Fatigue-driven model for mode II interlaminar delamination propagation of fibre/epoxy-reinforced composite laminates under three-point end-notched flexure

Abstract

This paper deals with mode II interlaminar delamination properties and failure mechanisms of composite laminates in three-point end-notched flexure (3-ENF) tests under fixed cyclic loading. The results for fracture toughness, fatigue driving force and delamination propagation behaviours of three different types of carbon and E-glass fibre/epoxy resin-reinforced composite laminates are evaluated and compared with each other for providing an insight into fatigue damage development in composites and for constituting a fundamental basis for the development of a characterization model. A new characterization formulation is presented for mode II interlaminar delamination propagation based on controlling fatigue stress. Finally, the model is applied to experimental data, demonstrating the practical and effective use of the proposed model. It is shown that fatigue-driven strength for mode II interlaminar delamination propagation can be obtained realistically from experimental data using the new formulae.

Keywords

Laminate delamination fatigue

Introduction

Composite materials are being increasingly used in structures due to the superior specific stiffness and strength. However, because of initial flaws formed during processing and assembling, delamination might occur and propagate under cyclic loading, and this could lead to catastrophic structure failure. Previous research^1–12 has shown that delamination predominantly appeared in mode I (tensile) and mode II (shearing). Strain energy release rate (SERR) based on fracture mechanics was adopted in characterizing mode I/II fracture behaviours albeit predominantly under static loading. Actually, a precondition for composites application in engineering structures (especially in primary aerospace structures) is the need to determine crack propagation properties in mode I/II delamination under cyclic loading for assessing damage tolerance and residual life of composite structure. A large body of research^13–19 exists to determine and to characterize crack propagation behaviour of fibre/epoxy-reinforced composites in mode I delamination. There is growing interest in crack propagation properties in mode II interlaminar delamination of composites subjected to cyclic loading. Trethewey et al.²⁰ implemented three-point end-notched flexure (3-ENF) tests on unidirectional carbon/epoxy laminates through the fixed displacement control method to investigate stable mode II delamination propagation behaviour. Experiments showed that mode II delamination size grew with the SERR in a power law, and the relationship between the delamination propagation rate and the SERR followed the Paris model. After this, the fixed displacement control method was applied in 3-ENF tests for investigating environmental effects^21,22 on mode II delamination propagation behaviour and microscopic structure^23–25 of fibre/epoxy composite laminates. Due to the unstable nature of delamination growth,⁷ four-point end-notched flexure (4-ENF) was presented for stable mode II delamination propagation behaviour.¹⁰ Nevertheless, the adverse effects of significant specimen shortening and variation in clamping limited its validity and application.²⁶

In practice, the 3-ENF test has received considerable attention recently in light of the simple loading condition and small interlaminar friction resulting from the compression of the middle loading pin. It is worth noting that the fixed load control in 3-ENF test holds superiority over the fixed displacement control in terms of much simpler test procedure and constant stress level at crack tip, etc.^27,28 Thus, there is a need for understanding the fixed load control method on 3-ENF test and for developing a characterization model to determine the mode II delamination propagation performance of fibre/epoxy composite structures for engineering designs, particularly in the aerospace field, which is the focus of this paper.

Fatigue-driven model for mode II delamination propagation behaviour of 3-ENF beam

According to the Euler beam theory and fracture mechanics,²⁹ the compliance C and SERR of a 3-ENF beam are, respectively

C = \frac{δ}{P} = \frac{2 L 3 + 3 a 3}{8 EWh 3}

(1)

G II = \frac{P 2}{2 W} \frac{\partial C}{\partial a} = \frac{9 P 2 a 2}{16 EW 2 h 3}

(2)

For an initial crack size a₀, from equation (1), it is possible to have the initial compliance C₀ as

C 0 = \frac{2 L 3 + 3 a_{0}^{3}}{8 EWh 3}

(3)

From equations (1) and (3), one has

a = [\frac{C}{C 0} a_{0}^{3} + \frac{2}{3} (\frac{C}{C 0} - 1) L 3] 1 / 3

(4)

Again, from equations (1) to (2), it is possible to allow

G II = \frac{9 P δ a 2}{2 W (2 L 3 + 3 a 3)}

(5)

Analogous to the J-integral or energy release rate G in fracture mechanics, the concept of an SERR is introduced and interpreted as the driving force and criterion for mode II interlaminar delamination propagation. From equation (5), the SERR $Δ G II$ under a fixed cyclic loading with constant amplitude P_a at a stress ratio of R is defined as

Δ G II = G II, max - G II, min = \frac{9 (1 - R 2) P max δ max a 2}{2 W (2 L 3 + 3 a 3)}

(6)

with

P max = \frac{2 P a}{1 - R}

(7)

Generalizing fatigue load-based fracture data is a time-consuming and expensive process, thus it is desirable to examine how fatigue-driven SERR characterization can be deduced from static loading test. From equation (5), the fatigue-driven force (i.e. normalized SERR) can be written as

(G IImax) normalized = \frac{G IImax}{G IIC} = \frac{P max δ max}{P c δ c} (\frac{a}{a c}) 2 (\frac{2 L 3 + 3 a_{c}^{3}}{2 L 3 + 3 a 3})

(8)

Again, from equation (1), it is possible to show

\frac{δ max}{δ c} = \frac{P max}{P c} (\frac{2 L 3 + 3 a 3}{2 L 3 + 3 a_{c}^{3}})

(9)

Substituting equation (9) into equation (8) shows

(G IImax) normalized = (\frac{aP max}{a c P c}) 2 = (\frac{a}{L}) 2 (\frac{L}{a c}) 2 (\frac{P max}{P c}) 2

(10)

Equation (10) characterizes the fatigue-driven force pattern based on static fracture properties and it is obvious that in the case of a fixed cyclic loading with constant amplitude and at a stress ratio of R, the driving force monotonically increases with the delamination length a.

Again, analogous to the Paris model³⁰ in fracture mechanics, the crack growth rate da/dN versus $Δ G II$ for mode II interlaminar delamination propagation can be written as

\frac{da}{dN} = C 1 (Δ G II) m 1

(11)

Substituting equation (6) into equation (11) and integrating yields

a = C 2 (N f - N) m 2

(12)

with

m 2 = \frac{1}{1 - 2 m 1}

(13)

C 2 = {C 1 (2 m 1 - 1) [\frac{9 P_{max}^{2} (1 - R 2)}{16 EW 2 h 3}] m 1} \frac{1}{1 - 2 m 1}

(14)

N f = \frac{a_{0}^{1 - 2 m 1}}{C 1 (2 m 1 - 1)} [\frac{16 EW 2 h 3}{9 P_{max}^{2} (1 - R 2)}] m 1

(15)

where N_f represents the cycle number of fatigue stress to fatigue failure. Equation (12) is termed as the fatigue-driven model to describe the law of the mode II interlaminar delamination propagation versus cycle number of fatigue stress. From equation (12), it is clear that if

N = 0

, then

a = a 0

, and if

N \to N f

, then

a \to \infty

ENF experiments and discussion

Three types of specimens were manufactured using the resin transfer moulding technique and constructed out of the following unidirectional FRP laminate, respectively:

GFRP1 Laminate: GFRP (glass-fibre-reinforced plastic) laminate made from Rc10.800/6002 (i.e., Rc10 800 glass fibre and 6002 bisphenol A-type epoxy resin).

GFRP2 Laminate: GFRP (glass-fibre-reinforced plastic) laminate made from Rc10.800/3232 (or Rc10 800 glass fibre and 3232 mid-temperature curing epoxy resin).

CFRP Laminate: CFRP (carbon-fibre-reinforced plastic) laminate made from T300/5405 (namely, T300 carbon fibre and 5405 modified bismaleimide resin).

The mechanical properties are listed in Table 1 and three types of specimens were prepared with the same length (140 mm), width (25 mm) and end-notch length (40 mm), but with different nominal thicknesses (GFRP1: 2.4 mm; GFRP2: 3.7 mm; CFRP: 3.5 mm). In order to pre-fabricate the end-notch, a poly(tetrafluoroethylene) film with 50 µm thickness and 40 mm length was placed on the mid-plane at one end of the pre-formed specimen before curing. During consolidation, the composite pre-form was first heated up to 160℃ at a rate of 5℃/min and held for 1 h at this temperature, and then heated up again at the same rate to the desired temperature of 180℃, at which the specimen was held for 2 h. Next, the mould was cooled to room temperature, and the consolidation was completed. After this, a thin blade knife was used to make a pre-crack with 3–7 mm length in mode I to avoid an artificially resin rich effect.³²

Table 1.

Mechanical property.³¹

Material	E (GPa)	G13 (GPa)	μxy	Smax (MPa)
GFRP1: Rc10.800/6002	54	6	0.31	63
GFRP2: Rc10.800/3232	58	6	0.31	80
CFRP: T300/5405	124	8.5	0.28	127

GFRP: glass-fibre-reinforced plastic; CFRP: carbon-fibre-reinforced plastic.

All 3-ENF tests were conducted on MTS 880-50 kN servo-hydraulic machine in a dry state at room temperature. The specimens were supported on two cylindrical rollers to simulate simply supported boundary conditions, and the bending loads were applied at the mid-span by a remotely controlled hydraulic jack. For the fracture toughness tests, the level of applied load was monitored with a load cell and the deflections were measured by dial gauges at a displacement rate of 2 mm/min. Unlike significant stick-slip behaviour resulted from the interlaminar friction effect in 4-ENF tests,¹⁰ no observable amount of stick-slip behaviour was detected during 3-ENF tests, consistent with the efforts of previous work.³³ The load–displacement $P - δ$ curves of three types of tested specimens were, respectively, recorded and reported as shown in Figure 1(a) to (c). It is apparent that the $P - δ$ curves of both types of GFRP specimens are almost identical in the linear elastic region without the shear interlaminar delamination. The CFRP specimens show a relatively large scatter in initial stiffness, mainly induced by a sudden jump during mode I pre-crack procedure and accordingly relatively larger error in controlling pre-crack length. As the load continues to increase, the loading curve gradually transits into nonlinear segments. This phase is accompanied with tiny cracking noise, indicating a possible subcritical crack growth. When the load reaches the peaks, an interlaminar delamination failure happened with a notable cracking noise, followed by a sharp decrease in the slope of that curve. The unstable crack process is noticed for all specimens at the point of peak load. The nonlinearity onset of CFRP material emerges at a relatively low displacement compared with that of the other two types of GFRP materials under the same bending span length. This is mainly due to the higher bending rigidity of CFRP (see Table 1) and correspondingly higher interlaminar shear stress induced at the same loading displacement.

Figure 1.

Load–displacement curves from 3-ENF test ((a) GFRP1, (b) GFRP2 and (c) CFRP).

Fixed load control (or cyclic loadings of constant stress amplitude) was applied on all mode II interlaminar delamination propagation tests at a stress ratio of 0.1 and at a frequency of 10 Hz for avoiding the hysteresis heating effect.³⁴ Owing to the difference of bending rigidity between different types of specimen, the cyclic loadings with constant amplitude were monitored from the loading response and controlled through adjusting the span distance between two cylindrical rollers during tests. So, the span distance was placed to be 80 mm for GFRP1 specimens while it was 100 mm for GFRP2 and CFRP specimens. In order to avoid the shifting of the specimen towards the notched end to cause a lower delamination resistance than the actual value, it is essential to add a restraint on the specimen during tests.^20,34,35 Thus, a longitudinal restraint was placed on the specimen by bonding a smooth and slender wood pin, which is so flexible that the stress concentration is negligibly small at the bonding location on the specimen (Figure 2). WZHD0850 long focal-length optical telemicroscope with 0.01 mm measurement accuracy was implemented to measure the delamination length a on side surfaces of specimen. The measurement was given at a fixed interval of cyclic loading, until a sinusoidal pattern of the loading response cannot be sustained by test machine, due to decreasing bending stiffness of a specimen. The experimental data of delamination length versus number of stress cycles $a - N$ for three types of specimens are shown in Figure 3(a) to (c), respectively.

Figure 2.

Shifting restraint in 3-ENF test.

Figure 3.

Interlaminar delamination size versus cycle number of constant cyclic loading ((a) GFRP1, (b) GFRP2 and (c) CFRP).

A 20 × optical microscope was employed to conduct the fractographic analysis of fracture specimen (Figure 4). From Figure 4, it is obvious that the mode II interlaminar delamination propagation region on fracture surface of CFRP shows ‘hackle patterns’, while the mode I pre-crack area appears ‘river markings’, demonstrating the fractographic feature of mode II delamination propagation resulted from shear stress distinct from that of mode I delamination propagation arising from tensile stress. The fractographic feature in this paper has a good agreement with those in literature.¹²

Figure 4.

Fracture surface for CFRP.

Application to a practical case

The peak loads on the

P - δ

curves of tested specimens were utilized to determine mode II fracture toughness

G IIC

from equations (4) to (5) (Table 2). From Table 2, it is noticeable that the CFRP has greater mean value but less coefficient of variation of mode II fracture toughness

G IIC

than the GFRPs.

Table 2.

Fracture toughness and coefficient of variation.

Specimen no.	$G IIC$ values (J/mm²)						Coefficient of variation (%)
Specimen no.	1	2	3	4	5	Ave.	Coefficient of variation (%)
GFRP1	1708.5	1781.3	1123.4	990.3	1371.6	1395.0	25.0
GFRP2	1092.4	1091.0	1393.3	1672.0	1475.4	1105.7	18.8
CFRP	1523.2	1506.9	1463.9	1607.7	1456.5	1511.7	4.0

GFRP: glass-fibre-reinforced plastic; CFRP: carbon-fibre-reinforced plastic.

Experimental propagation values of $G IImax$ under fixed mode II cyclic loading are calculated by equation (5) and then normalized by the $G IIC$ obtained from quasi-static 3-ENF test. Then, the characterization curves for fatigue-driven forces are plotted by equation (10) (Figure 5(a) to (c)). From this, it is evident that there is a reasonably good agreement between the experimental data and the characterization curves. Again it is worth noting that the characterization method proposed is semi-empirically based on fracture parameters (a_c, P_c) measured from the static loading test, which avoids the time-consuming fatigue tests to be conducted.

Figure 5.

Driving force curves at different loading ratios ((a) GFRP1, (b) GFRP2 and (c) CFRP).

By the linear regression parameter estimation method,³⁶ the parameters m₂, C₂ and N_f in model (12) are estimated from the experimental data (Figure 3(a) to (c)), and the delamination propagation curves are shown in Table 3 and Figure 3(a) to (c). One can see that there is good agreement between the experimental data and the fitted curves.

Table 3.

Delamination propagation curves.

Material	$P max / P c$	$a - N$ curve
GFRP1	0.94	$a = 35.01 (8.34 \times 103 - N) - 0.079$
	0.72	$a = 52.43 (4.38 \times 105 - N) - 0.088$
	0.63	$a = 35.01 (8.34 \times 103 - N) - 0.079$
GFRP2	0.65	$a = 67.65 (5.84 \times 103 - N) - 0.11$
	0.56	$a = 86.75 (3.22 \times 104 - N) - 0.12$
	0.47	$a = 59.72 (6.06 \times 105 - N) - 0.063$
CFRP	0.55	$a = 247.93 (2.46 \times 104 - N) - 0.21$
	0.50	$a = 112.52 (7.54 \times 104 - N) - 0.13$
	0.37	$a = 213.01 (3.49 \times 105 - N) - 0.16$

GFRP: glass-fibre-reinforced plastic; CFRP: carbon-fibre-reinforced plastic.

From the experimental data $[a i, N i]$ (Figure 3(a) to (c)), SERR $Δ G II, i$ and delamination propagation rate corresponding to the experimental data a_i can be obtained by using equation (6) and the secant method³⁷ (Figure 6(a) to (c)). Thus, the mode II delamination propagation rate models of three types of composites GFRP1, GFRP2 and CFRP are, respectively, obtained based on the linear regression parameter estimation method³⁶ and the data of mode II delamination propagation rate (Figure 6(a) to (c))

da / dN = 2.53 \times 10 - 24 (Δ G II) 6.74

(16)

da / dN = 8.86 \times 10 - 18 (Δ G II) 5.26

(17)

da / dN = 1.78 \times 10 - 13 (Δ G II) 3.34

(18)

Figure 6.

Interlaminar delamination propagation rate versus SERR ((a) GFRP1, (b) GFRP2 and (c) CFRP).

Equations (16) to (18) are shown in Figure 6(a) to (c). From this, it is evident that there is good agreement between the experimental data and the predicted curve. Thus, it is argued that the mode II delamination propagation rate models of equations (11) and (12) have adequately and logically characterized the physical characteristics and the phenomenological quantitative laws. It should be noted despite the accuracy of the proposed characterization method by equation (12) in characterizing delamination propagation, mode II delamination fatigue behaviour in terms of delamination propagation rate scattered. This is probably due to the inevitable microscopic deficiencies during manufacturing. Therefore, fatigue data are still needed for reliably fitting the Paris model parameters, which deserves further studies into it.

Conclusion

The focus of this paper has been to present mode II interlaminar delamination properties of composite laminates by using three-point end-notched flexure (3-ENF) tests under fixed cyclic loading to understand the failure mechanisms and to model the degradation process of mode II interlaminar delamination strength. Three different types of carbon and E-glass fibre/epoxy resin-reinforced composite laminates were used for the determination of mode II fracture properties. The results for the mode II fracture toughness, fatigue-driven force and delamination propagation behaviours were evaluated and compared with each other. The damage and failure mechanisms of the composite laminates have been discussed.

Some significative experimental and analytical results are obtained. A practical model for mode II interlaminar delamination strength and life prediction is presented. Its applicability for estimating mode II interlaminar delamination model using minimal experimental data has been proven successfully. Close correlation is achieved between the predictions and the actual experimental results.

Footnotes

Conflict of interest

None declared.

Funding

This project was supported by the NSFC (no. 51375033) and ASFC (no. 20095251024).

Notation

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