to analyse fracture of bone-like bamboo composites [16], concrete and granite [17–19], and polysilicon micro-components with nano-defects [20]. Despite completely different micro-mechanisms of cracking and fracture existing in laminar CFRP and granular/particle composites, this macro-mechanics approach considers a combined non-LEFM region through a combined fictitious crack length Δafic in front of the initial crack tip a0 as shown in Fig. 2(a), so that the overall effects of different micro-damage and cracking activities in various composites can be modelled through specific properties over the non-linear zone Δafic. The maximum fracture load Pmax of a CFRP panel with a shallow surface scratch is the primary concern. The common 3-p-b test geometry
Fig. 1. (a) GE9X engine, with fan case and 16 fan blades made from CFRP [3]. While the CFRP blades are protected at the leading edge, shallow scatch on the blade surface is still possible. (b) Shallow surface scratch in CFRP panel, dark blue indicating the depth of scratch and yellow areas showing surface lift due to edge delamination, as measured by Altisurf 520 (ALTIMET, France). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 2
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Fig. 2. (a) Stress distribution of CFRP panel under 3-p-b condition with a shallow surface scratch or initial crack a0. The standing-out ply thickness is selected as the characteristic com posite unit Cch. (b) Through-thickness micro-cracking in 2D woven fibre structures in front of the initial crack a0, modelled by the fictitious crack Δafic, covering the crack-tip damage zone. (c) X-ray μCT images of longitudinal section of a g-2.09-notched specimen after 3-p-b test.
� �2 KIC � 3Cch a*ch ¼ 0:25 ft
(3)
By combining Eqs. (1) and (3), σn is calculated as Eq. (4). f
σ n ¼ qffiffiffiffiffitffiffiffiffiffiffiffiffiffi 1 þ 3Caech
The fictitious crack Δafic in front of the initial crack tip a0 is linked to Cch through a discrete number βav as Eq. (5). The discrete number βav (e. g. 0, 0.5, 1.0, 1.5, 2.0) is used to model the discontinuous crack growth in composites. For a heterogeneous polycrystalline solid with an average grain size G, variation in the discrete number βav is due to randomly distributed coarse aggregate structures varying from specimen to spec imen [22]. For 2D twill-weaved CFRP panel, the uncertainty of βav is attributed to the fact that under an trans-laminar Mode-I fracture load, crack from a0 propagates much more effortlessly through transverse fi bres than through longitudinal fibres, as demonstrated in Fig. 2(b). In this study, we test the βav solution for polycrystalline solids, i.e., βav ¼ 1.5 [16,19], and check the applicability of the following equation to CFRP using experimental results from both 3-p-b and direct tensile tests.
Fig. 3. Non-LEFM fracture of CFRP panels with shallow surface scratch, which is strongly influenced by heterogeneous composite microstructures. LEFM fracture of homogeneous material defined by KIC is not applicable [4,16].
Δafic ¼ βav ⋅Cch ¼ 1:5Cch
strength ft (a material constant) is calculated by Eq. (1), where ae is the equivalent crack size and a*ch is the characteristic crack. f
1 þ a*e
ch
The equivalent crack size ae is defined by Eq. (2), where a0 is initial crack depth, α is ratio of a0 and W (sample thickness) and Y(α) is geo metric function. Since we concentrate on shallow surface scratch in this study, whose α is far smaller than 1, Y(α) is fairly close to 1.12. ae ¼
� ð1
�2
αÞ2 ⋅YðαÞ 1:12
⋅a0 ¼ ð1
αÞ4 ⋅a0
(5)
Following Fig. 2(a), σ n can be calculated by the equilibrium condi tions and the result is given by Eq. (6), where Pmax is fracture load, S is span length and B is sample width, considering the strain condition along the crack plane and two equilibrium conditions for bending stress and bending moment after Δafic is determined [23]. �� 1:5 BS ⋅Pmax 3Pmax ⋅S �¼ σn ¼ (6) ðW a0 Þ⋅ðW a0 þ 3Cch Þ 2BðW a0 Þ⋅ W a0 þ 2Δafic
(1)
σ n ¼ qffiffiffiffitffiffiffiffiffiaffiffiffi
(4)
By combining Eqs. (4) and (6), a linear relationship between Pmax and ft can be obtained as Eq. (7), where the equivalent area Ae is fully determined by W, α and Cch, i.e., Fracture load ¼ Tensile Strength x Ae. The closed-form solution of Eq. (7) on the linear relation between Pmax and Ae is easy to use as all parameters are well defined. The “composite” tensile strength ft determines the slope of the straight line.
(2)
The characteristic crack a*ch as given in Eq. (3) is a material constant related to the bulk composite material properties ft and KIC, which is the intersection point of strength criterion and toughness criterion in Fig. 3. And it is explicitly linked to characteristic composite unit Cch, according to previous study [16]. 3
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� � W 2 ⋅ð1 αÞ⋅ 1 α þ 3CWch � � qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pmax ¼ ft ⋅ Ae ðW; α; Cch Þ ¼ ft ⋅ 1:5 BS ⋅ 1 þ 3Caech
(7)
After ft is determined, the “composite” fracture toughness KIC (also a material constant) can be determined using ft and Cch as Eq. (8) ac cording to Eq. (3), so that the failure of CFRP with or without (α ¼ 0) shallow surface scratch can be predicted by the same strength criterion. (8)
KIC ¼ 2ft √ð3Cch Þ
It should be emphasized that a thin plate such as the 3-p-b sample illustrated in Fig. 2(a) can never be used to directly measure KIC following LEFM, which requires a sufficiently long crack and a suffi ciently thick CFRP panel to ensure the homogeneous material assump tion. The non-LEFM model adopted in this study has effectively removed the stringent requirements of LEFM, making theoretical prediction of the fracture load Pmax of a CFRP panel with a shallow surface scratch possible.
Fig. 4. (a) Dimensions (in mm) of dumbbell-shaped specimen for direct tensile tests (without shallow surface scratch). (b) Typical fracture location, indicating stress concentration from machining (only slight edge grinding using 120 Grit Sandpaper).
3. Sample preparation and experimental methods
specimen was easily determined by using Eq. (7), as listed in Table 2. Experimental scatters were noticed among specimens of different thickness and notch condition, so that a simple statistical analysis methodology (2D normal distribution) as in refs. [16,19] was adopted for ft measurements. Eq. (9) gives the basic formula, where μ is the mean value and σ is the standard deviation of the normal distribution.
2D woven CFRP panels (CarbonWiz Technology Limited, China) in 2.09 mm and 1.44 mm thick were used, whose ply thickness was around 140 μm as provided by the manufacturer and was adopted as the char acteristic composite unit Cch in the non-LEFM model. Panels were cut into 15 mm wide beams, whose length was 90 mm and 62 mm respec tively. The initial notch or surface scratch a0 was manually introduced along the centre line in the bottom surface (as illustrated in Fig. 2(a)) using a cutter. The average notch depth was 0.03 mm although actual depth varied, measured by Altisurf 520 (ALTIMET, France) as shown in Fig. 1(b). Corresponding notch-thickness ratios were listed in Table 1. 3-p-b tests were conducted using an Instron 5982 universal me chanical testing machine with a 100 kN load cell and 2.5 mm/min loading speed according to ASTM D 7264. Span-to-thickness ratio was 32. 10 samples were tested for each group, i.e., g-2.09, g-1.44, g-2.09notched and g-1.44-notched. Panels were also cut into dumbbell-shaped samples (dimensions given in Fig. 4(a)) for direct tensile tests (without shallow surface scratch), conducted using the same machine with 1 mm/ min loading speed. 10 specimens were tested for 2.09 mm panel and 5 specimens for 1.44 mm panel. A g-2.09-notched specimen with a0 ¼ 22 μm after 3-p-b test was scanned at 40 kV and 74 μA using an X-ray microcomputed tomography (X-ray μCT) system (Versa 520, Zeiss, Pleasanton, CA, USA) and images of longitudinal section were given in Fig. 2(c). Trans-laminar Mode-I crack growth from a0 was seen, around 2Cch long, so that the assumption of Δafic ¼ 1.5Cch before Pmax is reasonable.
1 f ðxÞ ¼ pffiffiffiffiffi e 2πσ
ft values from all 40 Pmax measurements and normal distribution analysis of them were plotted in Fig. 5(a), where μ ¼ 841.33 MPa and σ ¼ 106.77 MPa. Slope of the mean Pmax - Ae curve is ft ¼ μ ¼ 841.33 MPa, and that of the upper and lower Pmax - Ae curves are ft ¼ μ � 2σ ¼ 841.33 � 2*106.77 MPa. Commonly used linear curve fitting was also conducted, whose ft ¼ 857.03 MPa was close to that from normal distribution analysis. However, confidence bands were not predictable by curve fitting. According to the normal distribution analysis of ft, the non-linear σn – ae relation in Eq. (1) was illustrated in Fig. 5(b) as a predictive tool for safe design of CFRP fracture. An initial notch of 4 μm was assumed for g2.09 and g-1.44 to characterize machining defect. Trans-laminar "Composite" fracture toughness KIC ¼ 34.48 MPa√m was obtained based on Eq. (8). This estimation is very close to the values obtained by compact tension tests, KIC ¼ 38.9 MPa√m calculated from the “propa rffiq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 Ex Ex gation value” GIC ¼ 133 kJ/m2 using GIC ¼ pffiffiffiICffiffiffiffiffiffi vxy and Ey þ 2Gxy 2Ex Ey
KIC ¼ 32.3 MPa√m calculated from the “initiation value” GIC ¼ 91.6 kJ/ m2 [24,25]. Non-LEFM fracture applies to CFRP with shallow surface scratch, and LEFM fracture applies to very thick CFRP panel with deep notch. Although Pmax measurements of g-1.44 is smaller than that of g-1.44notched, which might be attributed to the poor sample preparation process, the average ft from 30 Pmax measurements (without g-1.44) is only 4.7% deviated from that of all 40 Pmax measurements. Furthermore, the lower limit of ft of g-1.44 and g-1.44-notched is consistent, consid ering the 95% reliability band predicted by the normal distribution.
With the analytically established Ae and experimentally measured Pmax from 3-p-b tests, the “composite” tensile strength ft value from each
Table 1 Specimen detailsa of 3-p-b tests with and without shallow surface scratch. g-1.44
g-2.09-notched
g-1.44-notched
Thickness W (mm) Notch Depth a0 (mm) Notch-Thickness Ratio α Span S (mm) Number of sample
2.09 0 0 66 10
1.44 0 0 46 10
2.09 0.03 0.014 66 10
1.44 0.03 0.021 46 10
a
(10)
Pmax ¼ ðμ � 2σ Þ⋅Ae ðW; α; Cch Þ
4.1. Determination of ft from 3-p-b fracture loads and normal distribution analysis
g-2.09
(9)
Upper and lower bounds with 95% reliability could be obtained by Eq. (10). In this way, experimental scatters were considered in this simple non-LEFM tensile strength model.
4. Results and discussion
Group
ðx μÞ2 2σ 2
4.2. Comparison between ft from 3-p-b tests and ft from tensile tests ft predicted from 3-p-b tests (40 specimens) using non-LEFM tensile
Four different types of 3-p-b specimens. 4
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Table 2 α, Pmax and ft from 3-p-b tests using non-LEFM tensile strength model. g-2.09
1 2 3 4 5 6 7 8 9 10
g-1.44
g-2.09-notched
g-1.44-notched
α
Pmax (N)
ft (MPa)
α
Pmax (N)
ft (MPa)
α
Pmax (N)
ft (MPa)
α
Pmax (N)
ft (MPa)
0 0 0 0 0 0 0 0 0 0
695 762 738 683 700 698 799 780 739 718
815.65 900.65 882.53 805.63 828.46 852.95 943.03 892.49 909.31 880.90
0 0 0 0 0 0 0 0 0 0
368 424 443 449 411 347 392 434 381 389
681.67 773.43 769.88 744.29 744.61 642.48 700.11 733.99 675.26 704.64
0.011 0.011 0.009 0.009 0.013 0.067 0.023 0.026 0.018 0.029
590 680 657 702 649 583 710 637 651 730
900.76 896.35 927.64 874.32 909.26 975.76 936.09 915.85 902.64 1012.29
0.014 0.018 0.017 0.029 0.029 0.010 0.017 0.028 0.016 0.029
475 457 462 434 344 581 544 598 385 350
881.19 884.94 845.24 863.93 709.01 987.58 961.42 1048.99 697.16 640.78
Fig. 6. Comparison between ft from 3-p-b tests (40 specimens), and curve fitted ft from 15 tensile tests. Ae ¼ A (cross-section area) for tensile tests.
tests with cross-section area ranging from 13 to 32 mm2. Average ft obtained from tensile tests was 784.82 MPa and the slope of linear curve fitting with fixed intercept 0 turned out to be 790 MPa. It is evident that although most of the Pmax measurements from 3-p-b tests gathered slightly above the linear fitted curve of tensile tests, indicating a small deviation in ft values, a marvellous overall consistency was achieved for tensile strength from two different testing and calculation approaches. Therefore, through the introduction of the composite microstructure measurement Cch in the closed-form model, the “composite” tensile strength ft from tensile tests can be used to describe 3-p-b failure of CFRP panel, and vice versa. Normal distribution of “composite” tensile strength ft from 3-p-b tests and tensile tests (total 55 measurements) was shown in Fig. 7(a), and mean curve, upper and lower bounds with 95% reliability were given in Fig. 7(b), where μ ¼ 825.92 MPa and σ ¼ 98.42 MPa. Magnified plotting of 40 Pmax measurements from 3-p-b tests was given in Fig. 7(c), which were well described by ft predicted from 55 measurements with 95% confidence band. Table 3 gave relative error between average ft estimated from 3-p-b tests using non-LEFM model and that from tensile tests, which was merely 6.7%. The flexural strength σf from 3-p-b tests were also evaluated by Eq. (11), where a0 ¼ 0 for g-2.09 and g-1.44, based on the classic SoM for homogeneous materials.
Fig. 5. (a) Predictions of ft from the non-LEFM model with normal distribution and its comparison with curve fitting (bold dotted line) based on all Pmax - Ae data from 3-p-b tests. (b) Non-LEFM prediction of CFRP fracture for safe design based on Eq. (1). All data are in strength and quasi-brittle fracture regions, specified by Fig. 3.
strength model was compared to ft obtained directly from tensile tests (15 specimens), as shown in Fig. 6, where Ae represented cross-section area for tensile tests. Coordinate axises were broken down so that left bottom corner showed Pmax measurements from 3-p-b tests with Ae ranging from 0 to 1 mm2 and right top corner presented that from tensile
σf ¼
3Pmax ⋅S 2BðW
a0 Þ2
(11)
Clearly, Cch is not involved in Eq. (11) as it is not for composites. According to the fundamental principles of SoM for homogeneous 5
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Table 3 Comparison of ft from 3-p-b tests using non-LEFM model, ft obtained directly from tensile tests, and flexural strength from 3-p-b tests using classic SoM. Fracture strength (MPa)
Minimum
Average
Maximum
Relative error (%)
non-LEFM model (3-pb tests)
All
640.79
841.33
1048.99
0
g-2.09 g-1.44 g-2.09notched g-1.44notched
805.63 642.48 874.32
871.16 717.04 925.09
943.03 773.43 1012.29
3.55 - 14.77 9.96
640.79
852.03
1048.99
1.27
tensile tests Strength of Materials ASTM D 7264 (3-p-b tests)
681.88
784.82
845.12
- 6.7
All
784.58
1026.46
1286.02
22
g-2.09 g-1.44 g-2.09notched g-1.44notched
978.26 817.71 1041.78
1057.84 912.59 1081.32
1145.11 984.36 1167.96
25.7 8.5 28.5
784.58
1054.09
1286.02
25.3
materials, both ft from direct tensile tests and σ f from bending tests should be the same. However, in most cases, experimental results indi cate σf > ft, which shows the inadequacy of SoM for composite fracture analysis. The flexural strength σf was 22% higher than estimated “composite” tensile strength ft, accordant to the flexural-tensile strength ratio for fibre composites, which is from 1.04 to 1.49 [26–28]. According to Fig. 2 (a), flexural strength or tensile strength in outer layer was derived as per linear stress distribution in tensile region under 3-p-b condition. Thus, it is higher than the estimated ft without consideration of the fictitious crack Δafic or crack-tip damage/micro-cracking. Eq. (7) with the consideration of micro-cracking shows both σf and ft are equal. 4.3. Failure mechanism analysis Specimens for 3-p-b tests and tensile tests were cut manually in lab with only slight edge grinding using 120 Grit Sandpaper. It was possible that some minor machining defects still existed, and cutting of curved sections might not be perfect. For example, tensile test specimens broke mainly in the junction area where stress concentration might exist due to variations in cross-section or polishing condition, as shown in Fig. 4(b). Therefore, tensile strength measurement from both test methods were underestimated while still comparable. It is noteworthy from Table 3 that ft estimation from g-1.44 was the smallest and that from g-1.44-notched saw largest scatter among all four groups, which might due to different failure modes as shown in Fig. 8, besides the aforementioned machining defects. Failure Mode 1 was defined with obvious fibre kinking on top surface, fibre tensile breakage on bottom surface, and trans-laminar crack growth in thickness direc tion, applied to specimens in large ellipse in Fig. 8. Failure Mode 2 saw less evident fibre fracture on both top and bottom surfaces, but distinct delamination on side face, for specimens in small ellipse. Therefore, relatively lower Pmax due to interfacial debonding was obtained for specimens failed by Mode 2 and thus tensile strength was slightly underestimated. 4.4. Extended application of non-LEFM model Fig. 7. (a) Normal distribution of “composite” tensile strength ft based on all Pmax - Ae data from all different 3-p-b tests and tensile tests, different to a normal distribution of scatters from identical specimens. (b) Normal distribution analysis of “composite” tensile strength ft based on all Pmax - Ae data from 3-p-b tests and tensile tests. (c) Comparison between “composite” tensile strength ft measurements and predictions with 95% reliability based on all tensile and bending tests.
The simple closed-form non-LEFM solution presented in this study was derived in recent years for quasi-brittle fracture of heterogeneous solids such as coarse-grained ceramics, rocks and concrete with the average grain size G as Cch [18,21], based on the previous boundary effect model [29–31]. This non-LEFM model has been well tested by a numerous test results from different sources for polycrystalline solids and concrete-like particle composites. 6
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composite unit Cch and “composite” tensile strength ft, i.e., KIC ¼ 2ft√(3Cch). (5) It is worthwhile emphasized again that the linear relation Pmax ¼ ft x Ae applies to non-LEFM fracture of CFRP with shallow surface scratch and ideal LEFM fracture of very thick CFRP structures with deep cracks/notches (which may never happen in reality). Since CFRP panels made from carbon fibre pre-pregs have compli cated hierarchical microstructures, further studies with systematic tests of CFRP with various woven structures are warranted. Declaration of competing interest None. CRediT authorship contribution statement Bingyan Yuan: Investigation, Formal analysis, Writing - original draft, Writing - review & editing. Yunsen Hu: Investigation. Xiaozhi Hu: Conceptualization, Methodology, Writing - review & editing, Supervision.
Fig. 8. “Composite” tensile strength ft estimated from 3-p-b tests as affected by failure modes.
The recent study on bone-like bamboo composites [16] has extended the non-LEFM solution to composites with fibre bundle microstructures. The current study on the carbon fibre composites made from 2D woven fabrics has further extended the model into fracture analysis of laminar CFRP structures made from carbon fibre pre-pregs, which is commonly used in aeronautical and aerospace industries. Probably, the extension of this closed-form solution to fracture of CFRP is the most important application of this non-LEFM model. It is encouraging to see that the simple closed-form model has worked well for all those different com posites with distinctly different microstructures. The characteristic composite unit Cch plays a significant role in modelling of composite fracture as shown in Eq. (7). Neither LEFM nor SoM contains Cch as they are strictly limited to homogeneous materials. Potentially, Eq. (7) with the characteristic composite unit Cch can open the door for future frac ture research of other composite systems as long as Cch responsible for bulk fracture properties such as “composite” tensile strength ft and “composite” fracture toughness KIC can be clearly identified.
Acknowledgement B. Yuan and Y. Hu thank the financial supports from Australian Government through “Australian Government Research Training Pro gram Scholarship”. References [1] A.R. Ravindran, R.B. Ladani, S. Wu, A.J. Kinloch, C.H. Wang, A.P. Mouritz, Multiscale toughening of epoxy composites via electric field alignment of carbon nanofibres and short carbon fibres, Compos. Sci. Technol. (2018), https://doi.org/ 10.1016/j.compscitech.2018.07.034. [2] C. Soutis, Fibre reinforced composites in aircraft construction, Prog. Aero. Sci. 41 (2) (2005) 143–151, https://doi.org/10.1016/j.paerosci.2005.02.004. [3] https://www.geaviation.com/commercial/engines/ge9x-commercial-aircraft-e ngine accessed May 2019. [4] A.G. Atkins, Y.-W. Mai, Elastic and Plastic Fracture: Metals, Polymers, Ceramics, Composites, Biological Materials, 1985. [5] P.D. Soden, M.J. Hinton, A.S. Kaddour, A comparison of the predictive capabilities of current failure theories for composite laminates, Compos. Sci. Technol. 58 (7) (1998) 1225–1254, https://doi.org/10.1016/S0266-3538(98)00077-3. [6] M.J. Hinton, A.S. Kaddour, P.D. Soden, A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence, Compos. Sci. Technol. 62 (12–13) (2002) 1725–1797, https://doi.org/ 10.1016/S0266-3538(02)00125-2. [7] R.M. Christensen, Failure criteria for fiber composite materials, the astonishing sixty year search, definitive useable results, Compos. Sci. Technol. (2019) 107718, https://doi.org/10.1016/j.compscitech.2019.107718. [8] R.M. Christensen, Timoshenko medal award paper—completion and closure on failure criteria for unidirectional fiber composite materials, J. Appl. Mech. 81 (1) (2013), 011011, https://doi.org/10.1115/1.4025177, 2014. [9] R.M. Christensen, K. Lonkar, Failure theory/failure criteria for fiber composite laminates, J. Appl. Mech. 84 (2) (2017), 021009, https://doi.org/10.1115/ 1.4035119. [10] R.M. Christensen, Lamination theory for the strength of fiber composite materials, J. Appl. Mech. 84 (7) (2017), 071007, https://doi.org/10.1115/1.4036825. [11] S.S.R. Koloor, M.R. Khosravani, R.I.R. Hamzah, M.N. Tamin, FE model-based construction and progressive damage processes of FRP composite laminates with different manufacturing processes, Int. J. Mech. Sci. 141 (2018) 223–235, https:// doi.org/10.1016/j.ijmecsci.2018.03.028. [12] P.F. Liu, J.Y. Zheng, Recent developments on damage modeling and finite element analysis for composite laminates: a review, Mater. Des. 31 (8) (2010) 3825–3834, https://doi.org/10.1016/j.matdes.2010.03.031. [13] X. Hu, Y. Mai, Mode I delamination and fibre bridging in carbon-fibre/epoxy composites with and without PVAL coating, Compos. Sci. Technol. 46 (2) (1993) 147–156, https://doi.org/10.1016/0266-3538(93)90170-L. [14] D. Shu, Y. Mai, Delamination buckling with bridging, Compos. Sci. Technol. 47 (1) (1993) 25–33, https://doi.org/10.1016/0266-3538(93)90092-U. [15] B.F. Sørensen, E.K. Gamstedt, R.C. Østergaard, S. Goutianos, Micromechanical model of cross-over fibre bridging–Prediction of mixed mode bridging laws, Mech. Mater. 40 (4–5) (2008) 220–234, https://doi.org/10.1016/j. mechmat.2007.07.007. [16] W. Liu, Y. Yu, X. Hu, X. Han, P. Xie, Quasi-brittle fracture criterion of bamboobased fiber composites in transverse direction based on boundary effect model,
5. Conclusion A simple closed-form solution for bending failure of CFRP panels with or without shallow surface scratch has been presented, and confirmed by preliminary tensile and bending test results presented in this study. A characteristic composite unit Cch has been identified as the key microstructure measurement for composite fracture, which is the ply thickness for CFRP made from pre-pregs. The following conclusions can be drawn from the current study: (1) The “composite” tensile strength ft of CFRP can be determined from both direct tensile tests and 3-p-b tests, but notched bending tests are much easier to perform. (2) Similar to the common relation for direct tensile tests, i.e., Fracture Load ¼ Strength x Area, an equivalent area Ae has been defined for 3-p-b conditions so that the fundamental relation that Fracture Load ¼ Strength x Ae still holds. (3) The closed-form non-LEFM model has previously been used suc cessfully for “particle composites” or heterogeneous poly crystalline solids such as coarse-grained ceramics, rocks and concrete. Without any modification, its direct application to CFRP panels appears to be successful based on the results ana lysed in this study. (4) The “composite” fracture toughness KIC of CFRP through the thickness direction is linked to the ply thickness or characteristic
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