Measurement of interlaminar fracture properties of composites using the J-integral method

Abstract

This study explored the use of J-integral approach for characterizing mode I, mode II and mixed-mode I/II interlaminar fracture toughness of composites materials. Delamination tests were conducted to measure the fracture toughness and the resistance curve (R-curve) for double cantilever beam, end-notched flexure and mixed-mode bending specimens. The J-integral approach and the well-established ASTM standard methods based on LEFM were compared in this work. The results obtained from both methods are in very good agreement. However, the J-integral method has the advantages of being applicable to materials with large fracture process zone (e.g., composites with fiber bridging) and provides a simpler experimental procedure with fewer inputs. More importantly, the presented method avoids the ambiguity in visual measurements of delamination length required by the ASTM standard methods. In this study, an image-processing program based on Hough transform was developed to obtain the rotation angles of the specimen. The method provides good accuracy and has lower requirements for image resolution, light and material surface compared to previous methods.

Keywords

Interlaminar fracture toughness delamination J-integral composite experimental methods

Introduction

Fiber reinforced polymer composites are increasingly used in applications including the aerospace, marine and civil industries over the past several decades. The superior strength and stiffness to weight ratio of composites are highly desirable in structures where weight is critical. However, composite laminates generally have low interlaminar strength and are highly susceptible to delamination initiation and propagation. Therefore, delamination is one of the major concerns in the use of composite structures. The initial delaminations in composite laminates are commonly induced by manufacturing defects or low-velocity impacts in service. The existing delaminations may propagate and severely reduce the in-plane strength, stiffness and durability of the composite laminates.^1–5

Consequently, the accurate measurement of interlaminar fracture toughness is of great importance in the design of composite structures, and it is critical to develop a reliable and practical characterization method. Standard test procedures have been well established for obtaining mode I, mode II and mixed-mode I/II interlaminar fracture toughness using the specimen configurations of double cantilever beam (DCB), end-notched flexure (ENF) and mixed-mode bending (MMB), respectively.^6–8 The fracture toughness is measured as the critical strain energy release rate (G_c) at the crack tip based on the assumptions of linear elastic fracture mechanics (LEFM). However, the LEFM is not applicable to those specimens containing large fracture process zone around the delamination front (e.g. composites with strong fiber bridging, adhesively boned joints). The fracture toughness measured by the current standard tests is also easily affected by the visual observation of delamination length. The problems involve difficulties in defining crack tip, crack tunneling effect and costly experimental set-up (e.g. travelling microscope).⁹ Alternatively, the J-integral-based approaches were proposed for characterizing interlaminar fracture toughness in recent years. The J-integral is a path independent integral proposed by Rice¹⁰ to evaluate the non-linear strain energy release rate. With properly selected integration paths, the fracture toughness of the specimens can be directly related to the load applied and the angular rotations of the beam at the loading points. Therefore, the J-integral methods provide simpler experimental procedure and avoid the ambiguity of visual measurements for delamination length. In addition, they are applicable to a wider selection of materials without the restrictions of small Fracture Process Zone (FPZ) by LEFM. A closed form solution of J-integral for the DCB specimen was derived by Anthony and Paris¹¹ to evaluate the mode I interlaminar fracture toughness. Following this work, Gunderson et al.⁹ performed the DCB tests on GFRP specimens using both the J-integral method and the ASTM standard procedure. A video camera and transducers were used to monitor the delamination length and record the beam angles at the loading points, respectively. Good agreements were found between the obtained G_IC and J_IC. Bradley et al.¹² and Nilsson¹³ derived a more general form of J for the DCB test considering large displacement, mid-plane stretching and additional bending moment generated by the loading point offset from the mid-plane. The use of J-integral method for the mode II end-load-split and ENF tests were also investigated analytically and experimentally.^12,14–16 Stigh et al.¹⁵ designed an integration path for the ENF specimen, and the mode II interlaminar fracture toughness was calculated by the forces and the rotation angles at the loading points. In Stigh’s study, an image-processing system was developed to record the rotation angles every 5 s, which were indicated by painted ribs glued to the specimen. Recently, with a similar procedure, the J-integral approach was applied on the MMB configuration by Sarrado et al.¹⁷ Good agreements were found between the results of ASTM standard and the J-integral method, and the latter showed a significant lower level of uncertainty. They also compared the use of the LEFM-based method and the J-integral method for analyzing adhesive joints under mode I and mode II loadings. The results showed that the LEFM-based method was not suitable for the analysis of adhesive joints fracture, as it depends heavily on the size of the FPZ.¹⁸

This study systematically explored the use of J-integral approach for characterizing mode I, mode II and mixed-mode I/II interlaminar fracture toughness of composites materials. The test configurations were DCB for mode I, ENF for mode II and MMB for mixed-mode I/II, which were the same as those used in the ASTM standards for comparison. With the advantages of wider selection of materials, less requirements on material properties and easier data acquisition, the experimental procedure presented in this work is very attractive and can be used as an alternative testing method for the measurement of interlaminar fracture toughness.

Theory

The J-integral developed by Rice¹⁰ is given as

J = \int_{Γ} (wdy - T \cdot \frac{\partial u}{\partial x} ds)

(1)

where

Γ

is the contour in counter-clockwise direction; w is the strain energy density;

T

is the stress vector on the contour; and

u

is the displacement vector, respectively. Considering an integration path along the surface of the specimen with the delamination and the loads in Figure 1(a), the J-integral along the path

Γ_{tip}

around the crack tip is equal to

J_{Γ} (Γ = Γ_{1} + Γ_{2} + \dots + Γ_{7})

, as the J-integral is path independent. Equation (1) is applied to evaluate each component of

J_{Γ}

. As shown in Figure 1(a), all the vertical paths along the edges of the specimen overhangs are in stress free region, which results in

J_{Γ_{1}} = J_{Γ_{3}} = J_{Γ_{6}} = 0

. For paths

Γ_{4}

and

Γ_{5}

on the free surfaces of the delamination,

T = 0

and

dy = 0

, hence

J_{Γ_{4}} = J_{Γ_{5}} = 0 .

Only

J_{Γ_{2}}

and

J_{Γ_{7}}

are non-zero, and the J-integrals are calculated as

J_{Γ_{2}} = \int (T_{A} (x) + T_{c} (x)) \frac{\partial v}{\partial x} d x

(2)

J_{Γ_{7}} = \int (T_{B} (x) + T_{D} (x)) \frac{\partial v}{\partial x} d x

(3)

where v is the vertical component of the displacement vector. If only the concentrated forces are considered in Figure 1(a), the J-integral can be expressed as Sarrado et al.¹⁷

J_{Γ} = (P_{A} θ_{A} + P_{B} θ_{B} + P_{C} θ_{C} + P_{D} θ_{D}) / b

(4)

where P_A to P_D are the acting forces at point A to D, and

θ_{A}

θ_{D}

are the corresponding beam rotation angles with clockwise as the positive direction; b is the width of the specimen. The expression is modified to take account for large displacement.¹³

J_{Γ} = (P_{A} \sin θ_{A} + P_{B} \sin θ_{B} + P_{C} \sin θ_{C} + P_{D} \sin θ_{D}) / b

(5)

Figure 1.

Definition of (a) the integration path and the loadings and rotation angles for (b) DCB specimen (c) ENF specimen and (d) MMB specimen.

According to this general expression, the J-integrals are evaluated separately for the closed form solutions of DCB, ENF and MMB tests. For the DCB test shown in Figure 1(b), the $J_{DCB}$ is expressed as

J_{DCB} = \frac{P}{b} (\sin θ_{A} - \sin θ_{B})

(6)

where P is the magnitude of the applied load at the crosshead.

For the ENF test shown in Figure 1(c), the forces on the specimen can be expressed by the loading P on crosshead based on the static equilibrium of the experimental setup, and the $J_{ENF}$ is written as

J_{ENF} = \frac{P}{2 b} (\sin θ_{B} - 2 \sin θ_{c} + \sin θ_{D})

(7)

Similarly, the $J_{MMB}$ for MMB in Figure 1(d) can be written as

J_{MMB} = \frac{P}{2 b} [\frac{c}{L} \sin θ_{A} + (\frac{1}{2} - \frac{c}{2 L}) \sin θ_{B} - (1 + \frac{c}{L}) \sin θ_{C} + (\frac{1}{2} + \frac{c}{2 L}) \sin θ_{D}]

(8)

where c and L are the length of level arm and half span, respectively. However, when testing low-stiffness material, the weight of the loading apparatus needs to be taken into consideration. Therefore, in present study, a revised expression is derived to make the lever weight corrections

J_{MMB} = \frac{1}{2 b} [(\frac{Pc + P_{g} c_{g}}{L}) \sin θ_{A} + (\frac{P + P_{g}}{2} - \frac{Pc + P_{g} c_{g}}{2 L}) \sin θ_{B} - (P + P_{g} + \frac{Pc + P_{g} c_{g}}{L}) \sin θ_{C} + (\frac{P + P_{g}}{2} + \frac{Pc + P_{g} c_{g}}{2 L}) \sin θ_{D}]

(9)

where the P_g and c_g are the loading apparatus weight and its center of gravity from the center roller as shown in Figure 1(d). In this case, the weight of the lever contributes to

J_{MMB}

by 2–4%, but the effect could be more significant if the fracture toughness of the material is lower. To be consistent with the ASTM standard, the weight correction should be made when the weight of attached loading apparatus (P_g) is more than 3% of the maximum applied load in the test, and it should be made for all tests if any test in a series of tests requires the correction.

As a result, the evaluation of interlaminar fracture toughness for mode I, mode II and mix-mode I/II only requires 2, 3 and 4 rotation angles values synchronized with the applied load, respectively. In this approach, inputs required by ASTM standards such as material elastic properties, specimen compliance and delamination length are not needed.

Specimens

The composite laminate used in this study was manufactured by the woven glass fiber-reinforced bismaleimide prepreg, Hexply F655. The fiberglass, HexForce 1581, is a weave type of 8-harness satin with a fabric weight of 298 g/m². The laminate consisted 16 layers of prepregs, and a thin polymer release film was placed in the mid-plane (between layers 8/9) as the initial delamination. The laminate was cured in an autoclave according to the manufacturer’s recommended curing cycle followed by a 6-h post-cure process at 243℃. The fabricated laminate had an average thickness of 4 mm and an insert length of 65 mm. The laminate was cut into rectangular specimens by a diamond wafer cutter. The specimen has a dimension of 200 mm in length and 20 mm in width. The delamination tips on both sides of the specimen were marked using an optical microscope as shown in Figure 2. All the specimens were desiccated in a vacuum oven at 80℃ for two days and stored in a dry cabinet before testing.

Figure 2.

Illustration of the insert and pre-crack on the side view of a specimen.

Experimental procedures

The interlaminar fracture toughness of mode I, mode II and mixed-mode I/II were measured using the standard methods and the J-integral approach, simultaneously. The mode I and mixed-mode tests were conducted on a Shimadzu universal testing machine, while the mode II tests were conducted on an Instron universal testing machine. Both machines had a load cell capacity of 10 kN. The experimental setups are summarized in Table 1. Both sides of the specimen were painted with thin layers of correction fluid from the delamination front as required by the ASTM standards to aid in visual detection of the delamination. A small paper ruler with an accuracy of 1 mm was pasted on the specimen below the mid-plane to indicate the delamination length. Rotation marks, made by thin aluminum plates, were attached on the specimen to indicate the rotation angles of the beams at loading points. A camera with 15 fps was set up to record the delamination length and the rotations of the marks during the test as shown in Figure 3(a). Pictures taken from the camera were able to capture the delamination front on the specimen with an accuracy of at least ±0.5 mm.

Figure 3.

(a) Experimental setup and processed images taken from (b) DCB test (c) ENF test and (d) MMB test (red lines indicate the marks detected by the program).

Table 1.

Test setup parameters.

Test method	Span (mm)	Initial delamination (mm)	Loading speed (mm/min)	Number of rotation angles	Image acquisition speed	Tab thickness (mm)
DCB	–	50	1.5	2	1 frame/2 s	2
ENF	100	35	0.5	3	1 frame/1 s	–
MMB	140	50	0.5	4	1 frame/2 s	2

DCB: double cantilever beam; ENF: end-notched flexure; MMB: mixed-mode bending.

Image-processing algorithm

The image-processing method described in Walander¹⁴ used different colors to distinguish the rotation of ribs glued to the specimen from the white background, which can be easily affected by aliasing and distortion of the image. Moreover, the method took only two mid-points from the ribs to calculate the rotation of the ribs. It also results in relatively long ribs to ensure the accuracy of the measurement. A simpler and more reliable image-processing program was developed using MATLAB in this study. Light rectangular marks printed with the pattern shown in Figure 3(b) to (d) were designed to indicate the rotation of the specimen. The program first captures all the boundaries of a greyscale image taken by the camera through detecting the discontinuities of brightness. All the straight lines of the edges are then extracted using Hough transform. Hough transform is a commonly used technique to find straight lines in a digital image by carrying out a voting procedure in the parameter space. Each point $(x_{i}, y_{i})$ on the edge image is discretized to a polar representation $(ρ_{i}, θ_{i})$ in parameter space, and an accumulator matrix that contains all the possible $(ρ, θ)$ values is created. For each edge point in the image, the corresponding element in the accumulator matrix is increased. As all points on the same line would have a common set of parameters $(ρ, θ)$ , the straight lines in the image can be obtained by searching for corresponding local maxima of the accumulator with certain threshold. The image is divided into certain bounding boxes, and each of them contains only one mark. Within a box, the printed straight lines on the mark can be easily picked out by the setting limits on the gradients and length of the detected lines. The undulate patterns on the mark were design to avoid the mark edges to be selected instead of the printed lines. The red lines were generated to show the detected lines on the marks by the program, where the rotation angles can be accurately calculated through its polar coordinates. The program was verified by measuring computer generated angle beams ranging from −14° to 14° with an interval of 1°. Figure 4 shows the results obtained from three $(ρ, θ)$ resolutions, and the percentage errors are generally below 2%. The setting of ρ = 1.5 and θ = π/180 was found to have a good balance between accuracy and computational cost, and hence was used for all the tests. This method is more accurate compared to the previous method as the angle is obtained through all the points on the line, and it has lower requirements on image resolution, light and material surface.

Figure 4.

Validation test results of the image-processing program.

Mode I DCB test

The mode I energy release rate for DCB (G_I) was measured using modified beam theory recommended by the ASTM standard 5528-13⁷

G_{IC} = \frac{3 P δ}{2 b (a + | Δ |)} F

(10)

where the P and δ are the applied force and load–point displacement; a and b are the delamination length and the specimen width, respectively. The

| Δ |

is a delamination correction term due to the rotation of DCB arms, which is found as an offset from the origin on horizontal axis by plotting the cube root of compliance as a function of the delamination length of the specimen. F is the large displacement correction factor

F = 1 - \frac{3}{10} (\frac{δ}{a}) 2 - \frac{3}{2} (\frac{δ t}{a 2})

(11)

where t is the sum of the half hinge thickness and the half DCB arm thickness (

h / 2

). The delamination length from the artificial insert was 45 mm, and the specimen was first loaded at 1.5 mm/min to create a pre-crack around 5 mm before unloading (see Figure 2). Then the specimen was loaded again at the same speed until the delamination propagated for at least 40 mm. The J_DCB was calculated using equation (6). The values of interlaminar fracture toughness for the DCB specimens to initiate from artificial insert and pre-crack and to propagate were recorded. The calculation of the initiation values was based on the 5%/max criteria described by the ASTM standard, which normally produces the most reproducible results.

Mode II ENF test

The ENF tests were conducted in accordance to the ASTM D7905/D7905M-14 to measure the mode II energy release rate (G_II), and the J_ENF was calculated using equation (7). The compliance calibration (CC) method was performed to obtain the G_IIC. A specimen was loaded at three different positions with delamination lengths of 40 mm, 20 mm and 30 mm, respectively. In the first two tests, the specimen was loaded up to 50% of its failure load while, in the third test, the specimen was loaded until failure to create a pre-crack (see Figure 2). The beam compliance (C) is linearly related with a³

C = A + ma 3

(12)

where a is the delamination length; A and m are the CC coefficients which can be obtained from linear regression analysis of the three crack lengths and the corresponding beam compliances. Therefore, the pre-crack length was calculated using the compliance obtained by reloading the specimen at the same position using the following expression.

a = (\frac{C - A}{m}) 1 / 3

(13)

The same procedure was performed on the specimen with 35 mm delamination including the pre-crack, and the G_II is given as

G_{II} = \frac{3 mPa 2}{2 b}

(14)

where b is the width of the specimen. The starting delamination length was set as 35 mm instead of 30 mm suggested by the ASTM standard in order to achieve a more stable crack propagation.

Mixed-mode MMB test

The interlaminar fracture toughness (G_T) at different mixed ratios was measured by MMB tests described by ASTM standard D6671/D6671M-13⁸ and J-integral approach using equation (9). According to the standard, the G_I, G_II and G_T can be calculated as

G_{I} = \frac{12 [P (3 c - L) + P_{g} (3 c_{g} - L)] 2}{16 b 2 h 3 L 2 E_{1 f}} (a + χ h) 2

(15)

G_{II} = \frac{9 [P (c + L) + P_{g} (c_{g} + L)] 2}{16 b 2 h 3 L 2 E_{1 f}} (a + 0.42 χ h) 2

(16)

G_{T} = G_{I} + G_{II}

(17)

where

P, c, L

and a are applied load, loading lever length, half span and delamination length, respectively. χ is the crack length correction parameter depending on material elastic properties. P_g and c_g are the loading apparatus weight and its center of gravity from the center roller. The bending modulus

E_{1 f}

and crack length correction parameter were obtained following the ASTM standard. The mode mixity in the MMB test is defined as

\frac{G_{II}}{G_{T}} = \frac{G_{II}}{G_{I} + G_{II}}

(18)

The artificial delamination length of the specimen was 45 mm and was extended using DCB test procedure for 5 mm prior to the MMB test to achieve a total length of approximately 50 mm including the pre-crack (see Figure 2).

Results and discussion

The interlaminar fracture toughness test results for mode I, mode II and mixed-mode at different mode mixities are summarized in Table 2.

Table 2.

Summary of fracture toughness tests result.

Test method	Specimens	Mode mixity	G_C (J/m²)		J_pre-crack (J/m²)	G_propagation (J/m²)		J_propagation (J/m²)
Test method	Specimens	Mode mixity	Insert	Pre-crack	J_pre-crack (J/m²)	Mean	Std dev	Mean	Std dev
DCB	DCB-D01	0.00	532.0	815.8	803.2	787.2	15.8	799.4	14.0
	DCB-D02	0.00	470.0	835.4	820.0	835.6	19.2	855.4	21.6
	DCB-D03	0.00	481.0	743.4	735.6	776.6	11.4	778.4	13.2
MMB	MMB-D01	0.33	–	869.7	835.5	–	–	–	–
	MMB-D02	0.41	–	968.1	954.5	–	–	–	–
	MMB-D03	0.42	–	959.2	946.8	–	–	–	–
	MMB-D04	0.44	–	985.9	996.1	–	–	–	–
	MMB-D05	0.51	–	1023.9	1016.1	–	–	–	–
	MMB-D06	0.56	–	1063.3	1063.6	–	–	–	–
	MMB-D07	0.63	–	1218.2	1214.9	–	–	–	–
	MMB-D08	0.70	–	1378.0	1315.7	–	–	–	–
	MMB-D09	0.80	–	1446.4	1385.9	–	–	–	–
ENF	ENF-D01	1.00	1465.0	1316.3	1299.6	1432.6	79.4	1385.8	48.7
	ENF-D02	1.00	1455.3	1310.9	1326.2	1393.0	70.9	1354.3	41.9
	ENF-D03	1.00	1534.5	1440.8	1472.6	1570.3	90.3	1581.3	78.8

DCB: double cantilever beam; ENF: end-notched flexure; MMB: mixed-mode bending.

Mode I interlaminar fracture toughness

Figure 5(a) shows a typical applied load vs. load–point displacement curve of a specimen from the DCB tests. The analytical solution of the DCB test based on LEFM is given by Jacques et al.¹⁹

P = \sqrt{\frac{8}{δ}} b (\frac{G_{IC}}{12}) \frac{3}{4} (h 3 E_{1 f}) \frac{1}{4}

(19)

where the G_IC was taken as the averaged value of all the DCB specimen results, and

E_{1 f}

was determined based on the modified beam theory. In Figure 5(a), good agreement was found between the experimental results and the analytical solution, as no fiber bridging was observed in all DCB specimens, and the FPZ was small. The rotations at two loading points were continuous monitored by the image-processing program and presented in Figure 5(b) with respect to the corresponding displacement. The G_I and J_I vs. load–point displacement were calculated applying equations (10) and (6), respectively and shown in Figure 5(c). The G_I,NL and G_IC are the mode I critical energy release rate obtained by non-linear and 5% offset/Maximum criteria described in the ASTM standard. The obtained delamination resistance curves (R-curve) showed consistent mode I interlaminar fracture toughness for initiation and propagation. However, the interlaminar fracture toughness from the inserts was found significantly lower than that from pre-cracks. As shown in Figure 5(c), both methods provided comparable results and the variations were very small (see Table 2). However, J-integral approach is able to take continuously accurate readings and avoid the need for visually determining the delamination length. In fact, J is interchangeable with G in the cases where the LEFM holds, and the delamination length is given as Gunderson et al.⁹

a = \frac{3 δ}{2 (\sin θ_{A} - \sin θ_{B})}

(20)

Figure 5.

(a) Load–displacement curves (b) rotation angles vs. crosshead displacement at loading point (c) Mode I interlaminar fracture toughness results and (d) observed and calculated delamination length.

In Figure 5(d), the calculated delamination length is compared with that from visual observation, and the results are in excellent agreement. It is worth noting that the visually obtained delamination was corrected with $| Δ |$ in Figure 5(d).

Mode II interlaminar fracture toughness

The load–displacement curve of an ENF test is given in Figure 6(a). The analytical solution of ENF is also presented in the figure for comparison, and it is given as

δ = \frac{P [3 (a + 0.42 χ) 3 + 2 L 3]}{8 E_{11} bh 3}

(21)

P = \sqrt{\frac{16 G_{IIC} b 2 h 3 E_{11}}{9 (a + 0.42 χ) 2}} (a < L)

(22)

where a is the delamination length and b is the specimen width, and the G_IIC was taken as the averaged value of all the ENF specimen results. Compared with DCB test result, the Mode II delamination result showed a bit deviation from the analytical solution due to the unstable propagation. The difference between the predicted maximum load and the maximum loaded measured was 10.3% for the ENF test compared to 2.7% for the DCB test. Moreover, the variation of the ENF test result (5.5%) was higher than that of the DCB test (2.0%). The recorded three rotation angles at the loading points with respect to crosshead displacement are presented in Figure 6(b). The G_II and J_II vs. crosshead displacement were calculated using equations (14) and (7), respectively and shown in Figure 6(c). As the delamination growth in the ENF tests had relatively shorter range and tended to be unstable, it was very hard to determine the delamination tips visually, and only a few points were obtained using the ASTM standard procedure. On the other hand, the J-integral showed the advantages of real-time monitoring the fracture toughness during the tests, as it has no requirements on knowing delamination length and even the information of specimen properties. As shown in Table 2, the variations of the test results obtained from the J-integral approach are almost half of those from the standard procedure. Unlike mode I test results, the mode II fracture toughness measured from pre-cracks is lower than that from inserts. It is because less energy is required for the delamination to grow within an already formed Failure Process Zone ahead of the pre-crack tip. However, no such zone exists in front of the insert, and it requires the zone to be created before the main crack can propagate, resulting in higher fracture toughness. During the delamination growth, the mode II interlaminar fracture toughness was found to be a constant from the delamination resistance curve (R-curve), but it was slightly higher than the initiation value.

Figure 6.

(a) Load–displacement curves (b) rotation angles vs. crosshead displacement at loading points and (c) Mode II interlaminar fracture toughness results.

Mixed-mode I/II interlaminar fracture toughness

MMB tests with mode mixity ranging from 0.33 to 0.8 (see Table 2) were conducted to measure the interlaminar fracture toughness of the specimens. Typical applied load vs. crosshead displacement curves at four mode mixities are presented in Figure 7(a). Stable delamination growth was not achieved even at the 0.33 mode mixity and a slow crosshead speed of 0.5 mm/min. All the specimens experienced a sudden failure as the loading curves dropped abruptly after the peak values. The total energy release rate (G_T) was calculated by equations (15), (16) and (17), and the ratio mixity given by equation (18). The J_MMB was evaluated using equation (9). The values of mixed-mode interlaminar fracture toughness obtained from both methods are presented in Figure 7(b), and good agreement was found in all mode mixities. A power law criterion²⁰ is often used to predict the mixed-mode fracture toughness in terms of the interaction of its mode I and mode II energy release rate

(\frac{G_{I}}{G_{IC}}) α + (\frac{G_{II}}{G_{IIC}}) α = 1

(23)

where α is a material dependent parameter. In most cases, α is taken as 1 or 2 for prediction. Benzeggagh and Kenane²¹ proposed another commonly used mixed-mode criterion (B–K law), and the interlaminar fracture toughness is expressed as

G_{C} = G_{IC} + (G_{IIC} - G_{IC}) (\frac{G_{II}}{G_{T}}) η

(24)

where η is a material dependent parameter. The criteria were applied to the experimental data in Figure 7(b). It is shown that both criteria are able to provide reasonable predictions while the power law criterion gives a better overall result on mixed-mode interlaminar fracture toughness. The parameter α for this material was found to be 1.2, which is very close to the linear failure criteria used in numerical simulation.²²

Figure 7.

(a) Load–displacement curves at various mode mixities and (b) total interlaminar fracture toughness vs. mode mixities.

Conclusions

This study explored the use of J-integral approach for characterizing interlaminar fracture toughness of composites materials under mode I, mode II and mixed-mode I/II loadings. The delaminations tests were conducted for evaluating interlaminar fracture toughness of DCB, ENF and MMB specimens. Fracture tests were conducted on DCB, ENF and MMB specimens to measure the interlaminar fracture toughness for initiation and propagation. The test results were obtained using the well-established ASTM methods and the J-integral method simultaneously for comparison, and the following conclusions can be drawn:

In the J-integral method, the fracture toughness of a specimen can be directly related to the applied load and the angular rotations of the beam at the loading points. Therefore, the experimental method allows the fracture toughness under mode I, mode II and mixed-mode I/II loading to be measured using a common and simple experimental procedure.

The results obtained from the J-integral method showed very good agreement with those from standard method due to the small FPZ of the specimen used in this study. However, the ENF results of the J-integral method exhibited significant lower variances compared to the method described in the standard procedure. The main reason is that the visual determination of delamination length was avoided in J-integral method for the relatively unstable propagation.

Without the restrictions of LEFM assumption, the J-integral method is applicable to more materials, especially for those with large FPZ and significant fiber bridging phenomenon. No elastic properties, system compliance and delamination length need to be measured, which reduces the sources of uncertainty.

An image-processing program based on Hough transform is developed in this study to calculate the rotation angles, which allows the resistance curve to be measured in real time. The method has good accuracy and lower requirements for image resolution, light and material surface compared to the previous method.

Footnotes

Acknowledgments

The authors would like to thank Dr. Liu Wei, Software Engineer, from Microsoft for providing help in the image-processing program. The authors also acknowledge the support of the school of Mechanical and Aerospace engineering, Nanyang Technological University.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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