Research ArticleFatigue Delamination Crack Growth in GFRP CompositeLaminates: Mathematical Modelling and FE Simulation

Hassan Ijaz ,1 Waqas Saleem ,1 Muhammad Zain-ul-abdein,1 Aqeel Ahmad Taimoor,2

and Abdullah Salmeen Bin Mahfouz3

1Department of Mechanical Engineering, University of Jeddah, Jeddah, Saudi Arabia2Department of Chemical and Materials Engineering, King Abdulaziz University, Jeddah, Saudi Arabia3Department of Chemical and Materials Engineering, University of Jeddah, Jeddah, Saudi Arabia

Correspondence should be addressed to Hassan Ijaz; hassan605@yahoo.com

Received 29 September 2017; Revised 9 January 2018; Accepted 22 January 2018; Published 28 March 2018

Academic Editor: Vaios Lappas

Copyright © 2018 Hassan Ijaz et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Glass fibre-reinforced plastic (GFRP) composite laminates are used in many industries due to their excellent mechanical andthermal properties. However, these materials are prone to the initiation and propagation of delamination crack growth betweendifferent plies forming the laminate. The crack propagation may ultimately result in the failure of GFRP laminates as structuralparts. In this research, a comprehensive mathematical model is presented to study the delamination crack growth in GFRPcomposite laminates under fatigue loading. A classical static damage model proposed by Allix and Ladevèze is modified as afatigue damage model. Subsequently, the model is implemented in commercial finite element software via UMAT subroutine.The results obtained by the finite element simulations verify the experimental findings of Kenane and Benzeggagh for the fatiguecrack growth in GFRP composite laminates.

1. Introduction

Composite laminates are frequently used in the modernautomobile, aerospace, and manufacturing industries due totheir excellent mechanical properties and lightweight charac-teristics. Moreover, the composite properties can be tailoredin the desired directions by adjusting the stacking sequencesand the directions of different plies [1]. The list of compositeis quite diversified, but carbon, Kevlar, and glass fibres aremost common in manufacturing of different structural parts.Among these, the glass fibre composite laminates are themost economical for structural parts, like the windmills,UAV bodies, and the filament tubes/cylinders.

Delamination is considered as a crack like entity betweenany two plies that can initiate and propagate in the compositelaminates under different loading conditions [2, 3], and thesituation may become severe since many structural partsmay fail in the real-life applications under the cyclic fatigueloading. Many authors proposed different mathematicalmodels based on the damage and the fracture mechanics

theories to simulate the delamination crack growth in theglass and the carbon fibre composite laminates. Fracturemechanics can predict the propagation of a crack that alreadyexists in the structural parts [4], while the damage mechanicsdeals with the propagation of the cracks as well as the simu-lation of the crack initiation process [5–8].

Many authors have also proposed different mathematicalformulations to analyze the delamination crack growth incomposite laminates under static loadings [9–12]. Martinand Murri [13], Paas et al. [14], Asp et al. [15], and Blancoet al. [16] performed the fatigue-driven delamination crackgrowth experiments. Robinson et al. [17], Tumino andCappello [18], Turon et al. [19], and Ijaz et al. [20] developedthe models to simulate the delamination crack growth usingfinite element (FE) analysis. Peerlings et al. proposed afatigue damage model for the analysis of crack growth inmetals [21]. Most of the proposed simulation work in litera-ture is related with the carbon fibre-reinforced plastic(CFRP) composites. In the present work, however, the glassfibre-reinforced plastic (GFRP) composite laminates under

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 2081785, 8 pageshttps://doi.org/10.1155/2018/2081785

the fatigue loading conditions are examined. A static damagemodel proposed by Allix and Ladevèze is modified to accom-modate the fatigue-driven delamination in the present study.

A delamination crack is, generally, represented by aninterfacial entity between two plies of the laminates. Failureof this interface is dictated by the three damage variables(d1, d2, and d3), which correspond to the three orthotropicdirections. Here, d1 represents mode I delamination crackgrowth, while d2 and d3 present mode II and mode IIIcrack propagations, respectively. The damage variable diis divided into the static (diS) and fatigue components(diF). Hence, the total damage (di) can be calculatedby taking the sum of the two aforementioned damagevariables (di = diS + diF, i = 1, 2, 3).

In this study, the FE simulation results for mode I andmode II fatigue delamination crack growth are comparedwith the experimental findings of Kenane [22–24].

This article is organized as follows: the classical staticdamage model is presented in Section 2, the proposed fatiguedamage model is detailed in Section 3, details of finite ele-ment analysis and its comparison with the experimentalresults are discussed in Section 4, and finally, the concludingremarks are given in Section 5.

2. Static Damage Model

The relative displacement of the laminate layers correspondsto the three mutually perpendicular vectors (N1, N2, and N3)in an orthotropic reference frame and can be expressed in thefollowing form:

U = U =U+ −U− = U1 N1 + U2 N2 + U3 N3 1

The interfacial stresses corresponding to the threedamage variables may be expressed as

σ13

σ23

σ33

=k01 1 − d1 0 0

0 k02 1 − d2 00 0 k03 1 − d3

U1

U2

U3

2

In the above equation, k01, k02, and k03 are the correspond-

ing interfacial stiffnesses. The damage model is derived byconsidering the thermodynamic forces. These are combinedwith the three damage variables [5, 6] as

Yd3= 12

σ332+

k03 1 − d32 ,

Yd1= 12

σ132

k01 1 − d12 ,

Yd2= 12

σ232k02 1 − d2

2 ,

3

where σ33 + represents that damage will grow only in thetensile loading (opening mode I) and will not grow duringthe compression. The above three damage variables are

combined to form a single equivalent damage energy releaserate for the mixed mode loading [6]:

Y t =Max τ≤t Yd3α + γ1Yd1

α + γ2Yd2α 1/α 4

Here, γ1 and γ1 are the coupling parameters and α isthe material parameter which governs the damage evolu-tion under the mixed mode loading conditions. For staticloads, the damage evolution law is defined in the followingform [5, 6]:

if d3S < 1 and Y < YR

thend1S = d2S = d3S = ω Y

elsed1S = d2S = d3S = 1,

5

where the damage evolution material function ω Y isdefined as [6]

ω Y = nn + 1

Y − YO +YC − YO

n

6

Here, YO and YC are the threshold and critical dam-age energy release rates, and n is termed as the charac-teristic function of material. YR is defined as the damageenergy associated with rupture and can be calculated byYR = YO + n + 1/n d1/nc YC − YO .

Identification of different parameters YC, γ1, and γ1 iscarried out by comparing the energy dissipation yielded bythe linear elastic fracture mechanics (LEFM) and the damagemechanics approaches for different failure modes [5, 6]:

GIC = YC,

GIIC = YCγ1

,

GIIIC =YCγ2

7

3. Fatigue Damage Model

In this section, the mathematical formulations and assump-tions used in the fatigue damage model are discussed. Theactual cyclic fatigue load varies between a maximum andminimum value as shown in Figure 1. But, for a high cyclicfatigue, the actual numerical load applied in FE analysiscorresponds to the maximum value of the cyclic loading;see Figure 1.

The original idea of the fatigue damage is adopted fromPaas et al. [14] and Peerlings et al. [21]. Peerlings proposeda strain-based damage model for cyclic loading for the crackpropagation in metals. Robinson adopted the idea ofPeerlings for CFRP composite laminates and used for thenormalized interfacial displacement [17].

In the present work, the damage evolution is governedby a single equivalent damage energy release rate Y t

2 International Journal of Aerospace Engineering

(see (4)); hence, the same is used to predict the fatiguedamage, as follows:

d•F =

∂dF∂t

=g d, Y

YC

Y•t

YC if Y

•≥ 0 and f ≥ 0

0 if Y•< 0 or f < 0

8

Here, f is a loading function and is defined as f = Y − Y∗.Y∗ is a threshold damage energy release rate. Here, g is adimensionless parameter that governs the fatigue damage.This parameter depends on the equivalent damage energyY t and the total damage d. This parameter is expressed as

g d, YYC

= Ceλ dYYC

β

9

Here, C, λ, and β are the fatigue damage material param-eters and will be identified by comparing the experimentalresults with the FE results. Equation (8) is in a rate form;therefore, it is integrated over time to get the increase in dam-age with respect to the time increment Δt [20]:

dF t + Δt = dF t +t+Δt

tdF•dt = dF t + 〠

N+ΔN

n=NΡ d, Y

10In the above equation, t corresponds to the number of

cycles N and Δt corresponds to the number of incrementcycles ΔN . Ρ d, Y is the evolution of damage due to thefatigue per cycle:

Ρ d, Y = C1 + β

eλdYYC

1+β11

Appl

ied

load

(Time)

Numerically applied load

Actually applied load

Figure 1: Envelope of the applied cyclic load.

L

a0

2h

(a) GFRP composite specimen nomenclature

M

M

(b) DCB specimen (mode I)

M

M

(c) ELS specimen (mode II)

Figure 2: Specimen nomenclature and boundary conditions.

3International Journal of Aerospace Engineering

The sum over the number of cycles presented in (10) isdone by means of the trapezoidal rule for definite integrals.The trapezoidal rule estimates the desired value by takingthe average of the initial and final values. At the end of theincrement, it is multiplied by ΔN . Hence, (10) takes the form

dF N + ΔN = dF N + 12 P d N + ΔN , Y N + ΔN

+ P d N , Y N ΔN12

As stated earlier, for the high-cycle fatigue, the total dam-age is a sum of the fatigue and static damage components.The following relation expresses the evolution of the staticdamage over a certain number of cycles:

1.0E + 01

Finite element resultsExperimental results (23)

1.0E + 02Gmax (KJ/m2)

1.0E + 03

Model I (DCB)

1.0E − 06

1.0E − 05

1.0E − 04

1.0E − 03

1.0E − 02

da/dN

(mm

/cyc

le)

Figure 4: Fatigue delamination crack growth (mode I).

0

10

20

30

40

50

−1.39E − 17 0.005

Yc = 0.43 kJ/m2

n = 0.5k3 = 9000 MPa/mm

0.01 0.015 0.02

𝜎33

(MPa

)

U3 (mm)

0

(a)

0

10

20

30

40

50

0 0.02 0.04 0.06 0.08U1 (mm)

𝜎13

(MPa

)

Yc = 0.43 kJ/m2

n = 0.5 , 𝛾1 = 0.25k3 = 2500 MPa/mm0

(b)

Figure 3: Evolution of stress with displacement (a) normal σ33 and displacement U3 and (b) shear σ13 and displacement U1.

dS N + ΔN = dS N + nn + 1

1YC − YO

n

Y N + ΔN − YOn+ − Y N − YO

n+

if Y N + ΔN ≥ Y N

13

4 International Journal of Aerospace Engineering

Now, the total damage due to the fatigue loading can becalculated by summing (11), (12) and (13) as

d N + ΔN = dF N + ΔN + dS N + ΔN 14

4. Finite Element Analysis and Results

Mode I and mode II delamination crack growth simula-tions of GFRP composite laminates are performed in FEsoftware CAST3M (CEA) [25]. The proposed fatigue dam-age model described in Section 3 is implemented viaUMAT subroutine and used for FE analysis. Double canti-lever beam (DCB) and end load split (ELS) specimens aremodelled for mode I and mode II delamination crackgrowths, respectively. This is shown in Figure 2. The spec-imen has a total length L, an initial crack length a0, andthe total height 2h.

The geometry of beam is modelled with 2D plane strainquadrangles. To simulate the debonding process, the inter-face between the specimen arms is modelled with an interfaceelement JOI2 [26].

Different parameters like YO, YC, γ1 , n, k01, and k03need to be identified for the FE analysis of mode I andmode II crack growths [27, 28]. Note that the value ofthreshold damage energy YO is taken as zero in all simu-lations. If the values of mode I and mode II critical energyrelease rates (GIC and GIIC) are already known from LEFMexperiments, then YC, γ1, and γ2 can be identified using(7). The value of n varies between 0 and 1.0, and a goodvalue can be estimated by comparing the experimentaland numerical results. The values of interfacial stiffnessescan be calculated by using the following relation [20]:

k03 =2n + 1 2n+1 /n

8n n + 1 Ycσ233,

k0i =γi 2n + 1 2n+1 /n

8n n + 1 Ycσ23i, with i ∈ 1, 2

15

In (15), σ33 and σ3i are the maximum interfacial normaland in-plane shear stresses. The energy release rate is calcu-lated by using the fracture mechanics theory for mode I [29]:

GI =M2

bEI, 16

where M is the applied moment, b is the width of the speci-men, E is the flexure modulus, and I is the second momentof area of bear arm. Similarly, the energy release rate for puremode II can be expressed as [29]

GII =34M2

bEI17

The proposed fatigue damage model is able to producethe linear crack growth rate as obtained by the classical Parislaw [23, 24]:

dadN

= BΔGGC

m

18

In the above equation, B and m are the material parame-ters and are determined from different fatigue tests (ΔG =Gmax − Gmin). Here, Gmax and Gmin correspond to the maxi-mum and minimum energy release rates during the cyclicvariation of load. GC is the fracture toughness of the material,which is determined through the static delamination crackgrowth experiments. In this study, the results of the experi-mental work of Kenane and Benzeggagh [22] on the fatiguedelamination growth of GFRP composite laminates is usedfor comparison with the FE simulation results. Nominaldimensions for DCB specimen are L = 150, h = 3 0, and a0 =25 andwidth is b = 20. Similarly, for theELS specimen, dimen-sions are L = 65, h = 3 0, a0 = 25, and b = 20. All the dimen-sions mentioned above are in millimeters for both types ofspecimens. Mode I and mode II critical energy release ratevalues are 0.429 kJ/m2 and 2.9 kJ/m2, respectively [22].

The modulus, E11, used in the fibre direction is 36.2GPa,and the major Poisson ratio is 0.33 for the GFRP compositelaminate [22]. The normal and in-plane shear stiffnesses, k03and k01, are found to be 9000MPa/mm and 1500MPa/mm,respectively, by using (15) for the maximum stress value of

1.0E + 02

Finite element resultsExperimental results (23)

1.0E + 03Gmax (KJ/m2)

1.0E + 04

Model II (ELS)

1.0E − 05

1.0E − 04

1.0E − 03

1.0E − 02

1.0E − 01

da/d

N (m

m/C

ycle

)

Figure 5: Fatigue delamination crack growth (mode II).

Table 1: Fracture toughness and associated fatigue parameters.

Testmethod

GC (kJ/m2)[23, 24]

InterfaceDamage fatigueparameters

Mode I 0.429

n = 0 5, YO = 0 kJ/m2 λ = 3 0Yc = 0 429 kJ/m2 β = 1 0 × 10−4

k30 = 9 3 × 103 MPa/mm C = 4 0 × 10−5

Mode II 2.9

n = 0 5, YO = 0 kJ/m2 λ = 3 0Yc = 2 9 kJ/m2 β = 5 0

k10 = 1 5 × 103 MPa/mm C = 75 0γ1 = 0 148

5International Journal of Aerospace Engineering

40MPa. The evolution of the normal and shear stresses withthe interfacial displacement of identified damage parametersis shown in Figure 3.

Figure 4 shows the linear Paris plot behaviour for mode Idelamination crack growth due to the fatigue loading. The FEresults obtained, using the proposed fatigue model, are foundto be in good agreement with the experimental results [23,24]. The suitable values of fatigue parameters (λ, β, and C)of the proposed model are selected to give the best fit of theexperimental results. Similarly, Figure 5 presents the Parisplot behaviour for mode II delamination crack growth underthe cyclic loading. The linear line of Paris plot behaviour is ingood agreement with experimental results. The identifiedfatigue parameters of GFRP composite laminates for a cyclicloading are given in Table 1.

Figure 6 shows the crack growth rates with a number ofcycles for different energy release rates, which correspondto the maximum amplitude values of the fatigue loading.The methodology of applied loading is explained earlier; seeFigure 2. From Figure 6, it can be observed that when appliedfatigue loading amplitude is very high, that is, Gmax = 0 35 kJ/m2, the crack growth rate is also high. Similarly, when thefatigue crack growth rate is slow, the low amplitude valuesare applied, that is, Gmax = 0 07 kJ/m2. The same phenome-non of the large crack growth rates with the larger loadingamplitudes is also depicted in Paris plot; see Figures 4 and 5.

Figures 7 and 8 present the damage evolution withrespect to the number of cycles for the first ten elements fromthe crack tip for maximum loading amplitudes correspond-ing to 0.3 and 0.1 kJ/m2 energy release rates for mode Ifatigue delamination crack growth. From the figures, onecan observe that damage initiates and finishes rapidly forhigh amplitude loading, that is, Gmax = 0 3 kJ/m2 (Figure 8).On the other hand, for Gmax = 0 1 kJ/m2 loading, damagefor the first element completes at a point that corresponds

to 1.0E5 cycles, and this point corresponds to the completedamage of 10th element for Gmax = 0 3 kJ/m2 loading.Figures 7 and 8 depict the reason of high delamination crackgrowth rates due to rapid initiation and completeness of thedamage variables for high amplitude fatigue loadings.

5. Conclusion

In this study, delamination crack growth simulations for theGFRP composite laminates under the fatigue loading are pre-sented. Details of the proposed mathematical model are alsoexplained. The model is implemented in CAST3M software

0

1

2

3

4

5

6

Crac

k gr

owth

(mm

)

Number of cycles

Gmax = 0.35

Gmax = 0.30Gmax = 0.25

Gmax = 0.20

Gmax = 0.15

Gmax = 0.10

Gmax = 0.07

0.0E + 00 1.0E + 04 2.0E + 04 3.0E + 04 4.0E + 04 5.0E + 04 6.0E + 04 7.0E + 04 8.0E + 04

Figure 6: Crack growth rate versus number of fatigue loading cycles (mode I).

0

0.2

0.4

0.6

0.8

1

1.2

0.0E + 00 7.0E + 04 1.4E + 05 2.1E + 05

Dam

age v

aria

ble

Number of cycles

Elem 1Elem 2Elem 3Elem 4

Elem 5Elem 6Elem 7Elem 8

Elem 9Elem 10

1st element

10th element

Gmax = 0.3 (KJ/m2)

Figure 7: Damage variable evolution due to mode I fatigue loading:maximum fatigue load corresponds to 0.3 kJ/m2.

6 International Journal of Aerospace Engineering

via UMAT subroutine. The crack growth rates obtained fromFE analysis are plotted against the energy release rates toobtain the linear Paris plot behaviour for mode I and modeII load cases. The simulation results are compared with theavailable experimental data of GFRP composite laminateand were found to be in good approximation. FE analysisresults predicted the large crack growth rates at highamplitude for the applied loading values.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

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Elem 1Elem 2Elem 3Elem 4

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0

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0.0E + 00 7.0E + 04 1.4E + 05 2.1E + 05 2.8E + 05 3.5E + 05

Dam

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aria

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1st element

10th element

Figure 8: Damage variable evolution due to mode I fatigue loading:maximum fatigue load corresponds to 0.1 kJ/m2.

7International Journal of Aerospace Engineering

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