Experimental research and model characterization on Mode I,Mode II and mixed-mode I/II delamination growth behavior of CFRP composites under quasi-static and fatigue loading

Abstract

This paper seeks to comprehensively investigate the interlaminar delamination growth behavior of carbon fiber reinforced polymer (CFRP) composites under quasi-static and fatigue loading by experimental research and model characterization. Mode I, Mode II and Mixed-mode I/II quasi-static and fatigue delamination growth tests were systematically conducted on double cantilever beam (DCB), end-notched flexure (ENF) and mixed-mode bending (MMB) specimens, respectively. The results show that the fatigue delamination growth (FDG) rate decreases with the increase of stress ratio under the same mode mixture and with the increase of mode mixture under the same stress ratio. Meanwhile, the delamination damage of CFRP composites is much more sensitive to fatigue loading in Mixed-mode I/II compared to pure Mode I and Mode II. A new FDG model considering both stress ratio and mode mixture effects is developed to characterize the FDG behavior of CFRP composites, showing good correlation with the experimental data. In addition, it can be found from fractographic analysis that mode mixture, loading condition and stress ratio significantly affect the microscale damage mechanisms during delamination growth process. The comprehensive experimental findings of the FDG behavior of CFRP composites and the new FDG model considering both stress ratio and mode mixture effects can provide effective theoretical and data references for engineering practice.

Keywords

CFRP composites delamination growth fatigue stress ratio mode mixture

Introduction

Fiber reinforced polymer (FRP) composites have been widely used in aircraft structures owing to their high specific strength, specific stiffness and corrosion resistance.¹ Most of the FRP composite laminates have poor out-of-plane properties due to a lack of reinforcement in the thickness direction. Thus, their interlaminar delamination has been a major damage mode which usually initiates from the free edge, ply drop and defects. The initiation and subsequent growth cannot be easily detected from surface signals, which could however lead to the direct deterioration of structural load-bearing capacity and service life.^2,3 The interlaminar delamination could be generally classified into Mode I, Mode II, Mode III and Mixed mode, the growth behavior of which have attracted great attention. Though the experimental standards for Mode I, Mode II and Mixed-mode I/II quasi-static delamination growth are mature (such as ASTM D5528,⁴ D7905⁵ and D6671⁶), available standards for fatigue delamination growth (FDG) are scarce. Therefore, it is necessary to further investigate the FDG behavior of FRP composites, providing foundation for accurate residual life prediction and damage tolerance design of composite aircraft structures.

Since fatigue crack growth rate is the main parameter for describing the crack growth behavior of materials controlled by the cyclic stress/strain field at crack front, fracture mechanics models relating the fatigue crack growth rate $d a / d N$ with stress intensity factor (SIF) $K$ or strain energy release rate (SERR) $G$ have been widely studied and used.^7,8 As one of them, the Paris-type model with power function has been proven to be valid for both metal and composite materials. Considering that it is difficult to precisely calculate the SIF at crack front of composites, SERR is commonly used in the Paris-type FDG model of composites as⁹

\frac{d a}{d N} = C {[f (G)]}^{m}

(1)

However, there is still no consensus on the function of controlling parameter $f (G)$ in equation (1). Some researchers stated that the maximum SERR plays a key role in FDG of composites, thus $f (G)$ could be $G_{\max}$ .^10,11 Some other researchers stated that it cannot reflect the cyclic characteristics of fatigue load and $f (G)$ should be the SERR range $Δ G$ with reference to the SIF range definition $Δ K$ of metal materials.^7,12 However, it was pointed out that $Δ G$ is a specific combination of stress amplitude and mean stress without physical basis, resulting in a potential pitfall in using it as a similitude as $Δ K$ . In order to avoid the misinterpretation of the FDG test results, the equivalent SERR range as $Δ G_{e q} = {(\sqrt{G_{\max}} - \sqrt{G_{\min}})}^{2}$ was deduced according to the principles of similarity.^13,14 The above Paris-type model can not only directly describe the relationship between FDG rate and SERR, but can also be integrated with the cohesive zone model (CZM),¹⁵ extended finite element method (XFEM),¹⁶ virtual crack closure technique (VCCT)¹⁷ and their coupled approaches,^18,19 to visually simulate the FDG path in composites.

Furthermore, the influence factors of FDG behavior of composites have also been extensively investigated by experiments. Firstly, as both the monotonic and cyclic parts of load cycles contribute to the delamination growth, significant stress ratio effect can be observed in the FDG curves of delamination growth rate versus maximum SERR or equivalent SERR range.^20,21 The delamination growth rate usually increases with the decrease of stress ratio under the same maximum SERR but with the increase of stress ratio under the same equivalent SERR range. The stress ratio effect was mainly attributed to the change of cyclic energy rather than only the crack shielding mechanisms such as crack closure and fiber bridging.^22,23 Moreover, a Paris-type FDG model considering stress ratio effect was developed by defining the controlling parameter as multiplication of the SIF contribution of both monotonic and cyclic parts of load cycles.⁸ This model was then extended by replacing the SIF related controlling parameter with the SERR related ones, such as $f (G) = {(Δ G)}^{1 - γ} G_{\max}^{γ}$ and $f (G) = {(Δ G)}^{1 - γ} G_{mean}^{γ}$ .²⁴ Yao et al.²⁵ further modified the exponent in equation (1) with the function of normalized maximum SERR to develop a total fatigue life model. Khan et al.^26,27 also proposed a new two-parameter model based on the SERR considering stress ratio effect. In recent years, the machine learning models have also been employed to characterize the FDG behavior of composites at different stress ratios.^28,29

Fiber bridging and crack closure also have influence on the FDG behavior of composites, especially for the Mode I. Due to the jump and bifurcation of delamination front among different fiber bundles, some fibers are pulled out between the upper and lower fracture surfaces, which reduce the stress concentration at delamination front and inhibit the delamination growth. This fiber bridging phenomenon usually increases the fracture toughness and reduces the delamination growth rate.^{9,14,23,30,31} Crack closure generally occurs at low stress ratios, which reduces the SERR at delamination front but has limited influence on the FDG rate.^23,32 In terms of the FDG models, a new definition of effective SERR range was proposed by subtracting the fiber bridging energy from the maximum SERR in $Δ G_{e q}$ to consider the fiber bridging effect, which is $Δ G_{e f f} = (G_{\max} - G_{f b}) {(1 - R)}^{2}$ .^33,34 Some similar definitions of effective SERR range were also developed by considering crack closure effect, such as $Δ G_{e f f} = G_{\max} - G_{c c}$ and $Δ G_{e f f} = {(\sqrt{G_{\max}} - \sqrt{G_{c c}})}^{2}$ .³² In contrast, some researchers stated that bridging fibers have little contribution to the SERR in FDG since they periodically store and release strain energy without permanent strain energy release.³⁵ Moreover, the models employing the normalized similitudes (including $G_{\max}$ , $Δ G$ , $Δ G_{e q}$ and $Δ G_{e f f}$ ) by fracture toughness $G_{c}$ ,^20,36 delamination resistance curve value $G_{c} (a)$ ^21,37,38 or fatigue delamination resistance curve value $G_{c f} (a)$ ^39–42 were developed to decrease the model parameters and the scatter of predictions.

Owing to the complex configuration, multidirectional layers and asymmetric crack orientation in actual composite structures, delamination often grows in Mixed-mode I/II. Mode mixture defined as the fraction of Mode II SERR to total SERR is another major factor affecting FDG behavior of composites. With the increase of mode mixture, the fracture toughness usually increases and the FDG rate usually decreases under the same SERR.^25,40,43,44 Meanwhile, complicated interaction fracture mechanisms happen in Mixed-mode I/II FDG, including fiber bridging, matrix shear cracking and matrix roller formation.⁴⁵ A lot of Mixed-mode I/II FDG models has also been developed, which could be generally divided into two categories. Firstly, some models were derived as the summation of individual Mode I and Mode II terms determined by Paris-type law.^7,46 On the other hand, some models were developed by modifying the controlling similitudes or model parameters of Paris-type function with introducing the mode mixture.^44,45,47 In addition, lots of other factors such as the composite material property,⁴⁸ ply thickness,⁴⁹ adjacent ply orientation,^3,49 interface configuration,⁵⁰ temperature⁵¹ and hygrothermal aging⁵² also have significant influence on the delamination growth behavior of composites under quasi-static and fatigue loading.

From the above review, it has been observed that most of the experimental studies on the quasi-static and fatigue delamination growth behavior of composites focus on a single mode. The existing Mixed-mode I/II FDG models scarcely consider the stress ratio effect, while the FDG models considering stress ratio effect are usually developed for pure Mode I and Mode II. However, composite structures often simultaneously undergo Mixed-mode I/II delamination damage growth and variable-amplitude fatigue loads containing different stress ratios. To the authors’ knowledge, the only FDG model considering both stress ratio and mode mixture effects to date⁴⁷ assumes that the model parameter C remains constant for all loading conditions, which might induce great predictive deviation in some conditions. It is also difficult for this model to distinguish the contributions of individual modes and to reflect the mode interaction effect, owing to its direct adoption of the total SERR as the controlling parameter. Hence, in this paper, Mode I, Mode II and Mixed-mode I/II delamination growth tests are systematically conducted on carbon fiber reinforced polymer (CFRP) composites under both quasi-static loading and fatigue loading with different stress ratios. Corresponding fractographic analysis is also carried out to achieve a comprehensive understanding of the delamination growth behavior of CFRP composites. Moreover, a new FDG model considering both stress ratio and mode mixture effects is developed using a weighted sum of two FDG rate components driven by Mode I and Mode II SERR. This model can adapt to various loading conditions and composite materials more flexibly while implicitly reflecting the mode interaction effect.

Materials and specimens

In this paper, the composite laminates are made of unidirectional CFRP composite material AC531/CCF800H (AVIC Composites Co., LTD) with a ply thickness of 0.14 ± 0.01 mm and a fiber volume fraction of 65%. The raw materials are unidirectional carbon fiber material CCF800H and epoxy material AC531. The diameter of a single carbon fiber is 5 μm. The mechanical properties of this composite material are listed in Table 1.⁵³ The stacking sequence of the composite laminates is [0₁₆//0₁₆] where “//” means the location of initial artificial delamination defect. During the manufacturing process, the prepregs were stacked according to the lay-up design with a non-adhesive Teflon film embedded between the middle layers, which were then cured in an autoclave to a nominal thickness of 4.64 mm. For the cure cycle, the composite laminates were firstly vacuumized to −0.092 MPa at room temperature and then pressurized to 0.6 ± 0.05 MPa. Subsequently, the temperature was raised to 180 ± 5°C at a speed of 1.5°C/min, which was then hold for 3°h. Finally, the temperature was reduced at a speed of 1.5°C/min and the pressure was relieved when temperature droped below 60°C. The cured composite laminates were detected by the ultrasonic C-scan technique to ensure almost no potential flaws.

Table 1.

Mechanical properties of composite material AC531/CCF800H.

$E_{11}$ /GPa	$E_{22} = E_{33}$ /GPa	$μ_{12} = μ_{13}$	$μ_{23}$	$G_{12} = G_{13}$ /GPa	$G_{23}$ /GPa	$X_{T}$ /MPa	$X_{C}$ /MPa
147	9.25	0.33	0.4	4.40	4.24	2276	1300

According to ASTM standards D5528,⁴ D7905⁵ and D6671,⁶ double cantilever beam (DCB), end-notched flexure (ENF) and mixed-mode bending (MMB) specimens were used for Mode I, Mode II and Mixed-mode I/II delamination growth tests, respectively. The above manufactured composite laminates were machined using a water-cooled steel blade to obtain the specimens as designed. Figure 1 presents the geometrical configurations of the three types of specimens. Additionally, a pair of piano hinges were adhesively attached to the upper and lower surfaces of the end of DCB and MMB specimens near the embedded artificial delamination. The lateral surfaces of the specimens, which were used for delamination observation, were coated with a thin layer of white correction liquid to enhance the visibility of crack front. Meanwhile, calibrated scale papers with a minimum scale of 1 mm were pasted on the painted lateral surfaces to facilitate the measurement of delamination growth length.

Experimental methods

Mode I delamination growth tests under quasi-static and fatigue loading

In accordance with ASTM standard D5528,⁴ Mode I quasi-static and fatigue delamination growth tests were carried out on DCB specimens in the lab environment. The experimental setups are present in Figure 2. A computer-controlled digital camera system was adopted to monitor and record the delamination growth process during the tests. Firstly, the DCB specimens were pre-cracked at a loading speed of 2 mm/min, which were suspended when the initial delamination crack grew by 3-5 mm. After checking that the difference in delamination length between the two lateral surfaces of specimen was within 2 mm, the specimens were unloaded at the same speed as the loading process. Then, the specimens for quasi-static tests were reloaded at a speed of 2 mm/min until the applied load decreased dramatically. The load-displacement curves and delamination length values were recorded accordingly. In addition, FDG tests were conducted at two stress ratios of 0.1 and 0.5 under displacement-control mode, with sinusoidal waveform loading applied at a frequency of 5 Hz. It has been reported that test frequency affects the fatigue behavior of FRP composites through two primary mechanisms, i.e., creep and hysteretic heating.^54,55 Nevertheless, previous research work^21,36,54 has demonstrated that the FDG behavior of composites is not significantly affected by test frequency if the self-heating temperature rise induced by cyclic loading remains minor. Therefore, low test frequencies (1∼10 Hz) are commonly employed in such studies. 1^{10,11,14,20,21,23,24,40,41,50,51,52,54} The load, displacement and delamination length data corresponding to the number of load cycles were recorded.

The Mode I SERR can be calculated using the compliance calibration (CC) method, which is written as⁴

G_{I} = \frac{m P δ}{2 b a} \frac{F}{N}

(2)

where

P

and

δ

are the applied load and displacement, respectively;

b

is the width of specimen;

a

is the delamination length which represents the horizontal distance between the delamination front and the load application point;

F

and

N

are the correction parameters, which can be given as

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

(3)

N = 1 - {(\frac{L^{'}}{a})}^{3} - \frac{9}{8} [1 - {(\frac{L^{'}}{a})}^{2}] (\frac{δ t}{a^{2}}) - \frac{9}{35} {(\frac{δ}{a})}^{2}

(4)

where

t

is the vertical distance between the center of hinge and the midplane of the specimen arm,

L^{'}

equals 0 when piano hinges are employed. As shown in Figure 3,

m

is the slope of the relationship curve between the logarithm of normalized compliance and the logarithm of delamination length, which can be obtained by the linear regression method.

C

is the specimen compliance which equals

δ / P

for quasi-static loading and equals

(δ_{\max} - δ_{\min}) / (P_{\max} - P_{\min})

for fatigue loading. For the quasi-static delamination growth tests, when

P

reaches the critical value for fracture,

G_{I}

equals the fracture toughness value

G_{Ic}

Mode II delamination growth tests under quasi-static and fatigue loading

In accordance with ASTM standard D7905,⁵ Mode II quasi-static and fatigue delamination growth tests were carried out on ENF specimens. The experimental setups are present in Figure 4. A three-point bending fixture was employed for loading. The ENF specimen was placed on two lower supporting rollers connected to the base which was attached to the lower disk of the testing machine. The upper loading roller was positioned at the middle span of the specimen and clamped to the upper jaw of the testing machine. Similarly, a computer controlled digital camera system was adopted. The quasi-static tests were conducted at a loading speed of 1 mm/min until the applied load suddenly dropped. The FDG tests were conducted at two stress ratios of 0.1 and 0.5 at a frequency of 5 Hz.

According to the classical Irwin-Kies theory, the relationship between SERR and specimen compliance can be expressed as^21,54,56

G_{II} = \frac{1}{2 b} P^{2} \frac{\partial C}{\partial a}

(5)

where

a

is the delamination length which represents the horizontal distance between the delamination front and the center of the supporting roller near the initial artificial delamination,

C

is the specimen compliance which can be calculated using the simple beam theory (SBT):

C = \frac{2 L^{3} + 3 a^{3}}{E_{1} b h^{3}}

(6)

where

L

is the half span of ENF experimental setup,

h

the thickness of specimen,

E_{1}

is the longitudinal modulus. Substituting equation (6) into equation (5) gives

G_{II} = \frac{9 P δ a^{2}}{2 b (2 L^{3} + 3 a^{3})}

(7)

For the quasi-static delamination growth tests, when $P$ reaches the critical value for fracture, $G_{II}$ equals the fracture toughness value $G_{IIc}$ .

Mixed-mode I/II delamination growth tests under quasi-static and fatigue loading

In accordance with ASTM standard D6671,⁶ the Mixed-mode I/II quasi-static and fatigue delamination growth tests were carried out on MMB specimens. The experimental setups are present in Figure 5. The piano hinges were connected to the fixture attached to the testing machine, while the opposite specimen end was placed on the lower supporting roller, guaranteeing the specimen was fixed in a horizontal position. The upper loading roller was positioned at the middle span of the specimen. The lever length of the MMB test apparatus (i.e., the horizontal distance between load application point on the lever and the center of upper loading roller) was set according to the mode mixture which was 0.5 in this paper. A mode mixture of 0.5 was selected as the representative Mixed-mode condition since both the Mode I and Mode II SERR account for half of the total SERR. This condition was deemed favorable for distinguishing the contributions of individual modes. The quasi-static tests were conducted at the speed of 1 mm/min until the applied load suddenly dropped. In order to precisely calculate the bending modulus of specimen which requires the system compliance value in the computational formula, a calibration specimen made of alloy steel with a known modulus was also loaded on the MMB apparatus to obtain its load-displacement curve. The FDG tests were conducted at two stress ratios of 0.1 and 0.3 at a frequency of 5 Hz. Considering that the Mode I and Mode II FDG data at the stress ratios of 0.1 and 0.5, as well as the Mixed-mode I/II FDG data at the stress ratio of 0.1 will be utilized as the fitting dataset for the developed FDG model, the FDG tests at the stress ratio of 0.3 different from the stress ratio of 0.5 for pure Mode I and Mode II tests were designed to validate the predictive capacity of the developed FDG model at the stress ratio not included in the fitting process. If more Mixed-mode I/II FDG tests could be carried out across a broader range of stress ratios and mode mixtures in the future, the fitting and validation of the developed FDG model would be more robust with sufficient test data.

The SERR of Mixed-mode I/II delamination damage growth is composed of Mode I and Mode II components, which can be expressed as⁶

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

(8)

where

G_{I}

and

G_{II}

are the Mode I and Mode II components of SERR. The mode mixture is defined as the fraction of Mode II to total SERR, i.e.,

φ = \frac{G_{II}}{G_{I} + G_{II}}

(9)

In MMB tests, the designed mode mixture $φ$ can be achieved by setting an appropriate lever length $c$ which is given as

c = \frac{12 β^{2} + 3 α + 8 β \sqrt{3 α}}{36 β^{2} - 3 α} L

(10)

where

L

is the half span of experimental setup, the expressions of

α

and

β

are as follows:

α = \frac{1 - φ}{φ}

(11)

β = \frac{a + χ h}{a + 0.42 χ h}

(12)

where

h

is the half thickness of specimen,

χ

is the correction parameter which yields

χ = \sqrt{\frac{E_{11}}{11 G_{13}} [3 - 2 {(\frac{Γ}{1 + Γ})}^{2}]}

(13)

Γ = 1.18 \frac{\sqrt{E_{11} E_{22}}}{G_{13}}

(14)

where

E_{11}

E_{22}

and

G_{13}

are the longitudinal, transverse and shear modulus, respectively. Additionally, the longitudinal bending modulus of the specimen

E_{1 f}

can be calculated as

E_{1 f} = \frac{8 {(a_{0} + χ h)}^{3} {(3 c - L)}^{2} + [6 {(a_{0} + 0.42 χ h)}^{3} + 4 L^{3}] {(c + L)}^{2}}{16 L^{2} b h^{3} (\frac{1}{m} - C_{s y s})}

(15)

where

m

is the slope of the load-displacement curve,

C_{s y s}

is the system compliance which can be calculated based on the load-displacement curve of the calibration specimen.

C_{s y s} = \frac{1}{m_{c a l}} - \frac{2 L {(c + L)}^{2}}{E_{c a l} b_{c a l} t_{c a l}^{3}}

(16)

where

m_{c a l}

is the slope of the load-displacement curve of the calibration specimen;

E_{c a l}

is the modulus of the calibration specimen;

b_{c a l}

and

t_{c a l}

are the width and thickness of the calibration specimen, respectively. Finally, the Mode I and Mode II components of SERR can be obtained 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}

(17)

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}

(18)

where

P_{g}

is the weight of lever and attached apparatus,

c_{g}

is the lever length to the center of gravity. For the quasi-static delamination growth tests, when

P

reaches the critical value for fracture,

G_{T}

equals the fracture toughness value

G_{Tc}

Results and discussion

Quasi-static delamination growth behavior of Mode I, Mode II and mixed-mode I/II

Figure 6(a) presents the load-displacement curves of Mode I DCB specimens. It can be seen that the load increases almost linearly with the increase of displacement at the initial stage during which the delamination scarcely grows. As the displacement continuously increases, the delamination begins to grow, leading to a reduction of the curve slope. Subsequently, the load decreases rapidly after reaching the peak value owing to the accelerated delamination growth. Finally, the load reduction with the increase of displacement gradually becomes slow corresponding to the stable delamination growth. The Mode I fracture toughness values were calculated according to equation (2), which are listed in Table 2 where CV represents the coefficient of variation. The fracture resistance curves (R-curves) are presented in Figure 6(b). It can be observed that the difference between the initial and stable values of fracture toughness is small, which is consistent with the experimental findings in literature. 5⁵⁷ This could be attributed to the thick interlayer of epoxy observed between adjacent plies, which leads to a higher initial fracture toughness. Another explanation could be the very small-scale fiber bridging during the delamination growth process.

Table 2.

Interlaminar fracture toughness results.

Mode	Data				Average	CV
Mode I $G_{Ic}$ (N/m)	245.6	275.8	283.2	245.7	262.6	7.5%
Mode II $G_{IIc}$ (N/m)	2917.1	2584.8	2694.4	2669.4	2716.4	5.2%
Mixed-mode I/II $G_{Tc}$ (N/m)	1184.4	1187.3	1149.2	1180.1	1175.3	1.5%

Figure 7(a) presents the load-displacement curves of Mode II ENF specimens. It is shown that at the initial stage, the increase of load with increasing displacement exhibits linear without obvious delamination growth. It subsequently becomes nonlinear with slight cracking sound, indicating that subcritical delamination growth happens. When the load reaches the peak value, the delamination instantaneously grows to the middle of the specimen length with remarkable cracking sound. Then, the load decreases drastically due to this unstable delamination growth which releases substantial strain energy. The Mode II fracture toughness values were calculated according to equation (7), which are also listed in Table 2. It should be noted that due to the aforementioned sudden and unstable delamination growth, R-curves are not typically constructed for Mode II ENF tests. Instead, a single-point evaluation of the critical SERR based on the onset of delamination growth is commonly used.^57,58 If more advanced observation techniques could be adopted in the future, more complete data would be obtained to provide a more comprehensive understanding of Mode II delamination growth behavior.

Figure 7(b) presents the load-displacement curves of MMB specimens at the mode mixture of 0.5. The observed delamination growth behavior is similar with Mode II. The Mixed mode I/II fracture toughness values were calculated according to equations (8), (17), and (18), which are also listed in Table 2. The results show that the Mixed-mode I/II fracture toughness is much higher than the Mode I fracture toughness but much lower than the Mode II fracture toughness. Moreover, the maximum CV value in Table 2 is 7.5%, which indicates that the data scatter of the interlaminar fracture toughness is within an acceptable range. Notably, this maximum CV corresponds to the Mode I fracture toughness, which can be attributed to the fact that the Mode I fracture toughness is considerably lower than the other two modes. Since the average fracture toughness serves as the denominator in the calculation formula of CV, this lower fracture toughness inherently leads to a higher CV value.

It can be observed from Figures 6(a) and 7 that a big difference exists among the Mode I, Mode II and Mixed-mode I/II load-displacement response after the peak loads. This could be primarily attributed to the fact that slow and stable delamination growth happens in Mode I whereas rapid and unstable delamination growth happens in Mode II and Mixed-mode I/II. As indicated by equations (2), (7), and (18), the Mode II SERR is much more sensitive to delamination length than the Mode I SERR, which means that it increases dramatically with the increase of delamination length. Once the Mode II SERR or Mixed-mode I/II SERR exceeds the initial fracture toughness, the delamination is prone to sudden growth. This accounts for the smooth downward trend after the peak loads in Mode I and dramatic decrease in Mode II and Mixed-mode I/II. Moreover, since the ENF specimen can still sustain the three-point bending load as an integral structure despite of containing a long delamination crack after unstable delamination growth, the Mode II curve exhibits a secondary ascending segment after the valley loads but with an extremely low slop, which means that the remaining load-bearing capacity of the specimen is very limited. In contrast, the MMB specimen containing a long delamination crack struggles to maintain structural integrity under combined opening and shearing loads. As a result, unstable delamination growth continues, leading to the sudden drop of the Mixed-mode I/II curve.

FDG model considering both stress ratio and mode mixture effects

The Paris-type model in equation (1) is commonly utilized to characterize the FDG behavior of composites. Using the normalized similitude by fracture toughness $G_{c}$ , the Mode I and Mode II FDG models can be expressed as

\frac{d a}{d N} = C {[\frac{f (G)}{G_{c}}]}^{m}

(19)

where

d a / d N

is the delamination growth rate. Since composite structures often undergo variable-amplitude fatigue loads containing different stress ratios, it is important to develop a FDG model considering the stress ratio effect. Given that both the monotonic and cyclic parts of load cycles have significant influence on the FDG behavior of composites, Hojo et al.⁵⁹ proposed a controlling parameter as multiplication of maximum SIF and SIF range (

{(K_{\max})}^{γ} {(Δ K)}^{1 - γ}, 0 \leq γ \leq 1

) in the Paris model to account for the respective contributions of maximum and cyclic stress. It has been widely used as an important reference to consider the stress ratio effect on FDG behavior of composites.^22,24,25 Based on the principles of similarity, the controlling similitude in equation (19) can be given as

f (G) = {(G_{\max})}^{λ} {(Δ G_{eq})}^{1 - λ}

(20)

where

λ

is the model parameter within the range of

[0, 1]

. The exponents

λ

and

1 - λ

reflect the weights of the monotonic and cyclic loading components in governing FDG behavior. The model parameter

λ

differs between Mode I and Mode II as the stress ratio effect on their FDG behavior is different.²² In contrast to Mode I, Mode II FDG involves no crack closure effect and is significantly governed by the shear friction of the fracture surfaces.

G_{\max}

is the maximum SERR.

Δ G_{eq}

is the equivalent SERR range, the expressions of which is given as follows:

Δ G_{eq} = {(\sqrt{G_{\max}} - \sqrt{G_{\min}})}^{2} = G_{\max} {(1 - R)}^{2}

(21)

where

R

is the stress ratio. Substituting equation (20) into equation (19) yields

\frac{d a}{d N} = C {[\frac{G_{\max} {(1 - R)}^{2 (1 - λ)}}{G_{c}}]}^{m}

(22)

Furthermore, the effect of mode mixture should also be considered in the FDG model as delamination often grows in Mixed-mode I/II. It can be summarized from previous studies that the existing Mixed-mode I/II FDG models are generally classified as two categories, i.e., the direct use of total SERR as the controlling parameter and the summation of FDG rate components driven by the individual Mode I and Mode II SERR.^{7,22,44–47,60} The former usually requires more model parameters and more experimental data for fitting and cannot decouple the individual mode contributions. The latter merely sums up the individual Mode I and Mode II terms without introducing weights related with mode mixtures, which might have great prediction deviations at varying mode mixtures. Therefore, a new model is developed based on the weighted summation of the two FDG rate components driven by Mode I and Mode II SERR.

\frac{d a}{d N} = w_{I} (φ) C_{I} {[\frac{G_{Imax} {(1 - R)}^{2 (1 - λ_{I})}}{G_{Ic}}]}^{m_{I}} + w_{II} (φ) C_{II} {[\frac{G_{IImax} {(1 - R)}^{2 (1 - λ_{II})}}{G_{IIc}}]}^{m_{II}}

(23)

where

w_{I} (φ)

and

w_{II} (φ)

are the weight functions, which are selected as exponential functions in this paper (

w_{I} (φ) = e^{k_{I} φ}

w_{II} (φ) = e^{k_{II} (1 - φ)}

). Then equation (23) becomes

\frac{d a}{d N} = e^{k_{I} φ} C_{I} {[\frac{G_{Imax} {(1 - R)}^{2 (1 - λ_{I})}}{G_{Ic}}]}^{m_{I}} + e^{k_{II} (1 - φ)} C_{II} {[\frac{G_{IImax} {(1 - R)}^{2 (1 - λ_{II})}}{G_{IIc}}]}^{m_{II}}

(24)

It can be observed that with the benefit of above exponential weight functions, equation (24) can be accurately degraded to the pure Mode I FDG model as $w_{I} (φ) = 1$ when $φ = 0$ , while it can also be accurately degraded to the pure Mode II FDG model as $w_{II} (φ) = 1$ when $φ = 1$ . Additionally, the above exponential weight functions can be nonlinearly increasing or decreasing and can be greater than or less than 1, depending on the specific values of $k_{I}$ and $k_{II}$ values. This characteristic enables the developed FDG model to flexibly adapt to different mode interaction mechanisms for various composite materials. Conversely, the fitted model from experimental data can also implicitly reflect the mode interaction effect. For example, a value of $k_{I} > 0$ implies that the mixture with Mode II promotes the FDG driven by Mode I SERR while $k_{I} < 0$ implies the opposite effect. Therefore, although the above exponential weight functions are empirical, they can provide significant advantages over alternative potential functions such as linear, power-law and polynomial forms. Nevertheless, the developed FDG model in equation (24) is still semi-empirical which cannot relate to the microscale physical mechanisms well. If the coupling terms could be further integrated into equation (24) without introducing excessive model parameters or great fitting difficulty in the future, the physical interpretability of the model would be further improved.

It should be noted that the developed FDG model in equation (24) does not account for fiber bridging effect. A lot of previous research results^{9,14,23,30,31,33,34} indicate that fiber bridging usually has great effect on the Mode I delamination growth behavior of composites under quasi-static and fatigue loading. However, this effect is much less for the unidirectional composites than the multi-directional composites.⁶¹ In this paper, only small-scale fiber bridging could be observed during the Mode I delamination growth of the unidirectional composites investigated. Additionally, previous research^9,35 reported that bridging fibers have little contribution to the SERR during the FDG process since they periodically store and release strain energy without permanent strain energy release, and the delamination growth can be independent on fiber bridging if a reasonable similitude parameter is used in the Paris-type FDG model. Therefore, it could be considered as reasonable not to explicitly account for fiber bridging effect in the developed FDG model.

Based on the experimental data of Mode I and Mode II FDG, the model parameters $C_{I}$ , $m_{I}$ , $λ_{I}$ , $C_{II}$ , $m_{II}$ and $λ_{II}$ in equation (24) can be firstly determined by the best fitting method. In the case of pure Mode I or Mode II, equation (24) is degraded to equation (22), taking logarithm of equation (22) gives

y = a + b x

(25)

where

y = \ln (\frac{d a}{d N})

x = \ln [\frac{G_{\max} {(1 - R)}^{2 (1 - λ)}}{G_{c}}]

a = \ln C

b = m

. Using the linear regression method,

a

and

b

are calculated as

b = \frac{L_{x y}}{L_{x x}}

(26)

a = \bar{y} - b \bar{x}

(27)

r = \frac{| L_{x y} |}{\sqrt{L_{x x} L_{y y}}}

(28)

where

{\begin{cases} \bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i} \\ \bar{y} = \frac{1}{n} \sum_{i = 1}^{n} y_{i} \\ L_{x x} = \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} \\ L_{y y} = \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2} \\ L_{x y} = \sum_{i = 1}^{n} (y_{i} - \bar{y}) (x_{i} - \bar{x}) \end{cases}

(29)

It can be observed that $\bar{x}$ , $L_{x x}$ and $L_{x y}$ are the functions of the undetermined constant $λ$ , which means that $a$ , $b$ and correlation coefficient $r$ are also related with $λ$ in the range of $(0, 1)$ . Based on the pure Mode I or Mode II FDG data, the undetermined constant $λ$ can be numerically solved by optimizing the correlation coefficient value. Then the unknown constants $a$ and $b$ as well as model parameters $C$ and $m$ are also obtained.

After the model parameters $C_{I}$ , $m_{I}$ , $λ_{I}$ , $C_{II}$ , $m_{II}$ and $λ_{II}$ are determined, the model parameters $k_{I}$ and $k_{II}$ in the weight functions can be fitted based on the experimental data of Mixed-mode I/II FDG. Taking logarithm of equation (24) and transformation, it gives

Y = A + B X

(30)

where

Y = \ln (\frac{d a}{d N}) - \ln C_{I} - m_{I} \ln [\frac{G_{Imax} {(1 - R)}^{2 (1 - λ_{I})}}{G_{Ic}}] - \ln C_{II} - m_{II} \ln [\frac{G_{IImax} {(1 - R)}^{2 (1 - λ_{II})}}{G_{IIc}}]

X = φ

A = k_{II}

B = k_{I} - k_{II}

. Similarly,

A

and

B

can be estimated using the linear regression method as shown in equations (26) to (29). Then the model parameters

k_{I}

and

k_{II}

can also be calculated. In the case of only experimental data at one mode mixture, the best fitting method with optimizing the correlation coefficient can be utilized to determine the model parameters

k_{I}

and

k_{II}

Model characterization of FDG behavior

Figure 8 presents the Mode I, Mode II and Mixed-mode I/II FDG test results, illustrating the relationship between the FDG rate $d a / d N$ and maximum SERR $G_{\max}$ . It is shown that both stress ratio and mode mixture have significant effect on the FDG behavior of CFRP composites. The FDG rate decreases with the increase of stress ratio under the same mode mixture and with the increase of mode mixture under the same stress ratio. In addition, it can be found from Table 2 and Figure 8 that the ratio of Mixed-mode I/II fatigue SERR to quasi-static fracture toughness is much lower than those of pure Mode I and Mode II. It indicates that the delamination damage of CFRP composites is much more sensitive to fatigue loading in Mixed-mode I/II compared to pure Mode I and Mode II, which should be considered during the damage tolerance design of composite aircraft structures. In addition, it can be seen from Figure 8 that the FDG rates fall within the range of 10^-6∼10^-2 mm/cycle. Considering that the low-cycle and high-cycle fatigue are usually distinguished according to the number of load cycles, and the practical detection precision in engineering practice is generally greater than 0.1 mm, the present research might cover the fatigue issues ranging from low-cycle to high-cycle regimes. Although the delamination measurement uncertainty is not considered in this paper, as the delamination growth during one load cycle is generally negligible, a single measurement of delamination length during one cycle could be considered as acceptable. If the camera system could be improved to capture more images within a single cycle in the future, the measurement uncertainty would be statistically analyzed to provide more reliable experimental data. Nevertheless, the data scatter of the FDG rate $d a / d N$ can be analyzed based on the assumption of logarithmic normal distribution. The 95% confidence bounds have been plotted in Figure 8.

The Mode I and Mode II FDG data at the stress ratios of 0.1 and 0.5, as well as the Mixed-mode I/II FDG data at the stress ratio of 0.1 in Figure 8 are utilized as the fitting dataset, while the Mixed-mode I/II FDG data at the stress ratio of 0.3 are employed as the validation dataset. The FDG model in equation (24) can be determined as

\frac{d a}{d N} = 3.692 \times e^{8.62 φ} {[\frac{G_{Imax} {(1 - R)}^{0.820}}{262.6}]}^{13.147} + 1.90 \times 10^{9} \times e^{(1 - φ)} {[\frac{G_{IImax} {(1 - R)}^{0.234}}{2716.4}]}^{25.532}

(31)

The coefficient of determination R² is commonly utilized as the prediction performance indicator, representing the degree to which the regression approximates the experimental data. Its expression is given as follows:

R^{2} = \frac{\sum_{i = 1}^{m} {({\hat{Y}}_{i} - \bar{Y})}^{2}}{\sum_{i = 1}^{m} {(Y_{i} - \bar{Y})}^{2}}

(32)

where

m

denotes the total number of experimental data points,

Y_{i}

denotes the i^th experimental data point which is the logarithm of

d a / d N

in this paper,

{\hat{Y}}_{i}

denotes the corresponding predicted value,

\bar{Y}

denotes the average value of all experimental data. For the fitted model in equation (31), the R² is calculated as 0.928, demonstrating good reliability of the fitted model parameters.

When the mode mixture is set as the specific values, equation (31) can be degraded to the Mode I, Mode II and Mixed-mode I/II FDG models considering stress effect as listed in Table 3. Furthermore, when the stress ratio is set as the specific values, they can be further degraded as the

d a / d N - G_{\max}

curves, which are also plotted in Figure 8. It can be seen from Figure 8 that the fitted

d a / d N - G_{\max}

curves have good agreement with the experimental data, demonstrating validity of the developed FDG model. There is a certain deviation between the predictions and experimental data at the mode mixture of 0.5 and the stress ratio of 0.3 as these experimental data are used for validation rather than fitting. In addition, the Mixed-mode I/II

d a / d N - G_{\max}

curves determined by unweighted summation of Mode I and Mode II components have also been plotted in Figure 8. Although the Mode I and Mode II

d a / d N - G_{\max}

curves coincide between the weighted and unweighted models, the weighted Mixed-mode I/II curve is higher than the unweighted counterpart. It means that the fitted weighted functions in equation (31) provide much higher predicted FDG rates than the direct summation, thereby showing better correlation with the experimental data. Table 3 also shows that the C values in the Mode I and Mode II components of the Mixed-mode I/II model at the mode mixture of 0.5 are greater than those in pure mode models, especially for the Mode I component. It implies that the mode mixture significantly promotes the FDG driven by the individual Mode I and Mode II SERR, and this mode interaction effect is more dominant in the Mode I component of delamination growth.

Table 3.

Degraded FDG model at specific mode mixtures.

Mode mixture	Degraded model
$φ = 0$	$\frac{d a}{d N} = 3.692 \times {[\frac{G_{Imax} {(1 - R)}^{0.820}}{262.6}]}^{13.147}$
$φ = 1$	$\frac{d a}{d N} = 1.90 \times 10^{9} {[\frac{G_{IImax} {(1 - R)}^{0.234}}{2716.4}]}^{25.532}$
$φ = 0.5$	$\frac{d a}{d N} = 274.83 {[\frac{G_{Imax} {(1 - R)}^{0.820}}{262.6}]}^{13.147} + 3.13 \times 10^{9} {[\frac{G_{IImax} {(1 - R)}^{0.234}}{2716.4}]}^{25.532}$

The above validation of the developed FDG model is only conducted in one condition (i.e., $φ = 0.5$ , $R = 0.3$ ) as the experimental data are limited, which is indeed insufficient. If more Mixed-mode I/II FDG tests could be carried out across a broader range of stress ratios and mode mixtures in the future, the fitting and validation of the developed FDG model would be more robust with sufficient test data. In order to further validate the developed FDG model in equation (24) with more abundant experimental data at different mode mixtures, the literature data of similar CFRP composites T800H/#3631^62–64 as shown in Figure 9 are adopted to fit the FDG model as

\frac{d a}{d N} = 0.567 e^{1.24 φ} {[\frac{G_{Imax} {(1 - R)}^{0.514}}{146.54}]}^{12.468} + 14.63 e^{20.17 (1 - φ)} {[\frac{G_{IImax} {(1 - R)}^{1.502}}{758.04}]}^{7.740}

(33)

In Figure 9, the experimental data at the mode mixtures of 0.637 and 0.840 are used as the validation dataset while the other data are used as the fitting dataset. It can be observed that the fitted $d a / d N - G_{\max}$ curves show good agreement with the experimental data in general, further demonstrating validity of the developed FDG model. The above validation indicates that the developed model can provide good or acceptable predictions at the known mode mixture and unknown stress ratio or the known stress ratio and unknown mode mixture. However, its predictive capacity in conditions where both the stress ratio and mode mixture are unknown requires further validation in the future work.

It should be noted that although the developed FDG model has only be validated with the experimental data of CFRP composites, it is also applicable for characterizing the FDG behavior of other FRP composites like glass fiber reinforced polymer (GFRP) composites. Future work is required to further validate the applicability of this model with additional FDG data of GFRP composites. Moreover, while the existing FDG models can effectively describe the FDG behavior of composites, most of them consider only stress ratio effect or only mode mixture effect or neither of them. It implies that after fitting the model parameters based on the experimental data at a specific stress ratio or a specific mode mixture, the model is only capable of predicting the FDG rates under this specific condition. To predict the FDG rates under other conditions, additional FDG tests are required to obtain the experimental data for refitting the model. The experimental cost would be high for the composite structures subjected to various stress ratios and mode mixtures in engineering practice. In contrast, based on the experimental data at specific known stress ratios and mode mixtures, the developed model in equation (24) can be fitted to predict the Mode I, Mode II and Mixed-mode I/II FDG rate of composites at unknown stress ratios or mode mixtures. Another FDG model considering both stress ratio and mode mixture effects⁴⁷ that we can search so far assumes that the model parameter C remains constant for all loading conditions, which might induce great predictive deviation in some conditions. It is also difficult for this model to distinguish the contributions of individual modes and to reflect the mode interaction effect, owing to its direct adoption of the total SERR as the controlling parameter. In contrast, the developed model can adapt to various loading conditions and composite materials more flexibly while implicitly reflecting the mode interaction effect through weighted summation of two FDG rate components driven by the individual Mode I and Mode II SERR, demonstrating great potential for engineering application.

Fractographic analysis

In order to reveal the influence of stress ratio and mode mixture on delamination damage growth mechanism of CFRP composites, fractographic analysis of representative fracture surfaces was undertaken by the scanning electron microscope (SEM) technique, which can provide evidence of microscale damage mechanisms that was not observable during tests. The Mode I, Mode II and mixed mode I/II fracture surfaces are shown in Figures 10–12.

It can be observed from Figure 10 that besides matrix cracks, apparent fiber/matrix debonding and fiber breakage occur at the interlaminar interfaces of Mode I DCB specimen. For the quasi-static fracture surface in Figures 10(a)–10(c), almost either clean exposed fibers or fiber imprints appear, which means that delamination damage generally grows across the fiber/matrix interfaces rather than the interlaminar resin-rich region. However, it tends to grow through the resin-rich region locally to form some large area of matrix surface without fiber imprints under cyclic loading, as shown in Figures 10(d)–10(i). Moreover, numerous transverse river-pattern and longitudinal wave-pattern matrix cracks as well as debonded fibers with remaining matrix appear, leading to the coarser fracture surfaces than quasi-static. The transverse river-pattern matrix cracks are stepwise stripes generally perpendicular to the crack growth direction, which are caused by cyclic loading (shown in Figures 10(f) and 10(i)). It also indicates that delamination damage grows more easily within the matrix than the fiber/matrix interfaces under long-term cyclic loading with low SERR. Furthermore, with the decrease of stress ratio, more irregular and serious matrix cracks and resin debris appear at the fatigue fracture surfaces. This could be attributed to more severe cyclic extrusion and injection of the resin under higher stress amplitude at the lower stress ratio.

Compared with the Mode I quasi-static fracture surface, the Mode II quasi-static fracture surface as shown in Figures 11(a)–11(c) is much coarser with irregular matrix cracks, oblique fiber fracture surfaces and numerous resin debris. A significant feature is the hackle pattern with a great deal of cusps caused by shear tearing (yielding) of the matrix, which are dependent on the local stress state at delamination front. Under Mode II fatigue loading, more damage grows through the interlaminar resin-rich region to form some large area of cloud-pattern matrix surface without fiber imprints due to the lower SERR and repeated sliding and friction between the upper and lower surfaces (shown in Figures 11(d)–11(i)). The cloud-pattern matrix surface means that the delamination damage grows across the resin-rich region to form a large fracture surface of matrix with irregular patterns caused by cyclic shear friction. The friction also planishes and even tears the shear cusps between neighboring fibers from the fracture surface to form the resin debris. Hence, the resin debris on fatigue fracture surface is smaller than that on quasi-static fracture surface. Meanwhile, less fiber breakage occurs under fatigue loading than quasi-static loading as a result of the lower SERR. Furthermore, the stress amplitude usually increases with the decrease of stress ratio, causing larger cloud-pattern matrix surface, more worn flat cusps and more but smaller resin debris resulted from more serious repeated sliding and friction.

The mixed mode I/II quasi-static fracture surface in Figures 12(a)–12(c) exhibits both aforementioned Mode I and Mode II characteristics under simultaneous opening and shear loading. Both clean and matrix cusp attached fibers, both transverse and oblique fiber fracture surfaces, as well as numerous resin debris can be observed. Under fatigue loading at the stress ratio of 0.3, more interlaminar resin-rich damage and less fiber breakage occur compared to quasi-static loading, as shown in Figures 12(d)–12(f). Meanwhile, both the typical Mode I fatigue characteristics (i.e., irregular transverse river-pattern and longitudinal wave-pattern matrix cracks) and Mode II fatigue characteristics (i.e., regular hackle-pattern matrix cracks) appear under coupled cyclic extrusion/injection and shear sliding. At the stress ratio of 0.1, the matrix cracks become more irregular as the Mode I and Mode II fatigue characteristics merge more tightly (shown in Figures 12(g)–12(i)). This could be attributed to the decrease of total SERR with the decrease of stress ratio. As the Mode I and Mode II SERR components decrease together while the Mode I fracture toughness is much lower than Mode II, the Mode I fatigue characteristics increases and the Mode II shear fatigue characteristics decreases.

Conclusions

This paper seeks to comprehensively investigate the Mode I, Mode II and Mixed-mode I/II delamination growth behavior of CFRP composites under quasi-static and fatigue loading. The conclusions are drawn as follows:

(i) The Mode I, Mode II and Mixed-mode I/II load-displacement curves exhibit linear at the initial stage. After the peak loads, the load decreases slowly with continuously increasing displacement and stable delamination growth under Mode I loading, while it decreases rapidly due to sudden and unstable delamination growth under Mode II and mode I/II loading. The Mode I, Mode II and Mixed-mode I/II fracture toughness of the CFRP composites AC531/CCF800H are 262.6 N/m, 2716.4 N/m and 1175.3 N/m, respectively.

(ii) Both stress ratio and mode mixture have significant effect on the FDG behavior of CFRP composites. The FDG rate decreases with the increase of stress ratio under the same mode mixture and with the increase of mode mixture under the same stress ratio. The delamination damage is much more sensitive to fatigue loading in Mixed-mode I/II compared to pure Mode I and Mode II. Moreover, a new FDG model considering both stress ratio and mode mixture effects is developed to characterize the FDG behavior of CFRP composites, showing good correlation with the experimental data. The developed model is also applicable for characterizing the FDG behavior of other FRP composites like glass fiber reinforced polymer (GFRP) composites.

(iii) The SEM results show that the Mode I quasi-static fracture surface is smooth having neat matrix cracks, clean debonded fibers and transverse fiber fracture surfaces, while Mode II quasi-static fracture surface is coarse with hackle-pattern matrix cracks, oblique fiber fracture surfaces and numerous resin debris. The mixed mode I/II quasi-static fracture surface exhibits both Mode I and Mode II characteristics under simultaneous opening and shear loading. In contrast to the quasi-static fracture surfaces, more irregular and serious matrix cracks and resin debris but less fiber breakage appears on the fatigue fracture surfaces, which are further enhanced with the decrease of stress ratio.

Footnotes

ORCID iDs

Aoshuang Wan

Feiyang Zhu

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundations of China (52205174), Fundamental Research Funds for the Central Universities (3122023PT13) and Natural Science Fund for Distinguished Young Scholars of Tianjin (23JCJQJC00100).

Declaration of conflicting interests

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

Data Availability Statement

Data will be made available on request.*

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